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Review ArticleHybrid Fundamental Solution Based Finite Element MethodTheory and Applications
Changyong Cao and Qing-Hua Qin
Research School of Engineering The Australian National University Acton ACT 2601 Australia
Correspondence should be addressed to Qing-Hua Qin qinghuaqinanueduau
Received 13 October 2014 Revised 23 December 2014 Accepted 24 December 2014
Academic Editor Luigi C Berselli
Copyright copy 2015 C Cao and Q-H Qin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its applicationin engineering problems is presented in this paper The framework and formulations of HFS-FEM for potential problem planeelasticity three-dimensional elasticity thermoelasticity anisotropic elasticity and plane piezoelectricity are presented In thismethod two independent assumed fields (intraelement filed and auxiliary frame field) are employedThe formulations for all casesare derived from the modified variational functionals and the fundamental solutions to a given problem Generation of elementalstiffness equations from the modified variational principle is also described Typical numerical examples are given to demonstratethe validity and performance of the HFS-FEM Finally a brief summary of the approach is provided and future trends in this fieldare identified
1 Introduction
A novel hybrid finite element formulation called the hybridfundamental solution based FEM (HFS-FEM) was recentlydeveloped based on the framework of hybrid Trefftz finiteelement method (HT-FEM) and the idea of the methodof fundamental solution (MFS) [1ndash5] In this method twoindependent assumed fields (intraelement filed and auxiliaryframe field) are employed and the domain integrals in thevariational functional can be directly converted to boundaryintegrals without any appreciable increase in computationaleffort as in HT-FEM [6ndash8] It should be mentioned that theintraelement field of HFS-FEM is approximated by the linearcombination of fundamental solutions analytically satisfyingthe related governing equation instead of 119879-complete func-tions as in HT-FEM The resulting system of equations fromthe modified variational functional is expressed in terms ofsymmetric stiffness matrix and nodal displacements onlywhich is easy to implement into the standard FEM It is notedthat no singular integrals are involved in the HFS-FEM bylocating the source point outside the element of interest anddo not overlap with field point during the computation [9]
The HFS-FEM mentioned above inherits all the advan-tages of HT-FEM over the traditional FEM and the boundaryelement method (BEM) namely domain decomposition andboundary integral expressions while avoiding the majorweaknesses of BEM [10ndash12] that is singular element bound-ary integral and loss of symmetry and sparsity [13] Theemployment of two independent fields also makes the HFS-FEM easier to generate arbitrary polygonal or even curve-sided elements It also obviates the difficulties that occur inHT-FEM [14 15] in deriving119879-complete functions for certaincomplex or new physical problems [16] The HFS-FEM hassimpler expressions of interpolation functions for intraele-ment fields (fundamental solutions) and avoids the coordi-nate transformation procedure required in the HT-FEM tokeep the matrix inversion stable Moreover this approachalso has the potential to achieve high accuracy using coarsemeshes of high-degree elements to enhance insensitivity tomesh distortion to give great liberty in element shape and toaccurately represent various local effects (such as hole crackand inclusions) without troublesome mesh adjustment [17ndash20] Additionally HFS-FEM makes it possible for a more
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 916029 38 pageshttpdxdoiorg1011552015916029
2 Advances in Mathematical Physics
flexible element material definition which is important indealingwithmultimaterial problems rather than thematerialdefinition being the same in the entire domain in BEMHowever we noticed that there are also some limitationsof HFS-FEM for example determining the positions ofsource points used for approximation interpolations It is alsoknown that fundamental solution based approximations canperform remarkably well in smooth problems but tend todeteriorate when high-gradient stress fields are presented
This paper is organized as follows in Section 2 thebasic idea and formulations of the HFS-FEM are presentedthrough a simple potential problem Then plane elasticityproblems are described in Section 3 Section 4 extends the 2Dformulations of the HFS-FEM to general three-dimensional(3D) elasticity problems The method of particular solutionand radial basis function approximation are shown to dealwith body force in this Section In Section 5 we extend theHFS-FEM to thermoelastic problems with arbitrary bodyforce and temperature change In Section 6 the HFS-FEMfor 2D anisotropic elastic materials is described based onthe powerful Stroh formalism Plane piezoelectric problemis discussed in Section 7 Finally typical numerical examplesare presented in Section 8 to illustrate applications and per-formance of the HFS-FEM Concluding remarks and futuredevelopment are discussed at the end of this paper
2 Potential Problems
21 Basic Equations of Potential Problems The Laplace equa-tion of a well-posed potential problem (eg heat conduction)in a general plane domainΩ can be expressed as [21 22]
nabla2119906 (x) = 0 forallx isin Ω (1)
with the boundary conditions
119906 = 119906 on Γ119906 (2)
119902 = 119906119894119899119894= 119902 on Γ
119902 (3)
where 119906 is the unknown field variable and 119902 represents theboundary flux 119899
119894is the 119894th component of outward normal
vector to the boundary Γ = Γ119906cup Γ119902 and 119906 and 119902 are specified
functions on the related boundaries respectively The spacederivatives are indicated by a subscript comma that is 119906
119894=
120597119906120597119909119894 and the subscript index 119894 takes values (1 2) for two-
dimensional and (1 2 3) for three-dimensional problemsAdditionally the repeated subscript indices imply summationconvention
For convenience (3) can be rewritten in matrix form as
119902 = A[
1199061
1199062
] = 119902 (4)
with A = [11989911198992]
22 Assumed Independent Fields In this section the pro-cedure for developing a hybrid finite element model withfundamental solution as interior trial function is described
based on the boundary value problem defined by (1)ndash(3)Similar to the conventional FEM and HT-FEM the domainunder consideration is divided into a series of elements [1516 21 23ndash30] In each element two independent fields areassumed in the way as described in [31] and are given inSection 22
221 Intraelement Field Similar to themethod of fundamen-tal solution (MFS) in removing singularities of fundamentalsolution for a particular element 119890 occupying subdomainΩ119890 we assume that the field variable defined in the element
domain is extracted from a linear combination of funda-mental solutions centered at different source points locatedoutside the element (see Figure 1)
119906119890(x) =
119899119904
sum
119895=1
119873119890(x y
119895) 119888119890119895= N
119890(x) c
119890
forallx isin Ω119890 y
119895notin Ω
119890
(5)
where 119888119890119895is undetermined coefficients 119899
119904is the number of
virtual sources and 119873119890(x y
119895) is the fundamental solution to
the partial differential equation
nabla2119873119890(x y) + 120575 (x y) = 0 forallx y isin R
2 (6)
as
119873119890(x y) = minus
1
2120587ln 119903 (x y) (7)
Evidently (5) analytically satisfies (1) due to the solutionproperty of119873
119890(x y
119895)
In implementation the number of source points is takento be the same as the number of element nodes which isfree of spurious energy modes and can keep the stiffnessequations in full rank as indicated in [21] The source pointy119904119895(119895 = 1 2 119899
119904) can be generated bymeans of themethod
employed in the MFS [32ndash35]
y119904= x0+ 120574 (x
0minus x119888) (8)
where 120574 is a dimensionless coefficient x0is the point on
the element boundary (the nodal point in this work) andx119888is the geometrical centroid of the element (see Figure 1)
Determination of 120574 was discussed in [31 36] and 120574 = 5ndash10 isusually used in practice
The corresponding outward normal derivative of 119906119890on Γ
119890
is
119902119890=120597119906119890
120597119899= Q
119890c119890 (9)
where
Q119890=120597N119890
120597119899= AT
119890(10)
with
T119890= [
120597N119890
1205971199091
120597N119890
1205971199092
]
119879
(11)
Advances in Mathematical Physics 3
1 2
34
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
cx
0xΓe
u = Nece
Frame field u(x) = edeNsx
(a) 4-node 2D element
1 3
57
2
4
6
8
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
Γe
Frame field u(x) = ede
u = Nece
N
cx
0x
sx
(b) 8-node 2D element
Figure 1 Intraelement field and frame field of a HFS-FEM element for 2D potential problems
222 Auxiliary Frame Field In order to enforce the con-formity on the field variable 119906 for instance 119906
119890= 119906
119891on
Γ119890cap Γ119891of any two neighboring elements 119890 and 119891 an auxiliary
interelement frame field is used and expressed in terms ofthe same degrees of freedom (DOF) d as those used in theconventional finite elements In this case is confined to thewhole element boundary as
119890(x) = N
119890(x) d
119890(12)
which is independently assumed along the element boundaryin terms of nodal DOF d
119890 where N
119890(x) represents the con-
ventional finite element interpolating functions For examplea simple interpolation of the frame field on a side with threenodes of a particular element can be given in the form
= 11199061+
21199062+
31199063 (13)
where 119894(119894 = 1 2 3) stands for shape functions in terms of
natural coordinate 120585 defined in Figure 2
23 Modified Variational Principle For the boundary valueproblem defined in (1)ndash(3) and (5) since the stationaryconditions of the traditional potential or complementaryvariational functional cannot guarantee the interelementcontinuity condition required in the proposedHFS FEmodelas in the HT FEM [21 26] a variational functional corre-sponding to the new trial functions should be constructedto assure the additional continuity across the common
N1
N2
N3
120585 = minus1 120585 = 0 120585 = +1
minus120585(1 minus 120585)
2
1 minus 1205852
120585(1 + 120585)
2
1 2 3
Figure 2 Typical quadratic interpolation for frame field
boundariesΓIef between intraelement fields of element ldquo119890rdquo andelement ldquo119891rdquo (see Figure 3) [36 37]
119906119890= 119906
119891(conformity)
119902119890+ 119902119891= 0 (reciprocity)
on ΓIef = Γ119890cap Γ119891
(14)
4 Advances in Mathematical Physics
e f
ΓIef
Figure 3 Illustration of continuity between two adjacent elementsldquo119890rdquo and ldquo119891rdquo
Amodified variational functional is developed as follows
Π119898= sum
119890
Π119898119890
= sum
119890
Π119890+ intΓ119890
( minus 119906) 119902dΓ (15)
where
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ (16)
in which the governing equation (1) is assumed to be satisfieda priori for deriving the HFS FE model The boundary Γ
119890of
a particular element consists of the following parts
Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)
where ΓIe represents the interelement boundary of the ele-ment ldquo119890rdquo shown in Figure 3
To show that the stationary condition of the functional(15) leads to the governing equation (Euler equation) bound-ary conditions and continuity conditions invoking (16) and(15) gives the following functional for the problem domain
Π119898119890
=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (18)
from which the first-order variational yields
120575Π119898119890
= intΩ119890
119906119894120575119906119894dΩ minus int
Γ119902119890
119902120575dΓ + intΓ119890
(120575 minus 120575119906) 119902dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(19)
Using divergence theorem
intΩ
119891119894ℎ119894dΩ = int
Γ
ℎ119891119894119899119894dΓ minus int
Ω
ℎnabla2119891dΩ (20)
we can obtain
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓ119906119890
119902120575dΓ
+ intΓIe
119902120575dΓ + intΓ119890
( minus 119906) 120575119902dΓ(21)
For the displacement-based method the potential confor-mity should be satisfied in advance
120575 = 0 on Γ119906119890
(∵ = 119906)
120575119890= 120575
119891 on ΓIef (∵ 119890=
119891)
(22)
then (21) can be rewritten as
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓIe
119902120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(23)
The Euler equation and boundary conditions can be obtainedas
119906119894119894= 0 in Ω
119890
119902 = 119902 on Γ119902119890
= 119906 on Γ119890
(24)
using the stationary condition 120575Π119898119890
= 0As for the continuous requirement between two adjacent
elements ldquo119890rdquo and ldquo119891rdquo given in (14) we can obtain it in thefollowing way When assembling elements ldquo119890rdquo and ldquo119891rdquo wehave
120575Π119898(119890+119891)
= intΩ119890+119891
119906119894119894120575119906dΩ minus int
Γ119902119890+Γ119902119891
(119902 minus 119902) 120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ + intΓ119891
( minus 119906) 120575119902dΓ
+ intΓIef
(119902119890+ 119902119891) 120575
119890119891dΓ + sdot sdot sdot
(25)
from which the vanishing variation of Π119898(119890+119891)
leads to thereciprocity condition 119902
119890+ 119902
119891= 0 on the interelement
boundary ΓIefIf the following expression
intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(26)
is uniformly positive (or negative) in the neighborhood of1199060 where the displacement 119906
0has such a value that
Π119898(119906
0) = (Π
119898)0and where (Π
119898)0stands for the stationary
value of Π119898 we have
Π119898ge (Π
119898)0
or Π119898le (Π
119898)0
(27)
inwhich the relation that 119890=
119891is identical on Γ
119890capΓ119891has
been used This is due to the definition in (14) in Section 23
Advances in Mathematical Physics 5
Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find
1205752Π119898= intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(28)
Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]
24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present
problem can be written as
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (29)
Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model
Π119890=1
2[intΓ119890
119902119906dΓ + intΩ119890
119906119896nabla2119906dΩ] minus int
Γ119902119890
119902dΓ
+ intΓ119890
119902 ( minus 119906) dΓ
= minus1
2intΓ119890
119902119906dΓ minus intΓ119902119890
119902dΓ + intΓ119890
119902dΓ
(30)
Then substituting (5) (9) and (12) into the functional (30)produces
Π119890= minus
1
2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890
(31)
in which
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
N119879119890Q119890dΓ
G119890= intΓ119890
Q119879119890N119890dΓ g
119890= intΓ119902119890
N119879119890119902dΓ
(32)
The symmetry ofH119890is obvious from the scalar definition (31)
of variational functional Π119890
To enforce interelement continuity on the common ele-ment boundary the unknown vector c
119890should be expressed
in terms of nodal DOF d119890Theminimization of the functional
Π119890with respect to c
119890and d
119890 respectively yields
120597Π119890
120597c119890
119879= minusH
119890c119890+ G
119890d119890= 0
120597Π119890
120597d119890
119879= G119879
119890c119890minus g119890= 0
(33)
from which the optional relationship between c119890and d
119890and
the stiffness equation can be produced
c119890= Hminus1
119890G119890d119890 K
119890d119890= g119890 (34)
whereK119890= G119879
119890Hminus1119890G119890stands for the element stiffness matrix
25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]
For theH119890matrix one can express it as
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x) dΓ (35)
by introducing the matrix function
F (x) = [119865119894119895(x)]
119898times119898= Q119879
119890N119890 (36)
Equation (36) can be further rewritten as
119867119894119895= intΓ119890
119865119894119895(x) dΓ =
119899119890
sum
119897=1
intΓ119890119897
119865119894119895(x) dΓ (37)
where
dΓ = radic(d1199091)2
+ (d1199092)2
= radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
d120585 = 119869d120585
(38)
and 119869 is the Jacobean expressed as
119869 = radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
(39)
where
[d1199091
d120585d1199092
d120585]
119879
=
119899119900
sum
119894=1
d119873119894(120585)
d120585
1199091119894
1199092119894
(40)
Thus the Gaussian numerical integration forHmatrix can becalculated by
119867119894119895=
119899119890
sum
119897=1
[int
+1
minus1
119865119894119895(x (120585)) 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
[
119899119901
sum
119896=1
119908119896119865119894119895(x (120585
119896)) 119869 (120585
119896)]
(41)
where 119899119890is the number of edges of the element and 119899
119901
is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G
119890
matrix using
119866119894119895=
119899119890
sum
119897=1
[int
1
minus1
119865119894119895[x (120585)] 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
119899119901
sum
119896=1
119908119896119865119894119895[x (120585
119896)] 119869 (120585
119896)
(42)
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
flexible element material definition which is important indealingwithmultimaterial problems rather than thematerialdefinition being the same in the entire domain in BEMHowever we noticed that there are also some limitationsof HFS-FEM for example determining the positions ofsource points used for approximation interpolations It is alsoknown that fundamental solution based approximations canperform remarkably well in smooth problems but tend todeteriorate when high-gradient stress fields are presented
This paper is organized as follows in Section 2 thebasic idea and formulations of the HFS-FEM are presentedthrough a simple potential problem Then plane elasticityproblems are described in Section 3 Section 4 extends the 2Dformulations of the HFS-FEM to general three-dimensional(3D) elasticity problems The method of particular solutionand radial basis function approximation are shown to dealwith body force in this Section In Section 5 we extend theHFS-FEM to thermoelastic problems with arbitrary bodyforce and temperature change In Section 6 the HFS-FEMfor 2D anisotropic elastic materials is described based onthe powerful Stroh formalism Plane piezoelectric problemis discussed in Section 7 Finally typical numerical examplesare presented in Section 8 to illustrate applications and per-formance of the HFS-FEM Concluding remarks and futuredevelopment are discussed at the end of this paper
2 Potential Problems
21 Basic Equations of Potential Problems The Laplace equa-tion of a well-posed potential problem (eg heat conduction)in a general plane domainΩ can be expressed as [21 22]
nabla2119906 (x) = 0 forallx isin Ω (1)
with the boundary conditions
119906 = 119906 on Γ119906 (2)
119902 = 119906119894119899119894= 119902 on Γ
119902 (3)
where 119906 is the unknown field variable and 119902 represents theboundary flux 119899
119894is the 119894th component of outward normal
vector to the boundary Γ = Γ119906cup Γ119902 and 119906 and 119902 are specified
functions on the related boundaries respectively The spacederivatives are indicated by a subscript comma that is 119906
119894=
120597119906120597119909119894 and the subscript index 119894 takes values (1 2) for two-
dimensional and (1 2 3) for three-dimensional problemsAdditionally the repeated subscript indices imply summationconvention
For convenience (3) can be rewritten in matrix form as
119902 = A[
1199061
1199062
] = 119902 (4)
with A = [11989911198992]
22 Assumed Independent Fields In this section the pro-cedure for developing a hybrid finite element model withfundamental solution as interior trial function is described
based on the boundary value problem defined by (1)ndash(3)Similar to the conventional FEM and HT-FEM the domainunder consideration is divided into a series of elements [1516 21 23ndash30] In each element two independent fields areassumed in the way as described in [31] and are given inSection 22
221 Intraelement Field Similar to themethod of fundamen-tal solution (MFS) in removing singularities of fundamentalsolution for a particular element 119890 occupying subdomainΩ119890 we assume that the field variable defined in the element
domain is extracted from a linear combination of funda-mental solutions centered at different source points locatedoutside the element (see Figure 1)
119906119890(x) =
119899119904
sum
119895=1
119873119890(x y
119895) 119888119890119895= N
119890(x) c
119890
forallx isin Ω119890 y
119895notin Ω
119890
(5)
where 119888119890119895is undetermined coefficients 119899
119904is the number of
virtual sources and 119873119890(x y
119895) is the fundamental solution to
the partial differential equation
nabla2119873119890(x y) + 120575 (x y) = 0 forallx y isin R
2 (6)
as
119873119890(x y) = minus
1
2120587ln 119903 (x y) (7)
Evidently (5) analytically satisfies (1) due to the solutionproperty of119873
119890(x y
119895)
In implementation the number of source points is takento be the same as the number of element nodes which isfree of spurious energy modes and can keep the stiffnessequations in full rank as indicated in [21] The source pointy119904119895(119895 = 1 2 119899
119904) can be generated bymeans of themethod
employed in the MFS [32ndash35]
y119904= x0+ 120574 (x
0minus x119888) (8)
where 120574 is a dimensionless coefficient x0is the point on
the element boundary (the nodal point in this work) andx119888is the geometrical centroid of the element (see Figure 1)
Determination of 120574 was discussed in [31 36] and 120574 = 5ndash10 isusually used in practice
The corresponding outward normal derivative of 119906119890on Γ
119890
is
119902119890=120597119906119890
120597119899= Q
119890c119890 (9)
where
Q119890=120597N119890
120597119899= AT
119890(10)
with
T119890= [
120597N119890
1205971199091
120597N119890
1205971199092
]
119879
(11)
Advances in Mathematical Physics 3
1 2
34
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
cx
0xΓe
u = Nece
Frame field u(x) = edeNsx
(a) 4-node 2D element
1 3
57
2
4
6
8
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
Γe
Frame field u(x) = ede
u = Nece
N
cx
0x
sx
(b) 8-node 2D element
Figure 1 Intraelement field and frame field of a HFS-FEM element for 2D potential problems
222 Auxiliary Frame Field In order to enforce the con-formity on the field variable 119906 for instance 119906
119890= 119906
119891on
Γ119890cap Γ119891of any two neighboring elements 119890 and 119891 an auxiliary
interelement frame field is used and expressed in terms ofthe same degrees of freedom (DOF) d as those used in theconventional finite elements In this case is confined to thewhole element boundary as
119890(x) = N
119890(x) d
119890(12)
which is independently assumed along the element boundaryin terms of nodal DOF d
119890 where N
119890(x) represents the con-
ventional finite element interpolating functions For examplea simple interpolation of the frame field on a side with threenodes of a particular element can be given in the form
= 11199061+
21199062+
31199063 (13)
where 119894(119894 = 1 2 3) stands for shape functions in terms of
natural coordinate 120585 defined in Figure 2
23 Modified Variational Principle For the boundary valueproblem defined in (1)ndash(3) and (5) since the stationaryconditions of the traditional potential or complementaryvariational functional cannot guarantee the interelementcontinuity condition required in the proposedHFS FEmodelas in the HT FEM [21 26] a variational functional corre-sponding to the new trial functions should be constructedto assure the additional continuity across the common
N1
N2
N3
120585 = minus1 120585 = 0 120585 = +1
minus120585(1 minus 120585)
2
1 minus 1205852
120585(1 + 120585)
2
1 2 3
Figure 2 Typical quadratic interpolation for frame field
boundariesΓIef between intraelement fields of element ldquo119890rdquo andelement ldquo119891rdquo (see Figure 3) [36 37]
119906119890= 119906
119891(conformity)
119902119890+ 119902119891= 0 (reciprocity)
on ΓIef = Γ119890cap Γ119891
(14)
4 Advances in Mathematical Physics
e f
ΓIef
Figure 3 Illustration of continuity between two adjacent elementsldquo119890rdquo and ldquo119891rdquo
Amodified variational functional is developed as follows
Π119898= sum
119890
Π119898119890
= sum
119890
Π119890+ intΓ119890
( minus 119906) 119902dΓ (15)
where
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ (16)
in which the governing equation (1) is assumed to be satisfieda priori for deriving the HFS FE model The boundary Γ
119890of
a particular element consists of the following parts
Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)
where ΓIe represents the interelement boundary of the ele-ment ldquo119890rdquo shown in Figure 3
To show that the stationary condition of the functional(15) leads to the governing equation (Euler equation) bound-ary conditions and continuity conditions invoking (16) and(15) gives the following functional for the problem domain
Π119898119890
=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (18)
from which the first-order variational yields
120575Π119898119890
= intΩ119890
119906119894120575119906119894dΩ minus int
Γ119902119890
119902120575dΓ + intΓ119890
(120575 minus 120575119906) 119902dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(19)
Using divergence theorem
intΩ
119891119894ℎ119894dΩ = int
Γ
ℎ119891119894119899119894dΓ minus int
Ω
ℎnabla2119891dΩ (20)
we can obtain
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓ119906119890
119902120575dΓ
+ intΓIe
119902120575dΓ + intΓ119890
( minus 119906) 120575119902dΓ(21)
For the displacement-based method the potential confor-mity should be satisfied in advance
120575 = 0 on Γ119906119890
(∵ = 119906)
120575119890= 120575
119891 on ΓIef (∵ 119890=
119891)
(22)
then (21) can be rewritten as
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓIe
119902120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(23)
The Euler equation and boundary conditions can be obtainedas
119906119894119894= 0 in Ω
119890
119902 = 119902 on Γ119902119890
= 119906 on Γ119890
(24)
using the stationary condition 120575Π119898119890
= 0As for the continuous requirement between two adjacent
elements ldquo119890rdquo and ldquo119891rdquo given in (14) we can obtain it in thefollowing way When assembling elements ldquo119890rdquo and ldquo119891rdquo wehave
120575Π119898(119890+119891)
= intΩ119890+119891
119906119894119894120575119906dΩ minus int
Γ119902119890+Γ119902119891
(119902 minus 119902) 120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ + intΓ119891
( minus 119906) 120575119902dΓ
+ intΓIef
(119902119890+ 119902119891) 120575
119890119891dΓ + sdot sdot sdot
(25)
from which the vanishing variation of Π119898(119890+119891)
leads to thereciprocity condition 119902
119890+ 119902
119891= 0 on the interelement
boundary ΓIefIf the following expression
intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(26)
is uniformly positive (or negative) in the neighborhood of1199060 where the displacement 119906
0has such a value that
Π119898(119906
0) = (Π
119898)0and where (Π
119898)0stands for the stationary
value of Π119898 we have
Π119898ge (Π
119898)0
or Π119898le (Π
119898)0
(27)
inwhich the relation that 119890=
119891is identical on Γ
119890capΓ119891has
been used This is due to the definition in (14) in Section 23
Advances in Mathematical Physics 5
Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find
1205752Π119898= intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(28)
Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]
24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present
problem can be written as
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (29)
Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model
Π119890=1
2[intΓ119890
119902119906dΓ + intΩ119890
119906119896nabla2119906dΩ] minus int
Γ119902119890
119902dΓ
+ intΓ119890
119902 ( minus 119906) dΓ
= minus1
2intΓ119890
119902119906dΓ minus intΓ119902119890
119902dΓ + intΓ119890
119902dΓ
(30)
Then substituting (5) (9) and (12) into the functional (30)produces
Π119890= minus
1
2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890
(31)
in which
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
N119879119890Q119890dΓ
G119890= intΓ119890
Q119879119890N119890dΓ g
119890= intΓ119902119890
N119879119890119902dΓ
(32)
The symmetry ofH119890is obvious from the scalar definition (31)
of variational functional Π119890
To enforce interelement continuity on the common ele-ment boundary the unknown vector c
119890should be expressed
in terms of nodal DOF d119890Theminimization of the functional
Π119890with respect to c
119890and d
119890 respectively yields
120597Π119890
120597c119890
119879= minusH
119890c119890+ G
119890d119890= 0
120597Π119890
120597d119890
119879= G119879
119890c119890minus g119890= 0
(33)
from which the optional relationship between c119890and d
119890and
the stiffness equation can be produced
c119890= Hminus1
119890G119890d119890 K
119890d119890= g119890 (34)
whereK119890= G119879
119890Hminus1119890G119890stands for the element stiffness matrix
25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]
For theH119890matrix one can express it as
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x) dΓ (35)
by introducing the matrix function
F (x) = [119865119894119895(x)]
119898times119898= Q119879
119890N119890 (36)
Equation (36) can be further rewritten as
119867119894119895= intΓ119890
119865119894119895(x) dΓ =
119899119890
sum
119897=1
intΓ119890119897
119865119894119895(x) dΓ (37)
where
dΓ = radic(d1199091)2
+ (d1199092)2
= radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
d120585 = 119869d120585
(38)
and 119869 is the Jacobean expressed as
119869 = radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
(39)
where
[d1199091
d120585d1199092
d120585]
119879
=
119899119900
sum
119894=1
d119873119894(120585)
d120585
1199091119894
1199092119894
(40)
Thus the Gaussian numerical integration forHmatrix can becalculated by
119867119894119895=
119899119890
sum
119897=1
[int
+1
minus1
119865119894119895(x (120585)) 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
[
119899119901
sum
119896=1
119908119896119865119894119895(x (120585
119896)) 119869 (120585
119896)]
(41)
where 119899119890is the number of edges of the element and 119899
119901
is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G
119890
matrix using
119866119894119895=
119899119890
sum
119897=1
[int
1
minus1
119865119894119895[x (120585)] 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
119899119901
sum
119896=1
119908119896119865119894119895[x (120585
119896)] 119869 (120585
119896)
(42)
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
1 2
34
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
cx
0xΓe
u = Nece
Frame field u(x) = edeNsx
(a) 4-node 2D element
1 3
57
2
4
6
8
Centroid
SourceNode
X2
X1
Intraelement field
Ωe
Γe
Frame field u(x) = ede
u = Nece
N
cx
0x
sx
(b) 8-node 2D element
Figure 1 Intraelement field and frame field of a HFS-FEM element for 2D potential problems
222 Auxiliary Frame Field In order to enforce the con-formity on the field variable 119906 for instance 119906
119890= 119906
119891on
Γ119890cap Γ119891of any two neighboring elements 119890 and 119891 an auxiliary
interelement frame field is used and expressed in terms ofthe same degrees of freedom (DOF) d as those used in theconventional finite elements In this case is confined to thewhole element boundary as
119890(x) = N
119890(x) d
119890(12)
which is independently assumed along the element boundaryin terms of nodal DOF d
119890 where N
119890(x) represents the con-
ventional finite element interpolating functions For examplea simple interpolation of the frame field on a side with threenodes of a particular element can be given in the form
= 11199061+
21199062+
31199063 (13)
where 119894(119894 = 1 2 3) stands for shape functions in terms of
natural coordinate 120585 defined in Figure 2
23 Modified Variational Principle For the boundary valueproblem defined in (1)ndash(3) and (5) since the stationaryconditions of the traditional potential or complementaryvariational functional cannot guarantee the interelementcontinuity condition required in the proposedHFS FEmodelas in the HT FEM [21 26] a variational functional corre-sponding to the new trial functions should be constructedto assure the additional continuity across the common
N1
N2
N3
120585 = minus1 120585 = 0 120585 = +1
minus120585(1 minus 120585)
2
1 minus 1205852
120585(1 + 120585)
2
1 2 3
Figure 2 Typical quadratic interpolation for frame field
boundariesΓIef between intraelement fields of element ldquo119890rdquo andelement ldquo119891rdquo (see Figure 3) [36 37]
119906119890= 119906
119891(conformity)
119902119890+ 119902119891= 0 (reciprocity)
on ΓIef = Γ119890cap Γ119891
(14)
4 Advances in Mathematical Physics
e f
ΓIef
Figure 3 Illustration of continuity between two adjacent elementsldquo119890rdquo and ldquo119891rdquo
Amodified variational functional is developed as follows
Π119898= sum
119890
Π119898119890
= sum
119890
Π119890+ intΓ119890
( minus 119906) 119902dΓ (15)
where
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ (16)
in which the governing equation (1) is assumed to be satisfieda priori for deriving the HFS FE model The boundary Γ
119890of
a particular element consists of the following parts
Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)
where ΓIe represents the interelement boundary of the ele-ment ldquo119890rdquo shown in Figure 3
To show that the stationary condition of the functional(15) leads to the governing equation (Euler equation) bound-ary conditions and continuity conditions invoking (16) and(15) gives the following functional for the problem domain
Π119898119890
=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (18)
from which the first-order variational yields
120575Π119898119890
= intΩ119890
119906119894120575119906119894dΩ minus int
Γ119902119890
119902120575dΓ + intΓ119890
(120575 minus 120575119906) 119902dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(19)
Using divergence theorem
intΩ
119891119894ℎ119894dΩ = int
Γ
ℎ119891119894119899119894dΓ minus int
Ω
ℎnabla2119891dΩ (20)
we can obtain
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓ119906119890
119902120575dΓ
+ intΓIe
119902120575dΓ + intΓ119890
( minus 119906) 120575119902dΓ(21)
For the displacement-based method the potential confor-mity should be satisfied in advance
120575 = 0 on Γ119906119890
(∵ = 119906)
120575119890= 120575
119891 on ΓIef (∵ 119890=
119891)
(22)
then (21) can be rewritten as
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓIe
119902120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(23)
The Euler equation and boundary conditions can be obtainedas
119906119894119894= 0 in Ω
119890
119902 = 119902 on Γ119902119890
= 119906 on Γ119890
(24)
using the stationary condition 120575Π119898119890
= 0As for the continuous requirement between two adjacent
elements ldquo119890rdquo and ldquo119891rdquo given in (14) we can obtain it in thefollowing way When assembling elements ldquo119890rdquo and ldquo119891rdquo wehave
120575Π119898(119890+119891)
= intΩ119890+119891
119906119894119894120575119906dΩ minus int
Γ119902119890+Γ119902119891
(119902 minus 119902) 120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ + intΓ119891
( minus 119906) 120575119902dΓ
+ intΓIef
(119902119890+ 119902119891) 120575
119890119891dΓ + sdot sdot sdot
(25)
from which the vanishing variation of Π119898(119890+119891)
leads to thereciprocity condition 119902
119890+ 119902
119891= 0 on the interelement
boundary ΓIefIf the following expression
intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(26)
is uniformly positive (or negative) in the neighborhood of1199060 where the displacement 119906
0has such a value that
Π119898(119906
0) = (Π
119898)0and where (Π
119898)0stands for the stationary
value of Π119898 we have
Π119898ge (Π
119898)0
or Π119898le (Π
119898)0
(27)
inwhich the relation that 119890=
119891is identical on Γ
119890capΓ119891has
been used This is due to the definition in (14) in Section 23
Advances in Mathematical Physics 5
Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find
1205752Π119898= intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(28)
Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]
24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present
problem can be written as
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (29)
Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model
Π119890=1
2[intΓ119890
119902119906dΓ + intΩ119890
119906119896nabla2119906dΩ] minus int
Γ119902119890
119902dΓ
+ intΓ119890
119902 ( minus 119906) dΓ
= minus1
2intΓ119890
119902119906dΓ minus intΓ119902119890
119902dΓ + intΓ119890
119902dΓ
(30)
Then substituting (5) (9) and (12) into the functional (30)produces
Π119890= minus
1
2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890
(31)
in which
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
N119879119890Q119890dΓ
G119890= intΓ119890
Q119879119890N119890dΓ g
119890= intΓ119902119890
N119879119890119902dΓ
(32)
The symmetry ofH119890is obvious from the scalar definition (31)
of variational functional Π119890
To enforce interelement continuity on the common ele-ment boundary the unknown vector c
119890should be expressed
in terms of nodal DOF d119890Theminimization of the functional
Π119890with respect to c
119890and d
119890 respectively yields
120597Π119890
120597c119890
119879= minusH
119890c119890+ G
119890d119890= 0
120597Π119890
120597d119890
119879= G119879
119890c119890minus g119890= 0
(33)
from which the optional relationship between c119890and d
119890and
the stiffness equation can be produced
c119890= Hminus1
119890G119890d119890 K
119890d119890= g119890 (34)
whereK119890= G119879
119890Hminus1119890G119890stands for the element stiffness matrix
25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]
For theH119890matrix one can express it as
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x) dΓ (35)
by introducing the matrix function
F (x) = [119865119894119895(x)]
119898times119898= Q119879
119890N119890 (36)
Equation (36) can be further rewritten as
119867119894119895= intΓ119890
119865119894119895(x) dΓ =
119899119890
sum
119897=1
intΓ119890119897
119865119894119895(x) dΓ (37)
where
dΓ = radic(d1199091)2
+ (d1199092)2
= radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
d120585 = 119869d120585
(38)
and 119869 is the Jacobean expressed as
119869 = radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
(39)
where
[d1199091
d120585d1199092
d120585]
119879
=
119899119900
sum
119894=1
d119873119894(120585)
d120585
1199091119894
1199092119894
(40)
Thus the Gaussian numerical integration forHmatrix can becalculated by
119867119894119895=
119899119890
sum
119897=1
[int
+1
minus1
119865119894119895(x (120585)) 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
[
119899119901
sum
119896=1
119908119896119865119894119895(x (120585
119896)) 119869 (120585
119896)]
(41)
where 119899119890is the number of edges of the element and 119899
119901
is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G
119890
matrix using
119866119894119895=
119899119890
sum
119897=1
[int
1
minus1
119865119894119895[x (120585)] 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
119899119901
sum
119896=1
119908119896119865119894119895[x (120585
119896)] 119869 (120585
119896)
(42)
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
e f
ΓIef
Figure 3 Illustration of continuity between two adjacent elementsldquo119890rdquo and ldquo119891rdquo
Amodified variational functional is developed as follows
Π119898= sum
119890
Π119898119890
= sum
119890
Π119890+ intΓ119890
( minus 119906) 119902dΓ (15)
where
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ (16)
in which the governing equation (1) is assumed to be satisfieda priori for deriving the HFS FE model The boundary Γ
119890of
a particular element consists of the following parts
Γ119890= Γ119906119890cup Γ119902119890cup ΓIe (17)
where ΓIe represents the interelement boundary of the ele-ment ldquo119890rdquo shown in Figure 3
To show that the stationary condition of the functional(15) leads to the governing equation (Euler equation) bound-ary conditions and continuity conditions invoking (16) and(15) gives the following functional for the problem domain
Π119898119890
=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (18)
from which the first-order variational yields
120575Π119898119890
= intΩ119890
119906119894120575119906119894dΩ minus int
Γ119902119890
119902120575dΓ + intΓ119890
(120575 minus 120575119906) 119902dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(19)
Using divergence theorem
intΩ
119891119894ℎ119894dΩ = int
Γ
ℎ119891119894119899119894dΓ minus int
Ω
ℎnabla2119891dΩ (20)
we can obtain
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓ119906119890
119902120575dΓ
+ intΓIe
119902120575dΓ + intΓ119890
( minus 119906) 120575119902dΓ(21)
For the displacement-based method the potential confor-mity should be satisfied in advance
120575 = 0 on Γ119906119890
(∵ = 119906)
120575119890= 120575
119891 on ΓIef (∵ 119890=
119891)
(22)
then (21) can be rewritten as
120575Π119898119890
= intΩ119890
119906119894119894120575119906dΩ minus int
Γ119902119890
(119902 minus 119902) 120575dΓ + intΓIe
119902120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ(23)
The Euler equation and boundary conditions can be obtainedas
119906119894119894= 0 in Ω
119890
119902 = 119902 on Γ119902119890
= 119906 on Γ119890
(24)
using the stationary condition 120575Π119898119890
= 0As for the continuous requirement between two adjacent
elements ldquo119890rdquo and ldquo119891rdquo given in (14) we can obtain it in thefollowing way When assembling elements ldquo119890rdquo and ldquo119891rdquo wehave
120575Π119898(119890+119891)
= intΩ119890+119891
119906119894119894120575119906dΩ minus int
Γ119902119890+Γ119902119891
(119902 minus 119902) 120575dΓ
+ intΓ119890
( minus 119906) 120575119902dΓ + intΓ119891
( minus 119906) 120575119902dΓ
+ intΓIef
(119902119890+ 119902119891) 120575
119890119891dΓ + sdot sdot sdot
(25)
from which the vanishing variation of Π119898(119890+119891)
leads to thereciprocity condition 119902
119890+ 119902
119891= 0 on the interelement
boundary ΓIefIf the following expression
intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(26)
is uniformly positive (or negative) in the neighborhood of1199060 where the displacement 119906
0has such a value that
Π119898(119906
0) = (Π
119898)0and where (Π
119898)0stands for the stationary
value of Π119898 we have
Π119898ge (Π
119898)0
or Π119898le (Π
119898)0
(27)
inwhich the relation that 119890=
119891is identical on Γ
119890capΓ119891has
been used This is due to the definition in (14) in Section 23
Advances in Mathematical Physics 5
Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find
1205752Π119898= intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(28)
Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]
24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present
problem can be written as
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (29)
Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model
Π119890=1
2[intΓ119890
119902119906dΓ + intΩ119890
119906119896nabla2119906dΩ] minus int
Γ119902119890
119902dΓ
+ intΓ119890
119902 ( minus 119906) dΓ
= minus1
2intΓ119890
119902119906dΓ minus intΓ119902119890
119902dΓ + intΓ119890
119902dΓ
(30)
Then substituting (5) (9) and (12) into the functional (30)produces
Π119890= minus
1
2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890
(31)
in which
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
N119879119890Q119890dΓ
G119890= intΓ119890
Q119879119890N119890dΓ g
119890= intΓ119902119890
N119879119890119902dΓ
(32)
The symmetry ofH119890is obvious from the scalar definition (31)
of variational functional Π119890
To enforce interelement continuity on the common ele-ment boundary the unknown vector c
119890should be expressed
in terms of nodal DOF d119890Theminimization of the functional
Π119890with respect to c
119890and d
119890 respectively yields
120597Π119890
120597c119890
119879= minusH
119890c119890+ G
119890d119890= 0
120597Π119890
120597d119890
119879= G119879
119890c119890minus g119890= 0
(33)
from which the optional relationship between c119890and d
119890and
the stiffness equation can be produced
c119890= Hminus1
119890G119890d119890 K
119890d119890= g119890 (34)
whereK119890= G119879
119890Hminus1119890G119890stands for the element stiffness matrix
25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]
For theH119890matrix one can express it as
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x) dΓ (35)
by introducing the matrix function
F (x) = [119865119894119895(x)]
119898times119898= Q119879
119890N119890 (36)
Equation (36) can be further rewritten as
119867119894119895= intΓ119890
119865119894119895(x) dΓ =
119899119890
sum
119897=1
intΓ119890119897
119865119894119895(x) dΓ (37)
where
dΓ = radic(d1199091)2
+ (d1199092)2
= radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
d120585 = 119869d120585
(38)
and 119869 is the Jacobean expressed as
119869 = radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
(39)
where
[d1199091
d120585d1199092
d120585]
119879
=
119899119900
sum
119894=1
d119873119894(120585)
d120585
1199091119894
1199092119894
(40)
Thus the Gaussian numerical integration forHmatrix can becalculated by
119867119894119895=
119899119890
sum
119897=1
[int
+1
minus1
119865119894119895(x (120585)) 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
[
119899119901
sum
119896=1
119908119896119865119894119895(x (120585
119896)) 119869 (120585
119896)]
(41)
where 119899119890is the number of edges of the element and 119899
119901
is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G
119890
matrix using
119866119894119895=
119899119890
sum
119897=1
[int
1
minus1
119865119894119895[x (120585)] 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
119899119901
sum
119896=1
119908119896119865119894119895[x (120585
119896)] 119869 (120585
119896)
(42)
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
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Advances in Mathematical Physics 5
Proof For the proof of the theorem on the existence ofextremum we may complete it by the so-called ldquosecond vari-ational approachrdquo [38] In doing this performing variation of120575Π119898and using the constrained conditions (26) we find
1205752Π119898= intΓ119902
120575119902120575d119904
minussum
119890
[intΓIe
120575119902119890120575119890d119904 + int
Γ119890
120575119902119890120575 (
119890minus 119906119890) d119904]
(28)
Therefore the theorem has been proved from the sufficientcondition of the existence of a local extreme of a functional[38]
24 Element Stiffness Equation With the intraelement fieldand frame field independently defined in a particular element(see Figure 1) we can generate element stiffness equation bythe variational functional derived in Section 23 Followingthe approach described in [21] the variational functionalΠ119890corresponding to a particular element 119890 of the present
problem can be written as
Π119890=1
2intΩ119890
119906119894119906119894dΩ minus int
Γ119902119890
119902dΓ + intΓ119890
119902 ( minus 119906) dΓ (29)
Appling the divergence theorem (20) to the functional (29)we have the final functional for the HFS-FE model
Π119890=1
2[intΓ119890
119902119906dΓ + intΩ119890
119906119896nabla2119906dΩ] minus int
Γ119902119890
119902dΓ
+ intΓ119890
119902 ( minus 119906) dΓ
= minus1
2intΓ119890
119902119906dΓ minus intΓ119902119890
119902dΓ + intΓ119890
119902dΓ
(30)
Then substituting (5) (9) and (12) into the functional (30)produces
Π119890= minus
1
2c119879119890H119890c119890minus d119879119890g119890+ c119879119890G119890d119890
(31)
in which
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
N119879119890Q119890dΓ
G119890= intΓ119890
Q119879119890N119890dΓ g
119890= intΓ119902119890
N119879119890119902dΓ
(32)
The symmetry ofH119890is obvious from the scalar definition (31)
of variational functional Π119890
To enforce interelement continuity on the common ele-ment boundary the unknown vector c
119890should be expressed
in terms of nodal DOF d119890Theminimization of the functional
Π119890with respect to c
119890and d
119890 respectively yields
120597Π119890
120597c119890
119879= minusH
119890c119890+ G
119890d119890= 0
120597Π119890
120597d119890
119879= G119879
119890c119890minus g119890= 0
(33)
from which the optional relationship between c119890and d
119890and
the stiffness equation can be produced
c119890= Hminus1
119890G119890d119890 K
119890d119890= g119890 (34)
whereK119890= G119879
119890Hminus1119890G119890stands for the element stiffness matrix
25 Numerical Integral for H and G Matrix Generally it isdifficult to obtain the analytical expression of the integral in(32) and numerical integration along the element boundaryis required Herein the widely used Gaussian integration isemployed [22]
For theH119890matrix one can express it as
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x) dΓ (35)
by introducing the matrix function
F (x) = [119865119894119895(x)]
119898times119898= Q119879
119890N119890 (36)
Equation (36) can be further rewritten as
119867119894119895= intΓ119890
119865119894119895(x) dΓ =
119899119890
sum
119897=1
intΓ119890119897
119865119894119895(x) dΓ (37)
where
dΓ = radic(d1199091)2
+ (d1199092)2
= radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
d120585 = 119869d120585
(38)
and 119869 is the Jacobean expressed as
119869 = radic(d1199091
d120585)
2
+ (d1199092
d120585)
2
(39)
where
[d1199091
d120585d1199092
d120585]
119879
=
119899119900
sum
119894=1
d119873119894(120585)
d120585
1199091119894
1199092119894
(40)
Thus the Gaussian numerical integration forHmatrix can becalculated by
119867119894119895=
119899119890
sum
119897=1
[int
+1
minus1
119865119894119895(x (120585)) 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
[
119899119901
sum
119896=1
119908119896119865119894119895(x (120585
119896)) 119869 (120585
119896)]
(41)
where 119899119890is the number of edges of the element and 119899
119901
is the Gaussian sampling points employed in the Gaussiannumerical integration Similarly we can calculate the G
119890
matrix using
119866119894119895=
119899119890
sum
119897=1
[int
1
minus1
119865119894119895[x (120585)] 119869 (120585) d120585]
asymp
119899119890
sum
119897=1
119899119901
sum
119896=1
119908119896119865119894119895[x (120585
119896)] 119869 (120585
119896)
(42)
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
The calculation of vector g119890in (32) is the same as that
in the conventional FEM so it is convenient to incorporatethe proposed HFS-FEM into the standard FEM programBesides the flux is directly computed from (9)The boundaryDOF can be directly computed from (12) while the unknownvariable at interior points of the element can be determinedfrom (5) plus the recovered rigid-body modes in eachelement which is discussed in the following section
26 Recovery of Rigid-BodyMotion Considering the physicaldefinition of the fundamental solution it is necessary torecover themissing rigid-bodymotionmodes from the aboveresults Following the method presented in [21] the missingrigid-body motion can be recovered by writing the internalpotential field of a particular element 119890 as
119906119890= N
119890c119890+ 1198880 (43)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
(N119890c119890+ 1198880minus 119890)2 10038161003816100381610038161003816 node119894
= min (44)
which finally gives
1198880=1
119899
119899
sum
119894=1
Δ119906119890119894 (45)
in which Δ119906119890119894
= (119890minus N
119890c119890)|node119894 and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (34) and then 1198880is evaluated from (45)
Finally the potential field119906 at any internal point in an elementcan be obtained by means of (43)
3 Plane Elasticity Problems
31 Linear Theory of Plane Elasticity In linear elastic theorythe strain displacement relations can be used and equilibriumequations refer to the undeformed geometry [39] In therectangular Cartesian coordinates (119883
1 1198832) the governing
equations of a plane elastic body can be expressed as
120590119894119895119895
= 119887119894 119894 119895 = 1 2 (46)
If written as matrix form it can be presented as
L120590 = b (47)
where 120590 = [12059011
12059022
12059012]119879 is a stress vector b = [119887
1 1198872]119879 is
a body force vector and the differential operator matrix L isgiven as
L =[[[
[
120597
1205971199091
0120597
1205971199092
0120597
1205971199092
120597
1205971199091
]]]
]
(48)
120576 = LTu (49)
where 120576 = [12057611
12057622
12057612]119879 is a strain vector and u = [119906
1 1199062]119879
is a displacement vectorThe constitutive equations for the linear elasticity are
given in matrix form as
120590 = D120576 (50)
where D is the material coefficient matrix with constantcomponents for isotropic materials which can be expressedas follows
D =[[[
[
+ 2119866 0
+ 2119866 0
0 0 119866
]]]
]
(51)
where
=2]
1 minus 2]119866 119866 =
119864
2 (1 + ])
] =
] for plane strain]
1 + ]for plane stress
(52)
The two different kinds of boundary conditions can beexpressed as
u = u on Γ119906
t = A120590 = t on Γ119905
(53)
where t = [11990511199052]119879 denotes the traction vector and A is a
transformation matrix related to the direction cosine of theoutward normal
A = [
1198991
0 1198992
0 11989921198991
] (54)
Substituting (49) and (50) into (47) yields the well-knownNavier partial differential equations in terms of displace-ments
LDL119879u = b (55)
32 Assumed Independent Fields For elasticity problem twodifferent assumed fields are employed as in potential prob-lems intraelement and frame field [1 31 36] The intraele-ment continuity is enforced on nonconforming internaldisplacement field chosen as the fundamental solution ofthe problem [36] The intraelement displacement fields areapproximated in terms of a linear combination of fundamen-tal solutions of the problem of interest
u (x) =
1199061(x)
1199062(x)
=
119899119904
sum
119895=1
[
[
119906lowast
11(x y
119904119895) 119906
lowast
12(x y
119904119895)
119906lowast
21(x y
119904119895) 119906
lowast
22(x y
119904119895)
]
]
1198881119895
1198882119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(56)
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
where 119899119904is again the number of source points outside the
element domain which is equal to the number of nodes ofan element in the present research based on the generationapproach of the source points [31] The vector ce and thefundamental solution matrix Ne are now in the form
Ne
= [
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
)
119906lowast
21(x y
1199041) 119906
lowast
22(x y
1199041) sdot sdot sdot 119906
lowast
21(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
)
]
]
ce = [11988811 11988821
sdot sdot sdot 1198881119899
1198882119899]119879
(57)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119883
1 1198832) The
components 119906lowast119894119895(x y
119904119895) are the fundamental solution that is
induced displacement component in 119894-direction at the fieldpoint x due to a unit point load applied in 119895-direction at thesource point y
119904119895 which are given by [40 41]
119906lowast
119894119895(x y
119904119895) =
minus1
8120587 (1 minus ]) 119866(3 minus 4]) 120575
119894119895ln 119903 minus 119903
119894119903119895 (58)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2 The virtual source points
for elasticity problems are generated in the same manner asthat in potential problems described in Section 2
With the assumption of intraelement field in (56) thecorresponding stress fields can be obtained by the constitutiveequation (50)
120590 (x) = [12059011 12059022
12059012]119879
= Tece (59)
whereTe
=[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
)
]]]
]
(60)
As a consequence the traction is written as
1199051
1199052
= n120590 = Qece (61)
in which
Qe = nTe n = [
1198991
0 1198992
0 11989921198991
] (62)
The components 120590lowast119894119895119896(x y) for plane strain problems are given
as
120590lowast
119894119895119896(x y) = minus1
4120587 (1 minus ]) 119903
sdot [(1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 2119903
119894119903119895119903119896]
(63)
The unknown ce in (56) is calculated by a hybrid tech-nique [31] in which the elements are linked through anauxiliary conforming displacement framewhich has the sameform as that in conventional FEM (see Figure 1) This meansthat in the HFS-FEM a conforming displacement fieldshould be independently defined on the element boundary toenforce the field continuity between elements and also to linkthe unknown c with nodal displacement d
119890 Thus the frame
is defined as
u (x) =
1
2
=
N1
N2
d119890= N
119890d119890 (x isin Γ
119890) (64)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateralelement (see Figure 1) as an example N
119890and d
119890can be
expressed as
N119890= [
[
0 sdot sdot sdot 0 1
0 2
0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
4
0 1
0 2
0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
]
]
de = [11990611 11990621
11990612
11990622
sdot sdot sdot 11990618
11990628]119879
(65)
where 1 2 and
3can be expressed by natural coordinate
as
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(66)
33 Modified Functional for the Hybrid FEM As in Section 2HFS-FE formulation for a plane elastic problem can also beestablished by the variational approach [36] In the absenceof body forces the hybrid functional Π
119898119890used for deriving
the present HFS-FEM can be constructed as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ (67)
where 119894and 119906
119894are the intraelement displacement field
defined within the element and the frame displacementfield defined on the element boundary respectively Ω
119890
and Γ119890are the element domain and element boundary
respectively Γ119905 Γ119906 and Γ
119868stand respectively for the specified
traction boundary specified displacement boundary andinterelement boundary (Γ
119890= Γ
119905+ Γ
119906+ Γ
119868) Compared
to the functional employed in the conventional FEM thepresent variational functional is constructed by adding ahybrid integral term related to the intraelement and elementframe displacement fields to guarantee the satisfaction ofdisplacement and traction continuity conditions on the com-mon boundary of two adjacent elements
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Advances in Mathematical Physics
By applying the Gaussian theorem (67) can be simplifiedto
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ) minus int
Γ119905
119905119894119894dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ
(68)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement field we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(69)
The variational functional in (69) contains boundary inte-grals only and will be used to derive HFS-FEM formulationfor the plane isotropic elastic problem
34 Element Stiffness Matrix As in Section 2 the elementstiffness equation can be generated by setting 120575Π
119898119890= 0
Substituting (56) (64) and (61) into the functional of (69)we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (70)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(71)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields respectively
(33) and (34)
35 Recovery of Rigid-Body Motion For the same reasonstated in Section 26 it is necessary to reintroduce thediscarded rigid-body motion terms after we have obtainedthe internal field of an elementThe least squares method canbe employed for this purpose and the missing terms can berecovered easily by setting for the augmented internal field[22]
u119890= N
119890c119890+ [
1 0 1199092
0 1 minus1199091
] c0 (72)
where the undetermined rigid-bodymotion parameter 1198880can
be calculated using the least square matching of 119906119890and
119890at
element nodes119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
] = min (73)
which finally gives
c0 = Rminus1119890re (74)
where
Re =119899
sum
119894=1
[[[
[
1 0 1199092119894
0 1 minus1199091119894
1199092119894
minus1199091119894
1199092
1119894+ 1199092
2119894
]]]
]
re =119899
sum
119894=1
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
(75)
in which Δ119906119890119895119894
= (119890119895119894minus 119890119895119894) (119895 = 1 2) and 119899 is the number
of element nodes Once the nodal field is determined bysolving the final stiffness equation the coefficient vector c
119890
can be evaluated from (56) and then 1198880is evaluated from (74)
Finally the displacement field u119890at any internal point in an
element can be obtained by (72)
4 Three-Dimensional Elastic Problems
In this section the HFS-FEM approach is extended to three-dimensional (3D) elastic problem withwithout body forceThe detailed 3D formulations of HFS-FEM are firstly derivedfor elastic problems by ignoring body forces and then aprocedure based on the method of particular solution andradial basis function approximation are introduced to dealwith the body force [42] As a consequence the homogeneoussolution is obtained by using theHFS-FEMand the particularsolution associated with body force is approximated by usingthe strong form of basis function interpolation
41 Governing Equations and Boundary Conditions Let(1199091 1199092 1199093) denote the coordinates in a Cartesian coordinate
system and consider a finite isotropic body occupying thedomain Ω as shown in Figure 4 The equilibrium equationfor this finite body with body force can be expressed as
120590119894119895119895
= minus119887119894
119894 119895 = 1 2 3 (76)
The constitutive equations for linear elasticity and thekinematical relation are given as
120590119894119895=
2119866V1 minus 2V
120575119894119895119890119896119896+ 2119866119890
119894119895
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(77)
where 120590119894119895is the stress tensor 119890
119894119895is the strain tensor and
120575119894119895is the Kronecker delta Substituting (77) into (76) the
equilibrium equation is rewritten as
119866119906119894119895119895
+119866
1 minus 2V119906119895119895119894
= minus119887119894 (78)
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 9
Γux1
x2
x3
b1
b2
b3
Γt
O
Figure 4 Geometrical definitions and boundary conditions for ageneral 3D solid
For a well-posed boundary value problem the boundaryconditions are prescribed as follows
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(79)
where Γ119906cupΓ119905= Γ is the boundary of the solution domainΩ 119906
119894
and 119905119894are the prescribed boundary values In the following
parts we will present the procedure for handling the bodyforce appearing in (78)
42 The Method of Particular Solution The inhomogeneousterm 119887
119894associated with the body force in (78) can be
effectively handled by means of the method of particularsolution presented in [22] In this approach the displacement119906119894is decomposed into two parts a homogeneous solution 119906ℎ
119894
and a particular solution 119906119901119894
119906119894= 119906
ℎ
119894+ 119906119901
119894 (80)
where the particular solution 119906119901119894should satisfy the governing
equation
119866119906119901
119894119895119895+
119866
1 minus 2V119906119901
119895119895119894= minus119887
119894(81)
without any restriction of boundary condition However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2V119906ℎ
119895119895119894= 0 (82)
with the modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894on Γ
119906
119905ℎ
119894= 119905119894minus 119905119901
119894on Γ
119905
(83)
From the above equations it can be seen that once theparticular solution 119906
119901
119894is known the homogeneous solution
119906ℎ
119894in (82) and (83) can be obtained using HFS-FEM The
final solution can then be given by (80) In the next sectionradial basis function approximation is introduced to obtainthe particular solution and the HFS-FEM is presented forsolving (82) and (83)
43 Radial Basis Function Approximation For body force 119887119894
it is generally impossible to find an analytical solution whichenables us to convert the domain integral into a boundaryone So we must approximate it by a combination of basis(trial) functions or other methods with the HFS-FEM Herewe use radial basis function (RBF) [33 43] to interpolate thebody force We assume
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (84)
where 119873 is the number of interpolation points 120593119895 arethe RBFs and 120572
119895
119894are the coefficients to be determined
Subsequently the particular solution can be approximated by
119906119901
119894=
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (85)
where Φ119895
119894119896is the approximated particular solution kernel
of displacement satisfying (86) below Once the RBFs areselected the problem of finding a particular solution isreduced to solve the following equation
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (86)
To solve this equation the displacement is expressed interms of the Galerkin-Papkovich vectors
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(87)
Substituting (87) into (86) we can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (88)
Taking the Spline Type RBF 120593 = 1199032119899minus1 as an example we get
the following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4) (89)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (90)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(91)
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Advances in Mathematical Physics
and 119903119895represents the Euclidean distance between a field point
(1199091 1199092 1199093) and a given point (119909
1119895 1199092119895 1199093119895) in the domain of
interestThe corresponding particular solution of stresses canbe obtained as
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(92)
where 120582 = (2V(1minus2V))119866 Substituting (90) into (92) we have
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897 (93)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(94)
44 HFS-FEM for Homogeneous Solution After obtainingthe particular solution in Sections 42 and 43 we can
determine the modified boundary conditions (83) Finallywe can treat the 3D problem as a homogeneous problemgoverned by (82) and (83) by using the HFS-FEM presentedbelow It is clear that once the particular and homogeneoussolutions for displacement and stress components at nodalpoints are determined the distribution of displacement andstress fields at any point in the domain can be calculated usingthe element interpolation functionHowever for 3D elasticityproblems in the absent of body force that is 119887
119894= 0 the
procedures in Sections 42 and 43 will become unnecessaryand we can employ the following procedures to find thesolution directly
441 Assumed Intraelement and Auxiliary Frame Fields Theintraelement displacement fields are approximated in termsof a linear combination of fundamental solutions of theproblem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (95)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(96)
in which x and y119904119895
are respectively the field point andsource point in the local coordinate system (119909
1 1199092) The
fundamental solution 119906lowast119894119895(x y
119904119895) is given by [40]
119906lowast
119894119895(x y
119904119895) =
1
16120587 (1 minus ]) 119866119903(3 minus 4]) 120575
119894119895+ 119903119894119903119895 (97)
where 119903119894= 119909
119894minus 119909
119894119904 119903 = radic119903
2
1+ 1199032
2+ 1199032
3 119899119904is the number of
source points The source point y119904119895(119895 = 1 2 119899
119904) can also
be generated by means of the following method [36] as intwo-dimensional cases
y119904= x0+ 120574 (x
0minus x119888) (98)
where 120574 is a dimensionless coefficient x0is the point on the
element boundary (the nodal point in this work) and x119888is
the geometrical centroid of the element (see Figure 5)According to (50) and (49) the corresponding stress
fields can be expressed as
120590 (x) = [12059011 12059022
12059033
12059023
12059031
12059012]119879
= Tece (99)
where
Te =
[[[[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
133(x y
1) 120590
lowast
233(x y
1) 120590
lowast
333(x y
1) sdot sdot sdot 120590
lowast
133(x y
119899119904
) 120590lowast
233(x y
119899119904
) 120590lowast
333(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
212(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]]]]
]
(100)
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 11
1
4 3
2
Source pointsNodeCentroid
5 6
8 7
8-node 3D element
X2
X1
X3
Intraelement field
Ωe
Γe
Frame field
u = Nece
u(x) = edeN
cx
0x
sx
(a)
1
7 5
2
13 15
1917
3
4
6
89 10
1112
1416
1820
20-node 3D element
X2
X1
X3
Ωe
(b)
Figure 5 Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of20-node element are omitted in the figure for clarity and clear view which is similar to that of the 8-node element)
The components 120590lowast119894119895119896(x y) are given by
120590lowast
119894119895119896(x y) = minus1
8120587 (1 minus ]) 1199032
sdot (1 minus 2]) (119903119896120575119894119895+ 119903119895120575119896119894minus 119903119894120575119895119896) + 3119903
119894119903119895119903119896
(101)
As a consequence the traction can be written in the form
1199051
1199052
1199053
= n120590 = Qece (102)
in which
Qe = nTe n =[[
[
1198991
0 0 0 11989931198992
0 1198992
0 1198993
0 1198991
0 0 119899311989921198991
0
]]
]
(103)
To link the unknown c119890and the nodal displacement d
119890
the frame is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (104)
where the symbol ldquosimrdquo is used to specify that the field isdefined on the element boundary only N
119890is the matrix
of shape functions and d119890is the nodal displacements of
elements Taking the surface 2-3-7-6 of a particular 8-nodebrick element (see Figure 5) as an example matrix N
119890and
vector d119890can be expressed as
N119890= [0 N
1N20 0 N
4N30]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(105)
where the shape functions are expressed as
N119894=[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(106)
where 119894(119894 = 1ndash4) can be expressed by natural coordinate
120585 120578 isin [minus1 1]
1=(1 + 120585) (1 + 120578)
4
2=(1 minus 120585) (1 + 120578)
4
3=(1 minus 120585) (1 minus 120578)
4
4=(1 + 120585) (1 minus 120578)
4
(107)
and (120585119894 120578119894) is the natural coordinate of the 119894-node of the
element (Figure 6)
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Advances in Mathematical Physics
1 3
57
2
4
6
8
(0minus1)(minus1minus1) (1minus1)
(10)
(11)(01)
(minus10)
(minus11)
120578
120585
Figure 6 Typical linear interpolation for the framefields of 3Dbrickelements
442 Modified Functional for Hybrid Finite Element MethodIn the absence of body forces the hybrid functionalΠ
119898119890used
for deriving the present HFS-FEM is written as [22]
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(108)
By applying the Gaussian theorem (108) can be simplified as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
minus intΓ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(109)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for HFS finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ
(110)
The functional (110) contains only boundary integrals of theelement and will be used to derive HFS-FEM formulationfor the three-dimensional elastic problem in the followingsection
443 Element Stiffness Matrix Substituting (95) (102) and(104) into the functional (110) we have
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (111)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(112)
To enforce interelement continuity on the common elementboundary the unknown vector c
119890should be expressed in
terms of nodal DOF d119890 The stationary condition of the
functional Π119898119890
with respect to c119890and d
119890yields again
respectively (33) and (34)
444 Numerical Integral over Element Considering a surfaceof the 3D hexahedron element as shown in Figure 6 thevector normal to the surface can be obtained by
V119899= V120585times V120578=
V119899119909
V119899119910
V119899119911
=
d119909d120585d119910d120585d119911d120585
times
d119909d120578d119910d120578d119911d120578
=
d119910d120585
d119911d120578
minusd119910d120578
d119911d120585
d119911d120585
d119909d120578
minusd119911d120578
d119909d120585
d119909d120585
d119910d120578
minusd119909d120578
d119910d120585
(113)
where V120585and V
120578are the tangential vectors in the 120585-direction
and 120578-direction respectively calculated by
V120585=
d119909d120585d119910d120585d119911d120585
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120585
119909119894
119910119894
119911119894
V120578=
d119909d120578d119910d120578d119911d120578
=
119899119889
sum
119894=1
120597119873119894(120585 120578)
120597120578
119909119894
119910119894
119911119894
(114)
where 119899119889is the number of nodes of the surface and (119909
119894 119910119894 119911119894)
are the nodal coordinates Thus the unit normal vector isgiven by
119899 =V119899
1003816100381610038161003816V1198991003816100381610038161003816
(115)
where
119869 (120585 120578) =1003816100381610038161003816V119899
1003816100381610038161003816 = radicV2119899119909+ V2119899119910+ V2119899119911
(116)
is the Jacobian of the transformation from Cartesian coordi-nates (119909 119910) to natural coordinates (120585 120578)
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 13
For the119867matrix we introduce the matrix function
F (x y) = [119865119894119895(119909 119910)]
119898times119898= Q119879
119890N119890 (117)
Then we can get
H119890= intΓ119890
Q119879119890N119890dΓ = int
Γ119890
F (x y) dΓ (118)
and we rewrite it to the component form as
119867119894119895= intΓ119890
119865119894119895(119909 119910) d119878 =
119899119891
sum
119897=1
intΓ119890119897
119865119894119895(119909 119910) d119878 (119)
Using the relationship
d119878 = 119869 (120585 120578) d120585 d120578 (120)
and the Gaussian numerical integration we can obtain
119867119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(121)
where 119899119891and 119899
119901are respectively the number of surface of
the 3D element and the number of Gaussian integral pointsin each direction of the element surface Similarly we cancalculate the G
119890matrix by
119866119894119895=
119899119891
sum
119897=1
int
1
minus1
119865119894119895[119909 (120585 120578) 119910 (120585 120578)] 119869 (120585 120578) d120585 d120578
asymp
119899119891
sum
119897=1
119899119901
sum
119904=1
119899119901
sum
119905=1
119908119904119908119905119865119894119895[119909 (120585
119904 120578119905) 119910 (120585
119904 120578119905)] 119869 (120585
119904 120578119905)
(122)
It should be mentioned that the calculation of vector g119890
in equation is the same as that in the conventional FEMso it is convenient to incorporate the proposed HFS-FEMinto the standard FEM program Besides the stress andtraction estimations are directly computed from (99) and(100) respectively The boundary displacements can bedirectly computed from (104) while the displacements atinterior points of element can be determined from (95) plusthe recovered rigid-body modes in each element which isintroduced in the following section
445 Recovery of Rigid-Body Motion Terms As in Section 2the least square method is employed to recover the missingterms of rigid-body motions The missing terms can berecovered by setting for the augmented internal field
u119890= N
119890c119890+[[
[
1 0 0 0 1199093
minus1199092
0 1 0 minus1199093
0 1199091
0 0 1 1199092
minus1199091
0
]]
]
c0
(123)
and using a least-square procedure tomatch 119906119890ℎand
119890ℎat the
nodes of the element boundary
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (1199063119894minus 3119894)2
] (124)
The above equation finally yields
c0 = Rminus1119890re (125)
where
Re
=
119899
sum
119894=1
[[[[[[[[[[[
[
1 0 0 0 1199093119894
minus1199092119894
0 1 0 minus1199093119894
0 1199091119894
0 0 1 1199092119894
minus1199091119894
0
0 minus1199093119894
1199092119894
1199092
2119894+ 1199092
3119894minus11990911198941199092119894
minus11990911198941199093119894
1199093119894
0 minus1199091119894
minus11990911198941199092119894
1199092
1119894+ 1199092
3119894minus11990921198941199093119894
minus1199092119894
1199091119894
0 minus11990911198941199093119894
minus11990921198941199093119894
1199092
1119894+ 1199092
2119894
]]]]]]]]]]]
]
re =119899
sum
119894=1
[[[[[[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ1199061198903119894
Δ11990611989031198941199092119894minus Δ119906
11989021198941199093119894
Δ11990611989011198941199093119894minus Δ119906
11989031198941199091119894
Δ11990611989021198941199091119894minus Δ119906
11989011198941199092119894
]]]]]]]]]]
]
(126)
5 Thermoelasticity Problems
Thermoelasticity problems arise in many practical designssuch as steam and gas turbines jet engines rocket motorsand nuclear reactors Thermal stress induced in these struc-tures is one of the major concerns in product design andanalysis The general thermoelasticity is governed by twotime-dependent coupled differential equations the heat con-duction equation and the Navier equation with thermal bodyforce [44] In many engineering applications the couplingterm of the heat equation and the inertia term in Navierequation are generally negligible [44] As a consequencemost of the analyses are employing the uncoupled thermoe-lasticity theory which is adopted in this topic [45ndash52] Inthis section the HFS-FEM is presented to solve 2D and3D thermoelastic problems with considering arbitrary bodyforces and temperature changes [53]Themethod used hereinis similar to that in Section 4
51 Basic Equations for Thermoelasticity Consider anisotropic material in a finite domain Ω and let (119909
1 1199092 1199093)
denote the coordinates in Cartesian coordinate system Theequilibrium governing equations of the thermoelasticity withthe body force are expressed as
120590119894119895119895
= minus119887119894 (127)
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Advances in Mathematical Physics
where 120590119894119895is the stress tensor 119887
119894is the body force vector
and 119894 119895 = 1 2 3 The generalized thermoelastic stress-strainrelations and kinematical relation are given as
120590119894119895119895
=2119866]1 minus 2]
120575119894119895119890 + 2119866119890
119894119895minus 119898120575
119894119895119879
119890119894119895=1
2(119906119894119895+ 119906119895119894)
(128)
where 119890119894119895is the strain tensor 119906
119894is the displacement vector 119879
is the temperature change 119866 is the shear modulus ] is thePoissonrsquos ratio 120575
119894119895is the Kronecker delta and
119898 =2119866120572 (1 + V)(1 minus 2V)
(129)
is the thermal constant with 120572 being the coefficient oflinear thermal expansion Substituting (128) into (127) theequilibrium equations may be rewritten as
119866119906119894119895119895
+119866
1 minus 2]119906119895119895119894
= 119898119879119894minus 119887119894 (130)
For a well-posed boundary value problem the followingboundary conditions either displacement or traction bound-ary condition should be prescribed as
119906119894= 119906
119894on Γ
119906
119905119894= 119905119894
on Γ119905
(131)
where Γ119906cup Γ119905= Γ is the boundary of the solution domain Ω
119906119894and 119905
119894are the prescribed boundary values and
119905119894= 120590119894119895119899119895 (132)
is the boundary traction in which 119899119895denotes the boundary
outward normal
52 The Method of Particular Solution For the governingequation (130) given in the previous section the inhomo-geneous term 119898119879
119894minus 119887
119894can be eliminated by employing
the method of particular solution [16 36 44] Using super-position principle we decompose the displacement 119906
119894into
two parts the homogeneous solution 119906ℎ
119894and the particular
solution 119906119901119894as follows
119906119894= 119906
ℎ
119894+ 119906119901
119894(133)
in which the particular solution 119906119901119894should satisfy the govern-
ing equation
119866119906119901
119894119895119895+
119866
1 minus 2]119906119901
119895119895119894= 119898119879
119894minus 119887119894
(134)
but does not necessarily satisfy any boundary condition Itshould be pointed out that its solution is not unique and canbe obtained by various numerical techniques However thehomogeneous solution should satisfy
119866119906ℎ
119894119895119895+
119866
1 minus 2]119906ℎ
119895119895119894= 0 (135)
with modified boundary conditions
119906ℎ
119894= 119906
119894minus 119906119901
119894 on Γ
119906 (136)
119905ℎ
119894= 119905119894+ 119898119879119899
119894minus 119905119901
119894 on Γ
119905 (137)
From above equations it can be seen that once the particularsolution is known the homogeneous solution 119906
ℎ
119894in (135)ndash
(137) can be solved by (135) In the following sectionRBF approximation is described to illustrate the particularsolution procedure and theHFS-FEM is presentedwhich canbe used for solving (135)ndash(137)
53 Radial Basis Function Approximation RBF is to be usedto approximate the body force 119887
119894and the temperature field 119879
in order to obtain the particular solution To implement thisapproximation we may consider two different ways one is totreat body force 119887
119894and the temperature field 119879 separately as
in Tsai [54] The other is to treat 119898119879119894minus 119887119894as a whole [53]
Here we demonstrated that the performance of the latter oneis usually better than the former one
531 Interpolating Temperature and Body Force SeparatelyThe body force 119887
119894and temperature 119879 are assumed to be by
the following two equations
119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895
(119894 = 1 2 in R2 119894 = 1 2 3 in R
3)
119879 asymp
119873
sum
119895=1
120573119895120593119895
(138)
where 119873 is the number of interpolation points 120593119895 are thebasis functions and 120572
119895
119894and 120573
119895 are the coefficients to bedetermined by collocation Subsequently the approximateparticular solution can be written as follows
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896+
119873
sum
119895=1
120573119895Ψ119895
119894 (139)
where Φ119895119894119896and Ψ
119895
119894are the approximated particular solution
kernels Once the RBF is selected the problem of findinga particular solution will be reduced to solve the followingequations
119866Φ119894119897119896119896
+119866
1 minus 2]Φ119896119897119896119894
= minus120575119894119897120593 (140)
119866Ψ119894119896119896
+119866
1 minus 2]Ψ119896119896119894
= 119898120593119894 (141)
To solve (140) the displacement is expressed in terms ofthe Galerkin-Papkovich vectors [43 55ndash57]
Φ119894119896=1 minus ]119866
119865119894119896119898119898
minus1
2119866119865119898119896119898119894
(142)
Substituting (142) into (140) one can obtain the followingbiharmonic equation
nabla4119865119894119897= minus
1
1 minus ]120575119894119897120593 (143)
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 15
If taking the Spline Type RBF 120593 = 1199032119899minus1 one can get the
following solutions
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1)2(2119899 + 3)
2(R2) for 119899 = 1 2 3
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
2) for 119899 = 1 2 3
(144)
where
1198600= minus
1
2119866 (1 minus V)1199032119899+1
(2119899 + 1)2(2119899 + 3)
1198601= 5 + 4119899 minus 2V (2119899 + 3)
1198602= minus (2119899 + 1)
(145)
for two-dimensional problem and
119865119897119894= minus
120575119897119894
1 minus V1199032119899+3
(2119899 + 1) (2119899 + 2) (2119899 + 3) (2119899 + 4)
(R3) for 119899 = 1 2 3
(146)
Φ119897119894= 119860
0(1198601120575119897119894+ 119860
2119903119894119903119897) (R
3) for 119899 = 1 2 3
(147)
where
1198600= minus
1
8119866 (1 minus V)1199032119899+1
(119899 + 1) (119899 + 2) (2119899 + 1)
1198601= 7 + 4119899 minus 4V (119899 + 2)
1198602= minus (2119899 + 1)
(148)
for three-dimensional problem where 119903119895represents the
Euclidean distance of the given point (1199091 1199092) from a fixed
point (1199091119895 1199092119895) in the domain of interest The corresponding
stress particular solution can be obtained by
119878119897119894119895= 119866 (Φ
119897119894119895+ Φ
119897119895119894) + 120582120575
119894119895Φ119897119896119896
(149)
where 120582 = (2V(1 minus 2V))119866 Substituting (147) into (149) onecan obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R2) for 119899 = 1 2 3
(150)
where
1198610= minus
1
(1 minus V)1199032119899
(2119899 + 1) (2119899 + 3)
1198611= (2119899 + 2) minus V (2119899 + 3)
1198612= V (2119899 + 3) minus 1
1198613= 1 minus 2119899
(151)
for two-dimensional problem and substituting (147) into(149) one can obtain
119878119897119894119895= 119861
01198611(119903119895120575119897119894+ 119903119894120575119895119897) + 119861
2120575119894119895119903119897+ 119861
3119903119894119903119895119903119897
(R3) for 119899 = 1 2 3
(152)
where
1198610= minus
1
4 (1 minus V)1199032119899
(119899 + 1) (119899 + 2)
1198611= 3 + 2119899 minus 2V (119899 + 2)
1198612= 2V (119899 + 2) minus 1
1198613= 1 minus 2119899
(153)
for three-dimensional problemTo solve (141) one can treat Ψ
119894as the gradient of a scalar
function
Ψ119894= 119880
119894 (154)
Substituting (154) into (141) obtains the Poissonrsquos equation
nabla2119880 =
119898 (1 minus 2])2119866 (1 minus ])
120593 (155)
Thus taking 120593 = 1199032119899minus1 its particular solution can be obtained
[55] as follows
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1)2
(R2) for 119899 = 1 2 3
119880 =119898 (1 minus 2])2119866 (1 minus ])
1199032119899+1
(2119899 + 1) (2119899 + 2)(R3) for 119899 = 1 2 3
(156)
Then from (154) we can get Ψ119894as follows
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
2119899 + 1(R2) for 119899 = 1 2 3
Ψ119894=119898 (1 minus 2])2119866 (1 minus ])
1199031198941199032119899
(2119899 + 2)(R3) for 119899 = 1 2 3
(157)
The corresponding stress particular solution can be obtainedby substituting (147) into
119878119894119895= 119866 (Ψ
119894119895+ Ψ
119895119894) + 120582120575
119894119895Ψ119896119896 (158)
Then we have
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 1)(1 + 2119899V) 120575
119894119895+ (1 minus 2V) (2119899 minus 1) 119903
119894119903119895
(R2) for 119899 = 1 2 3
119878119894119895=
1198981199032119899minus1
(1 minus V) (2119899 + 2)(1 minus 2]) (2119899 minus 1) 119903
119894119903119895+ 120575119894119895(1 + 2119899V)
(R3) for 119899 = 1 2 3
(159)
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Advances in Mathematical Physics
532 Interpolating Temperature and Body Force TogetherConsidering the temperature gradient plays the role of bodyforce we can approximate119898119879
119894minus 119887119894together by the following
equation
119898119879119894minus 119887119894asymp
119873
sum
119895=1
120572119895
119894120593119895 (160)
Thus the approximate particular solution can be written as
119906119901
119894=
3
sum
119896=1
119873
sum
119895=1
120572119895
119894Φ119895
119894119896 (161)
Consequently we can follow the same way as that for bodyforce in Method 1 and employing (140) and (142)ndash(150) toobtain the desiredΦ119895
119894119896and 119878
119897119894119895 which are the same as those for
body force case only It is noted that Method 2 has a relativelysmaller number of equations to solve for the coefficients andthe condition number of the coefficient matrix is smaller aswell which will be beneficial to the solution
Once we have obtained the particular solutions of (133)we can use them to get the modified boundary conditionsin (136) to obtain the homogeneous solution by considering(135) Then we can employ the HFS-FEM described inSection 3 for 2D problem and Section 44 for 3D problem toobtain the homogeneous solutions
6 Anisotropic Composite Materials
In materials science composite laminates are usually assem-blies of layers of fibrous composite materials which canbe joined together to provide required engineering proper-ties such as specified in-plane stiffness bending stiffnessstrength and coefficient of thermal expansion [58] Indi-vidual layers (or laminas) of the laminates consist of high-modulus high-strength fibers in a polymeric metallic orceramic matrix material From the viewpoint of microme-chanics the fiber and matrix in each lamina can be treatedas the inclusion and matrix respectively On the other handfrom the viewpoint of macromechanics both a lamina andthe whole laminate can be viewed as a general anisotropicbody by classical lamination theory Hence the analysis ofanisotropic bodies is important for understanding of themicro- or macromechanical behavior of composites [58]
In the literature there are two main approaches dealingwith generalized two-dimensional anisotropic elastic prob-lems One is Lekhnitskii formalism [59 60] which beginswith stresses as basic variables and the other is Stroh for-malism [61 62] which starts with displacements as basicvariables Both of them are formulated in terms of complexvariable functions The Stroh formalism which has beenshown to be elegant and powerful is used to find theanalytical solutions for the corresponding infinite bodies [6163] The formalism is also widely employed in the derivationof the inclusion or crack problems of anisotropic materials[64 65] Because of the limitations of the analytical solutionswhich are only available for some problems with simplegeometry and boundary conditions [66 67] numerical
methods such as finite element method (FEM) boundaryelement method (BEM) mesh free method (MFM) andHybrid-Trefftz (HT) FEM are usually resorted to solve morecomplex problems with complicated boundary constraintsand loading conditions [68ndash70] In this section we presentedthe HFS-FEM for analyzing anisotropic composite materialsbased on the associated fundamental solutions in terms ofStroh formalism [71]
61 Linear Anisotropic Elasticity
611 Basic Equations and Stroh Formalism In the Cartesiancoordinate system (119909
1 1199092 1199093) if we neglect the body force
119887119894 the equilibrium equations stress-strain laws and strain-
displacement equations for anisotropic elasticity are [61]
120590119894119895119895
= 0 (162)
120590119894119895= 119862
119894119895119896119897119890119896119897 (163)
119890119894119895=1
2(119906119894119895+ 119906119895119894) (164)
where 119894 119895 = 1 2 3 119862119894119895119896119897
is the fourth-rank anisotropicelasticity tensor The equilibrium equations can be rewrittenin terms of displacements by substituting (163) and (164) into(162) as
119862119894119895119896119897
119906119896119895119897
= 0 (165)
The boundary conditions of the boundary value problem(163)ndash(165) are
119906119894= 119906
119894on Γ
119906
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905
(166)
where 119906119894and 119905
119894are the prescribed boundary displacement
and traction vector respectively In addition 119899119894is the unit
outward normal to the boundary and Γ = Γ119906+ Γ
119905is the
boundary of the solution domainΩFor the generalized two-dimensional deformation of
anisotropic elasticity 119906119894is assumed to depend on 119909
1and 119909
2
only Based on this assumption the general solution to (165)can be written as [61 62]
u = 2Re Af (119911) 120593 = 2Re Bf (119911) (167)
where u = (1199061 1199062 1199063)119879 is the displacement vector 120593 =
(1205931 1205932 1205933)119879 is the stress function vector and f(119911) =
[1198911(1199111) 1198912(1199112) 1198913(1199113)]119879 is a function vector composed of
three holomorphic complex functions 119891120572(119911120572) 120572 = 1 2 3
which is an arbitrary function with argument 119911120572
= 1199091+
1199011205721199092and will be determined by satisfying the boundary and
loading conditions of a given problem In (167) Re standsfor the real part of a complex number 119901
120572are the material
eigenvalues with positive imaginary part andA = [a1 a2 a3]and B = [b1 b2 b3] are 3 times 3 complex matrices formed bythe material eigenvector associated with 119901
120572 which can be
obtained by the following Eigen relations [61]
N120585 = 119901120585 (168)
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 17
where N is a 6 times 6 foundational elasticity matrix and 120585 is a 6times 1 column vector defined by
N = [
N1 N2
N3 NT1] 120585 =
ab (169)
where N1 = minusTminus1RT N2 = Tminus1 N3 = RTminus1RT
minus Q and thematricesQR and T are 3 times 3 matrices extracted from119862
119894119895119896119897as
follows
119876119894119896= 119862
11989411198961 119877
119894119896= 119862
11989411198962 119879
119894119896= 119862
11989421198962 (170)
The stresses can be obtained from the derivation of stressfunctions 120593 as follows
1205901198941 = 2Re Lf1015840 (119911) 120590
1198942 = 2Re Bf1015840 (119911) (171)
where
119871 = [minus1199011b1 minus1199012b2 minus1199013b3 minus1199014b4] (172)
612 Foundational Solutions To find the fundamental solu-tion needed in our analysis we have to first derive the Greenrsquosfunction of the problem an infinite homogeneous anisotropicelastic medium loaded by a concentrated point force (or lineforce for two-dimensional problems) p = (119901
1 1199012 1199013) applied
at an internal point x = (1199091 1199092) far from the boundary The
boundary conditions of this problem can be written as
int119862
d120601 = p for any closed curve 119862 enclosing x
int119862
du = p for any closed curve 119862
limxrarrinfin
120590119894119895= 0
(173)
Thus the Greenrsquos function satisfying the above boundaryconditions is found to be [72]
119891 (119911) =1
2120587119894⟨ln (119911
120572minus 120572)⟩A119879p (174)
Therefore fundamental solutions of the problem can beexpressed as
u =1
120587Im A ⟨ln (119911
120572minus 120572)⟩AT
p
120601 =1
120587Im B ⟨ln (119911
120572minus 120572)⟩AT
p(175)
The corresponding stress components can be obtained fromstress function 120601 as
120590lowast
1198941= minus120601
2= minus
1
120587ImB⟨
119901120572
(119911120572minus 120572)⟩AT
p
120590lowast
1198942= 120601
1=1
120587ImB⟨
1
(119911120572minus 120572)⟩AT
p
(176)
where p are chosen to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879
respectively ⟨sdot⟩ stands for the diagonal matrix correspondingto subscript 120572 Im denotes the imagery part of a complexnumber and superscript 119879 denotes the matrix transpose
y
120593 x1
x2
x
Figure 7 Schematic of the relationship between global coordinatesystem (119909
1 1199092) and local material coordinate system (119909 119910)
613 Coordinate Transformation A typical composite lami-nate consists of individual layers which are usually made ofunidirectional plieswith the same or regularly alternating ori-entation A layer is generally referred to the global coordinateframe 119909
1 1199092 and 119909
3of the structural element rather than to
coordinates 119909 119910 and 119911 associated with the ply orientationSo it is necessary to transform the constitutive relationship ofeach layer from the material coordinate frame to the uniformglobal coordinate frame
For the two coordinate systems mentioned in Figure 7the angle between the axis-119909
1and axis-119909 is denoted by 120593
which is positive along the anticlockwise direction and therelationship for transformation of stress and strain com-ponents between the local material coordinates and globalcoordinates is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1 [120590119909119909 120590119910119910 120590119910119911 120590119911119909 120590119909119910]119879
[120576119909119909 120576119910119910 120576119910119911 120576119911119909 120576119909119910]119879
= (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(177)
where the transformation matrix T and its inverse matrix aredefined as
T =
[[[[[[[
[
1198882
11990420 0 2119888119904
1199042
11988820 0 minus2119888119904
0 0 119888 minus119904 0
0 0 119904 119888 0
minus119888119904 119888119904 0 0 1198882minus 1199042
]]]]]]]
]
Tminus1 =
[[[[[[[
[
1198882
1199042
0 0 minus2119888119904
1199042
1198882
0 0 2119888119904
0 0 119888 119904 0
0 0 minus119904 119888 0
119888119904 minus119888119904 0 0 1198882minus 1199042
]]]]]]]
]
(178)
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Advances in Mathematical Physics
with 119888 = cos(120593) 119904 = sin(120593) Therefore the constitutiverelationship in the global coordinate system is given by
[12059011 12059022 12059023 12059031 12059012]119879
= Tminus1C (Tminus1)119879
[12057611 12057622 12057623 12057631 12057612]119879
(179)
62 Formulations of HFS-FEM
621 Assumed Independent Fields The intraelement dis-placement fields for a particular element 119890 is approximated
in terms of a linear combination of fundamental solutions ofthe problem as
u (x) =
1199061(x)
1199062(x)
1199063(x)
= Nece (x isin Ω119890 y119904119895notin Ω
119890) (180)
where the matrix Ne and unknown vector ce can be furtherwritten as
Ne =[[[
[
119906lowast
11(x y
1199041) 119906
lowast
12(x y
1199041) 119906
lowast
13(x y
1199041) sdot sdot sdot 119906
lowast
11(x y
119904119899119904
) 119906lowast
12(x y
119904119899119904
) 119906lowast
13(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
23(x y
1199041) sdot sdot sdot 119906
lowast
12(x y
119904119899119904
) 119906lowast
22(x y
119904119899119904
) 119906lowast
23(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
32(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
13(x y
119904119899119904
) 119906lowast
32(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899
1198882119899
1198883119899]119879
(181)
in which 119899119904is the number of source points x and y
119904119895are
respectively the field point and source point in the coordinatesystem (119909
1 1199092) local to the element under considerationThe
fundamental solution 119906lowast
119894119895(x y
119904119895) is given by (175) for general
elements 120574 is a dimensionless coefficient for determiningsource points as described in previous sections
The corresponding stress fields can be expressed as
120590 (x) = [12059011 12059022
12059023
12059031
12059012]119879
= Tece (182)
where
Te =
[[[[[[[[[[
[
120590lowast
111(x y
1) 120590
lowast
211(x y
1) 120590
lowast
311(x y
1) sdot sdot sdot 120590
lowast
111(x y
119899119904
) 120590lowast
211(x y
119899119904
) 120590lowast
311(x y
119899119904
)
120590lowast
122(x y
1) 120590
lowast
222(x y
1) 120590
lowast
322(x y
1) sdot sdot sdot 120590
lowast
122(x y
119899119904
) 120590lowast
222(x y
119899119904
) 120590lowast
322(x y
119899119904
)
120590lowast
123(x y
1) 120590
lowast
223(x y
1) 120590
lowast
323(x y
1) sdot sdot sdot 120590
lowast
123(x y
119899119904
) 120590lowast
223(x y
119899119904
) 120590lowast
323(x y
119899119904
)
120590lowast
131(x y
1) 120590
lowast
231(x y
1) 120590
lowast
331(x y
1) sdot sdot sdot 120590
lowast
131(x y
119899119904
) 120590lowast
231(x y
119899119904
) 120590lowast
331(x y
119899119904
)
120590lowast
112(x y
1) 120590
lowast
231(x y
1) 120590
lowast
312(x y
1) sdot sdot sdot 120590
lowast
112(x y
119899119904
) 120590lowast
212(x y
119899119904
) 120590lowast
312(x y
119899119904
)
]]]]]]]]]]
]
(183)
The components 120590lowast
119894119895119896(x y) are given by (176) when p
119894is
selected to be (1 0 0)119879 (0 1 0)119879 and (0 0 1)119879 respectively
As a consequence the traction can be written as
1199051
1199052
1199053
= n120590 = Qece (184)
in which
Qe = nTe
n =[[
[
1198991
0 0 11989931198992
0 11989921198993
0 1198991
0 0 11989921198991
0
]]
]
(185)
The unknown c119890in (180) and (182) may be calculated using
a hybrid technique [31] in which the elements are linkedthrough an auxiliary conforming displacement frame whichhas the same form as in conventional FEM (see Figure 1)Thus the frame field is defined as
u (x) =
1
2
3
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (186)
where the symbol ldquosimrdquo is used to specify that the field is definedon the element boundary only N
119890is the matrix of shape
functions and d119890is the nodal displacements of elements
Taking the side 3-4-5 of a particular 8-node quadrilateral
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
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Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 19
element (see Figure 1) as an example N119890and d
119890can be
expressed as
N119890= [0 0 N
1N2N30 0 0]
d119890= [11990611 119906
2111990631
11990612
11990622
11990632
sdot sdot sdot 11990618
11990628
11990638]119879
(187)
where the shape functions are expressed as
Ni =[[[
[
119894
0 0
0 119894
0
0 0 119894
]]]
]
0 = [[
[
0 0 0
0 0 0
0 0 0
]]
]
(188)
and 1 2 and
3are expressed by natural coordinate 120585 isin
[minus1 1]
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(189)
622 Modified Functional for HFS-FEM With the assump-tion of two distinct intraelement field and frame field forelements we can establish the modified variational principlebased on (165) and (166) for the hybrid finite elementmethodof anisotropic materials [21 73] In the absence of the bodyforces the hybrid variational functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
=1
2∬Ω119890
120590119894119895120576119894119895dΩ minus int
Γ119905
119905119894119894dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ
(190)
where the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
(191)
and Γ119890119868is the interelement boundary of element 119890 Performing
a variation of Π119898 one obtains
120575Π119898119890
= ∬Ω119890
120590119894119895120575119906119894119895dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
(192)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (193)
and the definitions of traction force
119905119894= 120590119894119895119899119895 (194)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890
119905119894120575119906119894dΓ minus int
Γ119890119905
119905119894120575119894dΓ (195)
Then substituting (195) into (192) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
(196)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 (197)
we can finally obtain the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ + int
Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119868
119905119894120575119894dΓ
(198)
Therefore the Euler equations for (190) result in (165) and(166) because the quantities 120575119906
119894 120575119905119894 and 120575
119894may be arbitrary
As for the continuity condition between elements it can beeasily seen from the following variational of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ + int
Γ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894
(199)
This indicates that the stationary condition of the functionalsatisfies both the required boundary and interelement con-tinuity equations In addition the existence of extremum offunctional (190) can be easily proved by the so-called ldquosecondvariational approachrdquo as well which indicates functional(190) has a local extremeTherefore we can conclude that thevariational functional (190) can be used for deriving hybridfinite element formulations
623 Element Stiffness Equation Using Gaussian theoremand equilibrium equations all domain integrals in (190) canbe converted into boundary integrals as follows
Π119898119890
= minus1
2intΓ119890
119905119894119906119894dΓ + int
Γ119890
119905119894119894dΓ minus int
Γ119905
119905119894119894dΓ (200)
Substituting (180) (184) and (186) into the functional (200)yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (201)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ
(202)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34)
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
20 Advances in Mathematical Physics
7 Piezoelectric Materials
Piezoelectric materials have the property of convertingelectrical energy into mechanical energy and vice versaThis reciprocity in the energy conversion makes them veryattractive for using in electromechanical devices such assensors actuators transducers and frequency generators Toenhance understanding of the electromechanical couplingmechanism in piezoelectric materials and to explore theirpotential applications in practical engineering numerousinvestigations either analytically or numerically have beenreported over the past decades [73ndash81] In this sectionthe HFS-FEM is developed for modeling two-dimensionalpiezoelectric material [82 83]The detailed formulations andprocedures are given in the following sections
71 Basic Equations for Piezoelectric Materials For a linearpiezoelectric material in absence of body forces and electriccharge density the differential governing equations in theCartesian coordinate system 119909
119894(119894 = 1 2 3) are given by
120590119894119895119895
= 0 119863119894119894= 0 in Ω (203)
where 120590119894119895is the stress tensor 119863
119894is the electric displacement
vector andΩ is the solution domainWith strain and electricfield as the independent variables the constitutive equationsare written as
120590119894119895= 119888119894119895119896119897
120576119896119897minus 119890119896119894119895119864119896 119863
119894= 119890119894119896119897120576119896119897+ 120581119894119896119864119896 (204)
where 119888119894119895119896119897
is the elasticity tensormeasured under zero electricfield 119890
119894119896119897and 120581
119894119895are respectively the piezoelectric tensor
and dielectric tensor measured under zero strain 120576119894119895and
119864119894are the elastic strain tensor and the electric field vector
respectivelyThe relation between the strain tensor 120576119894119895and the
displacement 119906119894is given by
120576119894119895=1
2(119906119894119895+ 119906119895119894) (205)
and the electric field component 119864119894is related to the electric
potential 120601 by119864119894= minus120601
119894 (206)
The boundary conditions of the boundary value problem(203)ndash(206) can be defined by
119906119894= 119906
119894on Γ
119906 (207)
119905119894= 120590119894119895119899119895= 119905119894
on Γ119905 (208)
119863119899= 119863
119894119899119894= minus119902
119899= 119863
119899on Γ
119863 (209)
120601 = 120601 on Γ120601 (210)
where 119906119894 119905119894 119902119899 and 120601 are respectively the prescribed bound-
ary displacement the prescribed traction vector the pre-scribed surface charge and the prescribed electric potentialIn addition 119899
119894is the unit outward normal to the boundary
and
Γ = Γ119906+ Γ119905= Γ119863+ Γ120601 (211)
is the boundary of the solution domainΩ
For the transversely isotropic material if 1199091-1199092is taken
as the isotropic plane one can employ either 1199091-1199093or 119909
2-1199093
plane to study the plane electromechanical phenomenonThus choosing the former and considering the plane strainconditions (120576
22= 120576
32= 120576
12= 0 and 119864
2= 0) (204) can be
reduced to
12059011
12059033
12059013
=[[
[
11988811
11988813
0
11988813
11988833
0
0 0 11988844
]]
]
12057611
12057633
212057613
minus[[
[
0 11989031
0 11989033
11989015
0
]]
]
1198641
1198643
1198631
1198633
= [
0 0 11989015
11989031
11989033
0]
12057611
12057633
212057613
+ [
12058111
0
0 12058133
]
1198641
1198643
(212)
For the plane stress piezoelectric problem (12059022= 120590
32= 120590
12=
0 and 1198632= 0) the constitutive equations can be obtained by
replacing the coefficients 11988811 11988813 11988833 11988844 11989015 11989031 11989033 12058111 and
12058133in (212) as
119888lowast
11= 11988811minus1198882
12
11988811
119888lowast
13= 11988813minus1198881211988813
11988811
119888lowast
33= 11988833minus1198882
13
11988811
119888lowast
44= 11988844 119890
lowast
15= 11989015
119890lowast
31= 11989031minus1198881211989031
11988811
119890lowast
33= 11989033minus1198881311989031
11988811
120581lowast
11= 12058111 120581
lowast
33= 12058133+1198902
31
11988811
(213)
72 Assumed Independent Fields For the piezoelectric prob-lems HFS-FEM is based on assuming two distinct displace-ment and electric potential (DEP) fields intraelement DEPfield u and an independent DEP frame field u along elementboundaries [21 29]
721 Intraelement Field The intraelement DEP field u iden-tically fulfills the governing differential equations (203) andis approximated by a linear combination of foundationalsolutions at different source points located outside of theelement domain
ue =
1199061
1199062
120601
=
119899119904
sum
119895=1
[[[
[
119906lowast
11(x y
119904119895) 119906
lowast
21(x y
119904119895) 119906
lowast
31(x y
119904119895)
119906lowast
12(x y
119904119895) 119906
lowast
22(x y
119904119895) 119906
lowast
32(x y
119904119895)
119906lowast
13(x y
119904119895) 119906
lowast
23(x y
119904119895) 119906
lowast
33(x y
119904119895)
]]]
]
sdot
1198881119895
1198882119895
1198883119895
= Nece (x isin Ω119890 y119904119895notin Ω
119890)
(214)
where the fundamental solution matrix Ne is now given by
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 21
Ne =
[[[[[
[
119906lowast
11(x y
1199041) 119906
lowast
21(x y
1199041) 119906
lowast
31(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
12(x y
1199041) 119906
lowast
22(x y
1199041) 119906
lowast
32(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
119906lowast
13(x y
1199041) 119906
lowast
23(x y
1199041) 119906
lowast
33(x y
1199041) sdot sdot sdot 119906
lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
) 119906lowast
33(x y
119904119899119904
)
]]]]]
]
ce = [11988811 11988821
11988831
sdot sdot sdot 1198881119899119904
1198882119899119904
1198883119899119904
]119879
(215)
in which x and y119904119895
are respectively the field point (iethe nodal points of the element in this work) and sourcepoint in the local coordinate system (119909
1 1199092) The component
119906lowast
119894119895(x y
119904119895) is the induced displacement component (119894 = 1 2)
or electric potential (119894 = 3) in 119894-direction at the field point119909 due to a unit point load (119895 = 1 2) or point charge(119895 = 3) applied in 119895-direction at the source point y
119904119895 The
fundamental solution 119906lowast119894119895(x y
119904119895) is given as [76 82 84]
119906lowast
11=
1
12058711987211
3
sum
119895=1
11990411989511199051
(2119895)1ln 119903119895
119906lowast
12=
1
12058711987212
3
sum
119895=1
11990411989521199051
(2119895)2arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
13=
1
12058711987213
3
sum
119895=1
11990411989531199051
(2119895)3arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
21=
1
12058711987211
3
sum
119895=1
11988911989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
22=
1
12058711987212
3
sum
119895=1
11988911989521199051
(2119895)2ln 119903119895
119906lowast
23=
1
12058711987213
3
sum
119895=1
11988911989531199051
(2119895)3ln 119903119895
119906lowast
31=
1
12058711987211
3
sum
119895=1
11989211989511199051
(2119895)1arc 119905119892
1199091minus 1199091199041
119904119895(1199093minus 1199091199043)
119906lowast
32=
1
12058711987212
3
sum
119895=1
11989211989521199051
(2119895)2ln 119903119895
119906lowast
33=
1
12058711987213
3
sum
119895=1
11989211989531199051
(2119895)3ln 119903119895
(216)
where 119903119895= radic(119909
1minus 1199091199041)2+ 1199042
119895(1199093minus 1199091199043)2 and 119904
119895is the three
different roots of the characteristic equation 1198861199046119894minus 119887119904
4
119894+ 119888119904
3
119894minus
119889 = 0 The source point y119904119895(119895 = 1 2 119899
119904) can be generated
by the following method [36]
y119904= x0+ 120574 (x
0minus x119888) (217)
Making use of (205) and the expression of intraelementDEP field u in (214) the corresponding stress and electricdisplacement in (212) can be written as
120590 = Tece (218)
where 120590 = [12059011
12059022
12059012
11986311198632]119879 and
Te =
[[[[[[[[[[[[[
[
120590lowast
11(x y
1199041) 120590
lowast
12(x y
1199041) 120590
lowast
13(x y
1199041) sdot sdot sdot 120590
lowast
11(x y
119904119899119904
) 120590lowast
12(x y
119904119899119904
) 120590lowast
13(x y
119904119899119904
)
120590lowast
21(x y
1199041) 120590
lowast
22(x y
1199041) 120590
lowast
23(x y
1199041) sdot sdot sdot 120590
lowast
21(x y
119904119899119904
) 120590lowast
22(x y
119904119899119904
) 120590lowast
23(x y
119904119899119904
)
120590lowast
31(x y
1199041) 120590
lowast
32(x y
1199041) 120590
lowast
33(x y
1199041) sdot sdot sdot 120590
lowast
31(x y
119904119899119904
) 120590lowast
32(x y
119904119899119904
) 120590lowast
33(x y
119904119899119904
)
120590lowast
41(x y
1199041) 120590
lowast
42(x y
1199041) 120590
lowast
43(x y
1199041) sdot sdot sdot 120590
lowast
41(x y
119904119899119904
) 120590lowast
42(x y
119904119899119904
) 120590lowast
43(x y
119904119899119904
)
120590lowast
51(x y
1199041) 120590
lowast
52(x y
1199041) 120590
lowast
53(x y
1199041) sdot sdot sdot 120590
lowast
51(x y
119904119899119904
) 120590lowast
52(x y
119904119899119904
) 120590lowast
53(x y
119904119899119904
)
]]]]]]]]]]]]]
]
(219)
in which 120590lowast
119894119895(x y
119904119895) denotes the corresponding stress
components (119894 = 1 2 3) or electric displacement (119894 = 4 5)along 119894-direction at the field point 119909 due to a unit point
load (119895 = 1 2) or a unit point charge (119895 = 3) applied in119895-direction at the source point 119910119904 and can be derived from(216) and are listed below which are derived by substituting
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
22 Advances in Mathematical Physics
the fundamental solutions into constitutive equations[85]
120590lowast
11=
1
12058711987211
3
sum
119895=1
[(119888111199041198951minus 119888131198891198951119904119895minus 119890311198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
12=
1
12058711987212
3
sum
119895=1
[(119888111199041198952+ 119888131198891198952119904119895+ 119890311198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
13=
1
12058711987213
3
sum
119895=1
[(119888111199041198953+ 119888131198891198953119904119895+ 119890311198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
21=
1
12058711987211
3
sum
119895=1
[(119888131199041198951minus 119888331198891198951119904119895minus 119890331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
22=
1
12058711987212
3
sum
119895=1
[(119888131199041198952+ 119888331198891198952119904119895+ 119890331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
23= minus
1
12058711987213
3
sum
119895=1
[(119888131199041198953+ 119888331198891198953119904119895+ 119890331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
31=
1
12058711987211
3
sum
119895=1
[(119888441199041198951119904119895+ 119888441198891198951+ 119890151198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
32=
1
12058711987212
3
sum
119895=1
[(minus119888441199041198952119904119895+ 119888441198891198952+ 119890151198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
33= minus
1
12058711987213
3
sum
119895=1
[(minus119888441199041198953119904119895+ 119888441198891198953+ 119890151198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
41=
1
12058711987211
3
sum
119895=1
[(119890151199041198951119904119895+ 119890151198891198951minus 120582
111198921198951)
sdot 1199051
(2119895)1
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
42=
1
12058711987212
3
sum
119895=1
[(minus119890151199041198952119904119895+ 119890151198891198952minus 120582
111198921198952)
sdot 1199051
(2119895)2
1199091minus 1199091199041
1199032
119895
]
120590lowast
43= minus
1
12058711987213
3
sum
119895=1
[(minus119890151199041198953119904119895+ 119890151198891198953minus 120582
111198921198953)
sdot 1199051
(2119895)3
1199091minus 1199091199041
1199032
119895
]
120590lowast
51=
1
12058711987211
3
sum
119895=1
[(119890311199041198951minus 119890331198891198951119904119895+ 120582
331198921198951119904119895)
sdot 1199051
(2119895)1
1199091minus 1199091199041
1199032
119895
]
120590lowast
52=
1
12058711987212
3
sum
119895=1
[(119890311199041198952+ 119890331198891198952119904119895minus 120582
331198921198952119904119895)
sdot 1199051
(2119895)2
119904119895(1199093minus 1199091199043)
1199032
119895
]
120590lowast
53= minus
1
12058711987213
3
sum
119895=1
[(119890311199041198953+ 119890331198891198953119904119895minus 120582
331198921198953119904119895)
sdot 1199051
(2119895)3
119904119895(1199093minus 1199091199043)
1199032
119895
]
(220)
in which the coefficients 119904119894119895 119889119894119895 119892119894119895 1199051198941198951198721111987212 and119872
13are
defined as in literature [84]From (204) (208) and (209) the generalized traction
forces and electric displacement are given as
1199051
1199052
119863119899
=
Q1
Q2
Q3
ce = Qece (221)
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 23
where
Qe = nTe
n =[[
[
1198991
0 1198992
0 0
0 11989921198991
0 0
0 0 0 11989911198992
]]
]
(222)
722 Auxiliary Frame Field For the two-dimensional piezo-electric problem under consideration the frame field isassumed as
u (x) =
1
2
120601
=
N1
N2
N3
d119890= N
119890d119890 (x isin Γ
119890) (223)
where N119890is a matrix of the corresponding shape functions
For the side 3-4-5 of a particular quadratic element as shownin Figure 1 the shape function matrix N
119890and nodal vector de
can be given in the form
N119890
=
[[[[
[
0 sdot sdot sdot 0 1
0 0 2
0 0 3
0 0 0 sdot sdot sdot 0
0 sdot sdot sdot 0 0 1
0 0 2
0 0 3
0 0 sdot sdot sdot 0
0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
6
0 0 1
0 0 2
0 0 30 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟
9
]]]]
]
de = [11990611 11990621
1206011sdot sdot sdot 119906
1411990624
1206014sdot sdot sdot 119906
1811990628
1206018]119879
(224)
where the shape functions N119890are expressed by natural
coordinate 120585
1= minus
120585 (1 minus 120585)
2
2= 1 minus 120585
2
3=120585 (1 + 120585)
2
(120585 isin [minus1 1])
(225)
73 HFS-FEM Formulations
731 Variational Principles Based on the assumption oftwo distinct DEP fields the Euler equations of the pro-posed variational functional should also satisfy the followinginterelement continuity requirements in addition to (203)ndash(210) [82 83]
119906119894119890= 119906
119894119891120601119890= 120601
119891(on Γ
119890cap Γ119891 conformity) (226)
119905119894119890+ 119905119894119891= 0 119863
119899119890+ 119863
119899119891= 0 (on Γ
119890cap Γ119891 reciprocity)
(227)
where ldquo119890rdquo and ldquo119891rdquo stand for any two neighboring elementsEquations (203)ndash(210) together with (226) and (227) can nowbe taken as the basis to establish the modified variational
principle for the hybrid finite elementmethod of piezoelectricmaterials [21 29]
Since the stationary conditions of the traditional potentialor complementary variational functional cannot satisfy theinterelement continuity condition required in the proposedHFS-FEM new modified variational functional should bedeveloped In the absence of the body forces and electriccharge density the hybrid functional Π
119898119890for a particular
element 119890 is constructed as
Π119898119890
= Π119890+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ (228)
where
Π119890=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(229)
and the boundary Γ119890of the element 119890 is
Γ119890= Γ119890119906cup Γ119890119905cup Γ119890119868= Γ119890120601cup Γ119890119863
cup Γ119890119868
Γ119890119906= Γ119890cap Γ119906 Γ
119890119905= Γ119890cap Γ119905
Γ119890120601= Γ119890cap Γ120601 Γ
119890119863= Γ119890cap Γ119863
(230)
and Γ119890119868is the interelement boundary of element 119890 Compared
to the functional employed in the conventional FEM thepresent hybrid functional is constructed by adding twointegral terms related to the intraelement and element frameDEP fields to guarantee the satisfaction of displacement andelectrical potential continuity condition on the commonboundary of two adjacent elements
It can be proved that the stationary conditions of theabove functional (228) lead to (203)ndash(210) To this endperforming a variation of Π
119898 one obtains [86]
120575Π119898119890
= 120575Π119890+ intΓ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894(120575
119894minus 120575119906
119894)] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899(120575120601 minus 120575120601)] dΓ
(231)
in which the first term is
120575Π119890= ∬
Ω119890
120590119894119895120575120576119894119895dΩ +∬
Ω119890
119863119894120575119864119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
= ∬Ω119890
120590119894119895120575119906119894119895dΩ +∬
Ω119890
119863119894120575120601119894dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ
(232)
Applying Gaussian theorem
∬Ω119890
119891119894dΩ = int
Γ119890
119891 sdot 119899119894dΓ (233)
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
24 Advances in Mathematical Physics
and the definitions of traction force and electrical displace-ment
119905119894= 120590119894119895119899119895 119863
119899= 119863
119894119899119894 (234)
we obtain
120575Π119890= minus∬
Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ + int
Γ119890
119905119894120575119906119894dΓ
+ intΓ119890
119863119899120575120601dΓ minus int
Γ119890119905
119905119894120575119894dΓ minus int
Γ119890119863
119863119899120575120601dΓ
(235)
Then substituting (235) into (231) gives
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ minus int
Γ119890119905
119905119894120575119894dΓ
minus intΓ119890119863
119863119899120575120601dΓ + int
Γ119890
[(119894minus 119906119894) 120575119905
119894+ 119905119894120575119894] dΓ
+ intΓ119890
[(120601 minus 120601) 120575119863119899+ 119863
119899120575120601] dΓ
(236)
Considering the fact that
intΓ119890119906
119905119894120575119894dΓ = 0 int
Γ119890120601
119863119899120575120601dΓ = 0 (237)
we finally get the following form
120575Π119898119890
= minus∬Ω119890
120590119894119895119895120575119906119894dΩ minus∬
Ω119890
119863119894119894120575120601dΩ
+ intΓ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119868
119905119894120575119894dΓ + int
Γ119868
119863119899120575120601dΓ
(238)
Therefore the Euler equations for (238) result in (203)ndash(210)and (226) because the quantities 120575119906
119894 120575119905119894 120575120601 120575119863
119899 120575
119894 and 120575120601
may be arbitrary As for the continuity condition of (227) itcan easily be seen from the following variation of two adjacentelements such as 119890 and 119891
120575Π119898(119890cup119891)
= minus∬Ω119890cupΩ119891
120590119894119895119895120575119906119894dΩ minus∬
Ω119890cupΩ119891
119863119894119894120575120601dΩ
+ intΓ119890119905+Γ119890119905
(119905119894minus 119905119894) 120575
119894dΓ + int
Γ119890119863+Γ119891119863
(119863119899minus 119863
119899) 120575120601dΓ
+ intΓ119890+Γ119891
(119894minus 119906119894) 120575119905
119894dΓ + int
Γ119890+Γ119891
(120601 minus 120601) 120575119863119899dΓ
+ intΓ119890119891119868
(119905119894119890+ 119905119894119891) 120575
119894+ intΓ119890119891119868
(119863119899119890+ 119863
119899119891) 120575120601dΓ
(239)
which indicates that the stationary condition of the func-tional satisfies both the required boundary and interelementcontinuity equations In addition the existence of extremumof functional (228) can be easily proved by the ldquosecondvariational approachrdquo as well which indicates functional(228) has a local extreme
732 Element Stiffness Equation The element stiffness equa-tion can be generated by setting 120575Π
119898119890= 0 To simplify the
derivation we first transform all domain integrals in (228)into boundary ones With Gaussian theorem the functionalin (228) may be simplified as
Π119898119890
=1
2∬Ω119890
(120590119894119895120576119894119895+ 119863
119894119864119894) dΩ minus int
Γ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
=1
2(intΓ119890
119905119894119906119894dΓ minus∬
Ω119890
120590119894119895119895119906119894dΩ)
+1
2(intΓ119890
119863119899120601dΓ minus∬
Ω119890
119863119894119894120601dΩ)
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
+ intΓ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
(240)
Due to the satisfaction of the equilibrium equation withthe constructed intraelement fields we have the followingexpression for the HT finite element model
Π119898119890
=1
2intΓ119890
119905119894119906119894dΓ + 1
2intΓ119890
119863119899120601dΓ minus int
Γ119905
119905119894119894dΓ
minus intΓ119863
119863119899120601dΓ + int
Γ119890
119905119894(119894minus 119906119894) dΓ + int
Γ119890
119863119899(120601 minus 120601) dΓ
= minus1
2intΓ119890
(119905119894119906119894+ 119863
119899120601) dΓ + int
Γ119890
(119905119894119894+ 119863
119899120601) dΓ
minus intΓ119905
119905119894119894dΓ minus int
Γ119863
119863119899120601dΓ
(241)
Substituting (214) (223) and (221) into the above functional(241) yields the formulation as
Π119898119890
= minus1
2c119890
119879H119890c119890+ c119890
119879G119890d119890minus d119890
119879g119890 (242)
where
H119890= intΓ119890
Q119879119890N119890dΓ G
119890= intΓ119890
Q119879119890N119890dΓ
g119890= intΓ119905
N119879119890tdΓ + int
Γ119863
N119879119890DdΓ
(243)
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 25
x1
x2
x3
L = 10
W = 10
A
H = 10
P = 10
Figure 8 Cubic block under uniform tension and body force geometry boundary condition and loading
0 1000 2000 3000 400030
32
34
36
38
40
42
44
46
Number of degrees of freedomHFS-FEMC3D8Hybrid EAS
Reference
Disp
lace
men
tu1
(10minus4
m)
(a)
0 1000 2000 3000 400090
95
100
105
110
115
120
125
Number of degrees of freedom
Reference
Stre
ss12059011
(MPa
)
HFS-FEMC3D8Hybrid EAS
(b)
Figure 9 Cubic block with body force under uniform distributed load convergent study of (a) displacement 1199061and (b) stress 120590
11
X Y
ZMesh 1
(a)
Mesh 2
X Y
Z
(b)
Mesh 3
X Y
Z
(c)
Figure 10 Cubic block under uniform tension and body force (a) mesh 1 (4 times 4 times 4 elements) (b) mesh 2 (6 times 6 times 6 elements) and (c) mesh3 (10 times 10 times 10 elements)
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
26 Advances in Mathematical Physics
u124222181614121080604020
(a)
230220210200190180170160150140130120110100908070605040
(b)
Figure 11 Contour plots of (a) displacement 1199061and (b) stress 120590
11of the cube
a
bT =ln(rb)
ln(ab)T0
(a)
X
Y
Z
(b)
Figure 12 (a) Geometry and boundary conditions of the long cylinder with axisymmetric temperature change and (b) mesh configurationsof a quarter of circular cylinder (128 eight-node elements)
The stationary condition of the functional Π119898119890
with respectto c
119890and d
119890 respectively yields (33) and (34) Following
the procedure described in [21 22] the missing rigid-bodymotion can be recovered by setting the augmented internalfield of a particular element 119890 as
u119890= N
119890c119890+[[
[
1 0 1199092
0
0 1 minus11990910
0 0 0 1
]]
]
c0 (244)
where the undetermined rigid-bodymotion parameter c0can
be calculated using the least square matching of ue and ue atelement nodes
min =
119899
sum
119894=1
[(1199061119894minus 1119894)2
+ (1199062119894minus 2119894)2
+ (120601119894minus 120601119894)2
] (245)
which finally gives
c0 = Rminus1119890re (246)
Re =119899
sum
119894=1
[[[[[
[
1 0 1199092119894
0
0 1 minus1199091119894
0
1199092119894
minus1199091119894
1199092
1119894+ 1199092
21198940
0 0 0 1
]]]]]
]
(247)
re =119899
sum
119894=1
[[[[[
[
Δ1199061198901119894
Δ1199061198902119894
Δ11990611989011198941199092119894minus Δ119906
11989021198941199091119894
Δ120601119890119894
]]]]]
]
(248)
in whichΔu119890119894= (u
119890minus u119890)|nodei and 119899 is the number of element
nodes As a consequence c0can be calculated by (246) once
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 27
50 75 100 125 150 175 200minus25
minus20
minus15
minus10
minus05
00
05
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
120590r
r (m)
(a)
Method 1 (HFS-FEM)Method 2 (HFS-FEM)Exact solution
50 75 100 125 150 175 200minus12
minus9
minus6
minus3
0
3
6
1205900
r (m)
(b)
Figure 13 (a) Radial thermal stresses and (b) circumferential thermal stresses with the cylinder radius
120590r
0 minus02minus04minus06minus08 minus1 minus12 minus14minus16 minus18 minus2 minus22
20
15
10
5
020151050
y
x
(a)
120590t
4 3 2 1 0 minus1 minus2 minus3 minus4 minus5 minus6 minus7 minus8 minus9
20
15
10
5
020151050
y
x
(b)
Figure 14 Contour plots of (a) radial and (b) circumferential thermal stresses (the background mesh in (a) and (b) is used for plots of thecalculated results in postprocessing)
the nodal DEP fields d119890and the interpolation coefficients c
119890
are respectively determined by (214) and (223) Then thecomplete DEP fields u
119890can be obtained from (244)
74 Normalization The order of magnitudes of the materialconstants and the corresponding field variables in piezoelec-tricity have a wide spectrum as large as 1019 in SI unit This
will lead to ill-conditioned matrix of the system [72 87]To resolve this problem normalization of each quantity byits reference value is employed The reference values for thestiffness piezoelectric stress constant dielectric constantsand strain are selected to be 119888
0= 10
11(Nm2) 119890
0=
101(NmV) 119896
0= 10
minus9(CmV) and 120576
0= 10
minus3(Vm)
respectively The reference values of other quantities as
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
28 Advances in Mathematical Physics
1
1
1
T = 30x3
b3 = minus2000[(x minus 05)2 + (y minus 05)2]
x2
x3
x1
(a)
X
Y
Z
(b)
X
Y
Z
(c)
Figure 15 (a) Schematic of the 3D cube under arbitrary temperature and body force (b) mesh used by HFS-FEM (125 20-node brickelements) and (c) mesh for ABAQUS (8000 C3D20R elements)
shown in Table 1 are determined in terms of these fourfundamental reference variables and the characteristic length1199090= 10
0(m)of the problem so that the normalized governing
equations remain in exactly the same form as the originalequations
8 Numerical Examples
Several numerical examples are presented in this section toillustrate the application of theHFS-FEM and to demonstrateits effectiveness and accuracy Unless otherwise indicatedmesh convergence tests were conducted for the referencesolutions obtained fromABAQUS in the following examples
81 Cubic Block under Uniform Tension and Body ForceAn isotropic cubic block with dimension 10 times 10 times 10and subject to a uniform tension as shown in Figure 8 isconsidered in this example A constant body force of 10Mpaand uniform distributed tension of 100MPa are applied tothe cube Figure 9 presents the displacement component1199061and the stress component 120590
11at point 119860 of the block
which are calculated by the HFS-FEM on the three mesheswith distorted 8-node brick elements (Figure 10) The resultscalculated by ABAQUS with a very fine mesh (with 40 times 40times 40 C3D8 and EAS element with 68921 nodes) are taken asa reference value for comparisonThe results from C3D8 andEAS elements in ABAQUS are also presented in Figure 9 for
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 29
z
u3
ABAQUSHFS-FEM
00 02 04 06 08 10minus0035
minus0030
minus0025
minus0020
minus0015
minus0010
minus0005
0000
(a)
minus1000
minus500
0
12059033
ABAQUSHFS-FEM
z
00 02 04 06 08 10
(b)
Figure 16 (a) Displacement 1199063and (b) stress 120590
33along one cube edge when subjected to arbitrary temperature and body force
Table 1 Reference values for material constants and field variables in piezoelectricity derived from basic reference variables 1198880 1198900 1198960 1205760 and
1199090
Displacement 1199060= 119909
01205760= 10
minus3(m) Electric Potential 120601
0= 119909
01198640= 10
7(V)
Stress 1205900= 11988801205760= 10
8(Nm2) Electric induction 119863
0= 119896
01198640= 10
minus2(Cm2)
Compliance 1199040= 12057601205900= 10
minus11(m2N) Impermeability 120573
0= 119864
01198630= 10
9(mVC)
Electric field 1198640= 120590
01198900= 10
7(Vm) Piezoelectric strain constant 119892
0= 119864
01205900= 10
minus1(mVN)
comparison It can be seen from these figures that the resultsobtained from both the HFS-FEM and ABAQUS converge tothe benchmark value with the number of degree of freedom(DOF) increasing For Mesh 1 the hybrid EAS element hasthe best performance while for Mesh 2 and Mesh 3 it canbe seen that HFS-FEM with 8-node brick elements exhibitsbetter accuracy in both displacement and stress comparedwith EAS element in traditional FEM Contour plots of 119906
1
and 12059011obtained by HFS-FEM on Mesh 3 are also presented
in Figure 11 It should be noted that for problems involvingbody forces the accuracy of the RBF interpolation has to beconsidered for a satisfactory solution The details on the RBFinterpolation can be found in previous literatures [43 88ndash90]
82 Circular Cylinder with Axisymmetric TemperatureChange In this example a long circular cylinder with axi-symmetric temperature change in domain is consideredBoth inside and outside surfaces of the cylinder are as-sumed to be free from traction The temperature 119879 changeslogarithmically along the radial direction With the sym-metry condition of the problem only one quarter of thecylinder is modeled The configurations of geometry andthe boundary conditions are shown in Figure 12 In ourcomputation the parameters 119886 = 5 119887 = 20 119864 = 1000] = 03 120572 = 0001 and 119879
0= 10 The two approaches listed in
Section 4 to approximate the body force and temperature arediscussed and analyzed in this example
Figure 13 presents the variation of the radial and thecircumferential thermal stresses with the cylinder radiusrespectively in which the theoretical values are given forcomparison [39] It is seen from Figure 13 that the resultsfrom Method 2 are much better than those obtained fromMethod 1 for both radial and circumferential stress It can beinferred that the error may be to a large extent due to the RBFinterpolation for which the number of interpolation pointshas a significant influence on its accuracy
Figure 14 displays the contour plots of (a) radial and(b) circumferential thermal stresses (the meshes used forcontour plot are different from that for calculation dueto using quadratic elements) It demonstrates that treatingtemperature gradient and body force together is more supe-rior to dealing with them separately However we have torely on Method 1 when the temperature change is discretedistribution or the gradient of the temperature field is notavailable
83 3D Cube under Arbitrary Temperature and Body ForceAs shown in Figure 15 a 3D cube of 1 times 1 times 1 with centerlocated at (05 05 05) is considered in this example Thematerial properties of the cube are Youngrsquos modulus 119864 =
5000 Poissonrsquos ratio ] = 03 and linear thermal expansion
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
30 Advances in Mathematical Physics
L
B
A
W
2b
2a
x2
x1
1205900
1205900
(a)
Z
Y
X
X
Y
Z
(b)
Z
Y
X
X
Y
Z
(c)
Figure 17 (a) Schematic of an orthotropic composite plate with an elliptic hole under uniform tension and its mesh configurations for (b)HFS-FEM 1515 quadratic elements and (c) ABAQUS 9498 quadratic elements
coefficient 120572 = 0001 The bottom surface is fixed on theground and the temperature distribution and body force areassumed to be
119879 = 301199093 119887
1= 0 119887
2= 0
1198873= minus2000 [(119909 minus 05)
2+ (119910 minus 05)
2
]
(249)
Because there is no analytical solution available the resultsfrom ABAQUS herein are employed for comparison Themeshes used by HFS-FEM and ABAQUS are given inFigure 15
Figure 16 presents the displacement 1199063and stress 120590
33
along one edge of the cube which is coinciding with 1199093axis
It can be seen that the results from HFS-FEM again agreevery well with those by ABAQUS It is demonstrated that theproposedHFS-FEM is able to predict the response of 3D ther-moelastic problems under arbitrary temperature and bodyforce It is also shown that HFS-FEM with RBF interpolationcan give satisfactory results using coarse meshes
84 Orthotropic Composite Plate with an Elliptic Hole underTension A finite composite plate containing an ellipticalhole (Figure 17) is investigated in this example A uniform
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 31
0 10 20 30 40 50 60 70 80 90minus2
minus1
0
1
2
3
4
5
6
7
120593 = 0∘ (ABAQUS)120593 = 30∘ (ABAQUS)120593 = 45∘ (ABAQUS)120593 = 60∘ (ABAQUS)120593 = 90∘ (ABAQUS)
120593 = 0∘ (HFS-FEM)120593 = 30∘ (HFS-FEM)120593 = 45∘ (HFS-FEM)120593 = 60∘ (HFS-FEM)120593 = 90∘ (HFS-FEM)
Hoo
p str
ess120590
120579120579
(GPa
)
Angle 120579 (deg)
Figure 18 Variation of hoop stresses along the rim of the ellipticalhole for different fiber orientation 120593
tension of 1205900
= 1GPa is applied in 1199092direction The
material parameters of the orthotropic plate are taken as 1198641=
113GPa 1198642= 527GPa 119866
12= 285GPa and V
12= 045
The length and width of the plate are 119871 = 20mm and 119882 =
20mm the major and minor lengths of the ellipses 119886 and 119887
are respectively 2mm and 1mm In the computation planestress condition is used The mesh configurations used byHFS-FEM and ABAQUS are given in Figure 17
Figure 18 shows the corresponding variation of the hoopstress along the rim of the elliptical hole when the orien-tation angle 120593 of reinforced fibers is equal to 0∘ 45∘ and90∘ respectively It is found from Figure 18 that the resultsfrom HFS-FEM have a good agreement with the referencesolutions from ABAQUS This indicates that the proposedmethod is able to capture the variations of the hoop stressinduced by the elliptical hole in the plate The contour plotsof stress components 120590
11 12059022 and 120590
12around the elliptic hole
in the composite plate for several different fiber angles areshown in Figure 19
Figure 20 shows the stress concentration factor (SCF)along with the inclined angle 120593 of the reinforced fibers whichexhibits a good agreement with the solutions fromABAQUSIt is obvious that the SCF of the punched plate rises with theincreasing fiber angle 120593 It is found from Figure 20 that thelargest SCF occurs at 120593 = 90
∘ whereas the smallest appears at120593 = 0
∘ It indicates the effectiveness of the proposed methodin predicting the SCF for anisotropic composites as well
85 Isotropic Plate with Multianisotropic Inclusions In thisexample a multi-inclusion problem is investigated to showthe capability of theHFS-FEM to deal with both isotropic andanisotropic materials in a unified way As shown in Figure 21
an isotropic plate containing multianisotropic inclusions ofsquare geometry (edge length 119886 = 2) is considered Thedistance between any two inclusions is assumed to be 119887 =
3 The material parameters for the inclusions are chosen as1198641= 13445GPa 119864
2= 119864
3= 1103GPa 119866
23= 298GPa
11986631
= 11986612
= 285GPa and V23
= 049 V31
= V12
=
0301Thematerial parameters for isotropicmatrix are elasticmodules 119864 = 28Gpa and poisonrsquos ratio ] = 03 The meshconfiguration of the plate forHFS-FEM is shown in Figure 21which uses 272 quadratic general elements
In general the Stroh formalism is suitable for theanisotropic material with distinct material eigenvalues andit fails for the degenerated materials like isotropic materialwith repeated eigenvalues 119901
120572= 119894 (120572 = 1 2 3) [59] However
a small perturbation of the material constants such as 1199011=
(1 minus 0004)119894 1199012= 119894 and 119901
3= (1 + 0004)119894 can be applied to
make the eigenvalues distinct and the results can be appliedconveniently
Table 2 shows the displacement and stresses at points AB and C as indicated in Figure 21 It is observed that thereis a good agreement between the results by the HFS-FEMand those from ABAQUS using very fine mesh in whichthe maximum relative error for displacement and stress byHFS-FEM occur at Point B (ie 119909
2= 0) and are 07
and 13 respectively Additionally it can be found that theresults from HFS-FEM are better than those from ABAQUSusing the same mesh Although the stress at the vicinity ofthe inclusions (Point C) has a little degradation due to thehigh stress concentration and stress contrast at the adjacentelements the displacement still agrees well with the referencesolution The variations of displacement components 119906
1and
1199062along the right edge (119909 = 8) by HFS-FEM are shown in
Figure 22
86 Infinite Piezoelectric Medium with Hole Consider aninfinite piezoelectric plane with a circular hole as shownin Figure 23 The material parameters are given in Table 3Suppose that mechanical load 120590infin
119909119909= 1205900= 10 parallel to the 119909
axis is imposed at infinity with traction and electric charge-free at the boundary of the hole In our calculation the radiusof the hole is set to be 119903 = 1 and 119871119903 = 20 is employed in theanalysis
Figure 24 shows the distribution of hoop stress 120590120579and
radial stress 120590119903along the line 119911 = 0 for remote loading 120590infin
119909119909
and along the line 119909 = 0 for remote loading 120590infin119911119911 respectively
Figure 25 presents the variations of the normalized stress1205901205791205900and the normalized electric displacement119863
1205791205900times10
10
along the hole edge under remote mechanical loading It isobvious that the results obtained from HFS-FEM agree wellwith the results from ABAQUS and Sosa [74]
It can be seen from Figure 24 that hoop stress 120590120579has
maximum value on the rim of the hole and it decreasesdramatically with the increase of the distance from the holeedge It is also shown that 120590
120579tends to be equal to the remote
applied load 1205900when 119903 increases toward infinity Compared
with the hoop stress 120590120579 it is obvious that the radial stress 120590
119903
is much smaller and usually does not need to be consideredIt is obvious that loading along the poling direction willproduce smaller stress concentration due to coupling effect
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
32 Advances in Mathematical Physics
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
1312
1064
0816
0567
0319
0071
minus0178
minus0426
minus0674
minus0923
minus1171
minus1419
minus1668
Y
Z Z ZX
Y
X
Y
X
4624
4234
3844
3454
3064
2673
2283
1893
1503
1113
0723
0333
minus0057
1646
1372
1098
0824
0551
0277
0003
minus0271
minus0545
minus0819
minus1093
minus1366
minus1640
(a)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
4475
4029
3584
3138
2693
2247
1802
1356
0911
0465
0020
minus0425
minus0871
3241
2925
2610
2294
1979
1663
1348
1032
0716
0401
0085
minus0230
minus0546
2679
2489
2199
1959
1719
1480
1240
1000
0760
0520
0280
0041
minus0199
(b)
Y
Z Z ZX
Y
X
Y
X
S 12059011 S 12059022 S 12059012(Avg 75) (Avg 75) (Avg 75)
6497
5952
5407
4862
4316
3771
3226
2681
2136
1590
1045
0500
minus0045
0910
0767
0624
0480
0337
0194
0051
minus0092
minus0235
minus0379
minus0522
minus0655
minus0808
1583
1319
1055
0791
0526
0262
minus0002
minus0266
minus0531
minus0795
1059
minus minus1323
minus1588
(c)
Figure 19 Contour plots of stress components around the elliptic hole in the composite plate (a) 120593 = 0∘ (b) 120593 = 45
∘ and (c) 120593 = 90∘
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 33
40
45
50
55
60
65
70
ABAQUS HFS-FEM
0 10 20 30 40 50 60 70 80 90
SCF
(1205901205791205791205900)
Angle 120593 (deg)
Figure 20 Variation of SCF with the lamina angle 120593
W
L
bb
ab
aaa aa
a
1205900
A
x2
x1
BC
(a)
X
Y
8
6
4
2
0
minus2
minus4
minus6
minus886420minus2minus4minus6minus8Z
(b)
Figure 21 (a) Schematic and (b) mesh configuration of an isotropic plate with multianisotropic inclusions
Table 2 Comparison of displacement and stress at points A and B
Items Points HFS-FEM (272 elements) ABAQUS (272 elements) ABAQUS (30471 elements )
Disp 1199061
A 004322 004318 004335B 003719 003721 003744C 003062 003076 003091
Stress 12059011
A 100446 99219 99992B 98585 98304 99976C 206453 137302 236625
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
34 Advances in Mathematical Physics
ABAQUSHFS-FEM
0034
0036
0038
0040
0042
0044
0046
Disp
lace
men
tu1
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(a)
ABAQUSHFS-FEM
minus0015
minus0010
minus0005
0000
0005
0010
0015
Disp
lace
men
tu2
(mm
)
x2 (mm)minus8 minus6 minus4 minus2 0 2 4 6 8
(b)
Figure 22 The variation of displacement component (a) 1199061and
(b) 1199062along the right edge of the plate (119909 = 8) by HFS-FEM and
ABAQUS
Table 3 Properties of the material PZT-4 used for the model
Parameters Values Parameters Values11988811
139 times 1010 Nmminus2 119890
151344Cmminus2
11988812
778 times 1010 Nmminus2 119890
31minus698Cmminus2
11988813
743 times 1010 Nmminus2 119890
331384Cmminus2
11988833
113 times 1010 Nmminus2 120581
1160 times 10
minus9 CNm11988844
256 times 1010 Nmminus2 120581
33547 times 10
minus9 CNm
The maximum values of 120590120579appear at 120579 = 90
∘ for case of 120590infin119909119909
and at 120579 = 0∘ and 120579 = 180
∘ for case of loading 120590infin
119911119911 both
of which agree well with the analytical solution from Sosa[74]The electric displacement119863
1205791205900times10
10 produced by 120590infin119909119909
and 120590infin119911119911
is nearly the same and is symmetrical with respect to
x
z
rA
L
120590infin xx=1205900
120590infin xx=1205900
120579
(a)
X
Y
Z
(b)
X
Y
Z
(c)Figure 23 (a) An infinite piezoelectric plate with a circular holesubjected to remote stress (b) Mesh used by HFS-FEM (c) Meshused by ABAQUS
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 35
0 2 4 6 8 10 12 14 16 18 20
8
12
16
20
24
28
32
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
Stre
ss120590120579
(MPa
)
r (m)
(a)
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Stre
ss120590r
(MPa
)
r (m)
120590xx (HFS-FEM)120590zz (HFS-FEM)
120590xx (ABAQUS)120590zz (ABAQUS)
120590xx (Sosa)120590zz (Sosa)
(b)
Figure 24 Distribution of (a) hoop stress 120590120579and (b) radial stress 120590
119903along the line 119911 = 0 when subjected to remote mechanical load 120590infin
119909119909and
along the line 119909 = 0 when subjected to remote mechanical load 120590infin119911119911
000 025 050 075 100minus2
minus1
0
1
2
3
4
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
Stre
ss1205901205791205900
120579 (120587 rad)
(a)
000 025 050 075 100
minus3
minus2
minus1
0
1
2
3
120590infinxx (HFS-FEM)120590infinzz (HFS-FEM)
120590infinxx (ABAQUS)120590infinzz (ABAQUS)
120590infinxx (Sosa)120590infinzz (Sosa)
120579 (120587 rad)
Elec
tric
al d
ispla
cem
entD
1205791205900
(1010)
(b)
Figure 25 Variation of (a) normalized stress 1205901205791205900and (b) electrical displacement 119863
1205791205900times 10
10 along the hole boundary under remotemechanical loading
the 119909 axis It is found that the maximum values of 119863120579appear
at 120579 = 65∘ and 120579 = 115
∘ which also agrees well with theanalytical solution
9 Conclusions
In this paper we have reviewed the HFS-FEM and itsapplication in engineering applications The HFS-FEM isa promising numerical method for solving complex engi-neering problems The main advantages of this methodinclude integration along the element boundaries only easily
adopting arbitrary polygonal or even curve-sided elementsand symmetric and sparse stiffness matrix and avoidingthe singularity integral problem as encountered in BEMMoreover as in HT-FEM this method offers the attractivepossibility to develop accurate crack singular corner orperforated elements simply by using appropriate special fun-damental solutions as the trial functions of the intraelementdisplacements It is noted that the HFS-FEM has attractedmore attention of researchers in computational mechanicsin the past few years and good progress has been made inthe field of potential problems plane elasticity piezoelectric
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
36 Advances in Mathematical Physics
problems and so on However there are still many possibleextensions and areas in need of further development in thefuture
(1) to develop various special-purpose elements to effec-tively handle singularities attributable to local geo-metrical or load effects (holes cracks inclusionsinterface corner and load singularities) with thespecial-purpose functions warranting that excellentresults are obtained at minimal computational costand without local mesh refinement
(2) to extend the HFS-FEM to elastodynamics fluid flowthin and thick plate bending and fracture mechanics
(3) to develop efficient schemes for complex engineeringstructures and improve the related general purposecomputer codes with good preprocessing and post-processing capabilities
(4) to extend this method to the case of multifield prob-lems such as thermoelastic-piezoelectric materialsand thermomagnetic-electric-mechanical materialsand to develop multiscale framework across fromcontinuum to micro- and nanoscales for modelingheterogeneous materials
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wang and Q H Qin ldquoFE approach with Greenrsquos functionas internal trial function for simulating bioheat transfer in thehuman eyerdquo Archives of Mechanics vol 62 no 6 pp 493ndash5102010
[2] C Cao A Yu and Q H Qin ldquoEvaluation of effective thermalconductivity of fiber-reinforced compositesrdquo International Jour-nal of Architecture Engineering and Construction vol 1 no 1pp 14ndash29 2012
[3] W J Santos J A F Santiago and J C F Telles ldquoOptimalpositioning of anodes and virtual sources in the design ofcathodic protection systems using the method of fundamentalsolutionsrdquo Engineering Analysis with Boundary Elements vol46 pp 67ndash74 2014
[4] A Karageorghis D Lesnic and L Marin ldquoThe method offundamental solutions for an inverse boundary value problemin static thermo-elasticityrdquo Computers and Structures vol 135pp 32ndash39 2014
[5] C Y Cao QHQin andA B Yu ldquoMicromechanical analysis ofheterogeneous composites using hybrid trefftz FEM and hybridfundamental solution based FEMrdquo Journal ofMechanics vol 29no 4 pp 661ndash674 2013
[6] K Y Wang Q H Qin Y L Kang J S Wang and CY Qu ldquoA direct constraint-Trefftz FEM for analysing elasticcontact problemsrdquo International Journal for Numerical Methodsin Engineering vol 63 no 12 pp 1694ndash1718 2005
[7] F L S Bussamra E Lucena Neto and W M PoncianoldquoSimulation of stress concentration problems by HexahedralHybrid-Trefftz finite element modelsrdquo Computer Modeling inEngineering amp Sciences vol 99 no 3 pp 255ndash272 2014
[8] C J Pearce G Edwards and L Kaczmarczyk ldquo3D cohesivecrack propagation using hybrid-Trefftz finite elementsrdquo inComputational Modelling of Concrete Structures N Bicanic HMang GMeschke and R de Borst Eds vol 1 CRCPress NewYork NY USA 2014
[9] H Wang and Q H Qin ldquoSpecial fiber elements for thermalanalysis of fiber-reinforced compositesrdquo Engineering Computa-tions vol 28 no 8 pp 1079ndash1097 2011
[10] Q H Qin ldquoNonlinear analysis of reissner plates on an elasticfoundation by the BEMrdquo International Journal of Solids andStructures vol 30 no 22 pp 3101ndash3111 1993
[11] O Atak S Jonckheere E Deckers D Huybrechs B Pluymersand W Desmet ldquoA hybrid Boundary Element-Wave BASedMethod for an efficient solution of bounded acoustic problemswith inclusionsrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 1260ndash1277 2015
[12] S Natarajan J Wang C Song and C Birk ldquoIsogeometric anal-ysis enhanced by the scaled boundary finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol283 pp 733ndash762 2015
[13] H T Wang Z H Yao and P B Wang ldquoOn the preconditionersfor fast multipole boundary element methods for 2D multi-domain elastostaticsrdquo Engineering Analysis with Boundary Ele-ments vol 29 no 7 pp 673ndash688 2005
[14] Q H Qin and S Diao ldquoNonlinear analysis of thick plates onan elastic foundation by HT FE with p-extension capabilitiesrdquoInternational Journal of Solids and Structures vol 33 no 30 pp4583ndash4604 1996
[15] J Jirousek AWroblewski Q H Qin and X Q He ldquoA family ofquadrilateral hybrid-Trefftz p-elements for thick plate analysisrdquoComputer Methods in Applied Mechanics and Engineering vol127 no 1ndash4 pp 315ndash344 1995
[16] H Wang Q H Qin and D Arounsavat ldquoApplication ofhybrid Trefftz finite element method to non-linear problems ofminimal surfacerdquo International Journal for Numerical Methodsin Engineering vol 69 no 6 pp 1262ndash1277 2007
[17] S-W Gao Y-S Wang Z-M Zhang and X-R Ma ldquoDualreciprocity boundary element method for flexural waves in thinplate with cutoutrdquo Applied Mathematics and Mechanics vol 26no 12 pp 1564ndash1573 2005
[18] M Dhanasekar J J Han and Q H Qin ldquoA hybrid-Trefftzelement containing an elliptic holerdquo Finite Elements in Analysisand Design vol 42 no 14-15 pp 1314ndash1323 2006
[19] J A T de Freitas and Z-Y Ji ldquoHybrid-trefftz equilibriummodel for crack problemsrdquo International Journal for NumericalMethods in Engineering vol 39 no 4 pp 569ndash584 1996
[20] C Hennuyer N Leconte B Langrand and E MarkiewiczldquoInterpolation functions of a hybrid-Trefftz perforated super-element featuring nodes on the hole boundaryrdquo Finite Elementsin Analysis and Design vol 91 pp 40ndash47 2014
[21] Q H Qin The Trefftz Finite and Boundary Element MethodWIT Press Southampton UK 2000
[22] Q H Qin and H Wang MATLAB and C Programming forTrefftz Finite ElementMethods CRC Press New York NY USA2008
[23] J A de Freitas and M Toma ldquoHybrid-Trefftz stress elementsfor incompressible biphasic mediardquo International Journal forNumerical Methods in Engineering vol 79 no 2 pp 205ndash2382009
[24] J Jirousek andQH Qin ldquoApplication of hybrid-Trefftz elementapproach to transient heat conduction analysisrdquo Computers ampStructures vol 58 no 1 pp 195ndash201 1996
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 37
[25] N Leconte B Langrand and EMarkiewicz ldquoOn some featuresof a plate hybrid-Trefftz displacement element containing aholerdquo Finite Elements in Analysis and Design vol 46 no 10 pp819ndash828 2010
[26] I D Moldovan T D Cao and J A T de Freitas ldquoHybrid-Trefftz finite elements for biphasic elastostaticsrdquo Finite Elementsin Analysis and Design vol 66 pp 68ndash82 2013
[27] I D Moldovan T D Cao and J A de Freitas ldquoHybrid-Trefftz displacement finite elements for elastic unsaturatedsoilsrdquo International Journal of Computational Methods vol 11no 2 Article ID 1342005 31 pages 2014
[28] Q H Qin ldquoHybrid Trefftz finite-element approach for platebending on an elastic foundationrdquo Applied Mathematical Mod-elling vol 18 no 6 pp 334ndash339 1994
[29] QHQin ldquoVariational formulations for TFEMof piezoelectric-ityrdquo International Journal of Solids and Structures vol 40 no 23pp 6335ndash6346 2003
[30] Q H Qin ldquoTrefftz finite element method and its applicationsrdquoApplied Mechanics Reviews vol 58 no 5 pp 316ndash337 2005
[31] H Wang and Q H Qin ldquoHybrid FEM with fundamentalsolutions as trial functions for heat conduction simulationrdquoActa Mechanica Solida Sinica vol 22 no 5 pp 487ndash498 2009
[32] H Wang Q H Qin and Y-L Kang ldquoA meshless model fortransient heat conduction in functionally graded materialsrdquoComputational Mechanics vol 38 no 1 pp 51ndash60 2006
[33] W Hui and Q H Qin ldquoSome problems with the methodof fundamental solution using radial basis functionsrdquo ActaMechanica Solida Sinica vol 20 no 1 pp 21ndash29 2007
[34] H Wang Q H Qin and Y L Kang ldquoA new meshless methodfor steady-state heat conduction problems in anisotropic andinhomogeneous mediardquo Archive of Applied Mechanics vol 74no 8 pp 563ndash579 2005
[35] M A Golberg C S Chen and H Bowman ldquoSome recentresults and proposals for the use of radial basis functions in theBEMrdquoEngineering Analysis with Boundary Elements vol 23 no4 pp 285ndash296 1999
[36] H Wang and Q H Qin ldquoFundamental-solution-based finiteelement model for plane orthotropic elastic bodiesrdquo EuropeanJournal of Mechanics ASolids vol 29 no 5 pp 801ndash809 2010
[37] H Wang Q H Qin and X-P Liang ldquoSolving the nonlinearPoisson-type problems with F-Trefftz hybrid finite elementmodelrdquo Engineering Analysis with Boundary Elements vol 36no 1 pp 39ndash46 2012
[38] H C Simpson and S J Spector ldquoOn the positivity of the secondvariation in finite elasticityrdquo Archive for Rational Mechanics andAnalysis vol 98 no 1 pp 1ndash30 1987
[39] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 2nd edition 1951
[40] S A Sauter and C Schwab Boundary Element MethodsSpringer Berlin Germany 2010
[41] Q H Qin and Y Y Huang ldquoBEM of postbuckling analysis ofthin platesrdquo Applied Mathematical Modelling vol 14 no 10 pp544ndash548 1990
[42] C Cao Q H Qin and A Yu ldquoA new hybrid finite elementapproach for three-dimensional elastic problemsrdquo Archive ofMechanics vol 64 no 3 pp 261ndash292 2012
[43] A H-D Cheng C S Chen M A Golberg and Y F RashedldquoBEM for theomoelasticity and elasticity with body forcemdasharevisitrdquo Engineering Analysis with Boundary Elements vol 25no 4-5 pp 377ndash387 2001
[44] D P Henry and P K Banerjee ldquoA new boundary elementformulation for two- and three-dimensional thermoelasticityusing particular integralsrdquo International Journal for NumericalMethods in Engineering vol 26 no 9 pp 2061ndash2077 1988
[45] D Aubry D Lucas and B Tie ldquoAdaptive strategy for tran-sientcoupled problems Applications to thermoelasticity andelastodynamicsrdquo Computer Methods in Applied Mechanics andEngineering vol 176 no 1ndash4 pp 41ndash50 1999
[46] R Carrazedo and H B Coda ldquoAlternative positional FEMapplied to thermomechanical impact of truss structuresrdquo FiniteElements in Analysis and Design vol 46 no 11 pp 1008ndash10162010
[47] R J Yang ldquoShape design sensitivity analysis of thermoelasticityproblemsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 102 no 1 pp 41ndash60 1993
[48] A Chaudouet ldquoThree-dimensional transient thermo-elasticanalyses by the BIE methodrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 1 pp 25ndash45 1987
[49] L L Cao Q H Qin and N Zhao ldquoAn RBF-MFS modelfor analysing thermal behaviour of skin tissuesrdquo InternationalJournal of Heat andMass Transfer vol 53 no 7-8 pp 1298ndash13072010
[50] Z-W Zhang HWang andQH Qin ldquoTransient bioheat simu-lation of the laser-tissue interaction in human skin using hybridfinite element formulationrdquoMolecular amp Cellular Biomechanicsvol 9 no 1 pp 31ndash54 2012
[51] Q H Qin and J Q Ye ldquoThermoelectroelastic solutions forinternal bone remodeling under axial and transverse loadsrdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2447ndash2460 2004
[52] Q H Qin ldquoThermoelectroelastic analysis of cracks in piezo-electric half-plane by BEMrdquo Computational Mechanics vol 23no 4 pp 353ndash360 1999
[53] C Cao Q H Qin and A Yu ldquoA novel boundary-integralbased finite element method for 2D and 3D thermo-elasticityproblemsrdquo Journal of Thermal Stresses vol 35 no 10 pp 849ndash876 2012
[54] C C Tsai ldquoThe method of fundamental solutions with dualreciprocity for three-dimensional thermoelasticity under arbi-trary body forcesrdquo Engineering Computations (Swansea Wales)vol 26 no 3 pp 229ndash244 2009
[55] A H-D Cheng ldquoParticular solutions of Laplacian Helmholtz-type and polyharmonic operators involving higher order radialbasis functionsrdquo Engineering Analysis with Boundary Elementsvol 24 no 7-8 pp 531ndash538 2000
[56] X-R Fu M-W Yuan S Cen and G Tian ldquoCharacteristicequation solution strategy for deriving fundamental analyticalsolutions of 3D isotropic elasticityrdquo Applied Mathematics andMechanics vol 33 no 10 pp 1253ndash1264 2012
[57] D Palaniappan ldquoA general solution of equations of equilibriumin linear elasticityrdquo Applied Mathematical Modelling vol 35 no12 pp 5494ndash5499 2011
[58] V V Vasiliev and E V Morozov Advanced Mechanics ofComposite Materials Elsevier Science 2007
[59] S G Lekhnitskii Theory of Elasticity of An Anisotropic BodyMir Publishers Moscow Russia 1981
[60] Q H Qin ldquoNew solution for thermopiezoelectric solid with aninsulated elliptic holerdquo Acta Mechanica Sinica vol 14 no 2 pp157ndash170 1998
[61] T C T TingAnisotropic ElasticityTheory andApplications vol45 ofOxford Science Publications Oxford University Press NewYork NY USA 1996
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
38 Advances in Mathematical Physics
[62] A N Stroh ldquoDislocations and cracks in anisotropic elasticityrdquoPhilosophical Magazine vol 3 no 30 pp 625ndash646 1958
[63] Q H Qin Y-W Mai and S-W Yu ldquoSome problems in planethermopiezoelectric materials with holesrdquo International Journalof Solids and Structures vol 36 no 3 pp 427ndash439 1999
[64] Q H Qin and Y W Mai ldquoCrack growth prediction of aninclined crack in a half-plane thermopiezoelectric solidrdquo The-oretical and Applied Fracture Mechanics vol 26 no 3 pp 185ndash191 1997
[65] Q H Qin and M Lu ldquoBEM for crack-inclusion problemsof plane thermopiezoelectric solidsrdquo International Journal forNumerical Methods in Engineering vol 48 no 7 pp 1071ndash10882000
[66] Q H Qin ldquoThermoelectroelastic Greenrsquos function for a piezo-electric plate containing an elliptic holerdquoMechanics ofMaterialsvol 30 no 1 pp 21ndash29 1998
[67] Q H Qin ldquo2D Greenrsquos functions of defective magnetoelectroe-lastic solids under thermal loadingrdquo Engineering Analysis withBoundary Elements vol 29 no 6 pp 577ndash585 2005
[68] J Wang S G Mogilevskaya and S L Crouch ldquoA numericalprocedure for multiple circular holes and elastic inclusionsin a finite domain with a circular boundaryrdquo ComputationalMechanics vol 32 no 4ndash6 pp 250ndash258 2003
[69] K N Rajesh and B N Rao ldquoTwo-dimensional analysis ofanisotropic crack problems using coupled meshless and fractalfinite element methodrdquo International Journal of Fracture vol164 no 2 pp 285ndash318 2010
[70] Q H Qin and YWMai ldquoBEM for crack-hole problems in ther-mopiezoelectricmaterialsrdquoEngineering FractureMechanics vol69 no 5 pp 577ndash588 2002
[71] C Cao A Yu and Q H Qin ldquoA novel hybrid finite elementmodel for modeling anisotropic compositesrdquo Finite Elements inAnalysis and Design vol 64 pp 36ndash47 2013
[72] Q H Qin Fracture Mechanics of Piezoelectric Materials WITPress Southampton UK 2001
[73] Q H Qin ldquoSolving anti-plane problems of piezoelectric mate-rials by the Trefftz finite element approachrdquo ComputationalMechanics vol 31 no 6 pp 461ndash468 2003
[74] H Sosa ldquoPlane problems in piezoelectric media with defectsrdquoInternational Journal of Solids and Structures vol 28 no 4 pp491ndash505 1991
[75] Q H Qin and S-W Yu ldquoAn arbitrarily-oriented plane crackterminating at the interface between dissimilar piezoelectricmaterialsrdquo International Journal of Solids and Structures vol 34no 5 pp 581ndash590 1997
[76] Q H QinGreenrsquos Function and Boundary Elements ofMultifieldMaterials Elsevier Science Oxford UK 2007
[77] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part II Effective crack modelrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 279ndash288 1996
[78] Q H Qin and Y-W Mai ldquoThermoelectroelastic Greenrsquos func-tion and its application for bimaterial of piezoelectricmaterialsrdquoArchive of Applied Mechanics vol 68 no 6 pp 433ndash444 1998
[79] Q H Qin ldquoGeneral solutions for thermopiezoelectrics withvarious holes under thermal loadingrdquo International Journal ofSolids and Structures vol 37 no 39 pp 5561ndash5578 2000
[80] Q H Qin and X Zhang ldquoCrack deflection at an interfacebetween dissimilar piezoelectric materialsrdquo International Jour-nal of Fracture vol 102 no 4 pp 355ndash370 2000
[81] S W Yu and Q H Qin ldquoDamage analysis of thermopiezoelec-tric properties part Imdashcrack tip singularitiesrdquo Theoretical andApplied Fracture Mechanics vol 25 no 3 pp 263ndash277 1996
[82] C Cao Q H Qin and A Yu ldquoHybrid fundamental-solution-based FEM for piezoelectricmaterialsrdquoComputationalMechan-ics vol 50 no 4 pp 397ndash412 2012
[83] C Cao A Yu and Q H Qin ldquoA new hybrid finite elementapproach for plane piezoelectricity with defectsrdquo Acta Mechan-ica vol 224 no 1 pp 41ndash61 2013
[84] H Ding G Wang and J Liang ldquoGeneral and fundamentalsolutions of plane piezoelectroelastic problemrdquoActaMechanicaSinica vol 28 pp 441ndash448 1996
[85] W Yao and H Wang ldquoVirtual boundary element integralmethod for 2-Dpiezoelectricmediardquo Finite Elements inAnalysisand Design vol 41 no 9-10 pp 875ndash891 2005
[86] C CaoMicro-MacroModeling of AdvancedMaterials by HybridFinite Element Method College of Engineering and ComputerScience Australian National University Canberra Australia2013
[87] WG Jin N Sheng K Y Sze and J Li ldquoTrefftz indirectmethodsfor plane piezoelectricityrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 139ndash158 2005
[88] G S Fam and Y F Rashed ldquoThe method of fundamentalsolutions applied to 3D structures with body forces usingparticular solutionsrdquo Computational Mechanics vol 36 no 4pp 245ndash254 2005
[89] H J Al-Gahtani and F M Mukhtar ldquoRBF-based meshlessmethod for the free vibration of beams on elastic foundationsrdquoApplied Mathematics and Computation vol 249 pp 198ndash2082014
[90] F Toja-Silva J Favier and A Pinelli ldquoRadial basis function(RBF)-based interpolation and spreading for the immersedboundary methodrdquo Computers amp Fluids vol 105 pp 66ndash752014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of