An ultra-accurate dynamic isogeometric analysis with higher order mass formulation
WANG DongDong1,2*, LI XiWei1, LIU Wei1 & ZHANG HanJie1,2
1 Department of Civil Engineering, Xiamen University, Xiamen 361005, China; 2 Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University,
Xiamen 361005, China
Received December 31, 2013; accepted May 4, 2014
An ultra-accurate isogeometric dynamic analysis is presented. The key ingredient of the proposed methodology is the devel-opment of isogeometric higher order mass matrix. A new one-step method is proposed for the construction of higher order mass matrix. In this approach, an adjustable mass matrix is formulated through introducing a set of mass parameters into the consistent mass matrix under the element mass conservation condition. Then the semi-discrete frequency derived from the free vibration equation with the adjustable mass matrix is served as a measure to optimize the mass parameters. In 1D analysis, it turns out that the present one-step method can perfectly recover the existing reduced bandwidth mass matrix and the higher order mass matrix by choosing different mass parameters. However, the employment of the proposed one-step method to the 2D membrane problem yields a remarkable gain of solution accuracy compared with the higher order mass matrix generated by the original two-step method. Subsequently a full-discrete isogeometric transient analysis algorithm is presented by using the Newmark time integration scheme and the higher order mass matrix. The full-discrete frequency is derived to assess the accuracy of space-time discretization. Finally a set of numerical examples are presented to evaluate the accuracy of the pro-posed method, which show that very favorable solution accuracy is achieved by the present dynamic isogeometric analysis with higher order mass formulation compared with that obtained from the standard consistent mass approach.
isogeometric analysis, dynamics, semi-discrete frequency, full-discrete frequency, higher order mass matrix
Citation: Wang D D, Li X W, Liu W, et al. An ultra-accurate dynamic isogeometric analysis with higher order mass formulation. Sci China Tech Sci, 2014, doi: 10.1007/s11431-014-5570-9
1 Introduction
The usual gap between the computer aided geometry design and the finite element analysis has been primely eliminated through the isogeometric analysis approach proposed by Hughes et al. [1]. After its inception, the isogeometric anal-ysis has grown as a tremendously active research area with fast increasing advances [2–11]. In the isogeometric analy-sis, the B-splines or non-uniform rational B-splines (NURBS) are commonly employed as the basis functions
for geometric description as well as the field variable ap-proximation [1,2]. These basis functions enjoy the smooth-ness and convex approximation properties with well- behaved mass matrices for structural vibration analysis. The study of isogeometric structural vibration analysis was initi-ated by Cottrell et al. [12] and Reali [13]. It has been shown that much more accurate frequency spectra can be obtained by the isogeometric analysis compared with the conventional higher order finite elements. Thereafter, the dynamic isogeometric analysis was also demonstrated by Hughes et al. [14] for the structural dynamics and wave propagation problems with superior accuracy. Shojaee et al.
doi: 10.1007/s11431-014-5570-9
2 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
[15] utilized the isogeometric method to study the free vi-bration for thin plates, while the static, free vibration, and buckling analysis of laminated composite and functionally graded plates was investigated by Thai et al. [16] and Valizadeh et al. [17]. Weeger et al. [18] performed an iso-geometric analysis of nonlinear vibration of Euler-Bernoulli beam. In these studies very good performance of isogeo-metric analysis was found as well. Very recently, Hughes et al. [19] carried out a detailed study on eigenvalue, boundary value and initial value problems discretized by NURBS and conventional finite elements and the nice property of iso-geometric analysis was further confirmed.
One important fact associated with the excellent dynamic characteristic of isogeometric analysis is the employment of consistent mass matrix [12–14]. On the other hand, quite discouraged results were observed when the lumped mass matrix was used, i.e., the accuracy even did not increase with the elevation of basis order [12,13]. Consequently a combination of the consistent and lumped mass matrices would not give an expected higher order mass formulation, which was usually employed to obtain the higher order fi-nite element mass matrix [20–22]. To resolve this issue, a two-step method was systematically developed by Wang et al. [23] for the successive construction of higher order mass matrices in isogeometric analysis. In this method, a reduced bandwidth mass matrix was firstly designed with the same order of accuracy as that of the consistent mass matrix. Subsequently a higher order mass matrix was rationally established through an optimal combination of the con-sistent and the reduced bandwidth mass matrices. It has been shown that with regard to the 1D vibration frequency, the 6th and 8th orders of accuracy are achieved for the quadratic and cubic NURBS with the proposed higher order mass matrices, and in contrast, consistent mass matrices yield 4th and 6th orders of accuracy for quadratic and cubic approximations, respectively. Moreover, a tensor product generalization of the 1D formulation has been developed for 2D membrane vibration problem [23].
In this study we present a dynamic isogeometric analysis with higher order mass formulation to achieve an ul-tra-accurate algorithm. In the first part of this work, a re-fined new one-step method is proposed to construct the higher order mass matrix directly which bypasses the aforementioned two-step method involving the establish-ment of the reduced bandwidth mass matrix [23]. In this approach, a set of parameters are introduced into the con-sistent mass matrix to formulate an adjustable mass matrix which conserves the total element mass. Then a semi- discrete frequency error analysis is performed on the free vibration stencil equation at a generic control point. The resulting expression of semi-discrete frequency is used as a basis to optimize the introduced mass parameters for the highest accuracy. It is shown that for the 1D rod problem, the present one-step method perfectly recovers the reduced bandwidth mass matrix and the higher order mass matrix by
choosing different parameters. Nonetheless, for the 2D membrane problem, the quadratic higher order mass matrix by the present approach yields a remarkable 10th order of accuracy for the semi-discrete frequency, which is the 6th order for the higher order mass matrix given by the original two-step method with tensor product formulation [23]. Subsequently the full-discrete frequency is derived based on the Newmark time stepping algorithm and the proposed higher order mass matrix. The accuracy of the full-discrete scheme with higher order mass formulation is then meas-ured by the ratio between the full-discrete and the analytical frequencies. Meanwhile, the transient analysis is also per-formed to evaluate the proposed approach and very favora-ble solution accuracy is observed.
The rest of this paper is organized as follows. In Section 2, after a brief review of the isogeometric discretization a one-step method for the higher order mass formulation with particular reference to the optimal semi-discrete frequency is presented in detail. Subsequently the full-discrete isoge-ometric analysis with higher order mass formulation is dis-cussed in Section 3, where the full-discrete frequency with higher mass matrix is derived as an accuracy measure. Then numerical examples are presented in Section 4 to verify the proposed methodology. Finally conclusions are given in Section 5.
2 Isogeometric higher order mass formulation
2.1 Isogeometric basis functions
The isogeometric analysis often employs B-Spline and NURBS basis functions for geometric description and finite element analysis. A group of p-th order B-spline basis func-tions ( )p
aN ’s are defined on a knot vector ξk that con-
sists of non-decreasing knots in the parametric domain [0, 1] :
T1 1{ 0, , , , 1} ,a n p ξk (1)
where n represents the number of basis functions. ξk is
called an open knot vector when the first and last knots have a multiplicity of ( 1)p . Based upon ξk , the B-spline
basis function ( )paN is defined as [1,24]:
For 0p :
10 1, [ , ),( )
0, otherwise.
a aaN (2)
For 1p :
11 11
1 1
( ) ( ) ( ),a pp p paa a a
a p a a p a
N N N
(3)
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 3
a recursive construction of quadratic B-spline basis func-tions is depicted in Figure 1.
The multidimensional p-th order B-spline basis function ( )p
abN ξ can be conveniently built by using the tensor
product of two one-dimensional basis functions in and
directions:
( ) ( ) ( ),p p pab a bN N N ξ (4)
in which ( , ) ξ , ( )pbN is the 1D p-th order basis
function in direction.
Figure 2 shows a quadratic B-spline basis function and its derivative. Further introducing a weight into the B-spline basis function yields the 1D and 2D NURBS basis functions
( )paR and ( )p
abR ξ :
1 1 1
( ) ( ) ( )( ) , ( ) ,
( ) ( ) ( )
p p pp pa a a b ab
a abn n mp p p
c c c d cdc c d
N w N N wR R
N w N N w
ξ (5)
where aw and abw are the 1D and 2D NURBS’s weights
that can be used to control the geometry. It is noted that the B-spline basis functions and NURBS basis functions are identical when unit weights are employed. Moreover, both the B-spline and NURBS basis functions form a partition of unity approximation.
In isogeometric analysis, the B-spline or NURBS basis functions are used for geometry modeling as well as the field variable approximation:
, 1
, 1
( ) ( ) ,
( , ) ( ) ( ) ,
n
ab aba b
nh
ab aba b
R
u t R d t
x ξ ξ x
x ξ Rd (6)
where ( )x ξ is a field point in a given problem domain ,
abx is the control point associated with exact geometry
description, ( , )hu tx is the isogeometric approximation of
the field variable ( , )u tx , say rod displacement or mem-
brane deflection in this study, ( )abd t is the coefficient
associated with the ab-th control point, and t denotes the
Figure 1 1D Quadratic B-spline basis functions and their derivatives.
Figure 2 2D Quadratic B-spline basis function and its derivative.
4 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
time. R and d are the vectors of the NURBS basis functions and control point coefficients.
2.2 Isogeometric semi-discretization
Here we focus on two types of benchmark problems, i.e., the 1D rod problem and the 2D membrane problem. Based upon the isogeometric discretization described by eq. (6), the classical semi-discrete equation of motion becomes [12, 21]
,+ =Md Kd f (7)
in which f is the force vector, M and K are the mass
and stiffness matrices that can be constructed by assembling their corresponding element counterparts eM and eK :
T d ,e
e e e
M R R (8)
T d ,e
e e eE
K R R (9)
where e represents the element domain defined by iso-geometric approximation, is the material density, E is
the material Young’s modulus, is the gradient operator, i.e., d d/ x for the 1D rod problem and
T{ / , / }x y for the 2D membrane problem, re-
spectively. A standard free vibration argument on eq. (7) leads to the following eigenvalue problem:
,2[ ( ) ]h 0K M φ (10)
where h is the semi-discrete frequency and φ is the
associated vibration mode. When the quadratic B-spline basis functions are used for
uniform isogeometric discretization with an element size of h , the element mass matrix and stiffness matrix for the 1D rod problem can be obtained as [12,23]
6 13 1
13 54 13 ,120
1 13 6
ec Ah
M (11)
2 1 1
1 2 1 ,6
1 1 2
e EA
h
K (12)
where A is the rod cross section area. As for the 2D membrane problem with a uniform iso-
geometric discretization characterized by the element size of h, the quadratic consistent mass matrix ecM and stiff-ness matrix eK are given by [23]
2
14400ec hM
36 78 6 78 169 13 6 13 1
78 324 78 169 702 169 13 54 13
6 78 36 13 169 78 1 13 6
78 169 13 324 702 54 78 169 13
169 702 169 702 2916 702 169 702 169
13 169 78 54 702 324 13 169 78
6 13 1 78 169 13 36 78 6
13 54 13 169 702 169 78 324 78
1 13 6 13 169 78 6 78 36
,
(13)
36012 10 2 10 13 7 2 7 1
10 60 10 13 14 13 7 26 7
2 10 12 7 13 10 1 7 2
10 13 7 60 14 26 10 13 7
13 14 13 14 108 14 13 14 13
7 13 10 26 14 60 7 13 10
2 7 1 10 13 7 14 10 2
7 26 7 13 14 13 10 60 10
1 7 2 7 13 10 2 10 12
e E
K
,
(14)
where is the membrane thickness.
2.3 Isogeometric higher order mass formulation
In ref. [23], we have proposed a two-step method to develop the higher order mass matrix in which a reduced bandwidth mass matrix erM is first established with the constraint of mass conservation and then an optimal linear combination of erM and ecM yields the desired higher order mass matrix ehoM , while erM has the same order of accuracy as its consistent counterpart ecM . In the present work, we further develop a new one-step method to unify the devel-opment of the reduced bandwidth and higher order mass matrices. More importantly, the current one-step method leads to extraordinary accuracy for the 2D membrane prob-lem.
(1) 1D rod problem. To optimize the frequency accuracy, the consistent mass
matrix of eq. (11) can be rephrased as the following adjust-able mass matrix eaM by introducing two mass parame-ters 1c and 2c :
1 2 2 1
2 2 2
1 2 1 2
6 13 1
13 54 2 13 .120
1 13 6
ea
c c c cAh
c c c
c c c c
M (15)
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 5
The principles for the design of eaM is: 1) S is: 1) soundary Value Problems ymmetry of mass matrix; 2) mass
conservation, i.e., 3
, 1
eaaba b
M
3
, 1
ecaba b
M .
Based upon eqs. (12) and (15), the stencil equation cor-responding to eq. (7) is
1 2 2 1
1 2
2 1 1 2
2
2 1 1 2
(1 ) (26 2 )
(66 2 4 )20
(26 2 ) (1 )
2 6 2 0,
a a
a
a a
a a a a a
c d c dh
c c d
c d c d
cd d d d d
h
(16)
where /c E is the wave speed. Further assume a
harmonic form for the nodal solution [25,26]:
0( ) exp[ ( )],ha ad t d kx t (17)
where 0d is the wave amplitude, k is the wave number,
1 , then by using eq. (17), eq. (16) gives
21 2
1 2
2
( ) (1 ) cos(2 ) (26 2 )cos( )
33 2
20 3 cos(2 ) 2cos( ) .
hh c kh c kh
c c
c kh kh
(18)
Consequently the semi-discrete frequency h can be obtained as
1 2
1 2
20 3 cos(2 ) 2cos( )1,
(1 ) cos(2 ) (26 2 )cos( )
33 2
h kh kh
c kh c khkh
c c
(19)
where is the exact continuum frequency. Following the standard procedure [12–14,23], from eq.
(19) the frequency accuracy order can be derived as
2 41 2 1 22 1 81 ( ) ( )
120 1440
h c c c ckh kh
61 2
1 1 1+ + ( ) .
40320 1350 43200c c kh
(20)
By choosing different 1c and 2c , eqs. (15), (19) and
(20) yield various mass matrices and their associated accu-racy orders.
a) Consistent mass matrix [12,13,23]: 1 2 0c c , ecM
is given by eq. (11),
411 ( ) .
1440
h
kh
(21)
b) Reduced bandwidth mass matrix [23]: 1 1,c 2 2,c
5 15 0
15 50 15 ,120
0 15 5
er Ah
M (22)
471 ( ) .
1440
h
kh
(23)
c) Higher order mass matrix [23]: 1 21/ 6, 1/ 3,c c
37 76 7
76 328 76 ,720
7 76 37
eho Ah
M (24)
6111 ( ) .
120960
h
kh
(25)
It has been shown that the consistent and reduced band-width mass matrices have the 4th order of accuracy, while the 6th order accuracy is achieved by the higher order mass matrix.
(2) 2D membrane problem. As for the 2D membrane problem, we propose the fol-
lowing adjustable mass matrix eaM through embedding five mass parameters 1c to 5c into the consistent mass
matrix ecM :
1 5 3 5 4 2 3 2 1
5 2 5 4 5 4 2 3 2
3 5 1 2 4 5 1 2 3
5 4 2 5 3 5 4 22
36 78 6 78 169 13 6 13 1
78 324 78 169 702 169 13 54 13
6 78 36 13 169 78 1 13 6
78 169 13 324 702 54 78 169 13
16914400
ea
e c c c c c c c c
c e c c c c c c c
c c e c c c c c c
c c c e c c c c ch
M 4 5 4 5 3 5 4 5 4
2 4 5 3 5 2 2 4 5
3 2 1 5 4 2 1 5 3
2 3 2 4 5 4 5 2 5
1
702 169 702 2916 702 169 702 169
13 169 78 54 702 324 13 169 78
6 13 1 78 169 13 36 78 6
13 54 13 169 702 169 78 324 78
1 13
c c c c e c c c c
c c c c c e c c c
c c c c c c e c c
c c c c c c c e c
c c
2 3 2 4 5 3 5 1
,
6 13 169 78 6 78 36c c c c c c e
(26)
6 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
where 1 1 2 3 4 52 2 2e c c c c c , 2 2 3 4 52 2 3e c c c c ,
3 4 54 4e c c .
As a result, the 2D equation of motion corresponding to eq. (7) with the adjustable mass matrix eaM of eq. (26) and the stiffness of eq. (14) becomes
2
2( ) ( ) 0,40 ab ab
hd c d (27)
where ( )abd and ( )abd are defined as
1 ( 2)( 2) 2 ( 2)( 1)
3 ( 2) 2 ( 2)( +1)
1 ( 2)( + 2) 2 ( 1)( 2)
4 ( 1)( 1) 5 ( 1)
4 ( 1)( +
( ) (1 ) (26 2 )
(66 3 ) (26 2 )
(1 ) (26 2 )
(676 4 ) (1716 6 )
(676 4 )
ab a b a b
a b a b
a b a b
a b a b
a b
d c d c d
c d c d
c d c d
c d c d
c d
1) 2 ( 1)( + 2)
3 ( 2) 5 ( 1)
1 2 3 5 ( +1)
3 ( + 2) 2 +1 ( 2)
4 +1 ( 1) 5 ( 1)
(26 2 )
(66 3 ) (1716 6 )
(4356 4 4 ) (1716 6 )
(66 3 ) (26 2 )
(676 4 ) (1716 6 )
(676 4
a b
a b a b
ab a b
a b a b
a b a b
c d
c d c d
e e e d c d
c d c d
c d c d
（ ）
（ ）
4 +1 ( 1) 2 +1 ( 2)
1 + 2 ( 2) 2 + 2 ( 1)
3 ( 2) 2 + 2 ( 1)
1 + 2 ( 2)
) (26 2 )
(1 ) (26 2 )
(66 3 ) (26 2 )
(1 ) ,
a b a b
a b a b
a b a b
a b
c d c d
c d c d
c d c d
c d
（ ） （ ）
（ ） （ ）
（ ）
（ ）
and
( 2)( 2) ( 2)( 1) ( 2)
( 2)( 1) ( 2)( 2) ( 1)( 2)
( 1)( 1) ( 1) ( 1)( 1)
( 1)( 2) ( 2) ( 1)
( 1) ( 2)
( 1)(
( ) 14 30
14 14
52 12 52
14 30 12
396 12 30
14
ab a b a b a b
a b a b a b
a b a b a b
a b a b a b
ab a b a b
a b
d d d d
d d d
d d d
d d d
d d d
d
2) ( 1)( 1) ( 1)
( 1)( 1) ( 1)( 2) ( 2)( 2)
( 2)( 1) ( 2) ( 2)( 1)
( 2)( 2)
52 12
52 14
14 30 14
.
a b a b
a b a b a b
a b a b a b
a b
d d
d d d
d d d
d
(29)
In 2D case, the harmonic nodal solution takes the fol-lowing form [23,25]:
0( ) exp[ ( cos sin ) ],hab x a y bd t d k x k y t (30)
where xk and yk are the wave numbers in x and y direc-
tions, respectively. Here in order to analytically determine the accuracy of the semi-discrete frequency, we assume
x yk k k . Subsequently substitution of eq. (30) into eq.
(27) yields
1 2
23 4
2 5
1 2 3 4 5
2
(1 )cos(4 ) (52 4 )cos(3 )
(808 6 4 )cos(2 )( )(3484 4 12 )cos( ) 285540
8 6 4 12
cos(4 ) 28cos(3 ) 112cos(2 ) 0,
4cos( ) 145
h
c kh c kh
c c khhc c kh
c c c c c
kh kh khc
kh
(31)
thus the semi-discrete frequency h becomes
1 2
3 4
2 5
1 2 3 4 5
145 cos(4 ) 28cos(3 )40
112cos(2 ) 4cos( )1.
(1 )cos(4 ) (52 4 )cos(3 )2(808 6 4 )cos(2 )
(3484 4 12 )cos( )
2855 8 6 4 12
h
kh kh
kh kh
c kh c khkhc c kh
c c kh
c c c c c
(32)
By invoking the Taylor expansion [12,23], eq. (32) gives the following estimation of the semi-discrete frequency ac-curacy:
2 4 662 4
8 108 10
1 ( ) ( ) ( )14400 86400 18144000
( ) + ( ) ,14400 14400
h gg gkh kh kh
g gkh kh
(33)
with
2 1 2 3 4 58 20 12 8 6 ,g c c c c c (34)
4 1 2 3 4 560 64 82 24 16 3 ,g c c c c c (35)
6 1 2 3 4 53600 7168 5110 672 448 21 ,g c c c c c (36)
8 1 2 3 4 5
37 512 3281 4 8 1,
168 315 5040 105 315 3360g c c c c c (37)
10 1 2 3
4 5
91095929 4096 1181 83940453440 14175 18144 4725
16 1.
14175 302400
g c c c
c c
(38)
It is straightforward to verify that the five relationships in eqs. (34)–(38) are not fully linearly independent and
4 359 / 42 3 / 2c c . Then in order to achieve a su-
per-accuracy of order 10, let 3 2c c and we would like to
have
2 4 6 8 0,g g g g (39)
which leads to
1 2 3 4 5
17 59 59 59 143, , , , .
42 42 42 42 63c c c c c (40)
Given the coefficients in eq. (40), the semi-discrete fre-
(28)
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 7
quency accuracy of eq. (33) reduces to
10 1022507131 ( ) 1 ( ).
8106075648000
h
kh O h
(41)
Finally the present formulation gives a remarkable 10th order accurate algorithm. Alternatively, selecting 1c
2 3 4 5 6 7 0c c c c c c in eq. (33) yields a 4th order
accuracy for the semi-discrete frequency associated with consistent mass formulation [23]:
411 ( ) .
1440
h
kh
(42)
Meanwhile, it is noted that the higher order mass matrix obtained by tensor product formalism in ref. [23] is the 6th order accurate. Therefore, the present one-step method gives an extra harvest of the semi-frequency accuracy for the 2D membrane problem.
3 Full-discrete isogeometric analysis with higher order mass formulation
As for the temporal discretization, we consider the classical Newmark method [21] which is frequently used for struc-tural dynamics simulation. Accordingly, the time domain
[0, ]t T is partitioned into a set of time intervals
1[ , ]j jt t , the advancement of the variables such as dis-
placement, velocity and acceleration at time jt , i.e.,
{ , , }j j jd v a , to their corresponding counterparts at 1jt ,
say, 1 1 1{ , , }j j j d v a , follows the following formula [21]:
2
1 1
1 1
(1 2 ) 2 ,2
(1 ) ,
j j j j j
j j j j
tt
t
d d v a a
v v a a (43)
where 1j jt t t , and are the parameters to de-
fine various time stepping methods. Introducing eq. (43) into the semi-discrete equation of eq. (7) at time 1jt gives
the space-time discrete equation:
21 1 1[ ( ) ] ,j j jt M K a f Kd (44)
with the predictor phase:
2
1
1
( )(1 2 ) ,
2(1 ) ,
j j j j
j j j
tt
t
d d v a
v v a (45)
and the corrector phase:
21 1 1
1 1 1
( ) ,
( ) .j j j
j j j
t
t
d d av v a
(46)
In order to analyze the algorithmic accuracy, it is con-venient to consider the following equivalent homogeneous scalar problem that is obtained by performing the standard
mode decomposition of 1
n
i iiq
d φ on eqs. (43)–(44):
2( ) 0,hq q (47)
in which q is the generalized displacement and for brevity the subscript is ignored. Substituting eq. (43) into eq. (47) gives [25]
1 ,j j y Ay (48)
with T{ , }q qy and A is the amplification matrix given
by
2
2 2
2
2 (1 2 )( )
2[1 ( ) ] 1 ( ) .
( 1)( ) 1
h
h h
h
t t
t t
t
A (49)
It is noted that for the Newmark method, choosing 1/ 2 leads to a 2nd order accuracy which is commonly
used, thus we restrict ourselves to this case in the subse-quent development. The characteristic equation for the am-plification matrix A is
21 2det( ) 2 0,A A A I (50)
where I is the 2 by 2 identity matrix. 1A and 2A are
2
1 2
2
( ) / 21 ,
1 ( )
1,
h
h
tA
t
A
(51)
is the eigenvalue of A that has the following form [25]:
exp( ),h t (52)
in which h is the full-discrete frequency that is different
from semi-discrete frequency. The comparison of h and the exact continuum frequency is a useful index to measure the accuracy of the fully discrete algorithm. Sub-stituting eq. (52) into eq. (50) yields [21]
2
2
1sin ,
2 14
( )
h
h
t
t
(53)
eq. (53) gives the relationship between the space-time full- discrete frequency h and the space semi-discrete fre-
quency h . Based upon eqs. (19) and (53), the full-discrete frequen-
cy h can be finally expressed as
8 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
2
2
2
1 2
1 2
sin2
5 3 cos(2 ) 2cos( ),
20 3 cos(2 ) 2cos( )
(1 )cos(2 ) (26 2 )cos( )
33 2
h t
C kh kh
C kh kh
c kh c kh
c c
(54)
where /C c t h is the Courant number and kc is used. Meanwhile, a combination of eqs. (32) and (53) yields the expression of h for the 2D membrane problem:
2
2
2
1 2
3 4
2
sin2
cos(4 ) 28cos(3 )
10 112cos(2 ) 4cos( )
145
cos(4 ) 28cos(3 )
40 112cos(2 ) 4cos( )
145
(1 )cos(4 ) (52 4 )cos(3 )
(808 6 4 )cos(2 )
(3484 4
h t
kh kh
C kh kh
kh kh
C kh kh
c kh c kh
c c kh
c
5
1 2 3 4 5
.
12 )cos( )
2855 8 6 4 12
c kh
c c c c c
(55)
4 Numerical results and discussions
4.1 Accuracy of semi-discrete formulation
(1) Free vibration of 1D rod. For the completeness of this study, we reconsider the free
vibration of an elastic rod problem, the geometry and mate-rial properties for the elastic rod are: length L=10, cross section area A=1, material density 1 , and Young’s
modulus 1E . Figure 3 lists the fundamental frequency results for the vibrations of fixed-fixed, fixed-free, free-fixed, free-free elastic rods using quadratic basis func-tions, where four types of mass formulations are compared, i.e., the lumped mass matrix “LM”, the consistent mass ma-trix “CM”, the reduced bandwidth mass matrix “RBM”, and the higher order mass matrix “HOM”. The periodic basis functions are used to eliminate the boundary effect and the control variable transformation method [27] is used to impose the essential boundary conditions. The numerical results in Figure 3 apparently demonstrate that the proposed higher order mass matrix has a 6th order of accuracy, while both the accuracy orders for the reduced bandwidth matrix and the standard consistent mass matrix are 4. Besides, the frequency spectra using various mass matrices are also de-scribed in Figure 4 where n and N denote the frequency number and the total number of degrees of freedom. The spectrum results once again demonstrate that in general HOM gives the most accurate semi-discrete frequency.
Figure 3 Comparison of h and for 1D rod vibration problem.
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 9
Figure 4 Frequency spectra for 1D rod vibration problem.
(2) Free vibration of square membrane. A square membrane with unit length and thickness is
taken as the benchmark example to test the 2D higher order mass formulation. This membrane has a surrounding fixed boundary condition. Without loss of generality, the material properties for this problem are: Material density 1 ,
Young’s modulus 1E . The isogeometric discretization is realized by the periodic quadratic basis functions and the meshes are plotted in Figure 5. Figure 6 lists the conver-gence rates of the semi-discrete frequencies 11
h and 22h .
For comparison purpose, the previous higher order mass matrix generated by the tensor product formalism in ref. [23] is termed as “HOM-6”, while the present higher order mass matrix is termed as “HOM-10”. The results in Figure 6 re-veal that the present higher order mass matrix HOM-10 produces the desired 10th order accuracy for 11
h and 22h ,
in contrast to the 6th order accuracy of HOM-6 and 4th or-der accuracy of CM. It is noted that x yk k is a funda-
mental assumption for the 2D higher order mass formula-tion. The frequency spectra as shown in Figure 7 further indicate that the most favorable performance is associated with the present higher order mass formulation.
4.2 Accuracy of full-discrete formulation
(1) Impact of 1D rod on a rigid wall. Consider the benchmark problem of an elastic rod im-
pacting onto a rigid wall as shown in Figure 8. The rod has a uniform initial velocity v0 and a fixed boundary condition
Figure 6 Comparison of h and for 2D square membrane problem. (a)
11h ; (b) 22
h .
Figure 7 Frequency spectra for 2D square membrane problem.
at the right end. The analytical solution for this problem is [28]
1 02 2
1
8 2 1( , ) ( 1) cos
2(2 1)
2 1 sin ,
2
n
n
v L nu x t x
Ln c
nct
L
(56)
where c is the wave speed and L is the rod length. In the computation, the geometry and material properties for the elastic rod are: length L=10, cross section area A=1,
10 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
Figure 8 An elastic rod impacting on a rigid wall.
material density 1 , and Young’s modulus 1E . Ten
uniform isogeometric B-spline elements are used to discre-tize the rod.
Based upon eq. (54), the accuracy for the fully discrete algorithm with the proposed higher order mass formulation is shown in Figures 9 and 10. For convenience of presenta-tion, the Courant number /C c t h is employed in the discussion. In Figures 9 and 10, the 1D comparisons of the full-discrete frequency h and the continuum frequency are plotted with respect to the element size and Courant number, four typical Newmark methods, i.e., central differ-ence method ( 0 ), Fox-Goodwin method ( 1/12 ),
linear acceleration method ( 1/ 6 ) and average accelera-
tion method ( 1/ 4 ). The results demonstrate that in
general the higher order mass formulation gives the most favorable full-discrete frequency accuracy.
To further demonstrate the present method, transient
analyses are also carried out. With a time step of 0.01t , the displacement time history curves of points LP and MP
as indicated in Figure 8 are plotted in Figures 11–14, which clearly show that HOM produces the smallest error. More-over a direct computation of eq. (10) at the element level gives the element maximum frequencies:
max max
60 30, ,hec hehoc c
h h (57)
where maxhec and max
heho denote the element maximum fre-
quencies for the consistent mass matrix and the higher order mass matrix, respectively. eq. (57) states that max max
hec heho ,
since the global maximum frequency maxh is bounded up
by the element maximum frequency maxhe , i.e., max
he
maxh [21], meanwhile the same global assembly procedure
is used to construct the global mass matrix, we may have the relationship max max
hc hho , where maxhc and max
hho are
the global maximum frequencies for the consistent and higher order mass matrices. Furthermore, due to the fact that the critical time step size ct is inversely proportional
to the global maximum frequency maxh , the higher order
mass matrix may be capable of providing a larger stable time step size compared with that of its consistent counter-part for a conditionally stable time stepping algorithm. To
Figure 9 Comparison of h and with varying element size for 1D rod problem.
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 11
Figure 10 Comparison of h and with varying Courant number for 1D rod problem.
show this, the maximum semi-discrete frequencies and cor-responding critical time step sizes ct ’s for the 1D rod
problem with 10 periodic B-spline elements for the con-sistent and higher order mass matrices are tabulated in Ta-ble 1. It is clearly seen that the higher order mass formula-tion has better stability behavior. According to Table 1, three computations are performed, i.e., 0.431t for the central difference method with 0 , 0.53t for the
Fox-Goodwin method with 1/12 and 0.75t for
the linear acceleration method with 1/ 6 . The results as
shown in Figure 15 demonstrate a stable solution for HOM but an unstable solution for CM with the same time step size. This adds a bonus to the proposed higher order mass matrix.
(2) Vibration of square membrane with initial velocity. Reconsider the previous 2D square membrane problem
with same material and geometry parameters. The compar-
isons of h and with varying element size and Courant number are shown in Figures 16 and 17, again su-perior accuracy is observed for the present higher order formulation compared with the consistent mass method, where HOM means the present higher order mass matrix with an accuracy of order 10. In case of transient analysis, this membrane is given an initial velocity of 0v
2sin sin
c x y
L L L
and the analytical solution is giv-
en by
2
( , , ) sin sin sin ,x y ct
u x y tL L L
(58)
In the transient computation, the geometry and material properties for the elastic membrane are: L=1, material den-sity 1 , and Young’s modulus 1E . The membrane is
Table 1 Maximum frequencies and critical time step sizes for 1D rod problem
Method Maximum frequency max
h (c/h) Critical time step size ct (h/c)
CM HOM CM HOM
= 0 4.6496 4.2151 0.4301 0.4745
=1/12 4.6496 4.2151 0.5268 0.5811
=1/6 4.6496 4.2151 0.7450 0.8218
12 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
Figure 11 Transient analysis for 1D rod problem with = 0. (a) Displacement of PL; (b) displacement error of PL; (c) displacement of PM; (d) displacement error of PM.
Figure 12 Transient analysis for 1D rod problem with =1/12. (a) Displacement of PL; (b) displacement error of PL; (c) displacement of PM; (d) displace-ment error of PM.
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 13
Figure 13 Transient analysis for 1D rod problem with =1/6. (a) Displacement of PL; (b) displacement error of PL; (c) displacement of PM; (d) displace-ment error of PM.
Figure 14 Transient analysis for 1D rod problem with =1/4. (a) Displacement of PL; (b) displacement error of PL; (c) displacement of PM; (d) displace-ment error of PM.
14 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
Figure 15 Displacement time history curves of point PL for 1D rod problem. (a) Central difference method; (b) fox-Goodwin method; (c) linear accelera-tion method.
Figure 16 Comparison of h and with varying element size for 2D membrane problem.
discretized by 9×9 periodic quadratic uniform B-spline el-ements and the time step size is 0.001t . The center deflections and the related errors by the Newmark family of methods are drawn in Figures 18–21, which confirms the superiority of the proposed higher order mass dynamic iso-geometric analysis method as well.
5 Conclusions
An ultra-accurate dynamic isogeometric analysis was pre-sented. This method is featured by the higher order mass formulation. The higher order mass matrix is rationally
formulated by a new one-step method through optimizing the semi-discrete frequency of an adjustable mass matrix. The adjustable mass matrix is constructed by directly intro-ducing a set of mass parameters into the consistent mass matrix under the condition of element mass conservation. Obviously, the adjustable mass matrix is identical to the consistent mass matrix with vanishing mass parameters. Moreover, it was shown that in 1D case the present one-step method exactly reproduces the existing reduced bandwidth mass matrix and the higher order mass matrix by selecting different mass parameters. Thus the present new one-step mass construction method offers a unified way for the con-struction of various mass matrices. More importantly, this
Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7 15
Figure 17 Comparison of h and with varying Courant number for 2D membrane.
Figure 18 Comparison of the center deflection with =0 for 2D square membrane under given initial velocity. (a) Center deflection; (b) center deflection error.
Figure 19 Comparison of the center deflection with =1/12 for 2D square membrane under given initial velocity. (a) Center deflection; (b) center deflec-tion error.
16 Wang D D, et al. Sci China Tech Sci July (2014) Vol.57 No.7
Figure 20 Comparison of the center deflection with =1/6 for 2D square membrane under given initial velocity. (a) Center deflection; (b) center deflection error.
Figure 21 Comparison of the center deflection with =1/4 for 2D square membrane under given initial velocity. (a) Center deflection; (b) center deflection error.
present strategy yields a 2D membrane higher order mass matrix that has a remarkable 10th order accuracy with quadratic B-spline basis functions, which is in contrast to the 6th order accuracy associated with the previous higher order mass matrix obtained from the tensor product formal-ism. Subsequently the space-time discretization with the classical Newmark time stepping algorithm was discussed for the higher order mass formulation in detail and the full-discrete frequency was derived as an accuracy measure of the present fully discretized scheme. The superiority of the present dynamic isogeometric analysis with higher order mass formulation was systematically demonstrated through frequency spectra, free vibration as well as transient analy-sis examples.
This work was supported by the National Natural Science Foundation of China (Grant No. 11222221).
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