A Spectral/hp Element Method for Three-Dimensional
Nonlinear Elasticity Problems
S. Dong1∗ and Z. Yosibash2
1Center for Computational & Applied Math, Department of Mathematics
Purdue University, USA2Department of Mechanical Engineering, Ben-Gurion University of the Negev, Israel
We present a high-order method employing Jacobi polynomial-based shape functions, asan alternative to the typical Legendre polynomial-based shape functions in solid mechanics,for solving three-dimensional geometrically nonlinear elasticity problems. We demonstratethat the method has an exponential convergence rate spatially and a second-order accu-racy temporally for problems of geometrically nonlinear elastostatics/elastodynamics. Testproblems involving finite or large deformations have been discussed.
I. Introduction
During the past two decades p/hp−versions of the finite element method (FEM) have evolved consid-erably, shown to provide robust and efficient results for problems of solid mechanics (particularly linearelasticity),.1, 2 In linear elasticity the high-order methods possess many advantages over “classical FEMs”,such as considerably higher convergence rates, the flexibility of using large aspect ratios of elements withoutsignificant deterioration in accuracy, locking-free behavior in respect to thickness for plate and shell-likestructures, and in respect to Poisson ratio for nearly incompressible materials. The efficiency and advan-tages of p−FEMs were also extended in recent years to non-linear problems such as elasto-plasticity3, 4 andfinite-deformation problems with follower loads,5–8 demonstrating that p−FEM’s advantages carry over tononlinear solid mechanics problems.
Despite the significant growth in the applications of p−FEM, the area of elastodynamics has receivedrelatively less attention in p/hp−FEM for solid mechanics. To the best of our knowledge, the only applicationsof p−FEMs to dynamic problems are those in.9, 10 In9 linear elastodynamic problems were investigated andsome formulations towards future finite-deformation implementation were briefly mentioned. In10 p−FEMsfor curved elastic and isotropic beams taking into account geometric nonlinearities were used to investigatethe vibrations occurring due to harmonic excitations.
The shape functions based on Legendre polynomials1 have been the dominant bases in p−FEM im-plementations in solid mechanics, and have witnessed widespread applications. On the other hand, shapefunctions employing generalized/warped tensor products of the more general Jacobi polynomials,11 albeitpopular in high-order computational fluid dynamics (CFD),11–13 have largely been ignored in solid mechan-ics. (The method in high-order CFD is popularly termed spectral element method.) Compared to thosebased on Legendre polynomials, the Jacobi polynomial-based shape functions possess several advantages.For example, they result in mass and stiffness matrices with more favorable numerical conditioning.14 TheJacobi-based approach provides a unified treatment for polymorphic geometric shapes (i.e. hexahedrons,pentahedrons/prisms, tetrahedrons and pyramids), and is very flexible in generating hybrid meshes in hp-extensions. They also allow for very high element orders; up to the order p = 100 can be easily obtained.It is therefore highly desirable to exploit these advantages of Jacobi polynomial-based shape functions toconstruct a general high-order method capable of handling commonly encountered polymorphic elements forthree-dimensional solid mechanics problems.
Herein we enlarge the fields of the application of p−FEMs to problems of three-dimensional solid dy-namics, in particular, geometrically nonlinear hyper-elasto-dynamics (finite deformations). We employ the
hierarchical shape functions based on Jacobi polynomials11 for spatial discretizations, instead of the typicalLegendre polynomial-based shape functions.1 The resulting method can handle all types of commonly en-countered three-dimensional elements of high order, e.g. hexahedrons, pentahedrons/prisms, tetrahedronsand pyramids. For temporal discretization we have chosen to employ the the average acceleration variant ofthe Newmark-β scheme15 (with Newmark parameters γ = 1
2 and β = 14 ). This is an implicit scheme that
exhibits second-order convergence in time and is unconditionally stable under linear analysis.For simplicity of presentation we consider the so-called St. Venant-Kirchhoff constitutive equation as a
model problem for the finite-strain solid dynamic applications. This model, although non-physical at largestrains, is simple enough for detailed numerical investigations, and can be easily replaced by any other morerealistic hyper-elastic model. It is obtained from the strain energy density function:
Ψ(E) =λ
2(trE)
2+ µE : E, (1)
by using hyper-elasticity relations:
S =∂Ψ
∂E= λ(trE)I3 + 2µE, (2)
where S is the second Piola-Kirchhoff stress tensor, E is the Green-Lagrange strain tensor, λ and µ arematerial coefficients, and I3 is the third-order identity tensor.
II. Shape functions based on Jacobi polynomials
We briefly introduce the hierarchical shape functions based on Jacobi polynomials that are employed inthis paper. To facilitate the discussion we first define three principal functions on the interval −1 ≤ x ≤ 1denoted by ψa
i (x), ψbij(x) and ψc
ijk(x) (0 ≤ i ≤ I , 0 ≤ j ≤ J , 0 ≤ k ≤ K where I , J and K are positiveintegers), which form the basis for constructing the shape functions in three-dimensional space:
ψai (x) =
1−x2 , i = 0,
1−x2
1+x2 P 1,1
i−1(x), 1 ≤ i < I,1+x2 , i = I,
(3)
ψbij(x) =
ψaj (x), i = 0, 0 ≤ j ≤ J,(
1−x2
)i+1, 1 ≤ i < I, j = 0,
(
1−x2
)i+1 1+x2 P 2i+1,1
j−1 (x), 1 ≤ i < I, 1 ≤ j < J,
ψaj (x), i = I, 0 ≤ j ≤ J,
(4)
ψcijk(x) =
ψbjk(x), i = 0, 0 ≤ j ≤ J, 0 ≤ k ≤ K,
ψbik(x), 0 ≤ i ≤ I, j = 0, 0 ≤ k ≤ K,(
1−x2
)i+j+1, 1 ≤ i < I, 1 ≤ j < J, k = 0,
(
1−x2
)i+j+1 1+x2 P 2i+2j+1,1
k−1 (x), 1 ≤ i < I, 1 ≤ j < J, 1 ≤ k < K,
ψbik(x), 0 ≤ i ≤ I, j = J, 0 ≤ k ≤ K,
ψbjk(x), i = I, 0 ≤ j ≤ J, 0 ≤ k ≤ K.
(5)
In the above expressions P α,βn (x) (α, β > −1) are the Jacobi polynomials which represent the family of
polynomial solutions to a singular Sturn-Liouville problem.Assume that the coordinates of the standard domain are denoted by ξ1, ξ2 and ξ3. Then the hierarchical
shape functions in three-dimensional space are defined in the standard domain as follows in terms of theabove principal functions (φpqr(ξ1, ξ2, ξ3) denoting the shape function with p, q, r being appropriate indices).
• For a hexahedral element, (ξ1, ξ2, ξ3)|−1 ≤ ξ1, ξ2, ξ3 ≤ 1, the shape function is defined by φpqr(ξ1, ξ2, ξ3) =ψa
p(ξ1)ψaq (ξ2)ψ
ar (ξ3).
• For a prismatic element, (ξ1, ξ2, ξ3)| − 1 ≤ ξ1, ξ3; ξ1 + ξ3 ≤ 1; −1 ≤ ξ2 ≤ 1, the shape function is
defined by φpqr(ξ1, ξ2, ξ3) = ψap(η1)ψ
aq (ξ2)ψ
bpr(ξ3), where η1 = 2(1+ξ1)
1−ξ3
− 1.
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• For a tetrahedral element, (ξ1, ξ2, ξ3)| − 1 ≤ ξ1, ξ2, ξ3; ξ1 + ξ2 + ξ3 ≤ 1, the shape function is defined
by φpqr(ξ1, ξ2, ξ3) = ψap(η1)ψ
bpq(η2)ψ
cpqr(ξ3), where η1 = 2(1+ξ1)
−ξ2−ξ3
− 1 and η2 = 2(1+ξ2)1−ξ3
− 1.
• For a pyramidic element, (ξ1, ξ2, ξ3)| − 1 ≤ ξ1, ξ2, ξ3; ξ1 + ξ3 ≤ 1; ξ2 + ξ3 ≤ 1, the shape function isdefined by φpqr(ξ1, ξ2, ξ3) = ψa
p(η1)ψaq (η2)ψ
cpqr(ξ3).
The set of shape functions can be decomposed into vortex modes, edge modes, face modes and interior modesto facilitate implementations. Herein the basis functions defined above will be employed to discretize thethree-dimensional geometrically nonlinear elasto-dynamic equations.
III. Formulation and Discretization
Consider the finite deformation of a three-dimensional object initially occupying domain Ω0 ⊂ R3 withboundary ∂Ω0 = ∂Ω0D ∪ ∂Ω0N , where Dirichlet boundary conditions (BC) are provided on ∂Ω0D andNeumann-type (traction) BCs on ∂Ω0N . Let X denote the position vector of a material point in the initialconfiguration of the object, Ω0, at time t = 0, and let x denote its position vector at time t in the deformedconfiguration, Ω(t). Then the displacement vector u is a function of X: u = x (X, t) −X.
The weak form of the momentum equation represents the principle of virtual work. With respect tothe initial configuration, it is given as follows: Find the displacement field u(X, t) ∈ V(t) = w(X, t) ∈[H1(Ω0)]
3|w(X, t) = uD(X, t) on ∂Ω0D such that
P(u,v) =
∫
Ω0
S :1
2
(
∂v
∂X· F(u) + F(u) ·
∂v
∂X
)
dΩ0 −
∫
∂Ω0N
T · vdΓ −
∫
Ω0
ρ0f · vdΩ0
+
∫
Ω0
ρ0∂2u
∂t2· vdΩ0 = 0 ∀v ∈ V0,
(6)
where V0 = w(X, t) ∈ [H1(Ω0)]3|w(X, t) = 0 on ∂Ω0D. In the above equation S, f and ρ0 are the
second Piola-Kirchhoff stress tensor, external body force, and the structural mass density (in the initialconfiguration), respectively. The external traction force T is assumed to be deformation-independent (i.e.non-follower load). The deformation gradient tensor F(u) is defined by F(u) = ∂x
∂X= I3 + ∂u
∂X.
Consider the St. Venant-Kirchhoff constitutive law for the material (equation (1)). Equation (2) can bere-written as S = λ(trE)I3 + 2µE = C(4) : E, where C(4) is a fourth-order constant tensor (elasticity tensor)representing the material properties, and E is the Green-Lagrange strain tensor, E(u) = 1
2 (F(u) · F(u) − I3).We use P int(u,v) to denote the first term in equation (6), which represents the virtual work due to the internalstress, and is nonlinear with respect to the displacement u; The second and the third terms represent thevirtual work due to the external forces, and will be denoted by Pext(u,v); The last term is the virtual workdue to the inertia, and will be denoted by P inert(u,v). Equation (6) can therefore be symbolically writtenas
P(u,v) = P int(u,v) −Pext(u,v) + P inert(u,v) = 0, ∀v ∈ V0. (7)
We employ the Newmark scheme to solve the nonlinear elasto-dynamic equation (7). At the time step(n+ 1),
P(un+1,v) = P int(un+1,v) −Pext(un+1,v) + P inert(un+1,v) = 0, ∀v ∈ V0. (8)
The velocity and acceleration of time step (n+1) can be expressed in terms of variables of time step n basedon the constant-acceleration variant of the Newmark scheme:
un+1 = −un +2
∆t
(
un+1 − un)
, (9)
un+1 = −un −4
∆tun +
4
(∆t)2
(
un+1 − un)
. (10)
Substitute these expressions into equation (8), and we get
P(un+1,v) = P int(un+1,v) −Pext(un+1,v) + Pinert(un+1,un, un, un,v) = 0, ∀v ∈ V0 (11)
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where Pinert(un+1,un, un, un,v) = P inert
(
−un − 4∆t
un + 4(∆t)2
(
un+1 − un)
,v)
. Equation (11) need to be
solved for un+1 at every time step, and then equations (9) and (10) can be used to compute the velocity andthe acceleration.
Equation (11) is nonlinear with respect to un+1 due to the nonlinearity of the term P int(un+1,v), andcan be solved iteratively. Let un+1,(k) denote the solution at the k-th iteration, and assume that un+1,(k) isclose to the sought solution un+1. So equation (11) can be linearized about un+1,(k) in the direction of anincrement ∆u:
In the above equation the neglected terms represent higher order terms with respect to ∆u. The tangentialstiffness matrix DP(un+1,v) is determined by
DP(un+1,v)(∆u)
= DP int(un+1,v)(∆u) − DPext(un+1,v)(∆u) + DPinert(un+1,un, un, un,v)(∆u)
=
∫
Ω0
1
2
(
∂v
∂X· F(un+1) + F(un+1) ·
∂v
∂X
)
: C(4) :1
2
(
∂ (∆u)
∂X· F(un+1) + F(un+1) ·
∂ (∆u)
∂X
)
dΩ0
+
∫
Ω0
S(un+1) :
(
∂ (∆u)
∂X·∂v
∂X
)
dΩ0 +4
(∆t)2
∫
Ω0
ρ0 (∆u) · vdΩ0.
(13)
Since we consider only the deformation-independent external loads in this paper, the external loads do notcontribute to the tangential stiffness matrix, i.e. DPext(un+1,v) = 0. Equation (12) is linear with respectto ∆u, and indicates that if the known solution at the k-th iteration does not satisfy the principle of virtualwork, i.e. P(un+1,(k),v) 6= 0, one must find a change in P in the direction of ∆u. Therefore we can arriveat the following Newton-Ralphson iterative procedure to solve for un+1:
Loop over k until convergence(1) Solve the following equation for ∆u with a linear equation solver:
DP(un+1,(k),v)(∆u) = −P(un+1,(k),v) ∀v ∈ V0; (14)
(2) Update solution, un+1,(k+1) = un+1,(k) + ∆u, and index, k = k + 1.
To discretize Equation (14) in space, we expand ∆u and the test function v = (v1, v2, v3) in terms of the
shape functions of Section II, (∆u)i(x, t) =∑Nm
p=1(∆u)ip(t)φp(x), vi(x, t) =∑Nm
p=1 vip(t)φp(x) (i = 1, 2, 3)where Nm is the total number of modes and (∆u)ip are the expansion coefficients. Equation (14) can thenbe transformed into the following system of linear equations,
(
4
(∆t)2 M + K
)
∆U = R, (15)
which can be solved with an iterative linear equation solver such as the conjugate gradient solver. In theabove equation, ∆U is a vector of the expansion coefficients (∆u)ip. M is the mass matrix. The stiffnessmatrix K is a 3Nm × 3Nm matrix organized into a Nm × Nm matrix of blocks, each block being a 3 × 3submatrix Kpq,ms (p, q = 1, . . . , Nm; m, s = 1, 2, 3) given by
Kpq,ms =
3∑
i,j,k,l=1
∫
Ω0
1
2
(
∂φp
∂Xi
Fmj + Fmi
∂φp
∂Xj
)
C(4)ijkl
1
2
(
∂φq
∂Xk
Fsl + Fsk
∂φq
∂Xl
)
dΩ0+δms
3∑
i,j=1
∫
Ω0
Sij
∂φq
∂Xi
∂φp
∂Xj
dΩ0
where Fij , Sij , C(4)ijkl and Xi are the components of the deformation tensor F, second Green-Kirchhoff stress
tensor S, elasticity tensor C(4), respectively. δms is the Kronecker delta function. The right-hand-side (RHS)of equation (15), R, is the residual vector, with length 3Nm. Its elements can be represented by Rmp
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y
0
0.2
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0.6
0.8
1
z
00.2
0.40.6
0.81
Y
Z
X
(a)Order
Err
ors
2 4 6 8 1010-15
10-13
10-11
10-9
10-7
10-5
10-3
10-1
101
Linf-uL2-uH1-uLinf-vL2-vH1-vLinf-wL2-wH1-w
(b)
Figure 1. Geometrically nonlinear elastostatics: (a) Cubic object discretized with 2 prismatic elements. (b)L∞, L2 and H1 errors versus element order demonstrating exponential convergence rate.
(m = 1, 2, 3 and p = 1, . . . , Nm),
Rmp = −
∫
Ω0
Sij
1
2
(
∂φp
∂Xi
∂un+1,(k)m
∂Xj
+∂u
n+1,(k)m
∂Xi
∂φp
∂Xj
)
dΩ0
+
∫
∂Ω0N
TmφpdΓ +
∫
Ω0
ρ0fmφpdΩ0
+
∫
Ω0
ρ0
(
unm +
4
∆tun
m +4
(∆t)2un
m
)
φpdΩ0 −4
(∆t)2
∫
Ω0
ρ0un+1,(k)m φpdΩ0
(16)
In the above equation we have used the symbol Am (m = 1, 2, 3) to denote the three components of a vectorA.
Remarks: (1) The Newton-Ralphson iterative scheme typically converges to the machine accuracy withinonly a few iterations. For the test problems in Sections IV and V the typical number of Newton-Ralphsoniterations is 3 to 8 (to machine accuracy).
(2) In the implementation a Schur complement is employed to condense out the interior modes in Equation(15), which is then solved with some linear equation solver such as the conjugate gradient solver. Thus thereexist two levels of iterations in the solution process within a time step: Newton-Ralphson iteration at theouter level and conjugate gradient iteration at the inner level.
IV. Spatial and Temporal Convergence
We establish the spatial and temporal accuracies of the method by comparing simulation results againstanalytical solutions for two classes of problems: geometrically nonlinear elastostatics and elastodynamics.The primary purpose is not to study the physical structural behavior, but to verify the consistency of thescheme and investigate its convergence behavior. Therefore, for the test problems in the following sectionswe have used material properties with contrived values, and paid little attention to their correspondenceto the structures in physical reality. In this section and Section V, We have also omitted all the unitsfor physical parameters and variables in the test problems. We have assumed that for each test problema system of consistent units are used for all the physical variables and parameters. We consider the St.Venant-Kirchhoff material in the following tests, and use the Young’s modulus E and the Poisson ratio ν forthe material properties in some cases. They are related to the material properties λ and µ in equations (1)by µ = E
2(1+ν) , λ = νE(1+ν)(1−2ν) .
A. Geometrically nonlinear elastostatics
We first investigate the convergence of the scheme for geometrically nonlinear elastostatic problems. Considerthe finite deformation of the cubic object (Figure 1a) initially occupying the domain, 0 ≤ X ≤ 1, 0 ≤ Y ≤ 1and 0 ≤ Z ≤ 1, with a Young’s modulus E and a Poisson ratio ν. It is known that the face X = 0 isclamped, and there is a traction force field, T = (TX , TY , TZ), applying on the rest of the faces given by the
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1
y
0
0.2
0.4
0.6
0.8
1
z
0
0.2
0.4
0.6
0.8
1
X Y
Z
(a)Order
Err
ors
2 3 4 5 6 7 810-10
10-8
10-6
10-4
10-2
100
Linf-uL2-uH1-uLinf-vL2-vH1-vLinf-wL2-wH1-w
(b)delta-t
Err
ors
0.1 0.2 0.3
10-9
10-8
10-7
10-6
10-5
10-4
10-3
Linf-uL2-uH1-uLinf-vL2-vH1-vLinf-wL2-wH1-w
(c)
Figure 2. Geometrically nonlinear elastodynamics: (a) Cubic object discretized with 5 tetrahedral elements.(b) L∞, L2 and H1 errors at t = 0.1 as a function of the element order for a fixed ∆t = 0.01. (c) L∞, L2 and H1
errors at t = 0.5 as a function of ∆t for a fixed element order 4.
following function,
TX = nX(λ/2 + µ)Aα cos(αX)(A2α2 cos2(αX) − 1)
TY = nY (A2α2 cos2(αX) − 1)λ/2
TZ = nZ(A2α2 cos2(αX) − 1)λ/2,
(17)
where n = (nX , nY , nZ) is the outward-pointing unit vector normal to the surface in the initial configuration,A and α are prescribed constants. The following body force field is also applied on the object, ρ0f =(fX , fY , fZ),
For the above problem the displacements of the objects can be expressed by the following analytic functionsin terms of coordinates of the initial configuration:
uX = A sin(αX) −X
uY = 0
uZ = 0,
(19)
where uX , uY and uZ are the displacements in x, y and z directions, respectively.We compute the displacement fields by solving the weak form of the momentum equation with the
scheme in Section III (omitting the inertia term). In the initial configuration we discretize the domain with2 prismatic elements; In Figure 1a the thick solid lines mark the edges of these elements. Dirichlet BC isimposed on face X = 0 and traction BCs according to equation (17) are imposed on the other faces. Wesystematically vary the element order between 2 and 10, and at each element order calculate the L∞, L2
and H1 errors of the computed results against the exact solution (equation 19). In Figure 1b we plot theseerrors (logrithmic scale) versus the element order (linear scale) for a problem with the following parametervalues: A = 1.9, α = 1.0, E = 1.0, ν = 0.1. It is evident that the numerical errors decreases exponentiallywith increasing element order. The scheme has achieved a spatially exponential convergence rate for thisclass of problems.
B. Geometrically nonlinear elastodynamics
We next study the convergence of the scheme for geometrically nonlinear elastodynamic problems. Considerthe vibration of the cubic object, 0 ≤ X ≤ 1, 0 ≤ Y ≤ 1 and 0 ≤ Z ≤ 1, with an initial mass density ρ0,Young’s modulus E, and Poisson ratio ν. Assume that the deformation of the object is finite throughoutthe time so that geometrically nonlinear formulations of the momentum equation need to be used. The faceX = 0 of the object is clamped, and a time-dependent traction force field is applied on the other faces,T = (TX , TY , TZ),
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where A, B, C, α and β are prescribed constants, n = (nX , nY , nZ) is the outward-pointing unit vectornormal to the surface. The following body force field is applied on the object, f = (fX , fY , fZ),
fX = −Bα2X sin(αt) − 1ρ0
(λ/2 + µ)Cβ2 sin(βX)[
1 − 3(A+B sin(αt) + Cβ cos(βX))2]
,
fY = fZ = 0(21)
The displacements (uX , uY , uZ) and velocity (uX , uY , uZ) at t = 0 is known,
uX = (A− 1)X + C sin(βX), uY = uZ = 0,
uX = BαX, uY = uZ = 0.(22)
With these conditions the problem has the following analytic solution (t ≥ 0) for the displacements in termsof coordinates of the initial configuration,
uX = (A− 1 +B sin(αt))X + C sin(βX)
uY = uZ = 0.(23)
To simulate the time-dependent finite deformation of the object, we discretize the domain with 5 tetra-hedral elements (see Figure 2a). A Dirichlet BC is employed on the face X = 0, and traction/Neumann BCs(equation 20) are imposed on the rest of the faces. To study the spatial convergence, we fix the time step size∆t, and systematically vary the element order between 2 and 8. At each order we integrate the momentumequation (6) over time from t = 0 to t = tf , and compute the L∞, L2 and H1 errors of the displacement fieldsat t = tf against the exact solution. To study the temporal convergence, we fix the element order and system-atically vary the time step size ∆t. For each value of ∆t we integrate the momentum equation from t = 0 tot = tf , and compute the L∞, L2 andH1 errors of the displacement fields at t = tf . In Figure 2b we plot the er-rors of the computed results as a function of the element order for a fixed ∆t = 0.01 with the following parame-ter values: A = 1.2, B = 0.1, C = 1.0, α = 1.0, β = 1.4, ρ0 = 1.0, E = 1.0, ν = 0.1, tf = 0.1. Evidently, thenumerical errors decrease exponentially with increasing element order, suggesting a spatial exponential con-vergence rate. In Figure 2c we plot the errors as a function of ∆t, for a fixed element order 4 with the followingparameter values, A = 1.2, B = 0.1, C = 0.1, α = 1.0, β = 0.01, ρ0 = 1.0, E = 1.0, ν = 0.1, tf = 0.5. Asthe time step size ∆t is reduced by half, the numerical errors are reduced by a factor of 4, suggesting thatthe scheme has a second-order accuracy in time for geometrically nonlinear elastodynamic problems.
V. Test Problems
We next employ the method in Section III in several example problems of geometrically nonlinear elasto-statics/dynamics. We again assume that the materials follow the St. Venant-Kirchhoff constitutive equation.
A. Free vibration of a rectangular board
The first test problem is the free vibration of a three-dimensional long rectangular board, and we attempt todemonstrate the differences between linear and geometrically nonlinear solutions for elastodynamic problemsunder otherwise identical conditions.
Figure 3a shows the board at its equilibrium position (thin solid lines) and at the initial configurationfor dynamic simulations (thick solid lines). At equilibrium, the board has a length Lx = 4.0 in x direction,a width Ly = 0.6 in y direction, and a thickness Lz = 0.2 in z direction. The left end of the board, x = 0,is clamped. We assume that at equilibrium the board has a mass density, ρ0 = 100, a Young’s modulus,E = 1000, and a Poisson ratio, ν = 0.3. We study the vibration of the board under two situations: (1)assuming small displacement throughout the time so that linear elastodynamic equations can be used, and(2) assuming finite deformation so that geometrically nonlinear elastodynamic equations need to be used.The initial configuration of the board for the dynamic simulations is generated with the following steps:(1) At the equilibrium position apply a traction force in z direction, Tz = 0.2, on the right end of theboard, x = Lx. (2) Compute the deformation of the board using the elastostatic solver; Employ the linearelastostatic solver in this step if the initial configuration is for linear dynamic simulations, and employ thegeometrically nonlinear elastostatic solver in this step if it is for geometrically nonlinear dynamic simulations;(3) Use the deformed state of the board as the initial configuration for the dynamic simulations.
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0
1
2
3
4
y00.20.4
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0
0.5
1
X
Y
Z
A
(a)t
Z-d
ispl
acem
ent
0 1000 2000 3000 4000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Linear elasticityNonlinear elasticity
(b)Frequency
Pow
ersp
ectr
alde
nsity
10-4 10-3 10-2 10-1 10010-10
10-8
10-6
10-4
10-2
100
Linear elasticityNonlinear elasticity
(c)
Figure 3. Free vibration of a rectangular board – comparison between linear and geometrically nonlinear solu-tions. (a) Initial configuration for dynamic simulations (thick solid lines) and the configuration at equilibrium(thin solid lines). (b) Time histories of z displacements of vertex A from linear and nonlinear solutions. (c)Comparison of the power spectra of the z-displacement signals in (b).
t
X-d
ispl
acem
ent
0 1000 2000 3000 4000-0.4
-0.2
0
0.2
Linear elasticityNonlinear elasticity
(a)Frequency
Pow
ersp
ectr
alde
nsity
10-4 10-3 10-2 10-1 10010-12
10-10
10-8
10-6
10-4
10-2
Linear elasticityNonlinear elasticity
(b)
Figure 4. Free vibration of a rectangular board: (a) Time histories of x displacements of vertex A from linearand nonlinear solutions. (b) Comparison of the power spectra of the x-displacement signals in (a).
For the dynamic simulations, the board is assumed to be at rest in the initial configuration for t < 0.At t = 0 the board is released from its initial configuration and allowed to freely vibrate. We discretize thedomain of the board at equilibrium with 8 identical hexahedral elements (see Figure 3a) in the x direction.Dirichlet BC (zero displacements) is applied to the left face of the board, and traction-free BCs are imposedon the other faces of the board. The initial displacements are prescribed based on the initial configurationof the board, with zero initial velocities.
In Figure 3(b) we plot the time histories of the z displacements of the vertex A (see Figure 3a) from thelinear and geometrically nonlinear simulations. The two signals have been shifted in time so that they arealigned. The results are obtained using an element order 5 and a time step size ∆t = 0.005. Simulations havealso been conducted using an element order 4, and we observe no significant difference in the results fromthose of order 5. We have also done simulations using time step sizes ∆t = 0.01 and 0.05, and observed thatthe highest frequencies of the vibration are not well resolved with ∆t = 0.05, and only marginally resolvedwith ∆t = 0.01. An interesting result from Figure 3b is that the linear and the geometrically nonlinearelastodynamic simulations result in the same essential frequency for this problem. The vibration amplitudeof the geometrically nonlinear solution is observed to be slightly lower than that of the linear solution.
The spectral content of the displacement signals reveals additional characteristics of the vibrations. Wehave calculated the FFTs of the two z displacement signals in Figure 3b, and computed their power spectra.In Figure 3c we plot the power spectral densities of the two signals as a function of the frequency. The overlapof the primary peaks of the two signals confirms that they have the same basic frequency. Compared to thatof the linear solution, the spectrum of the geometrically nonlinear solution demonstrates significantly morepeaks in the high frequency range, and clusters of discrete peaks can be observed around certain frequencies,which is likely a manifestation of the nonlinear interactions between different frequency components.
We next investigate the temporal and spectral characteristics of the x displacements. In Figure 4a weplot the time histories of the x displacements of the vertex A (see Figure 3a) of the linear and geometrically
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x
0
1
2
z
0
1
2
3
X
Y
Z
Tz
-Tx
-Tz’
initial configuration
finalconfiguration
intermediateconfiguration
Figure 5. Large deformation of a rectangular bar under multi-stage forces. Left end of the bar is clamped.
nonlinear solutions, which have also been shifted in time for alignment. These solutions exhibit qualitativelydifferent characteristics. The x displacement signal of the linear solution is essentially a sinusoidal function.On the other hand, the x displacement signal of the nonlinear solution demonstrates an additional frequencycomponent with twice the basic frequency. Under the linear elasticity assumption, as the board is displacedin the +z direction the x displacement of vertex A increases with increasing z displacement; And as the boardis displaced in the −z direction the x displacement of vertex A decreases with increasing z displacementmagnitude. Therefore, one observes essentially a sinusoidal signal. For the geometrically nonlinear solution,as the board is displaced in the +z direction, however, with increasing z displacement the x displacementof vertex A increases first, reaching a maximum at some point, and then decreases as the z displacementincreases further; And as the board is displaced in the −z direction, the x displacement of vertex A decreaseswith increasing z displacement magnitude. As a result, the x displacement reaches the maximum twice withina period, which is why an additional high-frequency component can be observed in the nonlinear solution.
Figure 4b compares the power spectra of the x displacement signals of the linear and geometricallynonlinear solutions. It can be observed again that the linear and nonlinear solutions yield the same basicfrequency for this problem. The nonlinear solution exhibits a significantly more complicated spectrum. Inaddition to the second main frequency, we observe a substantially larger number of peaks at high frequenciesand also clusters of spectral peaks around certain frequencies, similar to that of the z displacement. Thecluster of peaks around some frequencies becomes so dense that the spectrum appears nearly continuous.
B. Large deformation of a rectangular bar
In this problem we try to demonstrate the capability of the method for problems involving large deformations.We consider the large deformation of a three-dimensional rectangular bar under multi-stage traction forces.In its initial configuration (see Figure 5), the bar has a length Lx = 4.0 in the x direction, a width Ly = 0.6in the y direction, and a thickness Lz = 0.2 in the z direction. The left end of the bar is in the plane x = 0,and is clamped. The back surface of the bar is initially in the plane z = 0, and the bottom surface is initiallyin the plane y = 0. The bar is assumed to have a Young’s modulus E = 1000 and a Poisson ratio ν = 0.3.Consider the following three-stage traction forces applying on the free end (right end) of the bar: (1) Firstapply a constant traction force in the z direction, Tz, to the free end of the bar at the initial configuration;(2) In the deformed configuration at the end of the previous stage, apply a constant traction force in the−x direction, −Tx, to the free end of the bar; (3) In the deformed configuration at the end of the previousstage, apply a constant traction force in the −z direction, −T ′
z, to the free end of the bar.Figure 5 shows the final and intermediate configurations, together with the initial configuration, of the
bar under such a three-stage force with values Tz = 1.0, −Tx = −2.0, −T ′
z = −11.0. The domain hasbeen discretized with 8 hexahedral elements in the x direction, and the element order 4 is used in thecomputations. It is evident that the bar has undergone extremely large deformations under these traction
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forces. The results demonstrate that the high-order scheme presented in Section III is capable of handlingproblems involving very large deformations.
VI. Concluding remarks
In this paper we have presented a high-order method (p-version) employing Jacobi polynomial-basedhierarchical shape functions, as an alternative to the popular Legendre polynomial-based shape functions insolid mechanics, for solving three-dimensional geometrically nonlinear elasticity problems. The method canhandle all commonly encountered types of elements (hexahedrons, tetrahedrons, prisms/pentahedrons, pyra-mids). We have extended the Newmark scheme to discretize the three-dimensional geometrically nonlinearmomentum equations, and employed a Newton-Ralphson iterative scheme within a time step. By compar-ing simulation results with analytic solutions, we have demonstrated that the method has an exponentialconvergence rate spatially and a second-order accuracy temporally for problems of geometrically nonlinearelastostatics and elastodynamics.
Acknowledgments
The first author (Dong) gratefully acknowledges the support from NSF. Computer time was provided bythe TeraGrid (PSC, TACC, NCSA, SDSC) through an MRAC grant and by the Rosen Center for AdvancedComputing at Purdue University.
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