Galerkin methods in time for semi-discrete viscoelastodynamics

Michael Groß and Peter Betsch ∗

Chair of Computational Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11,D-57068 Siegen, Germany

In semi-discrete nonlinear elastodynamics, higher order energy and momentum conserving time stepping schemes turnedout to be well suited for computing long time motions [1]. In comparison to standard ODE integrators, conserving schemesexhibit superior stability properties which are of utmost importance in a nonlinear finite element framework. We showthat conserving schemes are particularly well suited as starting point for the development of energy consistent schemesfor dissipative dynamical systems. In particular, viscoelastic material behaviour is considered. A key advantage of energyconsistent schemes lies in the fact that the equilibrium state of viscoelastic systems can be definitely reached, independentof the material parameters. In the paper, we compare two Galerkin methods for the temporal discretisation of semi-discretenonlinear viscoelastodynamics: the standard continuous Galerkin (cG) method and an enhanced continuous Galerkin (eG)method which fulfils the total energy balance exactly.

1 Constitutive Formulation

We use an internal variable formulation of viscoelasticity, where the total free energy function Ψ = Ψ(C,Γ ) depends on

the right Cauchy-Green tensor C and the tensor-valued internal variable Γ . The evolution equation Γ = f evo(C,Γ ) for

determining the internal variable follows from the local entropy production S : C/2 − Ψ ≥ 0. In consideration of the gra-

dient form of the second Piola-Kirchhoff stress tensor S = 2∇CΨ , we obtain the inequality −∇Γ

Ψ : Γ ≥ 0, which has to

be fulfilled by the evolution equation. In this paper, we restrict us to isotropic finite viscoelasticity according to [2]. The

elasticity part of the free energy function then depends on the invariants of the right Cauchy-Green tensor and the viscous

part of the free energy function depends on the invariants of the so-called ‘elastic’ right Cauchy-Green tensor C e = CΓ−1.

Thus we obtain the total free energy function Ψ = Ψela(IC , IIC , IIIC) + Ψ

vis(ICe , IICe , IIICe). The evolution equation

Γ = 2[V

−1 : Me]Γ is chosen such that the local entropy production is given by a positive definite quadratic form with

respect to the ‘elastic’ Mandel stress tensor M e = [Ce]tSe, where Se = 2∇CeΨvis, and the fourth-order inverse viscosity

tensor V−1 = 1/2ηdev I

dev + 1/3ηvol Ivol.

2 Dynamical Formulation

We perform a spatial finite element discretisation of the body in a total Lagrangian formulation. The deformation map is

approximated by Lagrange polynomials: ϕ(X , t) =∑nnode

A=1NA(X)xA(t). From this deformation map, we derive the de-

formation gradient F = ∇Xϕ and the material velocity v = ∂ϕ/∂t, respectively. The corresponding right Cauchy-Green

tensor is given by C = FF t. We choose a Hamiltonian description of this finite-dimensional system. In this formula-

tion, we consider the coordinate vector x := (x1, . . . ,xnnode), the momentum vector p := (p1, . . . ,pnnode) as well as the

tensor field Γ as separate state variables. The Hamiltonian H(x, p,Γ ) is given by the sum of the total kinetic energy

T ∗(p) = p · M−1p/2 and the total free energy F(x,Γ ) =∫B0

Ψ(C(x),Γ ). The resulting Hamilton’s equations of mo-

tion x = M−1 p and p = −Qx are first order ODE’s as the evolution equation, where M−1 and Q = Q(x,Γ ) are matrices.

These equations of motion fulfil the following balance laws: First, the completeness condition of the shape functions leads

to total linear momentum conservation P =∑

A pA = 0, the symmetry of the matrix Q leads to total angular momentum

conservation L =∑

A

[qA × pA + qA × pA

]= 0, and the structural form of Hamilton’s equations of motion fulfils the bal-

ance ˙T ∗ = −Pint between kinetic energy T ∗ and stress power Pint =∫B0

S : C/2, which is equal to the balance H = −Dint

between Hamiltonian H and internal dissipation Dint =∫B0

Me : ΓΓ−1/2.

3 Temporal discretisation

Now, we perform a temporal finite element discretisation. For a shorter description, we combine the canonical state variables

x and p into one variable z = (x, p). We use Lagrange polynomials with respect to the master element I = [0, 1] as tempo-

ral shape functions. We obtain the trial functions zh(α) =∑k+1

I=1MI(α)zI as well as Γ

h(α) =∑

I MI(α)Γ I and the test

∗ Corresponding author: e-mail: betsch@imr.mb.uni-siegen.de, Phone: +49 271 740 2224, Fax: +49 271 740 2436

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 5, 397–398 (2005) / DOI 10.1002/pamm.200510175

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

functions δzh(α) =∑k

J=1MJ(α)δzJ as well as δΓh(α) =

∑J MJ(α)δΓJ , respectively. Finally, we derive the time inte-

gration schemes by using standard arguments from the weak form∫ 1

0Jδzh · [[zh]′ − hn J H(zh)zh] dα = 0 of the equations

of motion in connection with the weak form∫ 1

0δΓh : [[Γh]′ − hn 2 V−1 : M e(Γh)Γh] dα = 0 of the evolution equation.

The matrix J denotes the symplectic unit matrix [1], the prime denotes differentiation with respect to α and hn indicates the

time step size.

First, we show the standard time approximations in the constitutive laws corresponding to the cG method. The trial func-

tions directly implicate the approximation F h =∑

I MI F I of the deformation gradient. The approximationsCh = [F h]tF h

and Ce,h = Ch[Γh]−1 of the strain tensors are based on the approximation of the deformation gradient. The approximations

Se,h = 2∇CeΨvis(Ce,h) and Me,h= [Ce,h]tSe,h of the ‘elastic’ stress tensors depend on the approximation of the ‘elastic’

right Cauchy-Green tensor. And the total second Piola-Kirchhoff stress tensor then depends on the time approximation of

both strain tensors: Sh = 2∇CΨela(Ch) + Isym : Se,h[Γh]−t. In the eG method we apply the approach of assumed strain

approximations for which we have shown in [1] that it is invariant with respect to superimposed rigid body motions. The

approximation F h of the deformation gradient has been retained unmodified from the cG method. However, the strain

tensors will be now interpolated over the strains at the time nodes. Thus, we obtain the approximations C =∑

I MI CI

and Ce =∑

I MI CI [Γ I ]−1, where CI = [F I ]

tF I . The stresses then depend on these assumed strains. The ‘elastic’

Mandel stress tensor is given by M e= [Ce]tSe, where Se = 2∇CeΨvis(Ce), and the total second Piola-Kirchhoff stress

tensor then read S = 2∇CΨela(C) + Isym : Se[Γh]−t. The main issue of the eG method, however, is its energy consis-

tency which is based on an additionally introduced algorithmic stress tensor. We add to the approximation of the physically

based stress tensor S an algorithmic stress tensor S alg. The algorithmic stress tensor is a closed form expression derived

from a constrained least-squares minimisation of the Lagrange functional L(S alg, λ) =∫ 1

0‖Salg‖2/2 dα + λG(S alg), where

G(Salg) = Ψα=1 − Ψα=0 −∫ 1

0[[S + Salg] : C ′/2 + ∇ΓΨ : Γ ′] dα is the constraint (compare in [1] the case of nonlinear elas-

ticity). The result is the following weighted time derivative of the right Cauchy-Green tensor: S alg = 2 [G(0)/∫ 1

0‖C ′‖2 dα]C ′.

4 Numerical Example

As numerical example, we consider a general free motion of a L-shaped body initiated by an initial translation velocity vector

vT = (6, 0, 0) and an initial angular velocity vector ω0 = (0, 0, 0.6). We used linear finite elements in time and 4-node spatial

finite elements, for instance. The free energy functions are both Neo-Hookean with λ = 30000, µ = 7500 and ρ = 8.93. The

viscosities are ηdev = 50000 and ηvol = 10000. In the figure below, the motion itself and the total energy H of the L-shaped

body versus the time t is depicted. After changing time step size at t = 20, the total energy of the cG-method is increasing

while the energy of the eG-method continued to decrease. A further disadvantage of the cG-method is the only time-averaged

decrease using a medium time step size. Taking large time steps from the outset, the total energy of the cG-method does not

decrease and increase until the cG-method diverged, while the eG-method shows the same behaviour as before.

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References[1] Groß M., Betsch P. and Steinmann P. Conservation properties of a time FE method–part IV: Higher order energy and momentum

conserving schemes. UKL/LTM J 04-01, Department of Mechanical Engineering, University of Kaiserslautern, 2004. Accepted forpublication in the Int. J. Numer. Methods Eng., 2005.

[2] Reese S. and Govindjee S. A Theory of Finite Viscoelasticity And Numerical Aspects, Int. J. Solids Structures, 35 (1998), 3455-3482.

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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