PAMM · Proc. Appl. Math. Mech. 9, 367 – 368 (2009) / DOI 10.1002/pamm.200910157

Higher-order energy-momentum consistent time-stepping schemes fordynamic finite thermo-viscoelasticity

Michael Groß1 and Peter Betsch ∗1

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

This paper is concerned with the energy consistent simulation of a finite thermo-viscoelastic continuum body. The algorithmis based on a four-field formulation in the Lagrangian description, in which the deformation field, the velocity field, thetemperature field and a strain-like viscous internal variable field are independent unknowns. The Lagrangian temperaturefield is determined by the entropy evolution equation associated with Fourier’s law of isotropic heat conduction. The viscousevolution equation is derived from a nonlinear internal dissipation. The coupled nonlinear differential equations are discretisedby a new space-time finite element method, consisting of continuous as well as discontinuous finite element approximationsin time. Owing to particular time approximations in the constitutive laws, the energy and momentum balances are exactlyfulfilled in the fully discrete case.

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Motivation and introduction

We aim at robust and physically meaningful numerical simulations of polymeric damping materials. Typical applicationsare springs and bushings of micro-cellular polyurethane elastomers. These components damp vibrations and noises in smallspaces. We attain this goal by constructing an energy-momentum consistent time integrator for thermo-mechanical problems.This algorithm fulfills the discrete balances of energy and momentum.

2 Lagrangian description of the problem

The motion results from the deformation mappings ϕt, t ∈ T = [t0, T ] between the reference configuration B0 and the cur-rent configurations Bt. The velocity vector vt = ϕt is the tangent vector at the motion. The traction load tt = Pt N0 onBt is a linear map of the first Piola-Kirchhoff stress tensor P t and the outward unit normal N0 on B0. The heat inputQt = −〈Qt, N0〉 is the negative scalar product of the heat flux Qt and the outward unit normal. The equation of mo-tion ρ0 vt = DIV[P t] equates the local acceleration with the divergence of P t = F t St, where St = 2 ∂Ψt/∂C denotes thethe second Piola-Kirchhoff stress tensor with respect to the free energy Ψt, and F t = GRAD ϕt the deformation gradient.The deformation enters Ψt through the right Cauchy-Green tensor Ct = F T

t F t. The reference density ρ0 > 0 and the Ja-cobi determinant Jt = detF t > 0 are both positive. The time evolution of the temperature θt is governed by the balanceθtηt = −DIV[Qt] + Dint

t of the entropy ηt = −∂ψt/∂θ, the divergence of Fourier’s heat flux Qt = −k0JtC−1t GRAD θt and

the viscous internal dissipation Dintt = 〈〈∂ψt/∂A, At〉〉. Here, Dint

t is formulated by using a symmetric internal variable At,and is always non-negative due to the nonlinear evolution equation 4 ∂ψt/∂A = 2 µvis dA−1

t /dt − λvis tr(A−1t dAt/ dt)A−1

t .We prescribe displacements and temperatures at the boundary. Furthermore, we assume time-dependent distributed heat

fluxes and traction loads. And finally, we state an initial velocity and temperature field. The considered initial boundaryvalue problem is accompanied by conservation of total linear and angular momentum, which are special cases of the gen-eralised momentum M(t) =

∫B0

〈ρ0vt, µt〉. We obtain the total linear or the total angular momentum by substituting thedirection vector ν0 of a straight line or the tangent vector νt of a circular curve, respectively, for the vector µt. Conserva-tion M(T ) = M(t0) is definitely reached for vanishing traction loads. Additionally, the motion fulfills an a priori stabilityestimate with respect to the total energy V(t) =

∫B0〈ρ0vt, vt〉+ψt+(θt−θ∞)ηt relative to the ambient temperature θ∞. Sup-

posing the integrability of V(t) in time, the inequality V(T ) � V(t0) holds, independent of the sum of Dintt and the dissipation

Dcdut = 〈GRAD θt, Qt/θt〉 ≥ 0 arising from conduction of heat. The latter is also non-negative owing to Fourier’s law.

3 Space-time finite element approximation

For deriving energy consistent weak equations of motion, we exploit the time integrability of the kinetic energy T (t). We ob-tain

∫T

∫B0〈ρ0vt, δϕt〉+〈〈P t, GRAD(δϕt)〉〉 =

∫T

∫∂T B0

〈tt, δϕt〉 and∫T

∫B0〈ρ0δvt, ϕt〉 =

∫T

∫B0〈ρ0δvt, vt〉 as weak equa-

tions of motion, which exactly connects the kinetic energy difference with the stress power and the external mechanical power(balance of mechanical energy). Analogously, assuming the integrability of the internal energy E(t) in time, we obtain the

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

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

368 Short Communications 7: Coupled Problems

weak evolution equation∫T 〈〈2µvis dA−1

t /dt−λvis tr(A−1t At)A−1

t −4 ∂ψt/∂A, δAt〉〉 = 0. This leads to a piecewise continu-ous approximation of the motion and the internal variable evolution in time. Further, we arrive at the weak entropy equation∫B0

(et0 − e+t0)/(θt0 − θ∞)δθt0 +

∫T

∫B0

ηt δθt =∫T

∫B0

θ∞/(θt)2〈GRAD(δθt), Qt〉+δθt Dintt /θt −

∫T

∫∂QB0

δθt Qt/θt, whichrenders a piecewise discontinuous approximation of the temperature evolution. The last two equations connect the internalenergy difference exactly with the total dissipation, the stress power and the external thermal power (internal energy balance).

For the sake of clearness, we restrict us to explain the common linear approximation of the trial functions. We approximatethe motion, the velocity and the internal variable by a piecewise linear continuous curve, and the temperature and entropyevolution by a piecewise linear discontinuous curve. The approximations of the mechanical test functions are piecewiseconstant and discontinuous, and the approximation of the thermal test function is piecewise linear and discontinuous. Thenumber of quadrature points in the weak forms has to coincide with the number of discrete equations. We choose a bilinearapproximation in space for the simulated planar problems. Since M(t) associated with the linear approximation can be exactlyintegrated only by a second-order quadrature rule, we have to choose the midpoint rule in the equations of motion. The kineticenergy T (t) is also exactly integrated by the midpoint rule. However, the general un-integrability in time of E(t) with themidpoint rule requires a modified approximation. We recommend the introduction of an algorithmic stress tensor. Finally, thedistinct number of quadrature points in the weak forms is compensated by an algorithmic entropy production term.

4 Representative numerical simulations

In the first simulation of a free flying quadratic polyurethane plate, we verify the accuracy order for linear, quadratic andcubic finite elements in time. Governed by the Gaussian quadrature in the mechanical weak forms, the accuracy order is two,four and six (see Fig. 1). In the next simulation, we verify the balance laws. We consider a free flying annular polyurethaneplate. We compare here the time integrator with and without algorithmic constitutive laws. Both methods conserve the totalmomenta, however only the energy-momentum consistent method (left) avoids energy rises and instable behaviour (right).

Finally, we demonstrate the performance of the energy-momentum consistent method by means of a polyurethane springunder a sinusoidal traction load ttn

and heat input Qtn, which both are indicated by bold arrows. The thin arrows denote the

nodal velocities. The colour denotes the current temperature. The material is stiff and of high density (see Fig. 2).

References

[1] Groß M., Betsch P., and Steinmann P. Conservation properties of a time FE method–Part IV: Higher order energy and momentumconserving schemes. Int. J. Numer. Methods Engng., 63:1849–1897, 2005.

[2] Meng X.N. and Laursen T.A. Energy Consistent Algorithms for Dynamic Finite Deformation Plasticity. Comput. Methods Appl. Mech.Engrg., 191:1639-1675, 2002.

[3] Noels L., Stainier L. and Ponthot J.P. An Energy-Momentum Conserving Algorithm Using The Variational Formulation of Visco-PlasticUpdates. Int. J. Numer. Methods Engng., 65:904-942, 2006

[4] Armero F. Energy-dissipative Momentum-conserving Time-Stepping Algorithms for Finite Strain Multiplicative Plasticity. Comput.Methods Appl. Mech. Engrg., 195:4862-4889, 2006.

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

10

10

10

10

101010

−10

−8

−6

−4

−3 −2 −1

2

4

6

linearquad.cubic

hn (s)

e ϕ(1

)

10

10

10

10

101010

−12

−10

−8

−6

−3 −2 −1

24

6

linearquad.cubic

hn (s)

e θ(1

)

0 1 2 3

4

4 5

x 104.464

4.462

4.46

4.458

4.466

4.464

4.452

4.45

tn (s)

V(t n

)(J

)

0 0.5 1 1.5

4x 10

4.466

4.464

4.462

4.46

4.458

4.456

4.454

tn (s)

V(t n

)(J

)

Fig. 1 Simulation of free flights of a quadratic and an annular plate with and without algorithmic constitutive laws.

0

0

1

1

2

2

3

3

44

4 5

−1

−2

−3

−4

x 10

tn (s)

[tA t n

]2(N

/m)

0

0

1

1

2

2

3

3

4

4

4 5

−1

−2

−3

−4

x 10

tn (s)

QA t n

(W/m

)

307

308

309

310

311

312

313

314

315

1

1

3

3

5

5

7

7

9

9 11−1

−1−3−5−7−9−11

Temperature at tn = 4.38 s

x (m)

y(m

)

Fig. 2 Simulation of a tensile test of a polyurethane spring with a prescribed heat flux on the top.

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com