Higher-order energy consistent time integrators for nonlinear thermo-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.

An advantage of the temporal fe method is that higher-order accurate time integrators can be constructed easily. A furtherimportant advantage is the inherent energy consistency if applied to equations of motion. The temporal fe method is thereforeused to construct higher-order energy-momentum conserving time integrators for nonlinear elastodynamics (see Ref. [1]).Considering finite motions of a flexible solid body with internal dissipation, an energy consistent time integration is alsoof great advantage (see the references [2, 3]). In this paper, we show that an energy consistent time integration is alsoadvantageous for dynamics with dissipation arising from conduction of heat as well as from a viscous material. The energyconsistency is preserved by using a new enhanced hybrid Galerkin (ehG) method. The obtained numerical schemes satisfythe energy balance exactly, independent of their accuracy and the used time step size. This guarantees numerical stability.

1 The motivation and introduction

We aim at a numerical long-term simulation of dynamically loaded bodies consisting of rubber, such as solid rubber tyres orrubber shock absorbers. Rubber is an elastomer, which has the ability to return to its original shape after being deformed.According to an internal friction, the return is delayed, and leads to a self-heating, especially for dynamically loaded rubber.Since the self-heating can lead to a damage, a numerical simulation of rubber dynamics ought to determine the correct amountof the energy loss. Further, a time integration algorithm for arbitrary motions have to be numerically stable, in order to beconvenient for long-term simulations. However, it is well-known that in numerical simulations of large motions of stiff solidsmay occur numerical instabilities. This means, that qualitatively correct numerical results are obtained only with a time stepsize, which is very small compared to the total simulation time. This problem already occur, while simulating a flexible rod,where the instability leads to an inaccurately calculated internal force. The occurrence of a numerical instability is identifiableby a baseless blow-up of a Lyapunov-like function, which coincides with the relative total energy. This behaviour ends in adivergence, because the algorithm does not find an unique solution.

2 The considered numerical problem

The problem of numerical instabilities also occur, while simulating large motions of stiff dissipative materials. For example,a free flying stiff thermo-viscoelastic body (see Fig. 1). The corresponding Lyapunov-like function only decrease in anaveraged sense, and finally blow-up, because an associated stability estimate is not fulfilled. For a free unloaded motion, thetotal linear and total angular momentum is conserved till the method diverged (see Fig. 2, top row). Therefore, we presenta stabilised method, which shows a steady decreasing Lyapunov-like function, and no blow up, because the correspondingstability estimate is fulfilled numerically exactly. The total linear and total angular momentum is also conserved (see Fig. 2,middle row). These properties are independent of the order of the temporal finite elements, whether linear finite elements intime or, for example, quadratic finite elements in time (see Fig. 2, bottom row).

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Fig. 1 Simulation with bilinear finite elements in space and linear finite elements in time, using the unmodified method (left hand) and thestabilised method with enhanced weak forms (right hand).

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

PAMM · Proc. Appl. Math. Mech. 7, 4070007–4070008 (2007) / DOI 10.1002/pamm.200700183

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 The derivation of the stabilised method in a Lagrangian description

From the balance of mechanical energy, we derive the local form of the equations of motion. The entropy inequality principleleads to the local form of the entropy balance. We assume an isotropic visco-elastic material according to Ref. [4], and a heatflux according to Fourier’s law of isotropic heat conduction. The Clausius-Planck inequality associated with a viscosity tensorleads to a viscous evolution equation. These differential equations fulfil conservation laws for an unloaded motion. The totallinear momentum is conserved for a free translational motion. The total angular momentum is conserved for a free rotationalmotion, by taking into account the symmetry of the Kirchhoff stress tensor for all times. On the other hand, the differentialequations fulfil an ‘a priori’ stability estimate, which is based on the application of the fundamental theorem of calculus to therelative total energy. By means of this stability estimate, we derive the appropriate weak forms of the differential equations.In general, we obtain a Bubnov-Galerkin method in space. The temporal approximation of the equations of motion leadsto a continuous Galerkin method in time. Since the relative temperature is an admissible test function for the local entropybalance, we here obtain a discontinuous Galerkin method in time with an energy-consistent jump term. Hence, we obtainjumps at the element boundaries in the approximation of the thermal time evolutions. We determine the internal variablelocally by using a temporally weak form, because the internal variable evolution is only an initial value problem. In order tosatisfy the conservation laws and the stability estimate also with numerical quadrature, the weak forms has to be enhanced.The total linear momentum conservation is already satisfied. The total angular momentum conservation requires a certainnumber of Gauss points, and a symmetric approximation of the Kirchhoff stress tensor. The balance of kinetic energy iscalculated exactly by this required integration rule. In general, the balance of relative internal energy cannot be calculatedexactly by using numerical quadrature, and requires a modified stress approximation. We enhance the second Piola-Kirchhoffstress tensor, and obtain an additional term in the weak form of the second equation of motion (compare Ref. [1]). The distinctnumber of Gauss points in the weak forms requires an enhanced viscosity tensor in the weak form of the entropy evolution.

Fig. 2 Stability properties and conservation laws of the unmodified method (top row) and of the stabilised method with linear finite elementsin time (middle row) as well as with quadratic finite elements in time (bottom row).

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, 639-1675 (2002).

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

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

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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