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Variational time integrators for finite‐dimensional thermo‐elasto‐dynamics without heat conduction

This paper focuses on the formulation of variational integrators (VIs) for finite‐dimensional thermo‐elastic systems without heat conduction. The dynamics of these systems happen to have a Hamiltonian structure, after thermal displacements are introduced. It is then possible to formulate integrators by taking advantage of standard methods in VI. The class of integrators we construct have some remarkable features: (a) they are symplectic, (b) they exactly conserve the entropy of the system, or in other words, they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes, (c) they nearly exactly conserve the value of the energy for very long times and (d) they exactly conserve linear and angular momentum. We first describe how to adapt any VI for the classical mechanical systems to integrate adiabatic thermo‐elastic ones, and then formulate three new types of integrators. The first class, based on a generalized trapezoidal rule, gives rise to two first order, explicit integrators, and a second order, implicit one. By composing then the two first‐order integrators we construct a second order, explicit one. Finally, we formulate a fourth order, implicit integrator, which is a symplectic partitioned Runge–Kutta method. The performance of these new algorithms is showcased through numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.