Thermodynamically consistent time-stepping algorithms for non-linear thermomechanical systems

We present the basic theory for developing novel monolithic and staggered time-stepping algorithms for general non-linear, coupled, thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems they preserve the total energy and the entropy never decreases. Furthermore, if the governing equations of the problem have symmetries, the proposed integrators preserve them too. The formulation of such methods is based on two ideas: expressing the evolution equation in the so-called General Equations for Non-Equilibrium Reversible Irreversible Coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields. The new methods can be considered as an extension of the energy-momentum integration algorithms to coupled thermomechanical problems, to which they reduce in the purely Hamiltonian case. In the article, the new ideas are applied to a simple coupled problem: a double thermoelastic pendulum with symmetry. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize. Copyright (C) 2009 John Wiley & Sons, Ltd.