Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part I: Monolithic integrators and their application to finite strain thermoelasticity

We propose discretizations for a general class of nonlinear, coupled, thermomechanical problems of evolution. The most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model. To formulate such methods we exploit the geometric structure afforded by the GENERIC formalism of non-equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the GENERIC structure. As an illustration, a complete discretization of finite strain thermoelasticity is presented, using finite elements in space and a monolithic integrator in time. Simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness.