Thermodynamically consistent integration of coupled thermoelastic systems

This work deals with the energy-momentum-entropy consistent integration of thermoelastic systems. While energy-momentum preserving integrators are well-known for conservative mechanical systems, Romero recently introduced in [6] a thermodynamically consistent (TC) integrator for coupled thermomechanical systems. TC integrators also respect symmetries of the underlying coupled system and are therefore capable of conserving associated momentum maps. A first step towards the systematic design of a TC integrator is to cast the evolution equations into the GENERIC framework. GENERIC stands for General Equation for Non-Equilibrium Reversible-Irreversible Coupling and has been originally proposed by Grmela and Öttinger for complex fluids [3]. As a second step applying the notion of a discrete gradient in the sense of Gonzalez [2] leads to a TC integrator. The GENERIC-based framework reveals additional underlying physical structures of the thermodynamical system due to the separation of irreversible and reversible driving forces. Using the entropy as the thermodynamical state variable as in [4,6] the GENERIC framework yields an easy structure. However, this choice of thermodynamical state variable leads to a restriction in the material model and, more importantly, only allows to prescribe entropy Dirichlet boundary conditions. This drawback can only be compensated by using Lagrange-multipliers to be able to handle temperature Dirichlet boundary conditions, which unfortunately extends the system of algebraic equations to be solved (see Krüger et al. [5]). Alternatively, the present contribution uses the temperature as the thermodynamical state variable (see also the recent work by Conde Martín et al. [1]). This temperature-based approach allows to set Dirichlet boundary conditions directly. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)