Structure-preserving time integration of non-isothermal finite viscoelastic continua related to variational formulations of continuum dynamics

Abstract

A structure-preserving time integration scheme is a modified standard time stepping scheme, which fulfills the balance laws of a dynamical problem also in a discrete setting. Hence, such a scheme represents a space-time discretization of a continuum with all of its central properties. This paper deals with a special second-order accurate structure-preserving time integration of thermo-mechanical couplings in a moving continuum, satisfying all discrete thermodynamical balance laws in contrast to the underlying second-order accurate midpoint rule. More accurately, we consider isotropic heat conduction and the Gough–Joule effect in moving continua. The considered time evolution is described by five differential equations with respect to five independent variables, including the entropy field of the continuum. The bodies in the numerical examples are subjected to Dirichlet- or Neumann boundary conditions in the displacement as well as the temperature field. Although this structure-preserving time integrator is based on a time discrete spatially weak finite element formulation, it fulfills the same balance laws as the underlying five differential equations. Namely, in addition to the balances of entropy, linear momentum and angular momentum, also the balances of total energy and Lyapunov’s function are fulfilled in the discrete case.