Galerkin-based energy–momentum consistent time-stepping algorithms for classical nonlinear thermo-elastodynamics

This paper presents energy–momentum consistent time-stepping schemes for classical nonlinear thermo-elastodynamics, which include well-known energy–momentum conserving time integrators for elastodynamics. By using the time finite element approach, this time-stepping schemes are not restricted to second-order accuracy. In order to retain the first and second law of thermodynamics in a discrete setting, the equations of motion are temporally discretised by a Petrov–Galerkin method, and the entropy evolution equation by a new Bubnov–Galerkin method. The new aspect in this Bubnov–Galerkin method is the used jump term, which is necessary to avoid numerical dissipation beside the local physical dissipation according to Fourier's law. The stress tensor in the obtained enhanced hybrid Galerkin (ehG) method is approximated by a higher-order accurate discrete gradient. As additional new features of a monolithic solution strategy, this paper presents a convergence criterion and an initializer routine, which avoids scaling problems in the primary unknowns and leads to a more rapid convergence for large time steps, respectively. Representative numerical examples verify the excellent performance of the ehG time-stepping schemes in comparison to the trapezoidal rule, especially concerning rotor dynamics.