Analytical and numerical treatment of electro-thermo-mechanical coupling

Single-field problems are well understood concerning their numerical treatment and their numerical problems. In coupled situations, this is not always the case. This article is an attempt to fill the gap for the three-field problem of small strain electro-thermo-elasticity. Apart from the quasi-static equilibrium conditions, also, the nonlinear heat equation and the stationary but temperature-dependent electrical current equation are treated. The following aspects are investigated: First, analytical equations for code verification purposes are provided for particular subproblems. Secondly, we apply the method of vertical lines, which leads to a system of differential-algebraic equations—after the spatial discretization—that can be solved using stiffly accurate, diagonally implicit Runge–Kutta methods. This enables to apply high-order time integration schemes and to make use of step-size control by embedded schemes. Here, the known problems of the connection between mesh- and the temporal step-size must be studied, also focusing on the order reduction caused by nonlinear Dirichlet boundary conditions. Additionally, the resulting system of nonlinear equations is treated with different methods such as the Newton–Raphson scheme, the Newton–Chord method and a Newton–Chord method with Aitken relaxation leading to very efficient computations.