Discontinuous Galerkin method in numerical simulation of two-dimensional thermoelasticity problem with single stabilization parameter

Abstract

The aim of the paper is the development of discontinuous Galerkin with finite difference rules (DGFD) to a two-dimensional stationary and non-stationary thermoelasticity problem. Displacement and temperature fields are approximated on the same mesh frame but with various approximation orders, which are set independently for each of the fields. Because the DGFD method does not use nodes, special attention needs to be paid to applying boundary conditions. Various types of thermal and mechanical boundary conditions are considered. In the presented approach only one stabilization parameter for the coupled problem needs to be evaluated in the DGFD method. The same parameter used in thermal and in mechanical part. The considered domain is discretized by a polygonal mesh in which the polygonal elements may have arbitrary shapes, such as e.g. a fish shape, as well as typical rectangular shapes. The orthogonality of Chebyshev basis functions may be utilized for rectangular elements. Very high-order approximate solution can be obtained in such case. In the coupled problem, the same element may be high-order for displacement field while low-order to approximate temperature. The argument contained in the paper is illustrated with few examples.