Numerical Analysis of a Unilateral Problem in Planar Thermoelasticity

Abstract

We consider in this paper the numerical approximation of a quasi-static contact problem in linear thermoelasticity that models the evolution of the temperature and displacement of an elastic, homogeneous, and isotropic body that may come in contact with an elastic obstacle. We propose a finite element method to numerically approximate the continuous solution. Convergence without any regularity assumptions is proved and error estimates are obtained if the continuous solution is sufficiently regular.

A Two-Level Crank-Nicolson Difference Scheme and Its Richardson Extrapolation Methods for a Magneto-Thermo-Elasticity Model

“…However, it is very difficult for us to obtain the analytical solutions of the generalized models with arbitrary initial-boundary conditions. Fortunately, a lot of numerical methods, such as, numerical integration methods [25][26][27], finite element methods [28][29][30][31], boundary element method [32], collocation methods [33][34][35][36] and finite difference methods [37][38][39][40][41], have been developed by scientists and engineers for them. However, mathematical analyses including convergence, stability, posteriori estimation, and energy conservation or dissipation, have not been studied for most of them.…”

A vibrating thermoelastic plate in a contact with an obstacle

“…It can be assured due to [12] by assuming of thermally isolated lower and upper faces of the plate. The quasistatic case of such an approach has been solved numerically in [9]. We shall use the model derived in [13] under the assumption of a small change of temperature compared with its reference temperature.…”