Discontinuous Galerkin method for the three-dimensional problem of thermoelasticity

Abstract

The paper is focused on the mathematical and numerical approaches for the thermoelasticity problem in the three-dimensional domain. The mathematical description of considered problem is based on the second order differential equations of elasticity with the term describing thermal deformations. The numerical model uses the discontinuous Galerkin method which is widely used to solve the problems of hydrodynamics. The presented paper shows the possibility of using the mentioned method to solve the problem of thermomechanics.

Analysis of the three-dimensional thermoelasticity problem with the use of the continuous Galerkin method

“…The obtained results confirm the correctness of the solver based on the continuous Galerkin formulation, which is an alternative to the discontinuous version of the method [9] used to solve the problem of thermoelasticity. The created original solver allows for numerical computations using complex three-dimensional meshes.…”

The comparison of results obtained from the continuous and discontinuous Galerkin Method for the thermoelasticity problem

“…Details of the mathematical description of both CG and DG methods are presented in the works [2,5]. This paper presents only the final equations resulting from the application of FEM.…”

“…The use of CGM entails the necessity of treating the analyzed area in a global way, building a huge stiffness matrix (especially in 3D cases) and solving a large number of linear equations. The application of the Discontinuous Galerkin method (DGM) [3,4] to solve the spatial problem of thermoelasticity has been discussed in [5].…”

“…Based on assumptions of the mathematical and numerical models presented in [2,5], two proprietary solvers were prepared. Both solvers can be used in numerical calculations of coupled physical phenomena.…”

Theoretical justification of Ventcel's boundary conditions for a thin layer three-dimensional thermoelasticity problem

“…Recently, numerical modeling of thermoelasticity problems in a three dimensional region have been studied in [27] and [28]. However, in our case and due to the small thickness of the layer, these numerical computations can become cumbersome, especially for three-dimensional problems.…”