A coupled electro-thermal Discontinuous Galerkin method

Abstract

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems.In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method.The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H 1 -norm and in the L 2 -norm are shown to be optimal in the mesh size with the polynomial approximation degree.

Numerical Properties of a Discontinuous Galerkin Fomulation for Electro-Thermal Coupled Problems

“…Let us define the vector of the unknown fields M M M(2 × 1) = f V f T , with f V = − V T and f T = 1 T , [1,3]. Indeed it can be shown that the fluxes j j j e , j j j y and the fields gradients ∇(− V T ), ∇( 1 T ) are conjugated pairs.…”

“…The demonstration of numerical properties such as the optimal error estimate, stability of the formulation and uniqueness of the solution for β high enough is reported in [1].…”

“…A DG method has also been used in [9] to solve the thermo-elastic coupling problems due to temperature and pressure dependent thermal contact resistance. We have recently extended the DG method to solve electro-thermal coupled problems in terms of energetically conjugated fields gradients and fluxes [1], which to the authors knowledge, has not been introduced yet. In [1], the numerical properties of the nonlinear elliptic problem, following the method proposed in [2,5], have been derived.…”

“…We have recently extended the DG method to solve electro-thermal coupled problems in terms of energetically conjugated fields gradients and fluxes [1], which to the authors knowledge, has not been introduced yet. In [1], the numerical properties of the nonlinear elliptic problem, following the method proposed in [2,5], have been derived. In particular, the convergence rates of the error in both the energy and L 2 -norms have been shown to be optimal with respect to the mesh size in terms of the polynomial degree approximation k (respectively in order k − 1 and k).…”

A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems: application to shape memory composite materials

“…In that work the DG method is used to simulate the temperature jump, and the mechanical sub-problem is solved by the DG finite element method with a penalty function. In [27], a coupled nonlinear Electro-Thermal DG method, has been derived by the authors in terms of energetically conjugated fields gradients and fluxes. This conjugated pair has been obtained by a particular choice of the test functions (δf T = δ( T ), where T is the temperature and V is the electrical potential [42,72,38], which has allowed developing a stable nonlinear DG formulation with optimal convergence rate.…”