Étude de la convergence de la solution faible de quelques problèmes avec des lois de frottement régularisées.

Abstract

This thesis proposes to study the convergence of the regularized solution denoted (uρ, ϕρ) to the solution (u, ϕ) when the parameter ρ tends to 0. At the beginning of this work, we set the physical framework and the mathematical model of an anti-contact problem with rubbing, thus all the conditions and friction laws on the three boundaries Γ1, Γ2 and Γ3 . Then, we construct the variational formulation associated with the continuous problem. Subsequently, we study the exsitence and uniqueness of the weak solution based on some tools of the functional analysis. Next, we also construct a variational formulation of the regularized model that is given by a coupled system of a variational equation whose displacement field u is considered as unknown with an equation that depends only on an electrical potential. Finally, and from a numerical point of view, we describe a regularization of the non-differentiable functional due to the friction term appearing in the variational formulation of this electromechanical problem. This regularization is obtained by substituting the functional j(.) With a regularized version denoted jρ(.) With ρ is a strictly positive parameter. We have obtained a result of existence, uniqueness and convergence. We close this thesis with a numerical simulation and some tests by making variations of the h discretization step.