Coupled variational inequalities and application in electroelasticity

Abstract

This work is devoted to the mathematical and numerical study of a framework handling a system of coupled variational inequalities. We prove both the existence and the uniqueness of a weak solution to the problem. Then, we introduce a convergent iterative scheme. Using this latter, we decouple the problem into further subproblems and derive their corresponding minimization problems. As a practical application of this class of coupled abstract variational inequalities, we consider a class of problems that model an electroelastic body coming into frictional contact with a rigid electrically conductive foundation. Both electrical and mechanical contacts are of Signorini type. In other words, our model prescribes the mechanical response produced by the foundation and the outflow of the free charges across the contact zone. The last part of this paper is mainly reserved for the numerical resolution of the problem at hand. For this purpose, we have developed an alternating direction method of multipliers and convex dualities to compute and illustrate the solutions.