Shape optimization in contact problems with Coulomb friction

Abstract

The paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems following the Coulomb friction law. Mathematical modelling of the Coulomb friction problem leads to a quasi-variational inequality. It is shown that for small coefficients of friction the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable describing the shape of the elastic body.The shape optimization problem belongs to a class of so-called mathematical programs with equilibrium constraints (MPECs). The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists of minimizing a nonsmooth composite function generated by the objective and the (single-valued) control-state mapping. In this paper, the control-state mapping is much more complicated than in most MPECs solved in the literature so far, and the generalization of the relevant results is by no means straightforward. Numerical examples illustrate the efficiency and reliability of the suggested approach.