Sharp Interface Limit in Phase-Field based Structural Optimization of Variational Inequalities

Topology optimization of bodies in unilateral contact with a given friction is considered in the paper. The contact phenomenon is governed by the second order elliptic variational inequality. The aim of this optimization problem is to find such distribution of the material density function to minimize the normal contact stress. The original optimization problem is reformulated in terms of Cahn-Hilliard model as well as of material density function. In this approach the interface between phases is dependent on a small parameter. The aim of the present paper is to study the behavior of phase field based optimization problem as the interface parameter tends to zero. The existence result in the space of bounded variations functions for the optimization problem in the sharp interface limit case is shown. The sharp interface limit of the original functional regularized using Ginzburg-Landau energy functional in terms of Gamma-convergence is provided.