Topology optimization of variational inequalities using Cahn-Hilliard approach

The paper is concerned with the topology optimization of bodies in unilateral contact with given friction. The contact phenomenon is governed by the second order elliptic variational inequality. The aim of the optimization problem is to find such distribution of the material density function to minimize the normal contact stress. The phase field approach is used to analyze and solve numerically this optimization problem. The original cost functional is regularized using Ginzburg-Landau free energy functional including the surface and bulk energy terms. These terms allow to control global perimeter constraint and the occurrence of the intermediate solution values. The Lagrangian approach is used to calculate the derivative of the regularized cost functional and to formulate a necessary optimality condition. The optimal topology is obtained as the steady state of the phase transition governed by modified Cahn-Hilliard equation. The finite element method is used as a discretization method. Numerical examples are provided and discussed.