Structural optimization of contact problems using Cahn-Hilliard model

•Topology optimization of unilateral elastic contact problems with Tresca friction.•Regularization of topology optimization problem using phase transition models.•Necessary optimality condition in the form of generalized Cahn-Hilliard equation.•Finite element approximation.•Optimal topologies leading to reduction of maximal contact stress.

The paper deals with an analysis and numerical solution of topology optimization problems for elastic bodies in unilateral contact using a phase field approach. The topology optimization problem consists in finding such distribution of the body material in a design domain to minimize the normal contact stress along the boundary of the contacting bodies. The optimization problem is formulated as multiphase problem in terms of material density function. The original cost functional is regularized using surface and bulk energy terms. These terms allow to control global perimeter constraint and the occurrence of the intermediate solution values. Using Lagrange multipliers approach the derivative of the cost functional with respect to density function is calculated. Necessary optimality condition is formulated as the steady state of the phase transition governed by this equation. The finite element and the finite difference methods are used as the approximation methods. Numerical examples are provided and discussed.