Shape design with phase field methods for structural hemivariational inequalities in contact problems

We develop mathematical models for topology optimization in structural contact problems involving friction between elastic and rigid bodies. The governing mechanical constraint is a nonlinear, non-smooth, and non-convex hemivariational inequality, which provides a more general and realistic description of frictional contact forces than standard variational inequalities, but is also more challenging due to its non-convexity. A regularization approach is applied to handle the non-smoothness of general shape functionals in the sensitivity framework. Three phase-field algorithms are developed: a gradient-flow phase-field method, a phase-field method with second-order regularization of the cost functional, and a phase-field method coupled with topological derivatives. Various numerical experiments confirm the accuracy and effectiveness of the proposed topology optimization algorithms.