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

Abstract: We develop mathematical models for shape design and 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. For energy-type shape functionals, the Eulerian derivative of the hemivariational inequality is derived through rigorous shape sensitivity analysis. The rationality of a regularization approach is justified by asymptotic analysis, and this method is further applied to handle the non-smoothness of general shape functionals in the sensitivity framework. Based on these theoretical results, a numerical boundary variational method is proposed for shape optimization. For topology optimization, 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. To the best of our knowledge, these approaches are new for shape design in hemivariational inequalities. Various numerical experiments confirm the accuracy and effectiveness of the proposed shape and topology optimization algorithms.