Efficient Techniques for Shape Optimization with Variational Inequalities Using Adjoints

Abstract

In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for semismooth shape optimization problems. In this paper, we consider shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems. Under appropriate assumptions, we prove existence of adjoints for regularized problems and convergence to adjoints of the unregularized problem. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically.