Shape Optimization for a Class of Semilinear Variational Inequalities with Applications to Damage Models

The present contribution investigates shape optimization problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are first derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimization problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from other optimization problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimization problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields