Shape optimisation for a class of semilinear variational inequalities with applications to damage models

TL;DR

This paper develops shape optimisation techniques for semilinear elliptic variational inequalities, incorporating dynamic obstacles, with applications to damage models in continuum mechanics, including sensitivity analysis, material derivatives, and optimality conditions.

Contribution

It introduces a novel framework for shape optimisation involving dynamic obstacles in variational inequalities, with rigorous sensitivity and derivative analysis, applied to damage models in elastic solids.

Findings

Established strong convergence of material derivatives.

Derived state-shape derivatives under regularity assumptions.

Applied results to shape optimisation in brittle damage models.

Abstract

The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly 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 minimisation 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 another optimisation 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 optimisation 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.