Shape optimization for variational inequalities: the scalar Tresca friction problem

TL;DR

This paper develops a shape optimization framework for a scalar Tresca friction problem, deriving shape derivatives without regularization and demonstrating practical numerical solutions for energy minimization.

Contribution

It introduces a novel shape derivative analysis for Tresca friction problems using convex analysis tools, without regularization, and provides explicit shape gradients and numerical validation.

Findings

Shape derivative exists and matches a boundary value problem with Signorini conditions.

Explicit shape gradient characterization is achieved.

Numerical simulations confirm the method's applicability.

Abstract

This paper investigates, without any regularization or penalization procedure, a shape optimization problem involving a simplified friction phenomena modeled by a scalar Tresca friction law. Precisely, using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability, we prove that the solution to a scalar Tresca friction problem admits a directional derivative with respect to the shape which moreover coincides with the solution to a boundary value problem involving Signorini-type unilateral conditions. Then we explicitly characterize the shape gradient of the corresponding energy functional and we exhibit a descent direction. Finally numerical simulations are performed to solve the corresponding energy minimization problem under a volume constraint which shows the applicability.