Shape optimization for contact problem involving Signorini unilateral conditions

TL;DR

This paper develops a shape optimization framework for contact problems with Signorini conditions, deriving shape derivatives and performing numerical simulations without penalization, advancing the mathematical and computational understanding of such problems.

Contribution

It introduces a novel shape sensitivity analysis for Signorini contact problems using convex analysis tools, explicitly characterizing the shape gradient without penalization.

Findings

Explicit shape gradient derived for Signorini problems

Solution admits directional derivatives coinciding with other Signorini problems

Numerical simulations validate the theoretical methodology

Abstract

This paper investigates a shape optimization problem involving the Signorini unilateral conditions in a linear elastic model, without any penalization procedure. The shape sensitivity analysis is performed using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability. We prove that the solution to the Signorini problem admits a directional derivative with respect to the shape which moreover coincides with the solution to another Signorini problem. Then, the shape gradient of the corresponding energy functional is explicitly characterized which allows us to perform numerical simulations to illustrate this methodology.