Shape optimization involving the Tresca friction law in a 2D linear elastic model

TL;DR

This paper conducts a shape sensitivity analysis for a 2D linear elastic model with Tresca friction, deriving explicit shape gradients without regularization, and discusses challenges in extending to higher dimensions.

Contribution

It provides the first explicit shape gradient formula for Tresca friction problems in 2D without regularization, using twice epi-differentiability and change of variables.

Findings

Derived the shape derivative of the Tresca energy functional.

Established the equivalence with a boundary value problem involving Signorini conditions.

Numerical simulations confirm the theoretical shape gradient expressions.

Abstract

The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed.