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Shape optimization for variational inequalities: the scalar Tresca friction problem

Samir Adly, Loïc Bourdin, Fabien Caubet, Aymeric Jacob de Cordemoy

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.

Shape optimization for variational inequalities: the scalar Tresca friction problem

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.

Paper Structure

This paper contains 20 sections, 19 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of the shape optimization problem and methodology
  4. Main theoretical results
  5. Application to shape optimization and numerical simulations
  6. Organization of the paper
  7. Preliminaries
  8. Reminders on proximal operator and twice epi-differentiability
  9. Reminders on differential geometry
  10. Reminders on three basic nonlinear boundary value problems
  11. A Neumann problem
  12. A scalar Signorini problem
  13. A scalar Tresca friction problem
  14. Main theoretical results
  15. Setting of the shape perturbation and preliminaries
  16. ...and 5 more sections

Key Result

Proposition 2.10

Let $\Psi : \mathbb{R}_{+}\times \mathcal{H}\rightarrow \mathbb{R} \cup \left\{+\infty\right\}$ be a function such that, for all $t\geq0$, $\Psi(t,\cdot) : \mathcal{H}\rightarrow \mathbb{R} \cup\left\{+\infty\right\}$ is a proper, lower semi-continuous and convex function. Let $F : \mathbb{R}_{+ for all $t\geq 0$. If the conditions are satisfied, then $u$ is differentiable at $t=0$ with

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2: Mosco convergence
  • Definition 2.3: Mosco epi-convergence
  • Remark 2.4
  • Definition 2.5: Twice epi-differentiability depending on a parameter
  • Remark 2.6
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 40 more