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Journal Articles SIAM Journal on Optimization Year : 2023

Shape optimization for variational inequalities: the scalar Tresca friction problem

Samir Adly
Loïc Bourdin

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 of our method and our theoretical results.
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Dates and versions

hal-03848645 , version 1 (10-11-2022)
hal-03848645 , version 2 (09-05-2023)
hal-03848645 , version 3 (10-05-2023)
hal-03848645 , version 4 (10-12-2023)

Identifiers

  • HAL Id : hal-03848645 , version 3

Cite

Samir Adly, Loïc Bourdin, Fabien Caubet, Aymeric Jacob de Cordemoy. Shape optimization for variational inequalities: the scalar Tresca friction problem. SIAM Journal on Optimization, 2023, 33 (4), pp.2512-2541. ⟨hal-03848645v3⟩
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