Shape optimization for variational inequalities: the scalar Tresca friction problem

Samir Adly , Loïc Bourdin , Fabien Caubet (1) , Aymeric Jacob de Cordemoy (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

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.

Mots clés: Shape optimization; shape sensitivity analysis; variational inequalities; scalar Tresca friction law; Signorini's unilateral conditions; proximal operator; twice epi-differentiability

Domaines: Optimisation et contrôle [math.OC]

Samir Adly, Loïc Bourdin, Fabien Caubet, Aymeric Jacob de Cordemoy. Shape optimization for variational inequalities: the scalar Tresca friction problem. 2023. ⟨hal-03848645v2⟩