%0 Unpublished work %T Shape optimization for variational inequalities: the scalar Tresca friction problem %+ Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP) %A Adly, Samir %A Bourdin, Loïc %A Caubet, Fabien %A Jacob de Cordemoy, Aymeric %8 2023-05-10 %D 2023 %K Shape optimization %K shape sensitivity analysis %K variational inequalities %K scalar Tresca friction law %K Signorini's unilateral conditions %K proximal operator %K twice epi-differentiability %Z 49Q10, 49Q12, 35J85, 74M10, 74M15, 74P10 %Z Mathematics [math]/Optimization and Control [math.OC]Preprints, Working Papers, ... %X 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. %G English %2 https://univ-pau.hal.science/hal-03848645v3/document %2 https://univ-pau.hal.science/hal-03848645v3/file/Shape_optimization_for_variational_inequalities__the_scalar_Tresca_friction_problem%20%282%29.pdf %L hal-03848645 %U https://univ-pau.hal.science/hal-03848645 %~ CNRS %~ UNIV-PAU %~ LMA-PAU %~ INSMI %~ TDS-MACS %~ UPPA-OA