https://univ-pau.hal.science/hal-03848645v3Adly, SamirSamirAdlyBourdin, LoïcLoïcBourdinCaubet, FabienFabienCaubetLMAP - Laboratoire de Mathématiques et de leurs Applications [Pau] - UPPA - Université de Pau et des Pays de l'Adour - CNRS - Centre National de la Recherche ScientifiqueJacob de Cordemoy, AymericAymericJacob de CordemoyLMAP - Laboratoire de Mathématiques et de leurs Applications [Pau] - UPPA - Université de Pau et des Pays de l'Adour - CNRS - Centre National de la Recherche ScientifiqueShape optimization for variational inequalities: the scalar Tresca friction problemHAL CCSD2023Shape optimizationshape sensitivity analysisvariational inequalitiesscalar Tresca friction lawSignorini's unilateral conditionsproximal operatortwice epi-differentiability[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]Jacob de Cordemoy, Aymeric2023-05-10 01:12:092023-05-13 04:13:112023-05-12 09:45:30enPreprints, Working Papers, ...https://univ-pau.hal.science/hal-03848645v2application/pdf3This 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.