hal-03848645 https://univ-pau.hal.science/hal-03848645 https://univ-pau.hal.science/hal-03848645v3/document https://univ-pau.hal.science/hal-03848645v3/file/Shape_optimization_for_variational_inequalities__the_scalar_Tresca_friction_problem%20%282%29.pdf [CNRS] CNRS - Centre national de la recherche scientifique [UNIV-PAU] Université de Pau et des Pays de l'Adour - E2S UPPA [LMA-PAU] Laboratoire de Mathématiques et de leurs Applications de Pau - IPRA [INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions [TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes [UPPA-OA] uppa-oa Shape optimization for variational inequalities: the scalar Tresca friction problem Adly, Samir Bourdin, Loïc Caubet, Fabien Jacob de Cordemoy, Aymeric [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] UNDEFINED Shape optimization shape sensitivity analysis variational inequalities scalar Tresca friction law Signorini's unilateral conditions proximal operator twice epi-differentiability 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. 2023-05-10 2023-05-10 en