%0 Unpublished work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