HAL CCSD
Shape optimization for variational inequalities: the scalar Tresca friction problem
Adly, Samir
Bourdin, Loïc
Caubet, Fabien
Jacob de Cordemoy, Aymeric
Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP) ; Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)
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
https://univ-pau.hal.science/hal-03848645
2023
en
Shape optimization
shape sensitivity analysis
variational inequalities
scalar Tresca friction law
Signorini's unilateral conditions
proximal operator
twice epi-differentiability
49Q10, 49Q12, 35J85, 74M10, 74M15, 74P10
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
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
info:eu-repo/semantics/OpenAccess