{
  "id": "2410.11750",
  "version": "v1",
  "published": "2024-10-15T16:25:09.000Z",
  "updated": "2024-10-15T16:25:09.000Z",
  "title": "Shape optimization for variational inequalities: the scalar Tresca friction problem",
  "authors": [
    "Samir Adly",
    "Loïc Bourdin",
    "Fabien Caubet",
    "Aymeric Jacob de Cordemoy"
  ],
  "comment": "30 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-15T16:25:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "49Q12",
      "35J85",
      "74M10",
      "74M15",
      "74P10"
    ],
    "keywords": [
      "variational inequalities",
      "scalar tresca friction problem admits",
      "scalar tresca friction law",
      "signorini-type unilateral conditions",
      "shape optimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}