Volume 28, 2022
|Number of page(s)||29|
|Published online||25 May 2022|
Sensitivity analysis of a Tresca-Type Problem leads to Signorini’s conditions
Institut de recherche XLIM, UMR CNRS 7252, Université de Limoges, Limoges, France
2 Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, UMR 5142, 64000 Pau, France
* Corresponding author: email@example.com
Accepted: 5 April 2022
The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H1(Ω) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.
Mathematics Subject Classification: 49Q12 / 46N10 / 74M15
Key words: Tresca-type problem / Signorini-type problem / variational inequality / convex subdifferential / proximal operator / sensitivity analysis / twice epi-differentiability
© The authors. Published by EDP Sciences, SMAI 2022
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