Sensitivity Analysis of a Scalar Mechanical Contact Problem with Perturbation of the Tresca’s Friction Law

Loïc Bourdin (), Fabien Caubet () and Aymeric Jacob de Cordemoy ()
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Loïc Bourdin: Université de Limoges
Fabien Caubet: Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, UMR 5142
Aymeric Jacob de Cordemoy: Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, UMR 5142

Journal of Optimization Theory and Applications, 2022, vol. 192, issue 3, No 4, 856-890

Abstract: Abstract This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca’s friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular, the friction threshold involved in the Tresca’s friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence, we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally, numerical simulations are provided in order to illustrate our main result.

Keywords: Mechanical contact problems; Tresca’s friction law; Signorini unilateral conditions; Variational inequalities; Convex subdifferential; Proximal operator; Sensitivity analysis; Twice epi-differentiability; 49Q12; 46N10; 74M15 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-021-01993-x

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