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Sensitivity analysis of a scalar mechanical contact problem with perturbation of the Tresca's friction law

Loïc Bourdin, Fabien Caubet, Aymeric Jacob de Cordemoy

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.

Sensitivity analysis of a scalar mechanical contact problem with perturbation of the Tresca's friction law

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.

Paper Structure

This paper contains 24 sections, 16 theorems, 117 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mechanical context and motivations.
  3. Objectives and methodology.
  4. Main result.
  5. Organization of the paper.
  6. Notions from Convex Analysis and Functional Framework
  7. Notions from Convex Analysis
  8. Functional Framework
  9. Main Result
  10. Some Required Boundary Value Problems
  11. A Problem with Dirichlet-Neumann Conditions
  12. A Signorini Problem
  13. A Tresca Friction Problem
  14. Sensitivity Analysis of the Tresca Friction Problem
  15. Parameterized Tresca Friction Functional and Twice Epi-Differentiability
  16. ...and 9 more sections

Key Result

Proposition 2.6

Let $(\phi_{t})_{t>0}$ be a parameterized family of functions $\phi_{t} : \mathrm{V}\rightarrow \mathbb{R}\cup\left\{\pm \infty \right\}$ for all $t>0$ and let $\phi : \mathrm{V}\rightarrow \mathbb{R}\cup\left\{\pm \infty \right\}$. Then $(\phi_{t})_{t>0}$ Mosco epi-converges to $\phi$ if and on are satisfied.

Figures (4)

  • Figure 1: Unit disk $\Omega$ and its boundary $\Gamma=\Gamma_{\mathrm{D}}\cup\Gamma_{\mathrm{N}}\cup\Gamma_{\mathrm{T}}$, with $\Gamma_{\mathrm{T}}=\Gamma^{u_{0},g_{0}}_{\mathrm{T}_{\mathrm{S}_{\mathrm{N}}}}\cup \Gamma^{u_{0},g_{0}}_{\mathrm{T}_{\mathrm{S}_{\mathrm{D}}}}\cup \Gamma^{u_{0},g_{0}}_{\mathrm{T}_{\mathrm{S-}}}\cup\Gamma^{u_{0},g_{0}}_{\mathrm{T}_{\mathrm{S+}}}$.
  • Figure 2: The representation in logarithmic scale of the map $t\in\mathbb{R}^{+} \mapsto \left \| u_{t}-u_{0}-tu_{0}' \right \|_{\mathrm{H}^{1}(\Omega)}\in\mathbb{R}^{+}$ (red) and of the map $t\in\mathbb{R}^{+} \mapsto t^{2}\in\mathbb{R}^{+}$ (blue).
  • Figure 3: The first figure is the representation of $u_{t}$ and the second its first-order approximation $u_{0}+~tu_{0}'$ for $t=0.1$.
  • Figure 4: Illustration of the boundary $\Gamma$

Theorems & Definitions (45)

  • Definition 2.1: Domain and epigraph
  • Definition 2.2: Convex subdifferential operator
  • Definition 2.3: Proximal operator
  • Definition 2.4: Mosco-convergence
  • Definition 2.5: Mosco epi-convergence
  • Proposition 2.6: Characterization of Mosco epi-convergence
  • Definition 2.7: Twice epi-differentiability
  • Example 2.8
  • Definition 2.9: Twice epi-differentiability depending on a parameter
  • Theorem 2.10
  • ...and 35 more