Sensitivity analysis and optimal control for a friction problem in the linear elastic model

Abstract

This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, 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 tangential Signorini's unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model.

Figures (3)

• Figure 1: Unit disk $\Omega$ and its boundary $\Gamma=\Gamma_{\mathrm{D}}\cup\Gamma_{\mathrm{N}}$.
• Figure 2: Values of the optimal control on the boundary $\Gamma_{\mathrm{N}}:=\left\{(\cos\theta,\sin\theta)\in\Gamma \mid \frac{\pi}{2}<\theta<2\pi\right\}$.
• Figure 3: Values of the cost functional $\mathcal{J}$ with respect to the iterations.

Theorems & Definitions (61)

• Remark 1.1
• Definition 2.1: Strong solution to the Dirichlet-Neumann problem
• Definition 2.2: Weak solution to the Dirichlet-Neumann problem
• Proposition 2.3
• Proposition 2.4
• Definition 2.5: Strong solution to the tangential Signorini problem
• Definition 2.6: Weak solution to the tangential Signorini problem
• Definition 2.7: Consistent decomposition
• Proposition 2.8
• proof