Shape optimization for contact problems in linear elasticity

TL;DR

This thesis develops a gradient-based shape optimization method for contact problems in linear elasticity, using regularized variational inequalities and level-set techniques, validated through numerical benchmarks.

Contribution

It introduces a novel approach to compute shape derivatives for non-differentiable contact formulations and integrates a mesh-cutting technique for explicit shape representation.

Findings

Effective shape optimization algorithm for contact problems

Successful application to 2D and 3D benchmarks

Enhanced boundary condition enforcement on contact zones

Abstract

This thesis deals with shape optimization for contact mechanics. More specifically, the linear elasticity model is considered under the small deformations hypothesis, and the elastic body is assumed to be in contact (sliding or with Tresca friction) with a rigid foundation. The mathematical formulations studied are two regularized versions of the original variational inequality: the penalty formulation and the augmented Lagrangian formulation. In order to get the shape derivatives associated to those two non-differentiable formulations, we suggest an approach based on directional derivatives. Especially, we derive sufficient conditions for the solution to be shape differentiable. This allows to develop a gradient-based topology optimization algorithm, built on these derivatives and a level-set representation of shapes. The algorithm also benefits from a mesh-cutting technique, which gives an explicit representation of the shape at each iteration, and enables to apply the boundary conditions strongly on the contact zone. The different steps of the method are detailed. Then, to validate the approach, some numerical results on two-dimensional and three-dimensional benchmarks are presented.