A shape optimization algorithm based on directional derivatives for three-dimensional contact problems

TL;DR

This paper introduces a shape optimization algorithm for 3D contact problems in elasticity, utilizing directional derivatives to handle non-differentiable formulations, and demonstrates its effectiveness through numerical benchmarks.

Contribution

It develops a gradient-based topology optimization method for contact problems using directional derivatives and a level-set approach, with explicit shape representation and boundary condition enforcement.

Findings

Successful application to 3D contact problems

Effective shape optimization with explicit shape updates

Numerical validation on benchmark problems

Abstract

This work 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 follow an approach based on directional derivatives \cite{chaudet2020shape,chaudet2021shape}. This allows us 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 us 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.