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

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 introduced in previous works. 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.

Automated summary

This work focuses on shape optimization in contact mechanics, using regularized versions of the variational inequality. By introducing shape derivatives based on directional derivatives, a gradient-based topology optimization algorithm is developed. Mesh-cutting technique is used to give an explicit representation of the shape at each iteration, facilitating the application of boundary conditions on the contact zone. Numerical results on benchmarks validate the proposed approach.