  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.
