Shape optimization for contact problem involving Signorini unilateral conditions

This paper investigates a shape optimization problem involving the Signorini unilateral conditions in a linear elastic model, without any penalization procedure. The shape sensitivity analysis is performed using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability. We prove that the solution to the Signorini problem admits a directional derivative with respect to the shape which moreover coincides with the solution to another Signorini problem. Then, the shape gradient of the corresponding energy functional is explicitly characterized which allows us to perform numerical simulations to illustrate this methodology.