Level Set Method for Optimization of Contact Problems

This paper deals with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field. The optimization problem consists in finding, in a contact region, such topology and shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Level set methods [

3

], [

4

] are numerically efficient and robust procedures for the tracking of interfaces, which allows domain boundary shape changes in the course of iteration. The evolution of the level set function is governed by the Hamilton Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative [

2

] of an appropriately defined cost functional with respect to the free boundary.

In this paper the necessary optimality condition for the shape and topology optimization problem of this contact problem is formulated. The paper extends results of [

1

] to contact problems with a given friction. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids.Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.