ABSTRACT
Introduction The boundary variations technique for unilateral problems is applied in [18] in the framework of nonsmooth analysis combined with the speed method. Nonsmooth analysis is necessary since the shape differentiability of solutions to unilateral problems is obtained only in the framework of Hadamard differentiability of metric projections onto polyhedric sets in the appropriate Sobolev spaces. However, to our best knowledge, there are no numerical methods for simultaneous shape and topology optimization [16] of unilateral problems. The main difficulty in analysis of
unilateral problems, e.g., the contact problems in elasticity, is associated with the nonlinearity of the nonpenetration condition over the contact zone which makes the boundary value problem nonsmooth. In the paper we propose a method for numerical evaluation of topological derivatives for such problems. The notion of topological derivative of a shape functional is introduced in [14]. The knowledge of topological derivatives is required for the optimality conditions of simultaneous shape and topology optimization. The topological derivative of a given shape functional can be determined from the variations of the shape functional created by the variations of the topology of geometrical domains. The topology variations are defined by nucleation of small holes or cavities or more generally small defects in geometrical domains.
