Topological derivatives for optimization of plane elasticity contact problems

Topological derivatives for plane elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions to elasticity boundary value problems in domains with small holes and determine the explicit asymptotic expansions of solutions with respect to small parameter, which describes the radius of internal hole. The elastic potentials of Muskhelishvili give us an explicit solution in the ring C(ρ,R)={ρ<|x|