Modelling of topological derivatives for contact problems

The problem of topology optimisation is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called {\it outer asymptotic expansion} for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the {\it topological derivatives} of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimisation