Topological Derivatives for Contact Problems - Conical Differentiability and Asymptotic Analysis

摘要

Numerical methods of evaluation of topological derivatives are proposed for contact problems in two dimensional elasticity. Problems of topology optimisation are investigated for free boundary problems of boundary 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 outer asymptotic expansion for solutions of contact problems with respect to singular perturbation of geometrical domain depending on small parameter are obtained by an application of nonsmooth analysis. The topological derivatives can be used in numerical methods of simultaneous shape and topology optimisation, in particular, in the level set type methods.