Topology optimization for unilateral problems

Jan Sokolowski (1) , Antoni Zochowski (2)

1 IECN - Institut Élie Cartan de Nancy
2 IBS PAN - Systems Research Institute [Warsaw]

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 {\it 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.

Mots clés: optimal design.; Shape optimization; topological derivative; contact problem; asymptotic expansion; energy functional; optimal design

Domaines: Equations aux dérivées partielles [math.AP]

Jan Sokolowski, Antoni Zochowski. Topology optimization for unilateral problems. John Cagnol and Jean-Paul Zolésio. Control and boundary analysis. Selected papers from the 21st Conference on System Modeling and Optimization held in Sophia Antipolis, July 21--25, 2003, Chapman & Hall/CRC, pp.97-105, 2005, Lecture Notes in Pure and Applied Mathematics, 240. ⟨hal-00102010⟩