Shape Optimization in Unilateral Boundary Value Problems

Abstract

The aim of this contribution is to present some results, concerning the shape optimization in unilateral boundary value problems. In classical (direct) problems, one looks for the response of a structure subjected to given forces (for details see e.g. [5]). In optimal shape design problems, we try to identify a shape of a structure, response of which is as close as possible to designer’s ideas. In contact problems one of the possible required criteria is to find a shape of contact surfaces along which stresses are distributed without undesirable stress concentration zones. Question arises, which cost functional guarantee such phenomena. In [4] authors used the total potential energy evaluated in the equilibrium state. Restricting ourselves to simple Dirichlet-Signorini problem for a scalar function we show that under some conditions the minimization of the total potential energy really leads to a constant distribution of a flux along contact part. Moreover the whole problem is smooth, regardless the mapping: control ↦ state is only directionally differentiable. Shape optimization for contact problems, including approximation results and numerical realization is widely discussed in [1].KeywordsContact ProblemTotal Potential EnergyDesign Sensitivity AnalysisOptimal Shape DesignNato Advance Study InstituteThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.