Contact Shape Optimization

Abstract

Shape optimization is a branch of the optimal control theory in which the control variable is connected with the geometry of the problem. The aim is to find a shape from an a priori defined class of domains, for wich the corresponding cost functional attains its minimum. Shape optimization of mechanical systems, behaviour of which is described by equations, has been very well analyzed from the mathematical, as well as from the mechanical point of view, see [1], [2], [3] and references therein. The aim of this contribution is to extend results to the case, in which the system is described by the so called variational inequalities. There are two reasons for doing that: 1) The behavior of many mechanical models can be described just using the framework of variational inequalities, see [4], [5]. 2) The optimal control of systems governed by variational inequalities has one specific feature, compared with problems governed by equations, namely the whole problem is nonsmooth. By nonsmoothness we mean the fact that the mapping: Control variable → state → cost functional is not continuously differentiable. Therefore we have to be very careful wi+h the choice of numerical minimization methods. In classical optimization problems we also often meet the situation when the resulting function which has to be minimized is nonsmooth in the above mentioned sense. But this is due to the nonsmoothness of the cost functional itself, or by the presence of constraints, etc. Such kind of explicit nonsmoothness can be usually overcome by introducing dummy variables, e.g. The source of the nonsmoothness in our kind of problems is different, namely it is due to the nonsmoothness of the inner mapping: control variable → state, i.e. this phenomena is more implicit.