ABSTRACT
Michal Kocvara Institute of Information Theory and Automation, Prague, Czech Republic
Jiri V. Outrata University of Bayreuth, Bayreuth, Germany
1 Introduction The subject of this paper is shape optimization of elasto-perfectly plastic bodies obeying Hencky’s law, in particular, numerical realization of this problem. As the state problem (o f elasto-plasticity) can be formulated as a variational inequality (V I), the shape optimization problem leads to an interesting nonstandard control problem of the form
<7(u,y) -> inf
subject to y solves V IU u e u ,
where u is the control variable, y the state variable, V IU is the state problem depending on the control u and U is a compact set. The main difficulty in solving such problems is that the map assigning to the control u the state y is generally only locally Lipschitz (under some assumptions which will be specified later), i.e., it is nondifferentiable (nonsmooth). The standard way of solving such problems is to use the so-called regularization technique: rewrite the V I as an optimization problem (if it is possible) and then penalize the constraints in this problem by a smooth penalty. This technique is widely used, e.g., in the book [5]. It is obvious that the solution obtained by means of the regularization technique may be far from the true solution of the original problem. It is the goal of this paper to solve such problems “directly” , by solving the V I (or the corresponding optimization problem) without any inexact penalization. Then, of course, we have to take care of the nondifferentiability of the problem and use a nondifferentiable optimization (ND O ) code to its numerical solution.
