Optimization Problems with Unknown Boundaries in the Theory of Elasticity

Abstract

This chapter is devoted to the study of optimization problems for systems of partial differential equations, specifically of finding the optimum shapes of elastic bodies. We consider the boundary of the domain in which our equations are defined as the control variable. In the first three sections, the quality criteria are integral functional. In Section 4.1, we formulate the problem of optimizing rigidity of a shaft in torsion and derive some optimality criteria. Using these criteria in Section 4.2, we derive the shape of the cross section such that the bar has the greatest torsional rigidity. An optimization problem for a thin-walled bar is solved analytically, using a perturbation technique. For the case in which the walls are not thin, we employ a technique utilizing the theory of functions of complex variables. In Section 4.3 we study some problems of optimization of piecewise-homogeneous bars subjected to torsion and consider the problem of optimal reinforcement.