Irregularity Of Shape Optimization Problems AndAn Improvement Technique

Abstract

Shape optimization problems of linear elastic bodies, flow fields, magnetic fields, etc. for equilibrium types can be generalized as optimization problems of domains in which elliptic boundary value problems are defined. This paper shows that ordinary domain optimization problems do not have sufficient regularity and proposes a technique to overcome this irregularity. It briefly describes the derivation of the shape gradient functions for a self-adjoint shape optimization problem, and shape identification problems of the Dirichlet type, Neumann type and subdomain gradient assigned type. Using these shape gradient functions, the irregularity of ordinary domain optimization problems is shown through a discussion of the ill-posedness that occurs when the gradient method in Hilbert space is applied directly. To overcome this irregularity, the idea of a smoothing gradient method in Hilbert space is proposed. It is conclusively shown that a numerical method based on this idea coincides with the traction method previously proposed by one of the authors and this conclusion is verified by numerical experiments.