The method of arbitrary lines in optimal shape design: problems with an elliptic state equation

Abstract

Optimal shape design problems with elliptic state equations are defined. The shape of an optimised domain is controlled by a vector of design variables uniquely denning a Bézier curve. The state equations are semi-discretised by the h- p method of arbitrary lines (MAL) and approximate shape design problems are derived. The existence and convergence of both semi-discrete state solutions and approximate optimal shapes is proved. In shape design sensitivity analysis, the material derivative method is used. The resulting formulae are expressed by boundary integrals which contain the normal derivative of the state solution. The derivative is approximated by means of the MAL semi-discrete state solution and the convergence properties of the approximation are analysed. Finally, numerical examples are given which demonstrate the performance of MAL solutions as compared with other approaches based on (conforming and nonconforming) finite element methods.