Non-linear optimization of the shape function when solving the 2-D Helmholtz equation by means of the finite element method

Abstract

Lord Rayleigh suggested in 1984, in connection with his now classical principle, the use of trial functions containing an undetermined exponential parameter k, to be optimized by minimization of the eigenvalues with respect to itself. In the past five years several authors have, basically, used Rayleigh's optimization concept to solve a variety of problems of elastic stability, vibrations, heat conduction, etc., and more recently, implemented it in finite element formulations. The present paper deals with an extension of a previous study performed by the authors, named “the k-optimization of the shape functions in the FEM”, to 2-D eigenproblems governed by the Helmholtz differential equation. It is concluded that the method leads to considerable economies both in computer memory requirements and in CPU time. Three-node triangular and four-node quadrilateral elements are investigated in the present study.