The performance of spheroidal infinite elements

A number of spheroidal and ellipsoidal in.nite elements have been proposed for the solution of unbounded wave problems in the frequency domain, i.e solutions of the Helmholtz equation. These elements are widely believed to be more e;ective than conventional spherical in.nite elements in cases where the radiating or scattering object is slender or >at and can therefore be closely enclosed by a spheroidal or an ellipsoidal surface. The validity of this statement is investigated in the current article. The radial order which is required for an accurate solution is shown to depend strongly not only upon the type of element that is used, but also on the aspect ratio of the bounding spheroid and the non-dimensional wave number. The nature of this dependence can partially be explained by comparing the non-oscillatory component of simple source solutions to the terms available in the trial solution of spheroidal elements. Numerical studies are also presented to demonstrate the rates at which convergence can be achieved, in practice, by unconjugated-(‘Burnett’) and conjugated (‘Astley-Leis’)-type elements. It will be shown that neither formulation is entirely satisfactory at high frequencies and high aspect ratios