CAT'S EYE RADIATION WITH BOUNDARY ELEMENTS: COMPARATIVE STUDY ON TREATMENT OF IRREGULAR FREQUENCIES

Abstract

This paper reviews a number of techniques developed to overcome the well-known nonuniqueness problem in boundary integral formulations of acoustic radiation. Although the problem has received much attention, comparative studies are hardly known in this field. Furthermore, the problem has often been studied using an unsuitable example, namely a simple radiating sphere. In this case, often the addition of one collocation point in the centre of the sphere suffices to remove the nonuniqueness problem for a large range of wavenumbers. In contrast to the radiating sphere, the radiating cat's eye structure is considered in this paper. Solution of the discretized ordinary Kirchhoff–Helmholtz integral equation, also known as the Surface Helmholtz Equation, reveals a large number of so-called irregular frequencies, i.e. frequencies where the BEM fails. The paper compares the performance of different methods in alleviating this failure. The CHIEF method and its variation due to Rosen et al. are found to encounter difficulties at high frequencies. A much better performance is obtained by combining the Kirchhoff–Helmholtz integral equation with its normal derivative. In particular the method of Burton and Miller and a modification of it which avoids evaluating the hypersingular operator at nonsmooth points are tested. Both methods seem to provide reliable solutions. The modified method encounters minor failures in the frequency response function at a geometric singularity, although performing surprisingly well in many cases. More tests need to be carried out to assess fully the effectiveness of this method which allows easy use of continuous quadratic elements. However, it is the Burton and Miller formulation which appears to be the most reliable for acoustic radiation analysis. The use of CHIEF and its variations cannot be recommended.