Eliminating the fictitious frequency problem in BEM solutions of the external Helmholtz equation

Abstract

The problem of the fictitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Green's function is replaced by a modified Green's function, it is shown that these fictitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these fictitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.

Non-Singular Burton–Miller Boundary Element Method for Acoustics

“…It appears that, even with a very small interval of 0.001 for ka, we still missed many of the fictitious frequencies. This confirms the findings of [16,28], where it was found that for moderate ka values, the non-singular boundary element method is already quite good at avoiding non-physical solutions at fictitious frequencies.…”

“…Clearly, the non-physical solutions at the classical internal resonance frequencies show up in the curve for the non-singular standard boundary element method, for example, at ka = 9.356 and ka = 10.417, corresponding to the theoretical values of ka = 9.3558 and ka = 10.41712, respectively (see also Table 1 in Ref. [16]).…”

“…However, a main drawback of external problems when using the boundary element method with large ka values (with a a typical dimension of the problem) is that for certain values of ka, the so-called fictitious wave numbers (or frequencies), the internal resonance solution of the object can appear in the outer solution and, thereby, will produce nonunique (spurious) results. Although it has been shown that it is possible to eliminate certain fictitious frequencies using a modified Green's function [16], unfortunately, not all fictitious modes can be eliminated in this manner. We have to revert to one of the two most commonly used methods, namely the CHIEF method of Schenck [17] or the method of Burton and Miller [18].…”

A non-singular boundary element method for interactions between acoustical field sources and structures

“…Since the desingularized boundary element method with high order elements, such as quadratic curved elements, is so accurate, a further advantage is that it is very unlikely to hit an internal resonance frequency (and get a so-called spurious solution), thus there is no need to apply Burton-Miller type of schemes (Burton and Miller, 1971). For a more elaborate discussion on the fictitious frequency problem see (Klaseboer et al, 2019). Also with the superior quadratic elements we only need a fraction of the nodes needed to get the same accuracy as with standard flat constant elements.…”

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

“…In fact with our desingularized boundary element method, the probability of hitting a spurious solution is actually very low. In fact, it is even difficult to find such a frequency when specifically looking for it (see the figures in [10]). The two most popular methods to deal with fictitious frequencies are the CHIEF method proposed by Schenck [23] and the Burton-Miller method [3].…”