A Comprehensive Review of Boundary Integral Formulations of Acoustic Scattering Problems

Abstract

This is a review presenting an overview of the developments in boundary integral formulations of the acoustic scattering problems. Generally, the problem is formulated in one of two ways viz. Green’s representation formula, and the Layer-theoretic formulation utilizing either a simple-layer or a double-layer potential. The review presents and expounds the major contributions in this area over the last four decades. The need for a robust and improved formulation of the exterior scattering problem (Neumann or Dirichlet) arose due to the fact that the classical formulation failed to yield a unique solution at (acoustic) wave-numbers which correspond to eigenvalues (eigenfrequencies) of the corresponding interior scattering problem. Moreover, this correlation becomes more pronounced as the wave-numbers become larger i.e. as the (acoustic) frequency increases. The robust integral formulations which are discussed here yield Fredholms integral equations of the second kind which are more amenable to computation than the first kind. However, the integral equation involves a hypersingular kernel which creates ill-conditioning in the final matrix representation. This is circumvented by a regularisation technique. An extensive useful list of references is also presented here for researchers in this area.

The locally corrected Nyström method applied to 3D scalar SIE in acoustic cavities using curvilinear coordinates

“…(11) can not be calculated by Gaussian quadrature. As we will state below, to regularize the problem when having functions with singularities of different orders, we use the well known technique of singularity extraction to separate the functions in one smooth, and one analytically integrable part that despite of being singular can be treated by taking a limit to determine their Cauchy principal value.…”

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

“…These methods appeared long before the ones suggested by Nowak and Hall [44,45], but in the acoustical and electromagnetic wave contexts. As explained in the excellent review articles [4,8,72] (see also [55,53,41,58,19,20,64,42,18,5,63,21]), all these methods are based on employing linear combinations of the extended boundary condition, first-kind integral equation and second-kind integral equations so as to result, after discretization, in a matrix equation whose matrix is not singular at any frequency. In particular, this was the procedure adopted by Brakhage and Werner [14], Schenck [52], Bolomey and Tabbara [11,12], Burton and Miller [15], Mautz and Harrington [38,39], just to name a few.…”

“…To demonstrate the universal nature of my demonstration, I will treat not only the stress-free (homogenous Neumann) boundary condition but also the rigid (homogeneous Dirichlet) boundary condition. Both of these problems have obvious counterparts in the fields of fluid acoustics [52,42,58,8,19,20], and electromagnetism [11,12,38,39,41,55,64,71,53,34] where they have been, and continue to be, intensely studied (in fact, much more so than in the elastic wave community), increasingly with the support of applied mathematicians [14,15,18,4,5,21,72,74]. This involvement of mathematicians explains why the subject of what turns out to be that of spurious resonances is becoming more and more abstract and therefore not necessarily familiar to engineers and geophysicists.…”

The Boundary Element Method in Acoustics: A Survey

“…In this paper, the first is termed the Schenck method [149] and the second is termed the combined boundary integral equation method (CBIEM). There are several references on a general review and evaluation of the methods [55,[150][151][152][153][154][155][156][157] and of software implementations [158][159][160]. The methods have been used in a range of applications: loudspeakers [38,50,51,[161][162][163][164][165][166][167][168][169][170][171][172][173][174][175][176][177][178], transducers [52,[179][180][181][182][183], hearing aid [184], diffusers [185,186], sound within or around the human body [159,[187][188]…”