# The Boundary Element Method in Acoustics: A Survey
## 1 Introduction
Again, there are a variety of methods for the numerical solution of Helmholtz problems; the finite difference method, the finite element method [23; 35; 36; 37; 38; 39; 40; 41; 42; 43] and, the subject of this paper, the boundary element method. The boundary element method is one of several established numerical or computational methods for solving boundary-value problems that arise in mathematics, the physical sciences and engineering [44]. Typically, a boundary-value problem consists of a domain within or outside a boundary in which a variable of interest or physical property is governed by an identified partial differential equation (PDE). A computational method for solving a boundary-value problem is tasked with finding an approximation to the unknown variable within the domain. Mostly, this is carried out by domain methods, such as the finite element method [45], in which a mesh is applied to the domain. However, the boundary element method works in an entirely different way in that in the first stage further information is found on the boundary only; the solution at the domain points is found in the second stage by integrating over the known boundary data. The boundary element method requires a mesh of the boundary only, and hence is generally easier to apply than domain methods. The number of elements in the BEM is therefore expected to be much less than in the corresponding finite element method (for the same level of required accuracy or element size), and there is therefore often a potential for significant efficiency savings. The BEM is not as widely applicable as the domain methods; when problems are non-linear, for example, the application of the development of a suitable BEM requires significant further adaption [46; 47; 48; 49]. Typically, the boundary element method has found application in sound reproduction modelling, such as loudspeakers [50; 51], sonar transducers [52] and in modelling noise from vehicles [53; 54; 55; 56] and, more recently, aircraft [57; 58; 59]. For the acoustic boundary element method to be accessible, and hence widely used, it has to be implemented in software. This was precisely the rationale behind the development of the software [60] and the monograph [61], the latter also serving as a manual. Several texts on the same theme were published at about the same time [62; 63], following on from the earlier collection of works [64]. A chapter of a recent book contains a modern introduction to the acoustic boundary element method [65]. Implementing the acoustic BEM as software continues to be challenging in terms of scoping, the choice of sub-methods and efficiency. The method requires matrices to be formed with each component being the result of an integration, with some of the integrals being singular or hypersingular. The standard BEM requires the solution of a linear system of equations that is formed from the matrices, or,if the BEM is used for modal analysis, a non-linear eigenvalue problem. There have been significant reliability issues with the BEM for exterior problems. These challenges have been met, but much of the research is focused on future-proofing the method, with the increasing expectations in scaling-up and maintaining reasonable processing time. There is a continual desire to progress to higher resolutions of elements, particularly as this is necessary for modelling high frequency problems.