# The Boundary Element Method in Acoustics: A Survey
## 2 Acoustic Model
### The Scope of the Boundary Element Method in the Solution of Acoustic/Helmholtz Problems  
In applying the boundary element method to acoustic or Helmholtz problems, the user is in effect adopting a model for the boundary/ies and domain(s), the nature of which determines the integral equation(s) that is/are employed. Traditionally, the BEM has been developed to solve the acoustic or Helmholtz problem interior or exterior to a closed boundary in the standard physical domains are two-dimensional, three-dimensional and axisymmetric three-dimensional space. The basis of the method is the integral equations that arise through applying Green's theorems (direct BEM) or by using layer potentials (indirect BEM). The exterior problem has received much more attention, because of its value in solving over an infinite domain from a surface mesh and because of the difficulty in resolving its reliability problem. Domain methods, such as the finite element method, may also be applied; more elements are required, but the matrices are sparse and structured and the FEM is more established than the BEM in general. If the FEM is applied to exterior problems then techniques such as infinite elements [66] can be used to complete the outer mesh or the perfectly matched layer may be used to absorb outgoing waves [67; 68; 69]. The solution of the interior and exterior acoustic problem by the BEM is considered in Sections 4.1 and 4.3. An outline of the equations that arise in the finite element and related methods is given in Section 5.3.1. In domain methods like the FEM, whether it is applied to a structure or an enclosed fluid, a linear eigenvalue problem is a natural consequence. With the FEM, it is routine to extract the modes and resonant frequencies and use the modal basis to determine the response under excitation. From that perspective, it is natural to include the eigenvalue problem within the boundary element library. However, the application of the BEM leads to a non-linear eigenvalue problem, which requires special solution techniques and hence the construction of solutions through the modal basis is not a natural pathway in the BEM. Acoustic modal analysis via the BEM is considered in Section 4.2. Although significantly restrictive, one of the simplest acoustic models is the Rayleigh integral [70]. The model is that of a vibrating plate set in an infinite rigid baffle and radiating into a half-space. The Rayleigh integral relates the velocity potential or sound pressure at any point to the velocity map on the plate. As the Rayleigh integral shares its one operator with the integral equation formulations in the boundary element method, the Rayleigh integral method (RIM) [71] can be adopted into the boundary element method fold. The Rayleigh integral is a special case of half-space problems and these are considered in Section 5.1  
An acoustic model that uses the same operators--and hence fits in the BEM context--is that of a shell discontinuity. For example, this can be used to model the acoustic field around a thin screen or shield. The integral equation formulation for the shell are derived from those used in the traditional BEM, but taking the limit as the boundary becomes thinner [72; 73], leaving a model that relates a discontinuity in the field across the shell. Shell elements in acoustics are considered in Section 5.2. The development and application of the BEM from the models described are reviewed in this paper. Although this is a fairly exhaustive list of basic models as things stand, hybrid models can also be developed by using superposition or applying continuity and they will also be considered. The BEM may be applied in vibro-acoustics, aero-acoustics and inverse acoustics and these areas are discussed in Sections 5.3-5.5. Hybrid boundary element methods that include half-space formulation are covered in Section 5.1.2 and shells in Section 5.2.2. There have been recent developments in the development of the BEM for periodic structures, with noise barriers being the usual application [74; 75; 76; 77; 78; 79; 80; 81].