# The Boundary Element Method in Acoustics: A Survey
## 7 Concluding Discussion  
The main purpose of this paper is to encapsulate the modern scope of the boundary element method in acoustics; how it can be adopted for general 2D, 3D and axisymmetric dimensions, interior, exterior and half-space problems, thin shells and modal analysis. The standard BEM can be directly applied to an enclosed domain with cavities or to multi-boundary problems in an exterior domain. Boundary element methods are founded on a boundary integral equations formulations and these can be fused to form additional methods: thin shells can be used to model discontinuities in an interior or exterior domain defined by closed boundaries, half-space problems can be modelled by the Rayleigh integral or modifying the Green's function, domain methods can be linked to the BEM so that vibro-acoustic and aero-acoustic problems can be tackled and--through regularization of the BEM equations--inverse problems can be addressed. The boundary element method is a numerical method that only becomes a potentially useful tool for solving real-world problems in acoustic engineering when it has been implemented in software. Clearly, it is important that the executing code fits within the memory of the computer and completes the computation in reasonable time, and efficiency issues have been considered in Section 6.1. There are issues of generality and, in that regard, this work has focused on the generalised Robin boundary condition and general topologies. Reliability and maintenance are also very important issues in software development, and hence much of the focus has been on the standard exterior acoustic problem, which has had issues and solution approaches documented for more than half a century. The robustness of the BEM, particularly considering the validity of a defined elemental boundary, has had little attention in the field, but methods for assisting with this are summarised in Section 3.4.2. There is much commonality across the various topologies to which the boundary element method can be applied in acoustic/Helmholtz problems; the equations for different dimensional problems are literally the same, with a change in the definition of the Green's function and line integrals become surface integrals when we move from 2D to 3D. All the core equations, whether they are interior, exterior, shell problems or the straightforward half-space problems, have the same essential operators within a chosen dimensional space. Hence there is significant scope for component-wise development of software, adopting a 'library' approach and unifying the method. Significant issues have always surrounded the \(N_{k}\) operator. This operator was initially included in the combined integral equation formulations for the exterior problem, outlined in Section 4.3.4. Firstly, for surface (collocation) points its integral definition is hypersingular, which is difficult to interpret and evaluate, and perhaps for that reason alone, the alternative Schenck/CHIEF method is the preference of many. Once this significant issue is surpassed, the \(N_{k}\) is not compact, it is a pseudo-differential operator and this causes further issues for solving over by iterative methods and hence preconditioning has been introduced to circumvent this. Preconditioning can eliminate the hypersingular operator \(N_{k}\) for the standard exterior problem (96). However, \(N_{k}\) is still required for shell elements and hence it has to be included in a general library. Much of the current research theme in the area, put simply, is to shunt the BEM in acoustics from its traditional comfort-zone of problems with \(\sim\)10\({}^{3}\) elements to modern problems, to include high frequencies, with \(\sim\)10\({}^{6}\) elements and from straightforward problems to complicated applications [190; 238; 405]. The computational bottle-neck in the traditional BEM is in the use of LU factorisation or similar methods for solving the linear system.