# The Boundary Element Method in Acoustics: A Survey
## 5 Extending the Boundary Element Method in Acoustics
### Meshless Methods  
One of the main advantages of the boundary element method over domain methods, such as the finite element method is the reduced burden of meshing. With meshless methods, a mesh is not required at all. The simplest concept of a meshless method is the equivalent sources method (ESM)in which it is presumed that the acoustic field is equivalent to a field generated by a finite set of point sources:  
\[\varphi(\mathbf{p})\approx\sum_{j=1}^{m}\gamma_{j}G_{k}\Big{(}\mathbf{p},\mathbf{q}_{j}\Big{)}\]  
where the \(\mathbf{q}_{j}\) are the positions of the sources that are outside of the acoustic domain and the \(\gamma_{j}\) are the unknown source strengths and \(G_{k}\) is the appropriate Green's function. For the Dirichlet boundary condition, by matching the Dirichlet data on the boundary \(\varphi\Big{(}\mathbf{p}_{j}\Big{)}\) for \(\mathbf{p}_{j}\in S\), gives a linear system of equations, provided there are at least as many source points as there are items of boundary data. Similarly, by differentiating the above equation with respect to the normal to the boundary  
\[\frac{\partial\varphi(\mathbf{p})}{\partial n}\approx\sum_{j=1}^{m}\gamma_{j} \frac{\partial G_{k}}{\partial n}\Big{(}\mathbf{p},\mathbf{q}_{j}\Big{)}\,\]  
then approximate solution can be found by matching the values with Neumann data, and using a combination. Usually, more internal points than items of boundary data are chosen and a least-squares solution is sought. The meshless methods avoid the problem of singular integration. However, the ESM is not based on a boundary integral equation formulation and hence it is an alternative to the boundary element method and is therefore beyond the scope of this paper. Lee [337] provides a recent review of the ESM in acoustics.  
An alternative method for developing meshless methods has more in common with the standard boundary element method. The method can be applied to the standard interior or exterior problem. For the exterior problem, the methods relates to the Schenck or CHIEF method, of Section 4.3.3, but only the integral equations for the internal points are applied, and the integrals are approximated by using only the midpoint value  
\[\overline{M}_{DS,k}\underline{\phi}_{S}=\overline{L}_{DS,k}\underline{\phi} _{S}.\]  
With the number of interior points exceeding the number of boundary points the solution can be found in the least-squares sense. For the exterior problem, the Burton and Miller form is expected to have superior numerical properties:  
\[(\overline{M}_{DS,k}+\mathrm{u}\overline{N}_{DS,k})\underline{\phi}_{S}=( \overline{L}_{DS,k}+\mathrm{u}\overline{M}_{DS,k}^{\dagger})\underline{\phi} _{S}.\]  
These methods still avoid integration, particularly singular integration and can also be used on the modal problem. There are several reports of implementations of these methods [144; 145; 338; 339; 340; 341].  
As methods for solving acoustic problems, the meshless methods, are well behind the standard boundary element method in the sense of developing robust software. As with the Schenck or CHIEF method of Section 4.3.3, there is the added issue of determining the placement of the equivalent sources. The methods outlined in Section 3.4.2 could be used in determining whether source points are interior or exterior.