# The Boundary Element Method in Acoustics: A Survey
## 3 The Boundary Element Method and the Laplace Equation Stem
### Properties and Further Details
#### 3.4.1 Integration  
In the previous Subsection, it was shown that up to four operators are involved in the boundary element method for Laplace problems, and these four operators extend to acoustic/Helmholtz problems. For the Laplace problem, it may be possible to develop analytic expressions for the integrals [92], but in general--and particularly for acoustic/Helmholtz problems--numerical integration is necessary. In the main, the integrals are regular, and these are most efficiently evaluated by Gauss-Legendre quadrature [93]. This is straightforward to apply to straight-line panels and in the generator and (typically in composite form) azimuthally. There are also published points and weights for Gaussian quadrature on a triangle [94].  
However, the integrals corresponding to the diagonal components of the L\({}_{\text{SS}}\) matrix are weakly singular and the diagonal components of the N\({}_{\text{SS}}\) matrix are hypersingular, that is when the point lies on the panel. Moreover, if the point is close to the panel, for example at the centre of a neighbouring panel then the integrals are said to be nearly singular and may require a more accurate numerical integration rule, or special treatment [95]. Special numerical integration methods can be applied in order to evaluate the singular integrals and expressions for the hypersingular integrals may be found through limiting process. The weakly singular integrals, corresponding to the diagonal components of the L\({}_{\text{SS}}\) matrix have a O(\(\ln r\)) singularity in 2D and an O\(\left(\frac{1}{r}\right)\) singularity in 3D, where r is the distance from the central collocation point. For the axisymmetric 3D case, the azimuthal integration resolves the O\(\left(\frac{1}{r}\right)\) singularity to an O\(\left(\ln r\right)\) singularity on the generator. For the simple elements discussed, analytic expressions for the diagonal components of the L\({}_{\text{SS}}\) and N\({}_{\text{SS}}\) matrices are available for the 2D and 3D (non-axisymmetic) cases for Laplace's equation. Further work on singular integration can be found in the following references [96; 97; 98; 99; 100; 101].  
Although for simplicity, the four operators are often lumped together as integral operators, \(N\) is not an integral operator: \(N\) is termed a pseudo-differential operator, and it therefore has distinct properties. The full expressions for the straight line and triangular panels are given in the author's book [61] (pages 49-50). For illustration the expressions are given for the straight-line panel of length \(h\) (2D) and for an equilateral triangular panel (3D) with each side of length \(h\):  
\[\left[N\varphi\right]_{\Delta}\left(p;n_{p}\right)=-\frac{2}{\pi h}\text{ for the straight-line panel of length }h\ (2\text{D}) \tag{45}\]\[\left\{Nq\right\}_{\Delta}\!\!\left(\mathbf{p};\mathbf{n_{p}}\right)=-\frac{3}{2\pi\hbar} \text{ for the equilateral triangular panel with side of length $h$ (3D)} \tag{46}\]  
There is one simple but noteworthy remark to be made about the expressions (45) and (46). Normally, if the domain of integration is reduced in size, the integral is similarly reduced, at least in the limit as the domain size converges to zero. However, with these hypersingular integrals, the opposite is found to be the case! A significant consequence of this property is considered in Section 6.2.