Singular meshless method using double layer potentials for exterior acoustics

Time-harmonic exterior acoustic problems are solved by using a singular meshless method in this paper. It is well known that the source points cannot be located on the real boundary, when the method of fundamental solutions (MFS) is used due to the singularity of the adopted kernel functions. Hence, if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived. Herein we present an approach to obtain the diagonal terms of the influence matrices of the MFS for the numerical treatment of exterior acoustics. By using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. We also maintain the prominent features of the MFS, that it is free from mesh, singularity, and numerical integration. The normal derivative of the fundamental solution of the Helmholtz equation is composed of a two-point function, which is one of the radial basis functions. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary based on the potential theory. The solutions of three selected examples are used to compare with the results of the exact solution, conventional MFS, boundary element method, and Dirichlet-to-Neumann finite element method. Good numerical performance is demonstrated by close agreement with other solutions.