Natural Boundary Element Method For Exterior Helmholtz Problem With Prolate Spheroid Boundary

In this paper, three dimensional exterior Helmholtz problem with inner spheroidal boundary is studied using natural boundary element method, the properties and the computational methods of spheroidal wave functions encountered in this paper are discussed in details.The Helmholtz equation comes from mathematical physics problems such as time harmonic radiation and scattering of acoustic or electromagnetics and so on, and it is widely used. In the before, researches on this problem are mainly based on simpler boundaries such as circle, ellipse and spherical surface and so on. In this paper, based on inner prolate spheroidal boundary, using method of separation of variables and spheroidal wave functions in spheroidal coordinate system, the Poisson integral formula and the natural integral equation corresponding to the exterior Helmholtz problem are given. And then the stiffness matrix is derived, the numerical technology is discussed. Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineering where the spheroidal coordinate system is used, it is difficult to calculate them but some results about them are given here.The boundary element method is very efficient when it is used to solve exterior boundary value problems and singular problems with special boundaries, such as circle, ellipse and spherical surface. But for general cases, only natural boundary element method is not enough, we need the coupling or domain decomposition methods. The natural integral equation is the exact artificial boundary condition and its integral operator is just the Dirichlet to Neumann (DtN) operator. This operator plays a key role in the coupling and domain decomposition methods. A way for further using of the DtN finite element to solve numerical problems is provided.