On the singularity of the axially symmetric Helmholtz Green's function, with application to BEM

Abstract

This paper presents some results concerning the singularities of the axially symmetric Helmholtz Green's function and its gradients, which is of some importance in the implementation of Boundary Element Methods (BEM). The treatment of the singular part here differs from others in that it is based on a higher-order Laurent series, which leads to integrals better suited for numerical work, particularly in evaluation of the gradients. Moreover, the usual elliptic integral forms are recast as hypergeometric functions. The transformation relations among various hypergeometric functions allow efficient forms to be derived for computational work. The leading-order singularity in the gradients is amenable to treatment by analytical means. Analytical expressions are presented for integrals involving this singular term in conjunction with quadratic interpolation for the fields. It is pointed out that the singular nature is modified for observation points on the symmetry axis.