Revisit of the degenerate scale for an infinite plane problem containing two circular holes using conformal mapping

Abstract

It is well known that a degenerate scale results in a non-uniqueness solution in the BEM/BIEM. Study on the degenerate scale mainly focused on interior problems. Exterior problems were rarely discussed. In this paper, we revisited the problem of an infinite plane with two identical circular holes by using the complex variables instead of using the degenerate kernel. The domain was mapped to an annulus and the points at infinity were mapped to a pole through the conformal mapping. A boundary value problem was transformed into a Green’s function. Hence, we needed to consider the pole’s influence to the field. The complex variables provide us another way to solve these problems and it was easier than the degenerate kernel to understand. The reason why we use the conformal mapping is that the degenerate kernel may not be available and cannot be employed to derive the degenerate scale of general geometric shapes. Finally, we analytically derived the degenerate scale and compare the present result with that of the degenerate kernel. The equivalence is proved.