An analytical Green’s function for Laplace operator in an infinite plane with two circular holes using degenerate kernels

Abstract

In this paper, Green’s function problem of an infinite plane containing two circular boundaries is analytically studied by using the boundary integral equation method (BIEM). The original problem is decomposed into two parts by introducing the superposition technique. The first part is a free field caused by the concentrated force. The second part is a Laplace problem subjected to the corresponding boundary conditions. The second part can be solved by using the null-field boundary integral equation in conjunction with the degenerate kernel. Since the geometry of problem under study is composed by two circular boundaries, the kernel function is naturally expanded to series form in terms of the bipolar coordinates. The Green’s function is analytically expressed in three regions instead of two regions for the eccentric domain. To the authors’ best knowledge, this is the first time that the Green’s function of the problem is presented in this form. To show the validity of the present method, the contour result of the present method is compared with those obtained by the image method, the null-field BIE in conjunction with the adaptive observer of polar degenerate kernel.