A Study on Half-Plane Laplace Problems with a Circular Hole

Abstract

This paper describes a numerical procedure for solving half-plane Laplace problems with a circular hole by using the null-field integral equation and degenerate kernels. The unknown boundary potential and flux are approximated by the truncated Fourier series. Degenerate kernels are utilized in the null-field integral equation. A linear algebraic system is obtained without boundary discretization. To avoid the integration along the infinite boundary of half-plane problem, image method is utilized. The present method is verified through two examples with the analytical solutions derived by Lebedev. In addition, the results of BEM and meshless method as well as exact solutions are also compared to show the accuracy and efficiency. This approach can be extended to problems with multiple circular holes without any difficulties. 本文以勢能理論為基礎，提出以退化核與傅立葉級數展開求解含孔洞半平面的問題，此方法可視為半解析法。邊界未知勢能與流通量使用有限項傅立葉級數來近似求得。利用退化核與傅立葉展開可導得一線性代數方法而無須對邊界離散。半平面無限邊界則採用映射法予以處理。文中以 Lebedev 導得解析解的兩個不同邊界條件的拉普拉斯問題進行測試。所得結果並與邊界元素法與無網格法作比較，驗證本方法的正確性。本文並可將單圓 孔洞問題推廣至多圓孔洞問題。