A NEW METHOD FOR LAPLACE'S EQUATION IN TWO-DIMENSIONAL REGIONS WITH CIRCULAR HOLES

Abstract

This paper describes a numerical procedure for solving the Laplace problems of circular domain containing multiple circular holes by using the null-field integral equation, Fourier series and degenerate kernels. The unknown boundary potential and flux are approximated by using the truncated Fourier series. Degenerate kernels for the fundamental solutions are utilized in the boundary integral equation. A linear algebraic system is obtained without boundary discretization. Degenerate scale in the multiply-connected domain is also examined. Several examples are illustrated and the results are compared well with the exact solution and those of Caulk's data.本文以勢能理論為基礎，提出以退化核與傅立葉級數展開搭配零場積分方程求解含多孔洞拉普拉斯問題。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，並對退化尺度進行探討。最後以幾個不同邊界條件的拉普拉斯問題進行測試。所得結果無論與解析解或是與Caulk 的數值結果比較，均可驗證本方法的正確性。