A study on the method of fundamental solutions using the image concept

Abstract

In this paper, analytical and semi-analytical numerical solutions for Green's functions are obtained by using the image method which can be seen as a special method of fundamental solutions (MFS). The image method is employed to solve the Green's function for the annular, eccentric and half-plane Laplace problems. In addition, an analytical solution is derived for the fixed-free annular case. For the half-plane problem with a circular hole and an eccentric annulus, semi-analytical solutions are both obtained by using the image concept after determining the strengths of two frozen image points and a free constant by matching boundary conditions. It is found that two frozen images terminated at the two focuses in the bipolar coordinates for the problems with two circular boundaries. A boundary value problem of an eccentric annulus without sources is also considered. Error distribution is plotted after comparing with the analytical solutions derived by Lebedev et al. using the bipolar coordinates. The optimal locations for the source distribution in the MFS are also examined by using the image concept. It is observed that we should locate singularities on the two focuses to obtain better results in the MFS. Besides, whether the free constant is required or not in the MFS is also studied. The results are compared well with the analytical solutions.本文使用可視為一種特別的基本解法的映像法來推導格林函數的解析解與半解析解。求解問題包含同心圓環、半平面含圓洞與偏心圓環的格林函數問題，及偏心圓環的邊界值問題。在同心圓環例子中，可用映像法推導出解析解。針對半平面含圓洞及偏心圓環的格林函數問題，可利用映像法的觀念透過滿足邊界條件就可決定最後兩個凝固點的源強度與常數項大小進而推導出半解析解。我們發現映像點映射到最後凝固的位置座落於雙極座標上的兩個焦點上。針對偏心圓環不含集中力的邊界值問題，我們以基本解法求解並與解析解比較作出誤差分佈圖，進而探討可能的源最佳佈點位置。基本解法的源最佳佈點位置可用映像法的觀念來尋找。而佈點位置如果包含到兩個焦點時，我們可以得到比較準確的結果。除此之外， 基本解法中是否需要自由常數項也一併作討論。求解的結果與解析解相比得到很好的結果。