複變數阿達馬主值觀念及其應用

Abstract

本文在複數域導得高階奇異積分的阿達馬主值，並可利用此觀念求得梁方程的基本解。基於這個發現，可將實數域對偶邊界元素法延伸到複變數對偶邊界元素法。此種方法係基於邊界元素法的精神，只對問題給定邊界做離散，可簡化超奇異積分之計算，且可導得影響係數積分的閉合解，並可推廣柯西積分公式(Cauchy integral formula)到阿達瑪積分公式(Hadamard integral formula)。以上兩式即為複變數對偶邊界積分方程式。經離散後，可導得複變數對偶邊界元素法。於計算影響係數矩陣時，本文採用常數場解、留數定理與力平衡條件加以檢核，並建立了CVDBEM程式可適用於二維Laplace方程之邊界值問題。於求解影響係數矩陣過程中，複變數對偶積分式的第二式，所出現高階數極點留數計算的問題，以廣義的留數定理予以詮釋。本文建立了實變數與複變數對偶邊界積分方程間的關係，另柯西核函數與阿達瑪核函數所造成的勢能在橫越邊界的跳躍行為亦有所討論。 The dual boundary element method in real domain proposed by Hong and Chen in 1986 is extended to complex dual boundary element method. This new method can simplify the calculation for hypersingular integral and exact integration for the imfluence coefficients can be obtained. Also, the Hadamard integral formula can be obtained by taking derivative of the Cauchy integral formula. The two equations(Cauchy and Hadamard integral formula) construct the basis for the complex variable dual boundary element method can be implemented. A general purpose program, CVDBEM, was developed for two-dimensiona; Laplace problems. Finally, one numerical examples is considered to verify the validity of the proposed formulation. The results are also compared with solutions and real dual boundary element method results.