以非奇異性暇積分改善三維邊界積分法

Abstract

本論文旨在利用理論解析推導三維無奇異性瑕積分的邊界積分方程式，並配合數值模擬加以探討。有鑒於一般三維邊界元素法對於高階元素的瑕積分處理模式，多以適應性積分或極座標轉換為主，本論文提出另一種計算模式，將瑕積分轉換為一般積分加上部分沿著元素邊界的線積分，使得在瑕積分的計算能夠更精確及更簡潔。 其中，利用高斯通量定理，消除格林函數之奇異性。作法是僅需對原有的奇異性核函數加入一項使其異性消除，同時再將函數的解析積分值減去原方程式。由於格林函數的奇異性經過數學上的技巧加以消除，因此當求解未知數時已經不需要再假設形狀函數，即可直接使用物體真實的外型來做計算，也就是直接將物體上分佈的積分點視為滿足邊界條件的節點。如此，不但在計算的過程中大幅地被簡化，從此奇異積分項也將不再需要特別處理，使得數值程式得以更簡潔的被撰寫。 最後，本文將探討本篇論文所提出的理論解析，在實際數值上使用平滑邊界模型及非平滑邊界模型的特性。

The main purpose of this thesis is to present an efficient analysis for singular integrals of three dimensional boundary integral equations. Numerical analysis is used for this discussion. In treating a high order element which has a singular integral, most three dimensional boundary element methods use adaptive integration or polar coordinates transformation. This thesis also includes an alternative method for calculating boundary integral equations. The method transforms the singular integral into a desingularized integral plus a linear integral which is along the boundary of the element; thus, it is more accurate and simple. Two kinds of eliminating the singularity are mentioned. One is for Green’s function in elements by using linear integrals; the other is for its normal derivative by the divergence theorem. The method is to add a term into the original singular kernel while it subtracts its analytical solution from the original equation. Since the singularity of Green’s function can be eliminated by such a mathematical technique, shape function does not need to be supposed while solving the unknown. The calculation can be directly applied to the real shape of the object by taking the integral points distributed on the object as nodes that satisfy boundary conditions. Hence, not only the calculation is significantly simplified but also singularity terms do not require special processing. Therefore, the numerical programming is much easier. Applications of numerical theoretic analysis on smooth and non-smooth boundary models will be proposed and discussed at the end.