對稱化邊界元素法理論推導與程式開發

Abstract

本研究以洪宏基教授與陳正宗教授所共同提出的對偶邊界積分方程為架構，發展對稱化邊界元素法。其原理主要是利用對偶架構中四個核函數間的對稱與轉置對稱關係， 配合雙重積分的能量觀念，取代原先點配置（Point Collocation）技巧導得的非對稱邊界元素法。本研究有下列數個優點：(1).傳統邊界元素影響係數矩陣不對稱的缺點將可避免。(2).對稱化的邊界元素法於互制問題中將易於與有限元素法（FEM）結合。(3).由於未知邊界自由度的係數矩陣為對稱，故可減少計算機的記憶空間與計算時間並增加求解的精確度與速度。本研究為了要將未知邊界自由度的係數矩陣對稱化，故須在傳統邊界積分方程式再將邊點對邊界做一次積分，而這雙重的強奇異及超強奇異積分，為本研究的一項重點。最後，本研究發展一套以 FORTRAN 程式撰寫的二維 Laplace 場之對稱化邊界元素法程式，並舉數個二維例子，進行解析與數值驗證，並針對係數矩陣特性與計算精確度等方面探討非對稱與對稱 BEM 的差異。 Based on the dual framework derived by Hong and Chen, we developed symmetric boundary element method, instead of conventional BEM. Using the symmetry properties for the four kernels in the dual BEM, the symmetric BE formulation can be derived through double integrations. The main advantages are (1).The unsymmetric influence matrix in the conventional BEM can be avoided, (2).The coupling use with FEM can be easily implemented, and (3).The storage space in memory can be saved, and the solutions can be obtained more efficiently and accurately. The main challenge is that double integrations for the singular and hypersingular kernels should be dealt with. In order to check the influence matrices, not only the test of constant potential but also equilibrium condition were employed. A general program was developed for the Laplace equation. Finally, several examples were demonstrated. The comparisons with the conventional BEM and the symmetric BEM on memory storage, efficiency, and accuracy were discussed.