二次式變化之時間域邊界元素法於三維彈動力問題之研究

Abstract

本文旨在對出現在邊界積分式中的邊界值（位移場、曳引力場）於時間軸上所產生的變化作更精確的模擬。文中提出「兩個短時階內位移呈二次式變化，同時段，曳引力呈線性變化」之假設，如此再經褶積的步驟後，進而可推導得到一系列滿足因果律的精確解析褶積核心。我們也以一個當在處理邊界元素法空間奇異積分時所會使用到的重要性質來驗證上述核心之正確性，此性質描述出當場點與源點重合時或當一步很大的時階發生時，動力核心會退化成相對應的靜力核心。同樣地，文中使用二次元素來切割空間，故變數於空間上的變化也是呈現二次式。

In this paper, a time-domain boundary element method (BEM)formulation is presented. 3-D transient condensed convoluted kernels using quadratic temporal shape functions for elastodynamic displacement variables is derived in the formulations. Quadratic variation for displacement field in two consecutive time steps and linear variation for traction field in each time step are assumed in the BEM formulations. Therefore, so-call QL method for 3-D BEM in time domain is developed. The accuracy of all derived kernels are demonstrated through one important property dealing with divergent integral (strongly singularity) in BEM. The property is that the transient kernels will reduce to the corresponding static ones when the field point coincides with the source point or when a very large time step occurs. Also quadratic elements for spatial coordinates is employed in the numerical scheme in the presented method.