元素釋放法計算加速之研究

A Study on Accelerated computations in the Element Free Galerkin Method

朱峻平(Chun-Ping Chu)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程研究所/碩士(2003年)

https://doi.org/10.6840/cycu200300433

摘要

元素釋放法為無網格法(Meshless method)的一種，模擬問題時只需要節點資料，不需要描述節點關係的元素，因此可以免去有限元素法網格生成與重建的麻煩。元素釋放法在建立節點的形狀函數時運用了變動式最小平方近似法(Moving Least Square Method，簡稱MLS)，再配合適當的外力、位移和初始條件的處理，即可模擬初始界值問題。
由於元素釋放法各節點的形狀函數不盡相同，必需在程式中建立，所以模擬問題時所需的計算量較傳統的有限元素法多，因此如何有效的提昇元素釋放法計算效率是一個重要的課題。
改善元素釋放法處理問題的效率可以從三方面著手：(1)建立有效節點、高斯積分點、取樣點的搜尋法則，以減少資料搜尋的時間，(2)改善理論，減少建立節點形狀函數及其導數的計算量，(3)有效處理聯立方程式計算問題。本文主要在說明如何利用上面觀念配合元素釋放法處理二維彈性靜、動力問題，以達到減少計算量且有效模擬問題的目的。最後利用幾個數值算例說明這些處理方式的精確性與有效性，同時對未來可能發展的方向提出看法與建議。

關鍵字

元素釋放法 ； 節點搜尋 ； 最小平方近似法 ； 變寬帶

並列摘要

Element free Galerkin (EFG) method belongs to the class of a numerical analysis method so called meshless method. The main feature of this type of method is that the construction of shape functions is based on the moving least-square (MLS) interpolation technique, which requires only nodal data and not any connectivity requirement among these nodes as in conventional finite element method. The nodal shape function of the EFG method is constructed through some searching algorithms and the MLS technique and it may varies at each node. Previous experience shows that for the same problem to be solved, the computation time required by the EFG method is more than it required by the finite element method. Hence, how to improve the computational efficiency becomes an important issue in the further development of the EFG method.
There are three possible solutions to improve the computational efficiency of EFG method. Firstly, establish a more efficient searching scheme for finding corresponding nodal points when constructing shape function. Secondly, modify the theory and associated algorithm for constructing shape function. Thirdly, solve the system governing equations efficiently as presented in the present paper. The skyline solution technique is adopted to solve the system equations with large sparse coefficient matrix typically obtained in the EFG method. Numerical simulations of 2D elastostatic and elastodynamic problems demonstrate that the adoption of that three solutions technique into the EFG method can largely reduce the computation time than the one using conventional solving technique.

並列關鍵字

Meshless Method ； Moving Least Square Interpolation ； Skyline ； Element Free Galerkin Method

參考文獻

[18] 李政達，「元素釋放法於彈性動力之研究」，碩士論文，中原大學土木工程研究所，民國九十一年六月。

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