臺灣博碩士論文加值系統

本論文是探討在三角形網格上，應用梯度平滑法求解二維 Dirichlet邊界值問題的數值特性。
我們比較 Liu和 Xu[3]提出的最佳梯度平滑法，
以及 Lai[2]和 Huang[1]的修正方法在極不規則三角網格的數值解收斂特性和計算量。
此外，我們發現在上述方法中，m-GSD計算網格線中點梯度值的程式複雜度與計算量都比 c-GSD計算三角形重心處梯度值所需更為複雜與龐大，
我們對此提出以三角形重心梯度值為主的數值積分策略以省略網格線中點梯度值的計算。

1 簡介與文獻回顧..........................................1
2 梯度平滑法..............................................4
2.1 梯度平滑域............................................4
2.2 梯度平滑運算..........................................4
2.3 Liu和 Xu三種形式的 GSDs...............................7
2.3.1 n-GSD(node-associated GSD，簡稱 n-GSD).............7
2.3.2 c-GSD(centroid-associated GSD，簡稱 c-GSD).........8
2.3.3 m-GSD(midpoint-associated GSD，簡稱 m-GSD).........9
3 Laplace算子離散公式....................................10
3.1 Liu和 Xu的n-GSD方法..................................10
3.2 各類型梯度平滑法......................................14
4 Laplace算子 n-GSD線積分的簡化...........................15
4.1 Modified-GSD........................................15
4.1.1 局部保守場.........................................15
4.1.2 Laplace 算子公式...................................16
4.2 三角形重心為主的矩形法................................19
5 數值實驗與結論.........................................21
5.1 計算定義域與網格.....................................23
5.2 數據討論與結論.......................................30
5.2.1 25組隨機網格.......................................30
5.2.2 加細網格...........................................30
參考文獻.................................................41

[1] Chih-Ping Huang. Piecewise linear function solution space and modied-gradient smoothing domain method. August, 2012.
[2] Chen-Yao G. Lai. A note on the path integral of the gradient smoothing methods. January 2012.
[3] G. R. Liu and George X. Xu. A gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids, 58(10):1101-1133, 2008.
[4] G.R. Liu, Jian Zhang, K.Y. Lam, Hua Li, G. Xu, Z.H. Zhong, G.Y. Li, and X. Han. A gradient smoothing method (GSM) with directional correction for solid mechanics problems.
Computational Mechanics, 41(3):457-472, 2008.
[5] Jian Zhang, G.R. Liu, K.Y. Lam, Hua Li, and G. Xu. A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems.
Finite Elements in Analysis and Design, 44(15):889-909, 2008.