臺灣博碩士論文加值系統

本文推導出一種精確而有效的離散化非規則邊界有限元素法數值解，可用來解非規則邊界之熱傳遞問題。傳統的有限元素法可解任意幾何的邊界問題，但無法配合使用複層網格法計算求解。因此隨問題複雜性提高，而影響計算效率與儲存空間。利用三角元素導出的離散化計算方式，有別於傳統有限元素法組合成整體矩陣。數值問題經由複層網格法配合高斯-希德爾疊代法計算下，可改善計算效率與儲存空間的問題。
同時歸納並推導出所有三角元素組合的計算式，用以分別計算不同幾何邊界情況。在準確度方面，由於邊界外型是由三角元素所組成，因此可獲得與傳統有限元素法相同的結果。為說明數值的正確性，本研究以四分之一圓之熱傳問題為例，從一系列的計算例中，印證了數值的正確性與優良的計算效率。

An accurate effective finite element algorithm is developed for solving the heat conduction problem with an irregular boundary domain. The traditional finite element method can solve the arbitrary geometric problem, but it is unable to couple with multigrid computing technology. Therefore, as problem complexity increased,it will influence on computational efficiency and data memory space. A discrete computational models are derived by triangular element which are different from the global matrix composed of traditional finite element method. The numerical problem solved by the computational form is coupled with Gauss-Seidel iteration and multigrid computing technology to improve the computational efficiency and data memory space problem.
In this thesis, several of computational models composed of triangular element are derived for computing the differenct geometry of boundary conditions. Because of the computational forms of boundary is composed by the triangular element, the accuracy of numerical solutions can be obtained as same as traditional finite element method. In order to describe the exactness of the numerical computation, a series of computational cases of heat conduction with a quarter of circle region, have been confirmed with the exactness of numerical solution and the fine computational efficiency.

中文摘要
英文摘要
誌謝
目錄
表目錄
圖目錄
符號說明
第一章
緒論
1.1研究動機與背景
1.2文獻回顧
1.3研究目的
1.4論文架構
第二章
數值分析理論
2.1簡介
2.2離散化有限元素推導
2.3複層網格法
2.3.1修正法則
2.3.2全近似法則
2.3.3全複層網格法則
2.4複層網格法應用於非規則邊界問題之能量傳遞
2.5時間階梯法
第三章
複層網格法應用到非規則邊界溫度固定熱傳問題之公式推導
3.1簡介
3.2非規則網格分析推導過程
3.3離散化計算式
第四章
非規則邊界溫度固定熱傳問題數值案例分析
4.1簡介
4.2卜易生問題
4.3暫態問題
第五章
複層網格法應用到非規則邊界對流熱傳問題之公式推導
5.1簡介
5.2非規則網格分析推導過程
5.3離散化計算式
第六章非規則邊界對流熱傳問題數值案例分析
6.1簡介
6.2穩態熱對流問題
第七章
結論與建議
7.1結果與討論
7.2未來研究方向與建議
參考文獻

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