臺灣博碩士論文加值系統

本論文主要是研究在Dirichlet和Neumann邊界條件下之各種矩形組合領域Helmholtz特徵值問題。首先，我們將欲求解之矩形組合領域分割成多塊矩形領域，並採用特徵函數展開法求得各領域的勢函數表示式，然後利用各個相鄰領域之連續條件及特徵函數的正交性，建立各未知係數的關係式後即可得到解析解。然而，當求解領域形狀愈來愈複雜時，要推導出解析解就會更加困難，因此，採用選點法來建立聯立方程式，則可容易地求得本問題的半解析解。最後，為了証明本論文所提之解析解及半解析解方法的推導過程和所撰寫之程式的正確可靠，我們亦以自行編寫之Q9有限元素程式來驗証，其結果顯示：本論文所提之解析解及半解析解方法的推導過程和程式均正確無誤，而且可以提供給其他的數值方法作參考。

The purpose of this thesis is to solve the Helmholtz eigenvalue problems for the domain, which is connected with multiple rectangles, bounded by Dirichlet and Neumann boundary conditions. First, we divide the whole domain into several rectangular regions and utilized the eigenfunction expansion method to obtain the potential function from each region. Then, the relationship of unknown potential coefficients can be formulated to get the analytical solutions by applying the continuity condition of each neighboring region and the orthogonality of eigenfunctions. However, it is difficult to acquire the analytical solutions due to complicated geometry. Therefore, as the collocation method is adopted to construct the simultaneous equations, the semi-analytical solution can be easily determined. Finally, in order to confirm the correctness of the process for deriving the analytical and semi-analytical solutions, a finite element scheme is employed. The results show that all procedures of our two methods are right and good performance of our programming.

中文摘要
英文摘要
目錄
表目錄
圖目錄
第一章 緒論
1 - 1 前言
1 - 2 研究動機、目的及方法
1 - 3 研究範圍及內容
第二章 理論分析
2 - 1 緒論
2 - 2 問題描述與控制方程式
2 - 2 - 1 薄膜的自由振動問題
2 - 2 - 2 封閉聲場的聲壓振動問題
2 - 3 求解領域之形狀及邊界條件
第三章 Dirichlet邊界條件矩形組合領域之解析解
3 - 1 L形領域之解析解
3 - 2 U形領域之解析解
第四章 Dirichlet邊界條件矩形組合領域之半解析解
4 - 1 L形領域之半解析解
4 - 2 U形領域之半解析解
第五章 Neumann邊界條件矩形組合領域之解析解
5 - 1 L形領域之解析解
5 - 2 U形領域之解析解
第六章 Neumann邊界條件矩形組合領域之半解析解
6 - 1 L形領域之半解析解
6 - 2 U形領域之半解析解
第七章 模式驗証與數值計算例
7 - 1 Dirichlet邊界條件矩形組合領域之計算例
7 - 1 - 1 L形領域之數值計算例
7 - 1 - 2 U形領域之數值計算例
7 - 2 Neumann邊界條件矩形組合領域之計算例
7 - 2 - 1 L形領域之數值計算例
7 - 2 - 2 U形領域之數值計算例
第八章 結論
參考文獻
附表
附圖

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