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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/14920

Title: 邊界積分方程與多極Trefftz方法於含圓及球形邊界Helmholtz特徵值問題之研究
A study on eigenproblems for Helmholtz equation with circular and spherical boundaries by using the BIEM and the multipole Trefftz method
Authors: Shing-Kai Kao
Contributors: NTOU:Department of Harbor and River Engineering
Keywords: 退化核;零場積分方程;多極Trefftz方法;特徵值問題;假根
degenerate kernel;null-field integral equation;multipole Trefftz method;eigenproblem;spurious eigenvalue
Date: 2009
Issue Date: 2011-06-30T07:50:51Z
Abstract: 本文利用多極Trefftz方法與零場積分方程,分別處理二維與三維的特徵值問題。在第二章,零場積分方程引入退化核與球形諧和函數解同心圓球的特徵值問題。透過退化核函數展開基本解與利用球形諧和函數表示邊界物理量,則邊界積分便可以解析求得。真假特徵值分別透過奇異值分解的補充列與補充行技巧焠出。此外,真假邊界特徵向量可以在影響係數矩陣的奇異值分解結構中左酉與右酉矩陣的行向量發現。這些發現與二維例子吻合。第三章,在特徵值問題上成功地將傳統Trefftz方法推展到多極Trefftz方法。利用多極Trefftz方法處理多連通定義域的特徵值問題。因為引入了加法定理,所以無須佈點的技巧即可建構出一個線性代數系統。特徵值可以透過直接搜尋的技巧獲得。利用此法解特徵值問題將無假根的污染產生。
In this thesis, the multipole Trefftz method and the null-field integral equation are employed to deal with 2-D and 3-D eigenproblems, respectively. In the chapter 2, the null-field integral equation in conjunction with degenerate kernels and spherical harmonics are utilized to solve the eigenproblem of a concentric sphere. By expanding the fundamental solution into degenerate kernels and expressing the boundary density in terms of spherical harmonics, all boundary integrals can be analytically determined. By using the updating terms and updating document of singular value decomposition (SVD) technique, true and spurious eigenvalues can be extracted out, respectively. Besides, true and spurious boundary eigenvectors are obtained in the right and left unitary vectors in the SVD structure of the influence matrices. This finding agrees with that of 2-D cases. In the chapter 3, we succeed to extend the conventional Trefftz method to the multipole Trefftz method in eigenproblems. The multipole Trefftz method is used to deal with eigenproblems with a multiply-connected domain. By introducing the addition theorem, the collocation technique is not required to construct the linear algebraic system. The eigenvalues can be found by employing the direct searching technique. Solving eigenproblems by using this method is free of pollution of spurious eigenvalues.
URI: http://ethesys.lib.ntou.edu.tw/cdrfb3/record/#G0M96520009
Appears in Collections:[河海工程學系] 博碩士論文

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