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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/15063

Title: 零場邊界積分方程法求解含圓與橢圓邊界之自由振動與水波Helmholtz問題
Null-field boundary integral equations approach for solving Helmholtz problems of free vibration and water wave containing circular and elliptical boundaries
Authors: Jia-Wei Lee
李家瑋
Contributors: NTOU:Department of Harbor and River Engineering
國立臺灣海洋大學:河海工程學系
Keywords: Helmholtz問題;零場邊界積分方程式;退化核;橢圓座標;Mathieu函數;假根;虛擬頻率
Helmholtz problems;null-field boundary integral equation;degenerate kernel;elliptic coordinates;Mathieu functions;spurious eigenvalues;fictitious frequencies
Date: 2010
Issue Date: 2011-06-30T07:58:51Z
Abstract: 基於零場邊界積分方程五大優點((1)免除主值的計算,(2)指數收歛,(3)無需建構網格,(4)邊界層效應的消除,(5)良態代數系統)的成功經驗,本文使用零場邊界積分方程搭配退化核以及特徵函數展開來處理含橢圓形邊界的Helmholtz問題。本文不僅求解內域的自由振動問題,也考慮外域的水波問題。為了能夠充份地利用橢圓形邊界的幾何特性來求解Helmholtz問題,採用橢圓座標與Mathieu函數。閉合型的基本解以退化核的形式在橢圓座標下展開,邊界物理量則使用特徵函數展開。本文發現存在於退化核、邊界密度或邊界積分的Jacobian項,透過邊界通量Jacobian項的調縮,是可以互相對消。因此Mathieu函數的正交關係依然可以保留在橢圓邊界積分而以解析計算。經由滿足邊界條件的線性代數系統可以簡單地求得未知係數。在實際的運算過程中,誤差主要來自於邊界物理量擷取有限項所產生的,因此本法可視為一種半解析法。雖然使用本法處理含橢圓形邊界的問題會產生內域假根與外域虛擬頻率。但有趣的是我們發現到,這些假根與數值不穩定的波數值其實就是相對應滿足修正型Mathieu函數為零的值。針對假根與虛擬頻率的數值不穩定現象,本文分別採用CHIEF法、矩陣的補充列與補充行技巧以及Burton & Miller法來有效抑制這些假根與數值不穩定的現象。此外,針對特殊配置的橢圓柱陣列水波問題,本文發現到了近陷阱模態的物理現象。為了避免處理平移Bessel 與Mathieu函數的加法定理所造成的雙求和,而改採用自適性觀察座標系統,可同時求解含有圓形與橢圓形邊界的問題並得到一個半解析解。最後,本文將針對含任意數目、不同大小與任意分佈之不同種類邊界(含圓、橢圓與退化邊界)的特徵值問題與水波問題,發展一套系統性的程式。
Following the success of five advantages of the null-field boundary integral equation method (BIEM), the Helmholtz problems containing elliptical boundaries are solved by using the null-field BIEM in conjunction degenerate kernels and eigenfunction expansion in this thesis. Not only problems of interior free vibration and exterior water wave problems are considered. To fully utilize the property of ellipse for solving the Helmholtz problems, the elliptic coordinates and the associated Mathieu functions are adopted. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the elliptic coordinates. Besides, the boundary densities are expanded by using the eigenfunction expansion. A Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integration and they can cancel each other out. By scaling the boundary flux using a Jacobian term, the orthogonal relations can be reserved in the boundary contour integral and contour integration along elliptical boundaries can be analytically determined. By this way, the unknown coefficients can be easily determined through a linear algebraic system after matching boundary conditions. This approach is one kind of semi-analytical methods since errors only occur from the truncation of the number of the eigenfunction expansion terms in the real implementation. Although spurious eigenvalues of interior eigenproblem as well as fictitious frequencies for the exterior problem for elliptical boundaries may appear in the BIEM, it is interesting to find that those of them happen to be zeros of the modified Mathieu functions of the first kind or their derivatives. The appearances of spurious eigenvalues and fictitious frequencies are effectively suppressed by using three alternatives including the Combined Helmholtz Interior integral Equation Formulation (CHIEF) method, Burton & Miller approach and the singular value decomposition (SVD) updating technique. Besides, the near-trapped mode for an array of four elliptical cylinders is also observed. To avoid the addition theorem by translating the Bessel and Mathieu functions, the adaptive observer system is employed to solve the Helmholtz problems containing circular and elliptical boundaries at the same time in a semi-analytical manner. Finally, a general-purpose program was developed for solving eigenproblems or water wave problems containing arbitrary number, different sizes and various locations of circular and elliptical boundaries.
URI: http://ethesys.lib.ntou.edu.tw/cdrfb3/record/#G0M98520012
http://ntour.ntou.edu.tw/ir/handle/987654321/15063
Appears in Collections:[河海工程學系] 博碩士論文

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