Semi-Analytical Approaches for Multiply-Connected Eigenproblems Containing Circular and Elliptical Holes or Stringers

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Project title

Code/計畫編號

NSC99-2221-E019-015-MY3

Translated Name/計畫中文名

半解析法求解含橢圓形(圓形)孔洞與束條之多連通特徵值問題

Project Coordinator/計畫主持人

Jeng-Tzong Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Harbor and River Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=2380011

Year

2012

Start date/計畫起

01-08-2012

Expected Completion/計畫迄

31-07-2013

Bugetid/研究經費

785千元

ResearchField/研究領域

土木水利工程
海洋科學

在本三年期的計畫裡，我們擬使用兩種半解析法求解含圓、橢圓形孔洞與束條之多連通特徵值問題。一種是多極Trefftz法，另一種則是零場邊界積分方程法。關於多極Trefftz法，我們藉由引入多極展開的想法將傳統的Trefftz法延伸至多極Trefftz法。Trefftz法的基底函數將使用加法定理在極座標及橢圓座標下展開，此時邊界條件可以被解析描述。含有束條的特徵值問題在橢圓座標下，當徑向的座標參數縮減成零時可以被視為一種特殊的情況。因此使用橢圓座標系統來處理含有束條的特徵值問題應是可能。至於零場邊界積分方程法，我們將分別地在極座標與橢圓座標下使用加法定理把閉合型的基本解展開成退化核的形式。三年計畫架構圖，參見表A。在前兩年的計劃裡，我們將著重於使用多極Trefftz法分別求解含圓(第一年)與橢圓形邊界(第二年)之多連通特徵值問題。而在最後一年的計劃裡，將同時使用多極Trefftz法與零場邊界積分方程法求解同時包含圓與橢圓形邊界的特徵值問題。對於多極Trefftz法而言，極座標轉換到橢圓座標的加法定理或許是有困難的。因此，我們將嘗試把加法定理延伸至極座標轉換到橢圓座標。基於先前國科會計畫利用零場邊界積分方程法求解含圓形邊界的赫姆茲運算子問題的成功經驗，我們也使用零場邊界積分方程法來求解含圓、橢圓形孔洞與束條之多連通特徵值問題。當問題含有束條時(領域裡含有退化邊界)，而無需利用零場邊界積分方程法的超奇異方程式便可求解。主要的原因是，在橢圓座標下它可以視為一種特例。最後、我們將以一些數值算例來驗證本法的正確性，以及使用這兩種方法發展一個通用的程式，來求解含任意數目，不同大小與隨意位置的圓、橢圓形孔洞與束條之多連通特徵值問題。本計畫所提半解析法在解的高精準度與快速收斂性及無需網格的切割等，是否優於現有的數值方法將在本計劃中予以證實。 In this three-years proposal, we intend to employ two semi-analytical approaches to solve the multiply-connected eigenproblems containing circular and elliptical holes or stringers. One is the multipole Trefftz method and the other is the null-field boundary integral equation method (BIEM). Regarding the multipole Trefftz method, we will extend the conventional Trefftz method to the multipole Trefftz method by introducing the multipole expansion. The addition theorem will be employed to expand the Trefftz bases to the referred polar and elliptical coordinates centered at one circle or ellipse, respectively, where boundary conditions are specified. The eigenproblems with a stringer can be seem as a special case when the radial coordinate is shrunk to be zero in the elliptic coordinates. Therefore, the eigenproblems with stringers can be possibly solved by using the elliptic coordinates. Regarding the null-field BIEM, we will employ the addition theorem to expand the closed-form fundamental solution into the degenerate kernel in the polar and elliptic coordinates, respectively. The frame of this NSC proposal is shown below in Table A. In the former two years, we focus on the multiply-connected eigenproblems containing only circular or elliptical boundaries, respectively, by using the multipole Trefftz method. In the third year, multiply-connected eigenproblems containing the circular and elliptical boundaries at the same time will be solved by using the multipole Trefftz method and null-field BIEM. For the multipole Trefftz method, it may have difficulty to find the addition theorem for translating the polar coordinates to the elliptic coordinates. Therefore, we will attempt to extend the addition theorem for translating the polar coordinates to the elliptic coordinates. Following the success of previous NSC projects for solving the Helmholtz problems with circular boundaries by using the null-field BIEM, we also employ the null-field BIEM to deal with multiply-connected eigenproblems containing the circular, elliptical holes and/or stringers. When there is a stringer (degenerate boundary in the domain), it is possible that we need not to employ the hypersingular formulation in the null-field BIEM. The main reason is that it can be seem as a special case in the elliptic coordinates. Finally, several examples will be given to demonstrate the validity of the present approaches and develop a general-purpose program for solving multiply-connected eigenproblems containing any number, arbitrary size and various positions of circular, elliptical holes and/or stringers by using the two approaches. The high accuracy, fast rate of convergence and mesh-free advantages of the semi-analytical approach over other numerical methods will be verified in this project.

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Semi-Analytical Approaches for Multiply-Connected Eigenproblems Containing Circular and Elliptical Holes or Stringers

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