Program Development and Theoretical Study of Dual Boundary Integral Equation Method by Using the Singular Value Decomposition and Addition Theorem

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Project title

Code/計畫編號

NSC101-2221-E019-050-MY3

Translated Name/計畫中文名

奇異值分解法與加法定理在對偶邊界積分方程法的理論探討及程式開發

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=2641658

Year

2012

Start date/計畫起

01-08-2012

Expected Completion/計畫迄

31-07-2013

Bugetid/研究經費

786千元

ResearchField/研究領域

土木水利工程
數學

在本三年計劃中，第一年我們將利用奇異值分解法(SVD)來統一探討對偶邊界元素法求解工程問題時所造成的四個影響係數矩陣可能秩降的數學現象（退化邊界、退化尺度、假根與虛擬頻率），另亦有因物理現象（剛體運動、振動模態與陷阱模態）所可能產生的秩降問題。藉由使用SVD法，在對偶邊界元素法所建構之影響係數矩陣中數學與物理的資訊，將檢驗最小（零）奇異值所對應之左右酉向量以長條圖表示，並對域外零場解與非無聊的非零場解一併作圖，檢驗數學現象(虛擬)是否寄生在左酉向量中，而物理現象(真實)是否隱含於右酉向量裡。其中，退化邊界將以含板樁異向滲流與含裂縫扭轉桿為例，進行說明。退化尺度則將以橢圓、正N邊形與半圓為例，並擬與複變保角轉換作解析探討且與對數容量作一聯結並與數值結果比較。真假根與虛擬頻率將以橢圓邊界進行探討。在最後兩年的計畫裡，我們將以零場邊界積分方程法在先前國科會計畫中二維問題的成功經驗延伸到求解含球體、橢球體孔洞與退化邊界之三維Laplace與Helmholtz問題。零場邊界積分方程法中，我們將分別在球座標與橢球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，邊界物理量則使用特徵函數來模擬。且當問題的領域裡含有退化邊界時，我們無需利用零場邊界積分方程法的超奇異方程式便可求解。主要的原因在於橢球座標下它可以視為一種特例，即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。因此使用橢球座標系統來處理含有退化邊界的工程問題是可能的。第一年計畫裡，我們將驗證使用對偶邊界元素法求解四種退化問題的奇異值分解法的四個影響係數矩陣的酉架構。最後兩年，我們將舉出幾個數值算例來驗證零場邊界積分法在求解三維工程問題含球體與橢球體的正確性。最後，則以本法發展一套通用的程式，來求解含任意數目、不同大小與隨意位置的球體、橢球體孔洞與退化邊界之三維的工程問題。其中，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檢驗。 In this three-years proposal, we will employ the singular value decomposition (SVD) technique to examine the rank deficiency of influence matrices in the dual boundary element method (BEM) due to mathematics (degenerate boundary, degenerate scale, spurious eigenvalue and fictitious frequency) and physics (rigid body term, normal mode, and trapped mode). Information due to mathematics and physics in the left and right unitary vectors of SVD structure will be distinguished in the first year project. Anisotropic seepage flow and a cracked torsion bar for degenerate boundary will be considered. Degenerate scale of ellipse, regular N-gon and a half disc will be studied analytically through the conformal mapping and will be linked to logarithmic capacity and compared to BEM results. Spurious eigenvalus and fictitious frequencies due to elliptical boundaries will be emphasized. In the last two years, we will extend the successful experience of null-field boundary integral element method (BIEM) for 2D case to solve three-dimensional Laplace and Helmholtz problems containing spherical and spheroidal cavities or degenerate boundaries. Regarding the null-field BIEM, we will employ the addition theorem to expand the closed-form fundamental solution into the degenerate kernel for spherical and spheroidal boundaries in the spherical and spheroidal coordinates, respectively. Boundary densities are expanded by using eigenfunction expansions. If there is a 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 seen as a special case when the radial coordinate is shrunk to be zero in the spheroidal coordinates. Therefore, three-dimensional engineering problems containing degenerate boundaries can be solved by using the spheroidal coordinates. Several examples will be given to demonstrate the SVD structure for some engineering problems by using the dual BEM in the first year. The validity of the null-field BIEM for three-dimensional problems will be verified in the second and third year for Laplace and Helmholtz equations, respectively. Finally, a general-purpose program for solving some engineering problems (three-dimensional) containing any number, arbitrary size and various positions of spherical, spheroidal cavities and/or degenerate boundaries by using the null-field BIEM will be developed. High accuracy, fast rate of convergence and mesh-free advantages of the semi-analytical approach over other numerical methods will be examined in this project.

Keyword(s)

奇異值分解法
退化邊界
退化尺度
真假根
虛擬頻率
零場邊界積分方程法
橢球座標
SVD technique
degenerate boundary
degenerate scale
true and spurious eigrnvalues
fictitious frequency
null-field BIEM
spheroidal coordinates

DSpace CRIS

Program Development and Theoretical Study of Dual Boundary Integral Equation Method by Using the Singular Value Decomposition and Addition Theorem

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