# 臺灣博碩士論文加值系統

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 本文首先推導二維格林邊界方程式(Green’s boundary formula)及其法向導數之正規化公式，亦即完全不含奇異積分之積分方程，這樣的話，就可以在整個積分域內使用標準求積分公式( Standard quadrature formula )來求解。而正規化的方法就是將富利葉-雷建德(Fourier-Legendre)級數帶入未知函數內來打開，並且將積分上下限[a , b]轉換為[-1 , 1]，如此一來，級數內的係數就可以被決定，然後利用一個已知條件的橢圓柱來作分析，來確定這個方法的可行性。　　接著將本文的正規化化法應用到聲波散射的問題上，本文所採用的是廣為人知的Burton以及Miller方法，就是將平面亥姆霍茲( Helmholtz )積分方程式以及它的法向導數來作線性組合以克服無唯一解的問題，這樣的話就可以導出經過線性組合的無奇異點方程式。最後就來比較平行垂直X軸入射的平面波在全軟以及全硬圓柱表面所形成之散射場的數值解。
 This paper present the non-singular forms, in a global sense, of two-dimensional Green’s boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element-free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier-Legendre series, together with transforming the integration interval [a, b] to [-1, 1]; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite-part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well-known Burton-Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non-uniqueness problem. A general non-singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made.
 SUMMARY I摘要 II誌謝 III目錄 IV表目錄 V圖目錄 VI符號說明 VII第一章 緒論 11-1 研究動機及方法 11-2 文獻回顧 3第二章 格林邊界方程式以及其法向導數的正規化 6　　2-1 格林邊界方程式的正規化 6　　2-2 格林邊界方程式的法向導數的正規化 11　　2-3 數值模式的建立與計算研究 15　　2-3.1 數值模式的建立 15　　2-3.2 計算研究 16第三章 聲波散射問題 19　　3-1 Helmholtz積分方程式及其導數的正規化 19　　3-2 全軟物體積分方程式及其導數之正規化 23　　3-2.1 全軟物體積分方程式之正規化 23　　3-2.2全軟物體積分方程式法線方向導數之正規化 24　　3-3全硬物體積分方程式及其導數之正規化 26　　3-3.1全硬物體積分方程式之正規化 26　　3-3.2全硬物體積分方程式法線方向導數之正規化 27　　3-4聲波散射場模式之計算研究 28第四章 結論 32參考文獻 35附錄A 39附錄B 41附錄C 43附錄D 48附錄E 50
 [1]. Landweber L, Macagno M.Irrotational flow about ship forms. IIHR Report No.123, Iowa Institute of Hydraulic Research, University of Iowa, Iowa City, Iowa, 1969.[2]. Schenck HA. Improved integral formulation for acoustic radiation problems. Journal of the Acoustical Society of America 1968; 44: 41-58.[3]. Burton AJ, Miller GF. The application of integral equation methods to the numerical solution of some exterior boundary value problems. Proceeding of the Royal Society of London, Series A 1971; 323:201-210.[4]. Achenbach JD, Kechter JD, Xu Y-L. Off boundary approach to the boundary element method.Computer Methods in Applied Mechanics and Engineering 1988; 70: 191-201.[5]. Koopmann GH, Song L, Fahnline JB. A method for computing acoustic fields based on the principle of wave superposition. Journal of the Acoustical Society of America 1989; 86: 2433-2438.[6]. Benthien W, Schenck A. Nonexistence and nonuniqueness problems associated with integral equation methods in acoustics. Computers and structures 1997; 65(3):295-305.[7]. C. C. Chien, H. Rajiah, and S. N. Atluri “An Effective Method for solving Hypersingular Integral Equations in 3-D Acoustics,” J. Acoust. Soc. Am. 88, 918-937,1990.[8]. NM. Potential Theory and Its Application to Basic Problems of Mathematical Physics. Ungar : New York, 1967.[9]. Yang S. A. On the singularities of Green’s formula and its normal derivatives, with an application to surface-wave-body interaction problems. International Journal for Numerical Methods in Engineering 2000; 47: 1841-1864.[10]. Smirnov VI. A Course of Higher Mathematics, vol. IV. Pergamon: Oxford, 1964; 595.[11]. Sladek V, Sladek J. Singular Integrals in Boundary Element Methods. Computational Mechanics Publications: Southampton, 1998.[12]. Krishnasamy G, Rizzo FJ, Rudolphi TJ. Continuity requirements for density functions in the boundary integral equation method. Computational Mechanics 1922; 9: 267-284.[13]. Martin PA, Rizzo FJ, Hypersingular integrals: how smooth must the density be?. International Journal for Numerical Methods in Engineering 1996; 39: 687-704.[14]. Kaya AC, Erdogan F. On the solution of integral equations with strongly singular kernels. Quarterly of Applied Mathematics 1987; 45(1):105-122.[15]. Carstensen C, Stephan EP. Adaptive boundary element methods for some first kind integral equations. SIAM Journal on Numerical Analysis 1996; 33: 2166-2183.[16]. Chan RH, Sun HW, Ng WF. Circulant preconditioners for ill-conditioned boundary integral equations from potential equations. International Journal for Numerical Methods in Engineering 1998; 43: 1505-1521.[17]. Takahashi H, Mori M. Double exponential formulas for numerical integration. Publications, 9(3), Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 1974.[18]. Yang SA. A boundary integral equation method for two-dimensional acoustic scattering problems. Jounal of the Acoustical Society of America 1999; 105 : 93-105.[19]. Bowman JJ, Senior TBA, Uslenghi PLE (eds). Electromagnetic and Acoustic Scattering by Simple Shapes, Chapter 2. Hemisphere: New York, 1987.
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