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研究生:張凱博
研究生(外文):Kai-Po Chang
論文名稱:使用雷建德多項式來計算二維格林邊界方程式及其法向導數並應用於聲波繞射場
指導教授:楊世安楊世安引用關係
指導教授(外文):Yang Shih-An
學位類別:碩士
校院名稱:國立成功大學
系所名稱:造船及船舶機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:73
中文關鍵詞:聲波散射場雷建德多項式
外文關鍵詞:Acoustic ScatteringLegendre Polynomials
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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 1
1-1 研究動機及方法 1
1-2 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
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