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研究生:劉禹廷
論文名稱:使用多階層快速多極計算法於迭代法中加速求解散射問題
論文名稱(外文):Solving Scattering Problems Accelerated by Multilevel Fast Multipole Algorithm In Iterative Schemes
指導教授:林俊華林俊華引用關係
學位類別:碩士
校院名稱:國立海洋大學
系所名稱:電機工程學系
學門:工程學門
學類:電資工程學類
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:75
中文關鍵詞:動差法快速多極計算法多階層快速多極計算法
外文關鍵詞:moment methodfast multipole methodmultilevel fast multipole algorithm
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本論文是以數值動差法求解電磁散射問題。對任意形狀的三維導體表面以三角平板來近似,同時以動差法離散電場積分方程後,得到一個矩陣方程。在求解矩陣方程時,是以共軛梯度法(CGM)來得到電流係數值,但當處理的問題相當龐大時,共軛梯度法求解的運算量也相對的變得複雜且耗時。有鑑於此,在本論文特別使用一種數值方法-多階層快速多極計算法(Multilevel Fast Multipole Algorithm MLFMA)-旨在加快程式中矩陣-向量相乘的速度。此多階層快速多極計算法,可將原來矩陣-向量相乘的計算量由N平方縮減為NlogN,其中N為未知數個數。此MLFMA演算法可花費較少之記憶體量,因此在PC工作環境上,能處理較大尺寸的物體。

In this thesis, we use the method of moment (MOM) to solve the electromagnetic scattering problems. A three-dimensional conducting object of arbitrary shape is divided into triangular patches, and the electric field integral equation (EFIE) is discretized by the MOM. Then the conjugate gradient method (CGM) is used to solve the matrix equation for the unknown expansion coefficients of the surface current. But when the number of unknowns becomes very large, the CGM takes much time at each iteration. In view of this, we use the multilevel fast multipole algorithm (MLFMA) to speed up the matrix-vector multiplication in the CGM. The MLFMA reduces the complexity of a matrix-vector multiplication from N square to NlogN , where N is the number of unknowns. This algorithm requires less memory, and hence, large-sized objects and more practical problems can be solved on a PC.

摘要………………………………………………………………………i
目錄……………………………………………………………………iii
第一章 緒論……………………………………………………………1
1.1 研究動機與目的………………………………………………1
1.2 簡介……………………………………………………………1
1.3 文獻回顧……………………………………………………2
1.4 章節概要……………………………………………………3
第二章 數值理論分析與公式…………………………………………4
2.1 三角平板近似物體……………………………………………4
2.2 數值方法………………………………………………………4
2.2.1數值動差法……………………………………………4
2.2.1.1基底函數………………………………………5
2.2.1.2測試函數………………………………………6
2.2.2數值積分………………………………………………7
2.2.2.1奇異點的處理…………………………………9
2.3 共軛梯度法(CG)……………………………………………11
2.4 快速多極計算法(FMM)……………………………………12
2.4.1 FMM演算法…………………………………………13
2.5 多階層快速多極記算法(MLFMA)…………………………17
2.6 多極點展開式的極點數與方向取樣 ………………………19
2.6.1 極點個數的決定………………………………………19
2.6.2 方向取樣個數的決定…………………………………20
2.7 遠場雷達截面積 ……………………………………………20
第三章 數值程式實作…………………………………………………30
3.1 程式架構與類別 ……………………………………………30
3.1.1 基本類別………………………………………………31
3.1.2 應用類別………………………………………………32
3.2 程式流程 ……………………………………………………33
3.2.1 前續資料處理…………………………………………33
3.2.2 核心數值計算程式……………………………………35
3.2.3 後續處理………………………………………………36
3.3 記憶體管理…………………………………………………36
第四章 計算結果與討論………………………………………………46
4.1 FMM立方體分組測試………………………………………46
4.2 稀疏矩陣……………………………………………………47
4.3 正方形平板………………………………………………48
4.4 立方體………………………………………………………49
4.5 中空圓柱……………………………………………………49
4.6 單邊開口圓柱………………………………………………49
4.7 金屬導體球…………………………………………………50
4.8 MLFMA時間與記憶體的比較………………………………50
4.9 討論 …………………………………………………………51
第五章 結論……………………………………………………………71
參考文獻 ………………………………………………………………72

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