臺灣博碩士論文加值系統

本論文為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，同時以動差法離散化相關之積分方程，在求解所得之矩陣方程時，是以共軛梯度法來得到其電流係數值，但當求解之未知數變得相當龐大時其以共軛梯度法求解之運算量相對的也變得相當耗時。有鑑於此，在本論文特別使用一種數值方法-快數多極計算法(FMM)-旨在加快程式中矩陣-向量相乘之速度。此快速多極計算法，可將原來 之矩陣-向量相乘計算量縮減為 之計算量，其中N為未知數個數。本程式是以物件導向為觀念撰寫，以支援物件導向強大功能的Visual C++ 為工具軟體，將一些物體的參數值予以物件化，對於程式日後的擴充較為簡便。此FMM演算法也要求較少之記憶體，因此在PC工作環境上，就能處理更大較實際的物體。

In this thesis, we use the method of moment (MoM) to solve the electromagnetic scattering problems. A three-dimension arbitrary-shaped conductive object is divided into triangular patches, and the integral equation is discretized by MoM. Then a conjugate gradient method (CGM) is used to iteratively solve the resulting matrix equation for unknown expansion coefficients for the surface current. But when the number of unknowns is large, the CGM takes more time at each iteration. In view of this, we use the fast multipole method (FMM) to speed up the matrix-vector multiply in the CGM. The FMM reduces the complexity of a matrix-vector multiply from to , where N is the number of unknowns. The program makes use of the object-oriented programming technique and uses visual C++ as a tool to design some practical classes, which are convenient to expand programs further. This FMM algorithm also requires less memory, and hence, large and more practical problems can be solved on a PC computer.

摘要……………………………………………………………………I
感謝……………………………………………………………………ii
目錄……………………………………………………………………iii
第一章緒論…………………………………………………………1
1.1 動機與目的……………………………………………………1
1.2 文獻回顧………………………………………………………1
1.3 章節概要………………………………………………………2
第二章數值理論分析與公式………………………………………3
2.1三角平板近似物體………………………………………………3
2.2數值方法…………………………………………………………3
2.2.1數值動差法……………………………………………3
2.2.1.1基底函數………………………………………4
2.2.1.2測試函數………………………………………5
2.2.2數值積分………………………………………………7
2.2.2.1奇異點的處理…………………………………8
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
2.4.2 射線傳播多極計算法(RPFMA)………………………17
2.5 遠場雷達截面積 ………………………………………………18
2.5.1 Bistatic RCS近似monostatic RCS運算……………19
第三章數值程式實作………………………………………………26
3.1 程式架構與類別 ………………………………………………26
3.1.1 基本類別………………………………………………27
3.1.2 應用類別………………………………………………28
3.2 程式流程 ………………………………………………………29
3.2.1 前續資料處理…………………………………………29
3.2.2 核心數值計算程式……………………………………30
3.2.3 後續處理………………………………………………31
第四章計算結果與討論……………………………………………37
4.1正方形平板………………………………………………………37
4.2立方體……………………………………………………………38
4.3中空圓柱…………………………………………………………38
4.4單邊開口圓柱……………………………………………………39
4.5圓錐………………………………………………………………39
4.6 FMM時間與記憶體的比較………………………………………40
4.7 討論 ……………………………………………………………43
第五章 結論……………………………………………………………70
參考文獻 ………………………………………………………………71

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