本論文將針對週期性非平坦表面下之均勻介質物體進行探討，首先由TM(Transverse Magnetic)極化波照射經由表面透射至掩埋物體，接著利用馬克斯威爾方程式、二維期性格林函數和邊界條件推導出積分方程式，散射場積分方程式透過動差法求得散射場，將逆散射問題轉化成最佳化問題，使用傅立葉級數展開描述物體形狀，並配合自我適應之動態差異型演化法進行數值計算及物體重建。
    利用自我適應之動態差異性演化法重建出均勻介質物體，不論初始的猜測值如何，自我適應之動態差異性演化法總會收歛到整體的極值(global extreme)，因此在數值模擬顯示中，即使最初的猜測值遠大於實際值，我們仍可求得準確的數值解，成功的重建出物體形狀函數、週期長度和相對介電常數，並於數值模擬顯示中，量測的散射場即使加入雜訊，仍可得到良好的重建結果。我們發現，在週期性均勻介質物體中，週期長度的收斂速度總是優於形狀函數、介電常數。因此可知週期長度對於散射場的貢獻大於形狀函數及介電常數。

This thesis presents the reconstruction of a periodic homogeneous dielectric object buried in rough surfaces. First, a TM (Transverse Magnetic) polarized wave is transmitted through the surface to a buried object. Integral equations are derived by using the Maxwell equation, the two-dimensional periodic Green function, and the boundary condition. The integral equations are numerical solved by the method of moment (MOM) to obtain the scattering field. For inverse problem, the Fourier series expansion is used to describe the shape of the object, then the problem of inverse scattering is transformed into an optimization problem. Next, the self-adaptive dynamic differential evolution method is used for numerical calculation and object reconstruction. Numerical results show that the SADDE converges to the overall extreme value (global extreme) regardless of the initial guess. Even if the initial guess is far away from the actual value, SADDE can get the correct shape, periodic length and the relative permittivity of the periodic homogeneous dielectric object. We have found that the convergence speed of the periodic length is always better than the shape function and dielectric constant. Moreover, we can still obtain good reconstruction results even if the noise is added in the scattered field.

目錄
致謝	II
中文摘要	III
英文摘要	V
目錄	VII
圖目錄	IX
第一章 簡介	1
1.1 研究動機與相關文獻	1
1.2 本研究之貢獻	8
1.3 各章內容簡述	9
第二章 週期性均勻介質物體於非平坦表面下之逆散射理論	10
2.1 正散射的理論公式推導	10
2.2 動差法求正散射公式	16
2.3 正散射的理論數值驗證	20
第三章  自我適應之動態差異型演化法(Self-Adaptive Dynamic
Differential Evolution)	21
第四章  數值分析及模擬結果	29
4.1 模擬環境介紹	29
4.2 模擬結果	32
第五章 結論	45
參考文獻	47
附錄一 加快週期性格林函數收斂的方法	53
附錄二 當y≈y^'時改進週期性格林函數收斂的方法	55


圖目錄
圖2.1 週期性非平坦表面及掩埋物示意圖	11
圖2.2 驗證正散射模擬示意圖	20
圖3.1 自我適應之動態差異型演化法流程圖	23
圖3.2 自我適應之動態差異型進化法中突變方法一的示意圖 25
圖3.3 自我適應之動態差異型進化法中突變方法二的示意圖 26
圖4.1 模擬環境示意圖	30
圖4.2 SADDE於例子一的重建情形	34
圖4.3 例子一之形狀函數偏差量DF及週期長度偏差量DP的變化情形 35
圖4.4 例子一之形狀函數偏差量DF及介電常數偏差量DEPS的變化情形	35
圖4.5 例子一加入不同雜訊比之DF、DP、DEPS趨勢圖  36
圖4.6 SADDE於例子二的重建情形	38
圖4.7 例子二之形狀函數偏差量DF及週期長度偏差量DP的變化情形 39
圖4.8 例子二之形狀函數偏差量DF及介電常數偏差量DEPS的變化情形	39
圖4.9 例子二加入不同雜訊比之DF、DP、DEPS趨勢圖  40
圖4.10 SADDE於例子三的重建情形	42
圖4.11 例子三之形狀函數偏差量DF及週期長度偏差量DP的變化情形	43
圖4.12 例子三之形狀函數偏差量DF及介電常數偏差量DEPS的變化情形	43
圖4.13 例子三加入不同雜訊比之DF、DP、DEPS趨勢圖  44

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