二維內部傳輸特徵值問題的特徵值與散射場之數值研究

Abstract

此篇文章指出，根據不同的折射率和場域形狀下，最小正實數的數個傳輸特徵值和最小正實數的特徵值所對應的散射場的行為表現為何。我們也會將一些第二小正實數的特徵值所對應的散射場納入我們的討論。首先，我們先會介紹二維內部傳導問題，並講述如何轉換此問題成一廣義特徵值問題的細節。此外，我們也會講述關於如何轉換此廣義特徵值問題為二次特徵值問題的過程。然後，我們利用Persson和Strang所提供的MATLAB套件DistMesh生成網格，也利用矩陣的組集建構出此廣義特徵值問題所需的矩陣。我們以Jacobi-Davidson型方法解決我們的問題，方法有加入鎖定技巧分別針對廣義特徵值問題和二次特徵值問題的Jacobi-Davidson方法，以及加入壓縮技巧處理二次特徵值問題的Jacobi-Davidson方法。考量到方法的效能與所需資料的完整性，我們選定以鎖定技巧處理廣義特徵值問題的Jacobi-Davidson方法為我們的研究方法。根據數據結果與觀察，我們發現整體平均折射率與場域形狀會影響傳輸特徵值的行為。散射場的變化會與場域形狀和整體平均折射率與場域內折射率之差異程度的共同影響有關。而我們也許能從二維模型的結果，幫助我們了解如何設計出具備某些特定傳輸特徵值和散射場行為特徵的材料。

This thesis shows how the smallest positive real transmission eigenvalues and the scattered field related to the smallest positive real one behave corresponding to the index of refraction and the shape of the domain. Some scattered fields related to the second one are also included. At first, we will present the two-dimensional interior transmission eigenvalue problem and give the details about how to transform it into a generalized eigenvalue problem (GEP). The procedure of turning the GEP into a quadratic eigenvalue problem (QEP) is given as well. Then, we use a MATLAB package \texttt{DistMesh} provided by Persson and Strang to generate our desired meshes and the assembly of the matrices to construct to the matrices in the GEP. We consider Jacobi-Davidson type methods to solve our problem, and there are the Jacobi-Davidson method with locking for the GEP, that for the QEP, and the quadratic Jacobi-Davidson method with deflation presented. We choose the first one to complete our research because of its efficiency and the integrity of the desired data. From our numerical results and observation, we find that the behavior of the transmission eigenvalues is affected by the average index of refraction and the shape of the domain. The variation of the scattered fields we concern is influenced by the shape of the domain, and the combined effect of the average index of refraction and the degree of the difference of the index of refraction in a domain. These results may help us to get a better understanding of how to design a material with specific characteristics related to transmission eigenvalues and the scattered field by some modifications from a two-dimensional model.