The Applications of Hypersingular Meshless Method for Waveguide and Electromagnetic Wave Problems

在本文中，提出並且實作了一個用來求解電磁波波導管、空腔共振器或電磁波散射問題的數值方法，其中牽涉到求解二維以及三維的赫姆霍茲方程式。在基本解法數值方法中，為避開奇異性將源點佈在離開邊界的虛擬邊界上是必要的。而在本文中提出的數值方法，使用了雙層勢能核函數替代傳統的基本解法中的單層勢能核函數，並且將源點直接佈在物理邊界上，使源點和邊界點重合，導致在矩陣中發生超強奇異性。這麼作的目的是在於藉由去除奇異性技術將含奇異性以及超強奇異性的格林函數正規化，以推導出係數矩陣中的對角線項。如此本數值方法保留了傳統基本解法的無網格特色，且應用本數值方法可以得到可靠的答案。本文中完成了包括電磁波波導管、空腔共振器以及三維的聲波散射問題的數值模擬，並且藉由跟解析解以及其他數值方法的結果比較來證明本方法是可行而且精確的。In this thesis, a numerical algorithm for solving the ElectroMagnetic (EM) waveguide, EM resonator and electromagnetic wave scattering problems, involving 2-D and 3-D Helmholtz equation, is described and implemented. In the Method of Fundamental Solutions (MFS), seeding location of the source points on a fictitious boundary off-setting from the real boundary is necessary. However, in the proposed method the double-layer potential kernel functions are employed as the alternative radial basis functions (RBFs) in the conventional MFS which uses the fundamental solutions, seeding the source points on the real boundary, and the source points coincide with the boundary points, causing hypersingularity occurs. The purpose of above-mentioned statements is to derive the diagonal terms of the influence matrices by using a desingularization technique to regularize the singularity and hypersingularity of the Green’s functions. Applying the proposed method in which the meshless features of the MFS are maintained yields a reliable solution. Numerical simulations consist of the solutions of electromagnetic waveguide, resonator and 3-D electromagnetic wave scattering problems. Numerical examples are performed, and compared the present numerical results with the analytical solutions, results of conventional MFS and other numerical methods. The validity and accuracy of the proposed method are well demonstrated.中文摘要(I) Abstract(II) Table Caption(V) Figure Caption(VI) Chapter 1. Introduction(1) 1.1 Motivation(1) 1.2 Literature Review(3) 1.3 Scope of the Thesis(4) References(5) Chapter 2. Electromagnetism Preliminaries(8) 2.1 Maxwell’s Equations(8) 2.2 Wave Equations and Helmholtz Equations(9) 2.3 Boundary Conditions(11) 2.4 Waveguide and Resonator(12) 2.5 Derivation of Analytical Solutions(15) References(21) Chapter 3. Hypersingular Meshless Method (HMM)(23) 3.1 Theory of the HMM(23) 3.2 Conventional Method of Fundamental Solutions(24) 3.3 Single and Double Layer potentials(25) 3.4 HMM for Helmholtz Equation(27) 3.5 Derivation of the Diagonal Coefficients of Influence Matrix(30) References(35) Chapter 4. HMM for Waveguide and Resonator Problems(37) 4.1 Frequency Analysis by the SVD Technique(37) 4.2 Waveguide Problems(40) 4.3 Resonator Problems(42) References(44) Chapter 5. HMM for Wave Scattering Problems(46) 5.1 Formulations(46) 5.2 Numerical Results(49) References(53) Chapter 6. Conclusions and Further Researches(54) 6.1 Conclusions(54) 6.2 Further Researches(55) Appendix A The Detail Derivations of Equations for both Laplace Equation and Helmholtz Equation(57) Appendix B The HMM for Interior Wave Scattering Problems(58