臺灣博碩士論文加值系統

本論文提出一個含超強奇異性無網格法求解靜電及電磁波散射問題。藉由所提出的去除奇異性技術將含奇異與超強奇異的核函數正規化，本方法因此改善基本解法的缺點，同時將傳統基本解法的無網格的優良的性質被保留下來，而且不用遭遇奇異與數值積分的問題。因為核函數奇異與超奇異性被消除了，源點因此可被放置在真實邊界上且與邊界點重合，具爭議性的虛擬邊界也就可以不用管它了。再則，使用本法並配合領域分割法解決退化邊界的秩不足問題。最後在求解靜電與電磁波散射問題的數值結果例子中與解析解和對偶邊界元素法比較後，證明本法確實可行且精確的。

In this thesis, a hypersingular meshless method (HMM) is proposed to solve electrostatic and electromagnetic wave scattering problems. This method modifies the method of fundamental solutions (MFS) by using the desingularized technique to regularize the singularity and hypersingularity of the proposed kernel functions. In the meantime the meshless features of conventional MFS are preserved without singularity and numerical integration. The source points can be located on the real boundary coincident with boundary points since the diagonal terms of influence matrices are determined after the singularity and hypersingularity having being eliminated. So that testing to the controversial off-boundary distance can be avoided. Furthermore, by using the HMM in conjunction with domain decomposition technique, we also solve for the rank-deficiency problem with degenerate boundary. The numerical results are demonstrated it valid and accuracy in solving a number of testing cases for electrostatic and electromagnetic wave scattering problems after comparing with exact solutions and the results made by dual boundary element method.

誌謝…………………………………………………………………………………………………….. I
中文摘要...……………………………………………………………………………………………. II
Abstract……………………………………………………………………………………………......III
Table List…………………………………………………………………..………………………… VI
Figure List………………………………………………………………………………..…………..VII
Chapter 1. Introduction................................................................................................................. 1
1-1. Motivation........................................................................................................................................... 1
1-2. Contents of the thesis .......................................................................................................................... 3
Chapter 2. HMM for solving electrostatic problems.................................................................... 5
2-1. Introduction........................................................................................................................................ 5
2-2. Formulation ........................................................................................................................................ 6
2-2.1. Electrostatic preliminary.............................................................................................................................. 6
2-2.2. Conventional method of fundamental solutions.......................................................................................... 7
2-2.3. Formulation of HMM................................................................................................................................... 8
2-2.4. HMM for interior electrostatic problem ...................................................................................................... 9
2-2.6. Treatment of degenerate boundary problems ............................................................................ 12
2-3. Numerical simulation with HMM for interior electrostatic problems ......................................... 13
2-3.1. Case 1-Circular domain with discontinuous Dirichlet BCs...................................................................... 13
2-3.2. Case 2-Rectangular domain with discontinuous Dirichlet BCs ............................................................... 14
2-3.3. Case 3-Rectangular domain with mixed-type problem (discontinuous BC)........................................... 14
2-3.4. Case 4-Arbitrary domain cases (cases 4-1 to 4-3)...................................................................................... 15
2-3.5. Case 5-Degenerate boundary interior problems ....................................................................................... 16
2-4. Numerical simulation with HMM for exterior electrostatic problems......................................... 16
2-5. Remark.............................................................................................................................................. 17
Chapter 3. HMM for solving electromagnetic wave scattering problems................................. 18
3-1. Introduction...................................................................................................................................... 18
3-2. Formulation ...................................................................................................................................... 19
3-2.1. Governing equation .................................................................................................................................... 19
3-2.1.1. Maxwell’s equations ........................................................................................................................... 19
3-2.1.2. Wave equations .................................................................................................................................. 21

3-2.1.3. Helmholtz equation ............................................................................................................................ 21
3-2.2. Analytical solution for a circular conducting cylinder.............................................................................. 24
3-2.3. Formulation of HMM for electromagnetic wave scattering problems .................................................... 26
3-2.3.1. Formulation of double layer potentials............................................................................................. 27
3-2.3.2. Derivation of diagonal coefficients of influence matrices for arbitrary domain using a HMM.. 29
3-3. Numerical results for homogeneous electromagnetic wave scattering problems........................ 31
3-3.1. Scattering through a circular cylinder ....................................................................................................... 32
3-3.2. Scattering through a rectangular cylinder................................................................................................. 33
3-3.3. Scattering through a peanut-shaped cylinder............................................................................................ 34
Chapter 4. Conclusions and further researches......................................................................... 35
4-1. Conclusions ....................................................................................................................................... 35
4-2. Further studying............................................................................................................................... 35
Appendix A. The detail derivations of equations for both Laplace’s equation and Helmholtz
equation. ………………………………………………………………………………………37
Appendix B. Discretization process of MFS for single and double layer potentials.................... 38
References………………………………………………………………………………………......... 42

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