楊德良臺灣大學：土木工程學研究所劉子雍Liu, Tzu-YumgTzu-YumgLiu2007-11-252018-07-092007-11-252018-07-092007http://ntur.lib.ntu.edu.tw//handle/246246/50562本論文提出一個含超強奇異性無網格法求解三維勢能及外域聲學問題。其解以雙層勢能核函數代替傳統之基本解法所使用之單層勢能核函數來表示，藉由所提出的去除奇異性技術將含奇異與超強奇異性的核函數正規化，源點可直接佈置於實體邊界上，解決了傳統基本解法具爭議性的源點佈置問題。最後在求解勢能與外域聲學散射問題經一系列的各種不同型態的邊界條件之數值例子中，將其數值結果與解析解、基本解法、有限元素法、邊界元素法以及區域微分積分法等數值方法比較後，證明本方法確實可行且精確。In this thesis, a hypersingular meshless method is developed to solve the potential and exterior acoustic problems in three dimensions for arbitrary shapes. The solutions are represented by a distribution of double layer potentials instead of the single layer potentials as generally used in the conventional method of fundamental solutions. By using the desingularization technique to regularize the singularity and hypersingularity of double layer potentials, the source points can be located on the real boundary to avoid the sensitivity of locating fictitious boundary as used by the conventional method of fundamental solutions, and therefore the diagonal terms of influence matrices are determined. The main difficulty of the coincidence of the source and collocation points thus can be overcome. The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of analytical solution, the method of fundamental solutions, finite element method, boundary element method and local differential quadrature method for the Dirichlet, Neumann and mix-type boundary conditions problems with simple and complicated boundaries. The numerical results have demonstrated the validity and accuracy in solving a number of testing cases for potential and exterior acoustic problems after comparing with analytical solution and other numerical methods.Table of contents 口試委員會審定書 I 誌謝 II 中文摘要 III Abstract IV Table caption VII Figure caption VIII Abbreviation XI Chapter 1. Introduction 1 1-1. Motivation 1 1-2. Contents of the thesis 3 Chapter 2. Potential problems 6 2-1. Introduction 6 2-2. Formulation 7 2-3. Derivation of diagonal coefficients of influence matrices for arbitrary shape 10 2-4. Numerical results 11 2-4.1 Spherical shape case (case 1) 11 2-4.2 Cubic shape cases (cases 2-1 and 2-2) 13 2-4.3 Cylinder shape case (case 3) 14 2-4.4 Ring shape case (case 4) 16 2-4.5 Dual sphere shape cases (cases 5-1 to 5-3) 16 2-4.6 Pole shape cases (cases 6-1 to 6-3) 17 2-5. Remark 18 Chapter 3. Exterior acoustics problems 20 3-1. Introduction 20 3-2. Formulation 21 3-3. Derivation of diagonal coefficients of influence matrices for arbitrary shape 24 3-4. Numerical results 26 3-4.1 Case A. Scattering of a plane wave by a soft sphere (Dirichlet BC) 26 3-4.2 Case B. Scattering of a plane wave by a rigid sphere (Neumann BC) 27 3-4.3 Case C. Scattering by a bean shape obstacle (Dirichlet BC) 28 3-5. Remark 29 Chapter 4. Conclusions and further researches 30 4-1. Conclusions 30 4-2. Further works 31 Appendix A: The detail derivations of equation for both Laplace equation and Helmholtz equation. 32 Appendix B: Analytical derivation of diagonal coefficients of influence matrices for spherical domain by the separable kernels (Laplace problem). 34 Appendix C: Analytical derivation of diagonal coefficients of influence matrices for spherical domain by the separable kernels (Helmholtz problem) 37 References 40en-US含超強奇異性無網格法基本解法雙層勢能核函數徑向基底函數去除奇異性技術聲學散射三維hypersingular meshless methodmethod of fundamental solutionsdouble layer potentialradial basis functiondesingularized techniqueacoustic scattering3Dthesis