本論文主要是利用基本解無網格法來模擬流線函數、史托克斯方程式、赫姆霍滋方程式以及板震動問題。而我們主要的議題是利用基本解無網格法來求解双諧和方程式，而諧和方程式被廣泛的應用在薄板理論以及緩慢的流體問題之中。在本論文之中我們對流線函數的邊界條件做了一些假設，把部分的邊界條件移除，並和有限元素法模擬結果相比較，結果仍是相當精準。同時我們藉由一些假設來利用二薄維板震動的基本解求解二維流線函數，而數值結果也證實我們的假設是正確的。
在薄板震動問題之中我們會遭遇到邊界條件為零的情形，為了求得一有意義之非零解，本文藉由非奇異值分解法來求得其特徵值與特徵向量，並且模擬二維赫姆霍滋方程式與解析解做比較作為驗證，結果顯示在相當不錯且有效率。接著應用此模式在二維板震動的問題，在固定邊界條件下可以有效的求得特徵頻率。

In this study, we adopt the method of fundamental solutions (MFS) to simulate the two-dimensional stream function, the two-dimensional Stokes equations, the two-dimensional Helmholtz equation, and two-dimensional thin plate vibration problem. Our objective is to use the MFS to solve the Bi-Harmonic equations that are widely employed in the theory of thin plate and slow flow problems. Some reasonable assumptions have been made on the boundary conditions for the stream function. Comparison of the results by MFS to the results from FEM reveals that both are very close. On the other hand, we made some asymptotic assumptions of the fundamental solutions for thin plate vibration problem to simulate the two dimensional stream function and the numerical results show that our asymptotic behaviors have been justified.
In thin plate vibration problems, we will deal with the governing equation with the homogeneous boundary conditions. In order to find out meaningful non-trivial solutions, we utilize the singular value decomposition (SVD) method to detect the eigenvalues and eigenmodes. Therefore in order to justify the correct usage of the SVD method, we have simulated two dimensional Helmholtz equation with homogeneous boundary condition, and compared with the analytical solutions. The result indicates that the MFS is very accurate and efficient as far as computational aspects are concerned. Furthermore, we utilize singular value decomposition (SVD) method to solve the thin plate vibration problem and the eigenfrequencies have been obtained efficiently by employing the clamped boundary conditions for circular thin plates.

Table of Contents
摘要 Ⅰ
Abstract Ⅱ
Table of Contents Ⅲ
Figure List Ⅴ
Table List Ⅷ
Chapter 1 Introduction
1.1 Motivations and Objectives 1
1.2 Literature Review 2
1.3 Outline of the Thesis 4
Chapter 2 Method of Fundamental Solutions and Radial Basis Functions
2.1 Introduction to Method of Fundamental Solutions 5
2.2 Radial Basis Functions 8
Chapter 3 Stream Function and the Stokes Equations
3.1 Stream Function 12
3.1.1 Governing Equation 12
3.1.2 Numerical Results 14
3.2 Stokes Equations and their Fundamental Solutions 41
3.2.1 Governing Equations and Fundamental Solutions 41
3.2.2 Numerical Results 43
Chapter 4 Helmholtz Equations and Free Vibration Analysis of Plate
4.1 Helmholtz Equations 50
4.1.1 Governing Equation 50
4.1.2 Numerical Results 52
4.2 Free Vibration Analysis of Plate 64
4.2.1 Governing Equation 64
4.2.2 Numerical Results 65
Chapter 5 Conclusions and Future Studies
5.1 Conclusions 78
5.2 Future Studies 79
References 80

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