以基本解法求解波動方程式與自由液面問題

Abstract

本文採用基本解法並與其他數值技術結合發展成無網格數值模型，用以求解波動方程式與自由液面問題。本文所使用的尤拉-拉格朗日基本解法是結合尤拉-拉格朗日法與基本解法，用來求解對流-擴散方程式。並採用極小的擴散係數使獲得的數值解可以近似於對流方程式之解。其中，尤拉-拉格朗日法被用來描述對流現象，而基本解法則用來求取特徵線上的方程式數值解。尤拉-拉格朗日基本解法是一種真正的無網格數值方法，此方法可以將物理值在尤拉與拉格郎日座標中任意轉換。在本文中，雙曲線型方程式與方程式系統被用來證明尤拉-拉格朗日基本解法的求解能力。雖然基本解法可以輕易的求解對流-擴散方程式，但是卻難以用來直接求解波動方程式。原因在於波動方程式的基本解總是存在著脈衝函數(或是階梯函數)。所以達朗伯特格式在本文中被用來避免直接處理脈衝函數(或是階梯函數)，並結合尤拉-拉格朗日基本解法發展出一維波動模型。本文中所提出的一維波動模型難以處理複雜的邊界問題，且不易延伸至高維度空間問題。因此，本文引進了特解與齊性解的概念，發展出高維度空間之波動模型。其中，特解由基底函數負責描述，齊性解則由拉普拉斯方程式的基本解描述之。數值結果皆與解析解或數值解比對結果良好，證明本文所提出的無網格數值方法對於波動問題具有高度精度、一致性與效率。而更進一步的，本文使用尤拉-拉格朗日基本解法結合有限元素法，發展出自由液面模型。其本文所發展的自由液面模式之數值結果與其他數值模式和實驗數據比較皆能吻合。本文採用無網格數值方法成功求解對流、雙曲線型系統、自由液面與波動問題。從這些數值試驗案例，我們可以確信本文所提出的無網格數值方法一個有發展潛力的數值模擬工具。

In this dissertation, the method of the fundamental solutions (MFS) combined with the other techniques is adopted for developing the wave equation solver. The original Eulerian-Lagrangian method of fundamental solutions (ELMFS) for solving the advection-diffusion equation is based on the Eulerian-Lagrangian method (ELM) and the MFS, respectively. The ELMFS can approximate the solution of the pure advection equation with the extremely small diffusion coefficient. The ELM is applied to describe the advection phenomena and the numerical solutions along characteristic path are obtained by the MFS. The proposed ELMFS is the purely meshless method which can easily transport the solutions between the Eulerian and Lagrangian coordinates. The problems of hyperbolic equation and system are selected to prove the ability of the proposed ELMFS. Although the MFS can easily handle the advection-diffusion problems, it is difficult to solve the wave equation directly, because the fundamental solution of wave equation always accompanies with the Dirac delta function (or Heaviside step function). The D’Alembert formulation is used to avoid the problems of the Dirac delta function (or Heaviside step function). The one-dimensional wave model based on the D’Alembert formula and ELMFS is proposed in this dissertation. However, the proposed ELMFS wave model can not deal with the complex boundary conditions and difficult to extend for solving the multi-dimensional problems. Therefore, the concepts of the particular solution and homogeneous solution are used to develop the multi-dimensional wave model. In the proposed multi-dimensional wave model, the describing works of the particular solution is based on the radial basis functions and the fundamental solution of Laplace equation is used to describe the homogeneous solution. There are several examples of the advection, hyperbolic system and wave problems discussed in this present work. The numerical simulations of examples are compared well with the analytical solution or other numerical solutions. In the numerical tests, we demonstrate the high accuracy, consistency and efficiency of the proposed meshless numerical methods for wave problems. Finally the ELMFS is used to combine with the finite element method for dealing the free surface problem. The numerical results of the proposed free surface model are compared well with the other numerical model and experimental data. From the numerical comparisons, it is convinced that the proposed hyperbolic models are the promising meshless numerical tools for engineering and science applications.