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研究生： 張昶盛
Chang-sheng Zhang
論文名稱： 用有限元素法解二維不可壓縮之拉格朗日式奈維－斯托克斯方程式的一些數值結果
Some Numerical Results for Two Dimensional Incompressible Navier-Stokes Equations in Lagrangian Formulation Using Finite Element Method
指導教授： 鄭經斅
Arthur Cheng
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 100
語文別： 英文
論文頁數： 19
中文關鍵詞： 自由邊界問題 、奈維－斯托克斯方程式 、有限元素法 、拉格朗日式
外文關鍵詞： Lagrangian formulation, Finite element method, Free boundary problems, Navier-Stokes equations
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    • 這篇論文提出不同觀點來解固定定義域(fixed domain)的奈維－斯托克斯方程式(Navier-Stokes Equations)數值解。以往用空間座標(spatial coordinate)建模流體的運動，在這裡我們改用物質座標(material coordinate)，並藉有限元素法(finite element method)來研究方程式；這個方法最大的好處是許多自由邊界問題(free boundary problems)理論上可因此解出其數值解，但相對地需要花很多時間去計算。
    我們會用空間及物質兩種座標算出方程式的數值解，比較兩者的結果，並了解兩者之計算網格(mesh)越小時，差異會越趨近到 0 。

    In this thesis we propose a different point of view in solving Navier-Stokes equations on a fixed domain numerically. Instead of using the spatial coordinate to model the motion of the fluids, we formulate using the material coordinate and study the corresponding PDE by standard finite element method. The most important benifit of using the material coordinate is that a lot of free boundary problems can be theoretically solved in this fashion, while a main drawback of doing this is that it is very time-consuming. We compare the numerical results produced by these two different formulations, and conclude that the error between two sets of numerical results gets smaller as the mesh size approaches zero.

    摘要...................................................i Abstract..............................................ii Contents.............................................iii 1. Introduction........................................1 2. Lagrangian formulation..............................3 3. Discritization......................................5 3.1. The Eulerian method and Eulerian solution........5 3.2. The Lagrangian method and Lagrangian solution....5 4. Some results........................................8 4.1. Example 1.......................................10 4.2. Example 2.......................................12 4.3. Example 3.......................................14 4.4. Example 4.......................................16 5. Conclusion.........................................18 6. Reference..........................................19

    1. Klaus-Jürgen Bathe, Finite Element Procedures, Prentice hall, New Jersey, United State, 1996.
    2. Jean Donea and Antonio Huerta, Finite Element Methods for Flow Probelms, John Wiley & Sons Ltd, Chichester, England, 2003.
    3. Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge university press, Lund, Sweden, 1994.

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