https://scholars.lib.ntu.edu.tw/handle/123456789/171006
標題: | 利用基本解法產生符合邊界的二維正交網格 2D Orthogonal Grid Generation of an Irregular Region Using Method of fundamental solutions |
作者: | 李楊弘 Li-Yang, Hong |
關鍵字: | 網格轉換;柯西里曼條件;拉普拉斯方程式;無網格法;基本解法;複變映射理論;符合邊界正交網格;grid generation;Cauchy-Riemann condition;Laplace equation;Meshless method;Method of fundamental solutions;Boundary-fitted orthogonal grids | 公開日期: | 2012 | 摘要: | 本文旨為發展數值計算方法,以便在二維圖形中產生符合邊界正交數值網格,以應用於後續其他需要網格的數值模式使用。需要網格的數值方法發展久遠且完善,有其不可被忽略的地位,但建置網格的前置作業困難,尤其在不規則區域,且網格有無符合邊界和是否正交都會影響後續的數值計算結果,因此網格建置符合邊界之正交網格是數值計算模式中有很重要的角色。 藉由柯西里曼條件(Cauchy-Riemann condition)的正交性,形成拉普拉斯方程式(Laplace equation),可將二維不規則圖形轉成矩形區域,於矩形區域中建置網格後再經反向轉換回原幾何圖形,即可在原幾何圖形中得到正交網格。本文所採用之無網格數值方法的優點是不僅易於求得函數值,也可以很簡單求得其偏導數。而無網格法可以很容易產生計算點,其基礎點(Base points)的分佈可以很靈活,尤其是在不規則地區。當遇到角落為鈍角者,可以先利用複變轉換技巧將不規則的幾何圖形轉換成超矩形(Super-rectangle),由於超矩形映射到矩形區域乃以保角映射理論為基礎,經研究證明經過複變轉換成超矩形後再映射到矩形區域的結果可以增加正交性的準確度。 本文使用的方法為無網格法中的基本解法(Method of fundamental solutions, MFS),利用二維線性基本解(Fundamental solution)以解二維拉普拉斯方程式,優點為其偏導數可以準確地由直接微分求解而得到,有利於正交網格的形成。 在驗證方面,本文以半圓環狀、三角標形、圓形環狀、星形、花瓣形以及台灣形狀六個範例為例,與解析解比較或計算相對誤差驗證正交性。本方法的計算成果,除了在計算區域內奇異點的角落附近計算會有誤差外,其餘皆可得到正確的正交網格,且內角為零度的情況亦可。 The purpose of this study is to develop numerical method to generate two dimensional orthogonal grids in irregular regions for further computations of grid-based numerical models. This is because grid-based numerical methods have been fully developed and most numerical models in common uses are still coded with grid-based methods. Grid generation techniques to provide input information of orthogonal, boundary-fitted grids are essential. Making use of the orthogonality of the Cauchy-Riemann conditions, grid generation of the forward and inverse transformations were formulated by solving Laplace equations. The numerical method in this study is a meshless numerical method, based on the Method of fundamental solutions, MFS. For collocation, this method uses 2-D fundamental solution of Laplace equation as the solution form needed in the collocation. The advantage of this approach is not only the values of the function values but also the values of its derivatives can be easily obtained. The meshless numerical method is easier to generate computational points especially in irregular regions for its flexibility in distribution of the base points.. When irregular domain consists with the corner with obtuse angle, a complex mapping technique is employed to convert the irregular geometry into a hyper-rectangle. It is shown that accuracy of orthogonality can be improved since the hyper-rectangle mapped to rectangular domain is performed on the basis of conformal mapping theorem. There are six benchmark problems examined in this study, including a semi-annulus, an area bounded by two triangles, a full annulus, a four-pointed star, a flower-like irregular region, and the surrounding area of Taiwan. Correctness of present model is verified by checking the orthogonality of the generated results or comparing with exact solutions. In present model, except at the corner of singular points, the generated results are very accurate. |
URI: | http://ntur.lib.ntu.edu.tw//handle/246246/255418 |
顯示於: | 土木工程學系 |
檔案 | 描述 | 大小 | 格式 | |
---|---|---|---|---|
ntu-101-R99521309-1.pdf | 23.32 kB | Adobe PDF | 檢視/開啟 |
在 IR 系統中的文件,除了特別指名其著作權條款之外,均受到著作權保護,並且保留所有的權利。