利用修正有限配點法產生符合邊界的二維正交網格

Abstract

本文旨為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方法使用。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，因此網格建置很重要。 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轉換回原幾何圖形，即可在原幾何圖形中得到正交網格。 使用的方法為無網格法中的修正有限配點法(Modified finite point method, MFPM)，為利用相對位置的關係，藉由局部多項式(Local polynomial method)來近似連續函數，以解二維拉普拉斯方程式，優點為其偏導數可直接得到且精準，有利於正交網格的形成，且對於不規則的圖形，無網格法比較容易佈點計算。而修正有限配點法中的參數選擇，使用多目標規劃(Multiple Objective Programming, MOP)挑選出最適合的組合，本文以有解析解的半圓形環狀當做率定的案例。雖然採用無網格法產生網格，但仍不影響本文的價值。 在驗證方面，本文以半圓形環狀、三角標形、圓形環狀、星形、幸運草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。本模式的計算成果，除了計算區域內角為鈍角時其鈍角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 generate two dimensional orthogonal grids in irregular regions for further computations of grid-based numerical methods. 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 grid information is essential. Making use of the orthogonality of Cauchy-Riemann condition, grid generation of the forward and inverse transformations were formulated by solving Laplace equation . The numerical method in this study is a meshless numerical method, namely, modified finite point method (MFPM). Based on collocation, this method uses polynomials as the local solution form needed in the collocation approach. The advantage of this method is not only the values of the solution 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 grid. In modified finite point method, there are parameters to be determined, and Multiple Objective Programming (MOP) is used. Though the method used in this study to solve the coordinate transformation equation is a meshless one, it shows at least merits of present work with other numerical methods. There are six benchmark problems tested 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 an obtuse angle, the generated results are very accurate.