利用修正有限配點法產生複連通區域之符合邊界正交網格

Abstract

本文利用修正有限配點法(MFPM)，求解任意幾何形狀內二維正交網格的產生，其控制方程式為拉普拉斯方程式。 修正有限配點法(MFPM)是利用多項式為基底函數，搭配移動最小二乘法的無網格數值計算方法。比較配點法(FPM)，在局部邊界端配點時加入控制方程式還有邊界條件的因子，有助於改善邊界端還有角落端的不穩定；有別於傳統之網格數值方法，數值點跟數值點之間是沒有直接的關係式，所以在邊界上要如何給定也是相對自由的。 利用無網格數值方法高度自由的特性，傳統數值方法邊界條件容易受到分支切割的限制只能求解特定型態網格；傳統數值方法在不規則的計算區域在邊界上或角落上直接求解容易出現不穩定的現象，導致必須先在簡單的區域求解在利用各種數學方法轉換成想要的區域，過程相當繁雜，本模式可直接求解出穩定的正交網格。 本文利用圓型、橢圓、機翼做為複連通的挖空區域，來求解正交網格，由於控制方程式為拉普拉斯方程式，所以全區域都符合柯西里曼條件式；在流線場邊界條件上我們借助勢能場的偏導數解，搭配柯西里曼條件式，利用數值積分的方式來確認上邊界跟內邊界的狄氏邊界條件值。 挖空區域的邊界是平滑且變化不劇烈，如圓型、橢圓，可以得到相當穩定的未知函數解、跟偏導數解，但當我們把把橢圓調整的很扁的時候，我們發現開始有不穩定的現象，尤其在頭尾的兩端我們認為在進行數值模擬時，存在距離很近可是卻相關性很低的點，所以加入區域連結的概念，把區域做切割，進行配點時，篩選排除掉相關性低區域中的點，利用此一觀念，我們在很扁的橢圓案例上，獲得良好的結果，甚至在有尖點的機翼案例上，也都表現出色。

In this study, the Modified Finite Point Method (MFPM) is used to solve the governing equations which are Laplace equations and to obtain multi-connected region 2-dimensional orthogonal grid. Modified Finite Point Method (MFPM) is a mesh-less (mesh-free) numerical method. MFPM’s base functions are polynomials and collocate with moving least square(MLS). Compared with Finite Point Method (FPM), collocation of MFPM at boundary takes into account of both governing equations and boundary conditions. It helps to improve the unstable numerical phenomena at boundaries and corners. Unlike traditional grid-base numerical methods, there is no direct relationship between a point and another nearby point so the boundary conditions are relatively free to be applied. Previously, boundary conditions in traditional numerical methods were restricted to the branch cut so we can only solve certain specific-type grids. Also, it has been found that traditional numerical methods become unstable when solving governing equations directly at the boundary or corners. The MFPM model has been found that with regional connectivity approach it can directly solve the problem stably and obtain orthogonal grids. In this study, examples using circle, Rankine oval and NACA airfoil as hollowed-out domain are used to generate orthogonal grids. Based on Cauchy-Riemann conditions, Dirichlet’s boundary condition of the internal boundary of hollowed geometry has been derived to improve the accuracy of orthogonal computations. When hollowed area's boundary is smooth and no dramatic changes, such as circle, Rankine oval, very stable solutions can be obtained, with very accurate partial derivatives of the solutions. For a very flat Rankine oval, phenomena of numerical instability occurred, especially, near the head and tail of geometry, due to inappropriately including low-related points in different regions into collocation processes. By applying the concept of regional connectivity and dividing the whole domain into sub-domains, accurate collocation only allowed by including grids in appropriate regions. Using this concept, in the case of flat Rankine ovals and NACA airfoils with sharp points, present MFPM performed very well.