應用修正有限配點法產生邊界符合正交網格

Application of Modified Finite Point Method to Boundary-fitted Orthogonal Grid Generation

郭穎彰(Ying-Chang Kuo)

指導教授 ： 蔡丁貴

國立臺灣大學/工學院/土木工程學研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.03264

摘要

本文利用修正有限配點法(Modified finite point method, MFPM)的無網格法搭配複變轉換技巧產生二維正交網格。應用複變映射理論以及保角映射理論將現實區域的不規則幾何圖形轉換成由四個直角及四條平滑曲線組成之超矩形，再利用修正有限配點法將超矩形區域映射到符合邊界條件的矩形區域並解控制方程式－拉普拉斯方程式(Laplace equations)，之後再將此邊界符合的座標反轉換回原本的物理區域。
在先前的研究中，修正有限配點法透過多項式近似的配點法已能夠有效率的計算節點上的函數值及其偏導數。本文再利用複變轉換技巧將不規則的幾何圖形轉換成超矩形，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。
本文以三角鏢形的幾何圖形為範例，分析比較有無經過複變轉換之網格產生方法的結果。因為經過複變映射之後，任兩點間的距離都將被以指數比例拉伸或者壓縮，這將對配點法造成不小的影響。所以，不同的複變轉換順序將產生不一樣的超矩形形狀，本文發現不同轉換順序存在某種規則能使修正有限配點法的結果更好。本文使用的修正有限配點法能提供正確之座標轉換參數以得到準確的數值解，同時也能彈性的調整網格佈點及邊界條件以滿足在物理區域進行計算的需求。

關鍵字

網格轉換 ； 複變映射理論 ； 保角映射理論 ； 超矩形 ； 正交性 ； 柯西黎曼條件 ； 拉普拉斯方程式 ； 無網格法 ； 修正有限配點法 ； 局部多項式近似

並列摘要

In this thesis, the process of a 2-D orthogonal grid generation by the meshless method called Modified Finite Point Method(MFPM) and complex mapping technique is presented. Applying a complex mapping and orthogonal mapping theorem, a physical domain which is irregular geometrical shape into a hyper-rectangular shape can be transformed, the intermediate transformed domain, which is composed of four right angles and four smooth curved lines, and then MFPM is used to solve the governing equation, Laplace equations, with appropriate boundary conditions to map onto the final transformed rectangular domain. The boundary-fitted coordinates are then mapped backward onto the physical domain.
In previously research, MFPM can efficiently calculate the solutions and the partial derivatives by approximating the exact values at the nodes with polynomial collocation. A complex mapping technique is applied to transform an irregular geometrical shape to a hyper-rectangle. It is shown that orthogonality has increased since the hyper-rectangle mapped to rectangular domain is performed on the basis of conformal mapping theorem.
This paper takes the case of “an area bounded by two triangles” as a testing example and analyses comparisons between grid generations with and without complex transformation. Due to the distance between any two nodes will be stretched or shrunk in exponential ratio after complex mapping transformation, so it has a large influence on collocation. Therefore, different transforming order will result in different mapped shapes. It is shown that there exists some particular rules of transforming order to obtain more accurate approximation by MFPM. Present MFPM provides accurate solutions of scaling factors in coordinate transformations. It is also flexible to adjust nodal distribution and boundary conditions to satisfy computational needs in physical domain.

並列關鍵字

grid generation ； complex mapping theorem ； conformal mapping theorem ； orthogonal ； Cauchy-Riemann condition ； Laplace equation ； meshless ； modified finite point method ； local polynomial approximation

參考文獻

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