- 學位論文
應用邊界元素法產生邊界符合保角網格系統及相關奇異性問題研析
On the Application of BEM to Boundary-fitted Conformal Grid Generation and Associated Singularity Problems
摘要
本論文的研究課題為邊界符合保角網格系統以及其所牽涉到的奇異性問題探討。在產生本論文中的邊界符合網格系統時，需要用到邊界元素法，而運用此方法時，會遭遇到數學奇異性(Mathematical singularity)問題；此外，網格產生的過程，也會遇到幾何奇異性(Geometrical singularity)問題。因此，本論文將首先就邊界元素法中的「數學奇異性」以及計算邊界幾何邊界變化大時所引致的「物理奇異性」進行討論，之後再就兩者對網格產生的影響深入研究，最後以產生的邊界符合座標網格系統來進行計算流體力學計算。 數學奇異性 在奇異性問題的基礎研究上，邊界元素法的數學奇異性方面，本論文利用幾何解析方法及數個演算範例證明前人所認定的數值邊界層(Numerical boundary layer)(Liggett and Liu, 1983)，其實是不存在而可以在實際計算時加以避免。以前的觀念是：應用邊界元素法計算勢能，於邊界附近會有較大的誤差，認定這是由於積分式的核函數具有強奇異性(Strong singularity)所致，並稱此現象為數值邊界層。然而本研究發現在經過積分降低奇異性後，如果正確處理積分項中的極限角度(Limiting angle term)，利用替代弧形路徑逼近法(Contour approach method)及直接逼近邊界法(Direct approach method)可證明數值邊界層並不存在，也就是，正確處理角度的話，即使應用傳統的邊界元素法，也可以計算計算域中任何點的勢能，即使到達非常靠近邊界(10-9單位長度)的地方，也沒有特別大的誤差。 物理奇異性 於物理奇異性方面，由於幾何奇異性或邊界條件的突然變化，所帶來的勢流中流線與勢能線不正交的問題，可分為強奇異性(Strong singularity)問題與弱奇異性問題(Weak singularity)，強奇異性將導致勢能集中，而弱奇異性則使得勢能發散。然真正影響數值計算精度者為強奇異性。為了解決物理奇異性問題，本研究採用複變轉換 (Complex mapping)(Tsay and Hsu, 1997)技術來改變幾何形狀，去除邊界形狀帶來的奇異性。運用數值保角轉換(Numerical conformal mapping)，可一步一步地將幾何奇異點去除，於最後四個角正交，而邊界平滑的超矩形(Hyper-rectangular)區域上進行計算。利用數個計算實驗驗證，此方法可以有效降低物理奇異性所帶來的計算誤差。 保角網格產生法以及相關之奇異性問題研究 前面所述的兩種奇異性，會在本論文所使用的網格產生法中發生影響。在運用解拉普拉斯方程式(Laplace equation)產生保角網格系統(Conformal grids)時，由於前人並不了解(至少在文獻中沒有發現)物理奇異性問題，因此認定保角網格的產生極為不可能，進而加入微調項於拉普拉斯方程式中，犧牲保角性而維持正交性(Thompson, 1977)。然而，真正的保角網格系統將可以使得座標轉換後，拉普拉斯項維持拉普拉斯型態，避免控制方程式複雜化，增加運算效率。其實，只要在運用邊界元素法產生網格前，利用之前所述的複變轉換將物理奇異性去除，產生保角網格並非難事。然而，即便複變轉換有此優點，但在運用時，卻要注意branch-cut問題，其有轉換時計算域壓縮或是拉伸的問題，如果轉換後計算域為拉伸，超越branch-cut的部分便會產生多值(multi-value)的問題，使得網格扭曲交錯，這也是文獻中所記載完全保角的網格不易產生的原因。本研究中找出複變轉換的限制式，作為實際運用時，安排物理奇異點先後轉換順序之依據，也就是說，將不符合限制式的轉換點放在較後面的順序來轉換，符合限制式的角度先轉換的話，常常原本不符合限制式的奇異點也可以成功轉換。在研究論文中，於蘭嶼島網格例子中，可連續轉換255次將255個物理奇異點去除，而保證為保角轉換，便足以說明限制式的可行性。另外，由於在邊界元素法基礎研究中成功去除掉數值邊界層，因此利用此產生邊界符合網格系統，將可產生極為靠近邊界(10-9)的網格系統。 保角網格於計算流體力學上之應用 在應用邊界符合網格系統於計算流體力學計算方面，除了引進美國聯邦緊急事故處理局(FEMA)所發展的暴潮計算模式(FEMA Model)計算台灣環島暴潮水位，同時將FEMA模式中使用的矩形網格系統改為邊界符合座標網格系統，藉以消除計算時，網格系統無法符合自然邊界所產生的誤差，另外，本研究引進預測-修正(Predictor-Corrector)式以改善計算速率。新的暴潮模式可將複雜的物理平面經由邊界符合座標系統轉換至矩形計算平面上，在矩形規則平面上利用預測式與修正式之有限差分法作數值計算。而由於在進行暴潮驗證時，所謂的「現場暴潮位」並非實際量測到的值，而是實際量測到的水位扣除掉計算的天文潮位(Astronomic tide)。因此，天文潮位計算的準確度將影響暴潮水位計算準確與否之判斷。本研究所採用的調和分析(Harmonic analysis)技術，經淡水河河口驗潮站紀錄之水位驗證，發現自1999至2002年，年平均相對誤差最高不超過1.65%，最低可達1.04%，顯示此天文潮模式的可信度。因此，只要現場水位量測資料準確，運用此天文潮模式，便可產生可靠的現場暴潮水位。本研究所進行的台灣環島十個測站的暴潮計算結果，雖然與實際暴潮位有相當誤差，但在趨勢上頗為一致。研判誤差的原因為本研究假設風場為理想之圓形對稱風場型態，但實際風場並非如此，其假設對於計算結果會產生影響，故在風場假設上未來應做適當的改善。而地形的遮蔽效應，對風場也會產生影響，本研究則以環島海域地形轉換至計算平面上，理想對稱之颱風模型沒有詳加考慮地形遮蔽效應所造成的結果，這也是將來暴潮模式欲進一步發展時所需考慮的方向。 參考文獻 1. Liggett, J. A. and Liu, P. L-F., The boundary integral equation method for porous media flow, Allen & Unwin Ltd, UK, 1983. 2. Thompson, J. F., Thames, F. C. and Mastin, C. W., TOMCAT-A code for numerical generation of boundary-fitted curvilinear coordinate system on field containing any number ofarbitrary two-dimensional bodies, Journal of Computational Physics, 24, pp.274 (1977). 3. Tsay, T. K. and Hsu, F. S., Numerical grid generation of a irregular region, International Journal for Numerical Methods in Engineering, 40, pp.343(1997).
並列摘要
In this thesis, the process of a conformal grid generation by the boundary element method and the complex mapping technique is presented in details. To elaborate the grid generation method, the associated singularity problems of mathematical singularity and geometrical singularity, which are related to the BEM and the complex mapping, are studied comprehensively. It is found that the improvement in eliminating the numerical boundary layer leads to the ability of present grid generation method producing grids near the boundary in $10^{-9}$ scale when the boundary element method is applied. The problem of the grids' overlapping at a sharp corner in a conformal grid system is dealt with a complex mapping technique, which can manage such a geometrical singularity adequately. However, there are still some limitations on the application of the complex mapping. In this dissertation, discussions are made to illustrate the practical way in generating a conformal grid system that was once thought difficult to construct. In the end of this thesis, the boundary-fitted curvilinear grid system is applied to calculate a case of computational fluid dynamics in connection with storm surge. In the present study, the fundamental researches on Boundary Element Method (BEM) are performed at first. Through the studies on the mathematical singularity and the geometrical singularity, the resolution of the conformal grids near the boundary and the overlapping of grids at a sharp corner can be improved. In Chapter ef{chap2}, detailed comparisons of the contour method and the direct method to evaluate the accuracy of singular boundary integrals are made. In contrast to conventional numerical integration methods which suffer from the numerical boundary layer, the so-called boundary layer can be shown to vanish when the contour and the direct methods are applied. The singularity problems, including mathematical singularities and geometrical singularities, are discussed respectively in Chapter ef{chap2} and Chapter ef{chap3}. Furthermore, in Chapter ef{chap3}, the geometrical singularities are overcome by applying a complex mapping technique. Several benchmarks are proposed to offer detailed discussions. In Chapter ef{chap4}, based on the BEM and the complex mapping method, a powerful numerical grid generation method is presented. By using the correct evaluation technique in the integrals of the BEM, the new grid generation method can avoid the effect of mathematical singularity mentioned in Chapter ef{chap2}. This enables us to produce conformal grids near the boundary. Moreover, by using a complex mapping technique in Chapter ef{chap3}, the methods serve to avert the overlapping of the grids at a sharp corner, which occurred in the past studies. The new grid generation can produce a boundary-fitted orthogonal grids. In addition, it maintains conformal properties, either angles or length ratios, so as to map a domain of Laplace equation to another domain of still the Laplace equation. This makes the governing equation most concise after the conformal transformation. Nevertheless, some defects of the grid generation arising from the mathematical singularity and the geometrical singularity are found. The grid generation method appears to be destined to have multi-value and the branch-cut lines' problems, thus limiting its application in natural geometry. In this study, angle constraints are presented to serve as a criterion in checking the overlapping of grids. It helps in the trial and error operation during the step-by-step complex mappings. In addition, various numerical examples are employed to examine the validity of this grid generation system. Finally, in Chapter ef{chap5}, further applications of the orthogonal grid system are made on the storm surge around Taiwan; the trend of the results is the same as the observed data. Possible ways for improvement are discussed.