- 學位論文
基本解法於二維非平滑邊界之勢能分析
Analysis of Two-dimensional Potential Flows with Non-Smooth Boundaries by the Method of Fundamental Solutions
摘要
本論文的主旨在於探討不同形式的基本解於基本解法的可行性並且應用於勢能問題上。以往,透過格林函數推導所求得的拉普拉斯方程的基本解,僅考慮極座標系統中的距離函數而忽略了角度函數。因而,角度型態的基本解往往被忽略且很少被提及。本論文將基本解推廣至複數平面而得到存在於實部的徑向基底函數與虛部的角度基底函數。因應角度基底函數的特性,本論文提供一套佈點方式來相互搭配進行求解。應用角度基底函數與徑向基底函數所探討的一系列勢能問題裡,兩項基底函數從平滑的圓形幾何邊界相互比較至帶有尖角與薄型邊界計算域的過程中,透過不同形式的邊界條件測試下可以得知,角度基底函數對於處理薄型邊界與尖角計算域的問題有其優勢存在。此外,對於空氣動力學中的機翼問題亦進行了一些初步的勢能分析。本論文中的數值計算結果透過解析解的驗證均相當吻合, 以證明該作法的可行性與數值模式的正確性。
並列摘要
The present thesis contributes to discuss the feasibility and properties of different forms of the basis functions of the method of fundamental solutions in potential problems. Formerly, the fundamental solutions of the Laplace equation derived from the Green’s function in the polar coordinate system consider only the function of radius. In order to preserve the completeness of the solution, the complex variable theory is employed to reconsider the fundamental solution. The real part of the complex analytic function is a function of radius and the imaginary part is the function of argument. We assume that the imaginary part to be the angular basis of the Laplace equation. For the properties of the angular basis function, we promote a nodes distribution way corresponding to the angular basis function. In a series of the approximation of the potential problems, the radial basis and the angular basis functions are compared in the same computational domains from the circular to cusp domains and thin-boundary geometries with different types of boundary conditions. From these numerical experiments, the angular basis function is found to be favorable of simulating the domains with acute and narrow regions. Furthermore, the basic aerodynamic problems of airfoils are also discussed in the present study through the potential problems. Numerical results in this thesis are compared favorably with the exact solutions. The results demonstrate the rationality and the feasibility of our assumptions and numerical model.
參考文獻
延伸閱讀
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