- 學位論文
微分積分法之改良並求解Euler-Bernoulli的問題
A Modification of Differential Quadrature Method and Solving the Problems of Euler-Bernoulli Beam
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摘要
微分積分法 (DQM) 應用於數值計算離散化已行之有年，其中權重(weighting coefficients)為微分積分法中最重要的參數。運用多項式測試函數( test functions) 為其中一種方法，其原理在於用多項式來近似所求得的解；當把整個區域劃分成n個格點，我們使用(n-1)個階層的多項式作為測試函數，其存在之Vandermonde矩陣為一著名的高度病態矩陣；而其矩陣的反矩陣求解一直為數值方法中困難的課題。然而區域化微分積分法解決其病態矩陣的情況，並成功地應用於二維問題。 在此篇論文中，我們對於微分積分法提出一個新的看法(NSDQ)，此方法為全域化微分積分法。當把整個區域劃分成n個格點，我們僅使用(m-1)個階層的多項式作為測試函數，而m的數值非常小於n。我們發現用(m-1)個階層的多項式已足夠用來表示所求得的解，並改善了病態矩陣的問題。文中，我們結合擬時間積分法，成功地求解二維橢圓形偏微分方程。另外，我們透過SBCGE方法的觀念，將此法運用於Euler-Bernoulli梁的變形問題，其非常小的誤差值亦說明了此新方法的精確度。
關鍵字
微分積分法的新看法(NSDQ) ； Euler-Bernoulli 梁 ； 擬時間積分法(FTIM) ； 橢圓形偏微分方程 ； SBCGE ； Dirichlet邊界條件 ； Neumann邊界條件並列摘要
When the localized differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by solving the inverse of ill-posed matrices. Before, when one uses (n-1)th order polynomial test functions to determine the weighting coefficients with n number of grid points, the resultant n×n Vandermonde matrix is highly ill-posed and its inverse is hard to solve. Now we introduce a new insight into the DQ method (NSDQ) that using (m-1)th order polynomial test functions by n number of grid points and the size of Vandermonde matrix is m×n, of which m is much less than n. We find that the (m-1)th order polynomial test functions are accurate enough to express the solutions, and the novel method significantly improves the ill-condition of algebraic equations. Such the NSDQ method as combined with the FTIM (Fictitious Time Integration Method) can solve 2-D elliptic type PDEs. By implementing the SBCGE method in the DQ, it is also successfully applied in free and forced vibrations problems of the Euler-Bernoulli beam with the CGM and the RK4 method. There are some examples tested in our thesis and the numerical errors are found to be very small.
並列關鍵字
A new insight into differential quadrature (NSDQ) ； Euler-Bernoulli beam ； Fictitious time integration method (FTIM) ； Elliptic partial differential equations ； A direct substitution of boundary condition into discrete governing equations (SBCGE) ； Dirichlet boundary conditions ； Neumann boundary conditions參考文獻
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