A Modification of Differential Quadrature Method and Solving the Problems of Euler-Bernoulli Beam
Date Issued
2011
Date
2011
Author(s)
Shen, Ying-Hsiu
Abstract
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.
Subjects
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
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