Localized Differential Quadrature (LDQ) Method
Although DQ is a numerical technique of high accuracy, it was realized from the very beginning that DQ is inefficient when the number of grid points is large (Civan and Sliepcevich, 1983). Later it was revealed that it is also sensitive to the grid point distribution. Quan and Chang (1989a,b) numerically compared the performance of different grid point distributions and found that the grid points given by the roots of the Chebyshev polynomials of the first kind are nearly optimum. Bert and Malik (1996) pointed out that the optimum distribution of grid points is also problem-dependent. Moradi and Taheri (1998) investigated the effect of various spacing schemes on the accuracy of DQ results for buckling behaviors of composites. Recently, a systematic error analysis was carried out by Shu et al. (2001) to assess the effect of the grid distribution. From the error analysis it is concluded that the optimal grid distribution may not be given by the roots of orthogonal polynomials. It is clearly shown from the previous works that the grid distribution exerts significant influence on the accuracy of DQ results and the selection of the grid distribution depends on the problems under consideration. To date, great efforts have been devoted to finding the optimum grid distributions for different problems analyzed by direct DQ method (Chen, 2001; Chen et al., 2000; Fung, 2001). Even today, the general rule for the grid distribution is still lacking.