Generalized Differential Quadrature Method (GDQM), introduced by Bellman

Generalized Differential Quadrature Method (GDQM),
introduced by Bellman et al 9, is Generalized based on the weighted sum of
function values as an approximation to the derivatives of that function.
Bellman stated that partial derivative of a function with respect to a space
variable could be approximated by a weighted linear combination of function
values evaluated at some intermediate points in the domain of that variable.
Compared to FEM or FDM, GDQM is relatively a new method used for solving a
system of differential equations. In addition to the Less complex algorithm, in
comparison to FEM, GDQM also offers increased efficiency of the solution by
demanding less number of grid points (hence, equations) to mode1 the problem.
Therefore, owing to the improved performances of GDQM, this method has gained
increasing popularity in solving a variety of engineering problems. Bert et al
10 used DQM for static and free vibration analysis of anisotropic plates,
while Laura and Gutierrez 18 used the method in vibration analysis of
rectangular plates with non-uniform boundary conditions. Sherboume and Padney 12
used DQM to analyse the buckling of composite beams and plates. They used
different number of grid spacing in their analysis. The same problem was
addressed by Wang 13, who also used different grid spacing. He found that
employing uniform grid spacing could result in an inaccurate solution;
therefore, caution should be exercised when using such spacing. Liew et al 14
used the method for the analysis of thick symmetric cross-ply laminates with
first order shear deformation while Kang et al 15 used it to address the vibration
and buckling analysis of circular arches. Bert et al 16 analysed the large
deflection problem of a thin orthotropic rectangular plate in bending. The
three nonlinear differential equations of equilibrium of the plate were
transformed into differential quadrature form and solved numerically using the
method of Newton-Raphson. Lin et al 17 used the same procedures to solve the
problem of large deflection of isotropic plates under thermal loading. In their
analysis they used the generalized differential quadrature of Shu and Richards
18. Bert also examined the equally spaced grids as well as unequally spaced
in several structural mechanic’s applications. The domain is divided into N
discrete points and cij are the weighting coefficients of the derivative.
i values are important factors control the quality of the approximation,
resulting from the application of GDQM.