The differential quadrature method is a numerical solution technique for initial and/or boundary problems. It was developed by the late Richard Bellman and his associates in the early 70s and, since then, the technique has been successfully employed in a variety of problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review of the differential quadrature method, which should be of general interest to the computational mechanics community.

Differential Quadrature Method in Computational Mechanics: A Review

Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey

Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments

Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates

Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads and under various boundary conditions is presented. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to temperature rise. Material properties are assumed to be temperature-dependent and graded in the thickness direction according to a power law distribution in terms of the volume fractions of the constituents. The temperature field considered is assumed to be uniform in the XY plane and through the plate thickness. The plate may be clamped or simply supported on two opposite edges with the remaining others being either free, simply supported or clamped. A semi-numerical approach, which makes use of multi-parameter perturbation technique, one-dimensional differential quadrature approximation and Galerkin method, is employed to calculate the nonlinear bending of the plates. Numerical results for silicon nitride/stainless steel rectangular plates are given in dimensionless graphical forms. The roles played by the constituent volume fraction index, temperature rise, character of out-of-plane boundary conditions and in-plane constraints, width-to-thickness ratio as well as plate aspect ratio are studied.

Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions

SUMMARY This paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate, efficient and simple numerical technique for structural analysis. Firstly, drawbacks existing in the method of differential quadrature (DQ) are evaluated and discussed. Then, an improved and simpler generalized differential quadrature method (GDQ) is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients. Subsequently, the generalized differential quadrature is systematically employed to solve problems in structural analysis. Numerical examples have shown the superb accuracy, efficiency, convenience and the great potential of this method. Numerical approximation methods for solving partial differential equations have been widely used in various engineering fields. Classical techniques such as finite element and finite difference methods are well developed and well known. These methods can provide very accurate results by using a large number of grid points. However, in a large number of cases, reasonably approximate solutions are desired at only a few specific points in the physical domain. In order to get results even at or around a point of interest with acceptable accuracy, conventional finite element and finite difference methods still require the use of a large number of grid points. Consequently, the requirement for computer capacity is often unnecessarily large in such cases. In seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy, the method of differential quadrature (DQ) was introduced by Bellman et a!.'. The method of DQ is a global approximate method. This method is based on the ideas that the derivative of a function with respect to a co-ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher-order polynomial in the overall domain. The DQ method differs from the finite element method (FEM) in two aspects. Firstly, the FEM uses lower-order polynomials to approximate a function on a local element level, while the DQ method approximates a function on the global area using higher-order polynomials. Secondly, the DQ method directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and the derivatives can then be derived from the approximate function. In this aspect, the DQ method is more similar to the finite difference method (FDM). However, the FDM is also a local approximation method based on lower-order polynomial approximation. In fact, it can be shown that the FDM is just a special case of the DQ method where it is applied locally on the range [.xi- xi + l]. Owing to the higher-order

/pdf/application-of-generalized-differential-quadrature-method-to-3m9k6sqxao.pdf

Application of generalized differential quadrature method to structural problems

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Application of generalized differential quadrature method to structural problems