Differential Quadrature Method And Differential Quadrature Element Method--Theory And Application

The Differential Quadrature Method (DQM) and the Differential Quadrature Element Method (DQEM) are numerical methods developed recently and still under developing. The theory and applications of these two methods in Structural Mechanics are systematically studied in this dissertation. The principles and properties of these two methods are described, studied, and summarized in detail. It is shown that DQM is equivalent to a special kind of mixed collocation method. It is also shown for the first time that DQEM could be regarded as the combinations of the sub-domain method with the special mixed collocation method. The effects of the application of boundary conditions and the selection of grid points on the final results are investigated. The generalized formulations to apply the boundary constrains are derived for the nonlinear plate bending problems, and numerical results show that the DQM exhibits behaviors of semi-analytical methods. The interpolation functions with degree-of-freedom of first derivatives at boundary points are formulated, the properties of their base functions are also studies, the explicit formulae to compute the weighting coefficient matrices for various orders of derivatives are given, and the properties of the weighting coefficient matrices are investigated in details. Several Differential Quadrature elements are established for the first time, such as the circular arch element, conical shell element, shallow spherical shell element, and piezo-electrical shallow spherical shell element. The effects of the number of elements and the number of grid points on the accuracy of the numerical results are investigated. Based on the numerical results, it is shown that increasing the number of grid points in each element yields more accurate results than increasing the number of elements if the total number of degrees-of-freedom is the same. Thus, one should use as smaller numbers of element as possible when one establishes the computational model for analysis. The DQM and DQEM are thus used to obtain static, buckling, and free vibration solutions for various plate and shell structures, and results may be useful in engineering practice. Based on the results reported in this dissertation, one may conclude that the DQM and DQEM are the numerical methods exhibiting behaviors of semi-analytical methods, thus they will find wide application areas in structural engineering.