1st Edition

Advanced Differential Quadrature Methods

By Zhi Zong, Yingyan Zhang Copyright 2009
    362 Pages 112 B/W Illustrations
    by Chapman & Hall

    362 Pages 112 B/W Illustrations
    by Chapman & Hall

    Modern Tools to Perform Numerical Differentiation
    The original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.

    After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge–Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to quickly acquire hands-on experience with DQ methods.

    Focusing on leading-edge DQ methods, this book helps readers understand the majority of journal papers on the subject. In addition to gaining insight into the dynamic changes that have recently occurred in the field, readers will quickly master the use of DQ methods to solve complex problems.

    Approximation and Differential Quadrature

    Approximation and best approximation

    Interpolating bases

    Differential quadrature (DQ)

    Direct DQ method

    Block marching in time with DQ discretization

    Implementation of boundary conditions


    Complex Differential Quadrature Method

    DQ in the complex plane

    Complex DQ method for potential problems

    Complex DQ method for plane linear elastic problems

    Conformal mapping-aided complex DQ


    Triangular Differential Quadrature Method

    Triangular DQ method in standard triangle

    Triangular DQ method in curvilinear triangle

    Geometric transformation

    Governing equations of Reissner–Mindlin plates on Pasternak foundation


    Multiple Scale Differential Quadrature Method

    Multi-scale DQ method for potential problems

    Solutions of potential problems

    Successive over-relaxation (SOR)-based multi-scale DQ method

    Asymptotic multi-scale DQ method

    DQ solution to multi-scale poroelastic problems


    Variable Order Differential Quadrature Method

    Direct DQ discretization and dynamic numerical instability

    Variable order approach

    Improvement of temporal integration


    Multi-Domain Differential Quadrature Method

    Linear plane elastic problems with material discontinuity

    A multi-domain approach for numerical treatment of material discontinuity

    Multi-domain DQ method for irregular domain

    Multi-domain DQ formulation of plane elastic problems


    Localized Differential Quadrature Method

    DQ and its spatial discretization of the wave equation

    Stability analysis

    Coordinate-based localized DQ

    Spline-based localized DQ method


    Mathematical Compendium

    Gauss elimination

    SOR method

    One-dimensional band storage

    Runge–Kutta method (constant time step)

    Complex analysis

    QR algorithm


    DQ for numerical evaluation of function cos(x)

    Complex DQ for harmonic problem

    Localized DQ method




    Yingyan Zhang, Zhi Zong