Advanced Differential Quadrature Methods  book cover
1st Edition

Advanced Differential Quadrature Methods

ISBN 9781138117907
Published June 14, 2017 by Chapman and Hall/CRC
362 Pages - 112 B/W Illustrations

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Book Description

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.

Table of Contents

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



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