# Differential Quadrature and Its Application in Engineering

Springer Science & Business Media, Jan 14, 2000 - Mathematics - 340 pages
In the past few years, the differential quadrature method has been applied extensively in engineering. This book, aimed primarily at practising engineers, scientists and graduate students, gives a systematic description of the mathematical fundamentals of differential quadrature and its detailed implementation in solving Helmholtz problems and problems of flow, structure and vibration. Differential quadrature provides a global approach to numerical discretization, which approximates the derivatives by a linear weighted sum of all the functional values in the whole domain. Following the analysis of function approximation and the analysis of a linear vector space, it is shown in the book that the weighting coefficients of the polynomial-based, Fourier expansion-based, and exponential-based differential quadrature methods can be computed explicitly. It is also demonstrated that the polynomial-based differential quadrature method is equivalent to the highest-order finite difference scheme. Furthermore, the relationship between differential quadrature and conventional spectral collocation is analysed.
The book contains material on:
- Linear Vector Space Analysis and the Approximation of a Function;
- Polynomial-, Fourier Expansion- and Exponential-based Differential Quadrature;
- Differential Quadrature Weighting Coefficient Matrices;
- Solution of Differential Quadrature-resultant Equations;
- The Solution of Incompressible Navier-Stokes and Helmholtz Equations;
- Structural and Vibrational Analysis Applications;
- Generalized Integral Quadrature and its Application in the Solution of Boundary Layer Equations.
Three FORTRAN programs for simulation of driven cavity flow, vibration analysis of plate and Helmholtz eigenvalue problems respectively, are appended. These sample programs should give the reader a better understanding of differential quadrature and can easily be modified to solve the readers own engineering problems.

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### Contents

 Mathematical Fundamentals of Differential Quadrature Method Linear Vector Space Analysis and Function Approximation 1 12 Derivative Approximation by Differential Quadrature DQ Method 3 121 Integral Quadrature 4 122 Differential Quadrature 5 1 3 Analysis of A Linear Vector Space 6 132 Properties of A Linear Vector Space 8 14 Solution of Partial Differential Equations PDEs and Function Approximation 11 142 High Order Polynomial Approximation 13
 6321 1mplementation of Boundary Condition for Vorticity 162 6323 Implementation of Boundary Condition for Temperature 167 633 Solution Procedures 168 634 Some Numerical Examples 170 6342 The Natural Convection in A Concentric Annulus 172 64 Solution of Incompressible NavierStokes Equations in Primitive Variable Form 175 642 Pressure Correction Method 176 643 Two Approaches to Specify Boundary Condition for p and to Enforce Continuity Condition on the Boundary 178

 143 Fourier Series Expansion 18 1432 Even Function 21 1433 Odd Function 23 Polynomialbased Differential Quadrature PDQ 25 22 Computation of Weighting Coefficients for the First Order Derivative 26 222 Quan and Changs Approach 28 223 Shus General Approach 29 23 Computation of Weighting Coefficients for the Second and Higher Order Derivatives 32 232 Shus Recurrence Formulation for Higher Order Derivatives 34 233 Matrix Multiplication Approach 36 24 Error Analysis 38 242 The Derivative Approximation 40 25 Relationship Between PDQ and Other Approaches 44 2512 Relationship Between PDQ and Highest Order Finite Difference Scheme 48 252 Relationship Between PDQ and Chebyshev Collocation Method 52 26 Extension to the Multidimensional Case 55 262 Differential Cubature Method 60 27 Specific Results for Typical Grid Point Distributions 62 272 ChebyshevGaussLobatto Grid 63 273 Coordinates of Grid Points Chosen as the Roots of Chebyshev Polynomial 64 28 Generation of Low Order Finite Difference Schemes by PDQ 65 Fourier Expansionbased Differential Quadrature FDQ 69 32 Cosine Expansionbased Differential Quadrature CDQ for Even Functions 70 33 Sine Expansionbased Differential Quadrature SDQ for Odd Functions 81 Quadrature FDQ for Any General Function 86 35 Some Properties of Fourier Expansionbased Differential Quadrature 91 Some Properties of DQ Weighting Coefficient Matrices 95 42 Determinant and Rank of DQ Weighting Coefficient Matrices 96 422 Determinant and Rank of DQ Weighting Coefficient Matrices 98 43 Structures and Properties of DQ Weighting Coefficient Matrices 100 431 Definition of Centrosymmetric and Skew Centrosymmetric Matrices 101 432 Properties of Centrosymmetric and Skew Centrosymmetric Matrices 102 4322 Properties of Skew Centrosymmetric Matrices 105 433 Structures of DQ Weighting Coefficient Matrices 107 4332 Structures of Higher Order DQ Weighting Coefficient Matrices 109 44 Effect of Grid Point Distribution on Eigenvalues of DQ Discretization Matrices 110 441 Stability of Ordinary Differential Equations 111 442 Eigenvalues of Some Specific DQ Discretization Matrices 112 4422 The Diffusion Operator 117 4423 The ConvectionDiffusion Operator We consider the convectiondiffusion operator 119 45 Effect of Grid Point Distribution on Magnitude of DQ Weighting Coefficients 120 Solution Techniques for DQ Resultant Equations 123 52 Solution Techniques for DQ Ordinary Differential Equations 124 53 Solution Techniques for DQ Algebraic Equations 128 531 Direct Methods 130 532 Iterative Methods 134 5322 Iterative Methods for Lyapunov System 137 54 Implementation of Boundary Conditions 140 55 Sample Applications of DQ Method 143 552 Twodimensional Poisson Equation 145 553 Hetmholtz Eigenvalue Problem 148 Application of Differential Quadrature Method to Solve Incompressible NavierStokes Equations 153 62 Governing Equations 154 622 Nondimensional Form 157 623 VorticityStream Function Formulation 159 63 Solution of VorticityStream Function Formulation 160 632 Implementation of Boundary Conditions 161