1 Mathematical Fundamentals of Differential Quadrature Method: Linear Vector Space Analysis and Function Approximation.- 1.1 Introduction.- 1.2 Derivative Approximation by Differential Quadrature (DQ) Method.- 1.2.1 Integral Quadrature.- 1.2.2 Differential Quadrature.- 1.3 Analysis of A Linear Vector Space.- 1.3.1 Definition of A Linear Vector Space.- 1.3.2 Properties of A Linear Vector Space.- 1.4 Solution of Partial Differential Equations (PDEs) and Function Approximation.- 1.4.1 Two Basic Types of Solution for PDEs.- 1.4.2 High Order Polynomial Approximation.- 1.4.3 Fourier Series Expansion.- 22.214.171.124 General function.- 126.96.36.199 Even function.- 188.8.131.52 Odd function.- 2 Polynomial-based Differential Quadrature (PDQ).- 2.1 Introduction.- 2.2 Computation of Weighting Coefficients for the First Order Derivative.- 2.2.1 Bellman's Approaches.- 2.2.2 Quan and Chang's Approach.- 2.2.3 Shu's General Approach.- 2.3 Computation of Weighting Coefficients for the Second and Higher Order Derivatives.- 2.3.1 Weighting Coefficients of the Second Order Derivative.- 2.3.2 Shu's Recurrence Formulation for Higher Order Derivatives.- 2.3.3 Matrix Multiplication Approach.- 2.4 Error Analysis.- 2.4.1 The Function Approximation.- 2.4.2 The Derivative Approximation.- 2.5 Relationship Between PDQ and Other Approaches.- 2.5.1 Relationship Between PDQ and Finite Difference Scheme.- 184.108.40.206 Generation of Finite Difference Scheme.- 220.127.116.11 Relationship Between PDQ and Highest Order Finite Difference Scheme.- 2.5.2 Relationship Between PDQ and Chebyshev Collocation Method.- 2.6 Extension to the Multi-dimensional Case.- 2.6.1 Direct Extension for Regular Domain.- 2.6.2 Differential Cubature Method.- 2.7 Specific Results for Typical Grid Point Distributions.- 2.7.1 Uniform Grid.- 2.7.2 Chebyshev-Gauss-Lobatto Grid.- 2.7.3 Coordinates of Grid Points Chosen as the Roots of Chebyshev Polynomial.- 2.8 Generation of Low Order Finite Difference Schemes by PDQ.- 3 Fourier Expansion-based Differential Quadrature (FDQ).- 3.1 Introduction.- 3.2 Cosine Expansion-based Differential Quadrature (CDQ) for Even Functions.- 3.3 Sine Expansion-based Differential Quadrature (SDQ) for Odd Functions.- 3.4 Fourier Expansion-based Differential Quadrature (FDQ) for Any General function.- 3.5 Some Properties of Fourier Expansion-based Differential Quadrature.- 4 Some Properties of DQ Weighting Coefficient Matrices.- 4.1 Introduction.- 4.2 Determinant and Rank of DQ Weighting Coefficient Matrices.- 4.2.1 Definition and Properties of Determinant and Rank.- 4.2.2 Determinant and Rank of DQ Weighting Coefficient Matrices.- 4.3 Structures and Properties of DQ Weighting Coefficient Matrices.- 4.3.1 Definition of Centrosymmetric and Skew Centrosymmetric Matrices.- 4.3.2 Properties of Centrosymmetric and Skew Centrosymmetric Matrices.- 18.104.22.168 Properties of Centrosymmetric Matrices.- 22.214.171.124 Properties of Skew Centrosymmetric Matrices.- 4.3.3 Structures of DQ Weighting Coefficient Matrices.- 126.96.36.199 Structure of First Order DQ Weighting Coefficient Matrix.- 188.8.131.52 Structures of Higher Order DQ Weighting Coefficient Matrices.- 4.4 Effect of Grid Point Distribution on Eigenvalues of DQ Discretization Matrices.- 4.4.1 Stability of Ordinary Differential Equations.- 4.4.2 Eigenvalues of Some Specific DQ Discretization Matrices.- 184.108.40.206 The Convection Operator.- 220.127.116.11 The Diffusion Operator.- 18.104.22.168 The Convection-Diffusion Operator.- 4.5 Effect of Grid Point Distribution on Magnitude of DQ Weighting Coefficients.- 5 Solution Techniques for DQ Resultant Equations.- 5.1 Introduction.- 5.2 Solution Techniques for DQ Ordinary Differential Equations.- 5.3 Solution Techniques for DQ Algebraic Equations.- 5.3.1 Direct Methods.- 5.3.2 Iterative Methods.- 22.214.171.124 Iterative Methods for Conventional System.- 126.96.36.199 Iterative Methods for Lyapunov System.- 5.4 Implementation of Boundary Conditions.- 5.5 Sample Applications of DQ Method.- 5.5.1 Burgers Equation.- 5.5.2 Two-dimensional Poisson Equati
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.
No other books on this subject in this area
Fortran programs attached allow easy practical application of the methods in the book