Differential Quadrature and Its Application in Engineering

Aimed at practicing engineers, scientists and graduate students, this volume provides an overview of differential quadrature and its implementation in solving problems of flow, structure and vibration. Sample programs provide readers with a better understanding of the topic, and can easily be modified to solve other engineering problems.

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

Inhalt
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.- 1.3 Analysis of A Linear Vector Space.- 1.4 Solution of Partial Differential Equations (PDEs) and Function Approximation.- 2 Polynomial-based Differential Quadrature (PDQ).- 2.1 Introduction.- 2.2 Computation of Weighting Coefficients for the First Order Derivative.- 2.3 Computation of Weighting Coefficients for the Second and Higher Order Derivatives.- 2.4 Error Analysis.- 2.5 Relationship Between PDQ and Other Approaches.- 2.6 Extension to the Multi-dimensional Case.- 2.7 Specific Results for Typical Grid Point Distributions.- 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.- 4.3 Structures and Properties of DQ Weighting Coefficient Matrices.- 4.4 Effect of Grid Point Distribution on Eigenvalues of DQ Discretization Matrices.- 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.- 5.4 Implementation of Boundary Conditions.- 5.5 Sample Applications of DQ Method.- 6 Application of Differential Quadrature Method to Solve Incompressible Navier-Stokes Equations.- 6.1 Introduction.- 6.2 Governing Equations.- 6.3 Solution of Vorticity-Stream Function Formulation.- 6.4 Solution of Incompressible Navier-Stokes Equations in Primitive Variable Form.- 7 Application of Differential Quadrature Method to Structural and Vibration Analysis.- 7.1 Introduction.- 7.2 Differential Quadrature Analysis of Beams.- 7.3 Differential Quadrature Analysis of Thin Plates.- 7.4 Differential Quadrature Analysis of Shells.- 8 Miscellaneous Applications of Differential Quadrature Method.- 8.1 Introduction.- 8.2 Application to Heat Transfer.- 8.3 Application to Chemical Reactor.- 8.4 Application to Lubrication Problems.- 8.5 Application to Waveguide Analysis.- 8.6 Solution of the Helmholtz Equation.- 8.7 Effect of Mesh Point Distribution on Accuracy of DQ Results.- 9 Application of Differential Quadrature to Complex Problems.- 9.1 Introduction.- 9.2 Multi-domain DQ Method.- 9.3 DQ Application in Curvilinear Coordinate System.- 9.4 Differential Cubature Method for Complex Problems.- 10 Generalized Integral Quadrature (GIQ) And Its Application to Solve Boundary Layer Equations.- 10.1 Introduction.- 10.2 Generalized Integral Quadrature (GIQ).- 10.3 DQ-GIQ Algorithm to Solve Boundary Layer Equations.- 10.4 Steady Boundary Layer Solutions.- 10.5 Unsteady Boundary Layer Solutions.- Appendices.- A. A Fortran Program for Simulation of Natural Convection in A Square Cavity.- B. A Fortran Program for Vibration Analysis of Rectangular Plates.- References.

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