Differential Quadrature and Its Application in Engineering

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.- 4.3.2.1 Properties of Centrosymmetric Matrices.- 4.3.2.2 Properties of Skew Centrosymmetric Matrices.- 4.3.3 Structures of DQ Weighting Coefficient Matrices.- 4.3.3.1 Structure of First Order DQ Weighting Coefficient Matrix.- 4.3.3.2 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.- 4.4.2.1 The Convection Operator.- 4.4.2.2 The Diffusion Operator.- 4.4.2.3 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.- 5.3.2.1 Iterative Methods for Conventional System.- 5.3.2.2 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 Equation.- 5.5.3 Helmholtz Eigenvalue Problem.- 6 Application of Differential Quadrature Method to Solve Incompressible Navier-Stokes Equations.- 6.1 Introduction.- 6.2 Governing Equations.- 6.2.1 Dimensional Form.- 6.2.2 Non-dimensional Form.- 6.2.3 Vorticity-Stream Function Formulation.- 6.3 Solution of Vorticity-Stream Function Formulation.- 6.3.1 Discretization of Governing Equations.- 6.3.2 Implementation of Boundary Conditions.- 6.3.2.1 Implementation of Boundary Condition for Vorticity.- 6.3.2.2 Implementation of Boundary Condition for Stream function.- 6.3.2.3 Implementation of Boundary Condition for Temperature.- 6.3.3 Solution Procedures.- 6.3.4 Some Numerical Examples.- 6.3.4.1 The Flow Past A Circular Cylinder.- 6.3.4.2 The Natural Convection in A Concentric Annulus.- 6.4 Solution of Incompressible Navier-Stokes Equations in Primitive Variable Form.- 6.4.1 Introduction.- 6.4.2 Pressure Correction Method.- 6.4.3 Two Approaches to Specify Boundary Condition for p' and to Enforce Continuity Condition on the Boundary.- 6.4.3.1 Approach 1.- 6.4.3.2 Approach 2.- 6.4.4 Computational Sequence.- 6.4.5 Sample Application and Comments on the Two Approaches.- 6.4.5.1 Importance of Enforcing Continuity Condition on the Boundary.- 6.4.5.2 Comments on Performance of Two Approaches.- 7 Application of Differential Quadrature Method to Structural and Vibration Analysis.- 7.1 Introduction.- 7.2 Differential Quadrature Analysis of Beams.- 7.2.1 Governing Equations and Boundary Conditions.- 7.2.2 Numerical Discretization.- 7.2.3 Implementation of Boundary Conditions.- 7.2.3.1 The ?-technique.- 7.2.3.2 Modification of Weighting Coefficient Matrices.- 7.2.3.3 Direct Substitution of Boundary Conditions into Discrete Governing Equations.- 7.2.4 Numerical Example: Free Vibration Analysis of A Uniform Beam.- 7.3 Differential Quadrature Analysis of Thin Plates.- 7.3.1 Governing Equations and Boundary Conditions.- 7.3.2 Numerical Discretization.- 7.3.3 Implementation of Boundary Conditions.- 7.3.3.1 The 8-technique.- 7.3.3.2 Modification of Weighting Coefficient Matrices.- 7.3.3.3 Direct Substitution of Boundary Conditions into Discrete Governing Equations.- 7.3.3.4 General Approach.- 7.3.4 Numerical Example: Free Vibration Analysis of Square Plates.- 7.4 Differential Quadrature Analysis of Shells.- 7.4.1 Governing Equations and Boundary Conditions.- 7.4.2 Numerical Discretization.- 7.4.3 Implementation of Boundary Conditions.- 7.4.4 Numerical Example: Free Vibration Analysis of A Composite Laminated Conical Shell.- 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.- 9.2.1 Topology of Interface.- 9.2.1.1 Patched Interface.- 9.2.1.2 Overlapped Interface.- 9.2.2 Multi-domain DQ Application in Fluid Mechanics.- 9.2.3 Multi-domain DQ Application in Solid Mechanics.- 9.2.4 Multi-domain DQ Application in Waveguide Analysis.- 9.3 DQ Application in Curvilinear Coordinate System.- 9.3.1 Coordinate Transformation.- 9.3.2 DQ Simulation of Incompressible Flows in Irregular Domains.- 9.3.3 DQ Vibration Analysis of Irregular Plates.- 9.3.3.1 Partial Transformation.- 9.3.3.2 Complete Transformation.- 9.3.3.3 Implementation of Boundary Conditions.- 9.3.3.4 Sample Application.- 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).- 10.2.1 One-dimensional Generalized Integral Quadrature.- 10.2.2 Error Analysis.- 10.2.3 Extension to Multi-dimensional Cases.- 10.2.4 Sample Application of GIQ Method.- 10.3 DQ-GIQ Algorithm to Solve Boundary Layer Equations.- 10.3.1 Stream Function as Dependent Variable.- 10.3.2 Primitive Variables as Dependent Variables.- 10.4 Steady Boundary Layer Solutions.- 10.4.1 One-dimensional Case.- 10.4.2 Two-dimensional Case.- 10.4.3 Three-dimensional Case.- 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.