## Differential Quadrature and Its Application in EngineeringIn 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 |

6432 Approach 2 | 180 |

644 Computational Sequence | 181 |

645 Sample Application and Comments on the Two Approaches | 182 |

6452 Comments on Performance of Two Approaches | 184 |

Application of Differential Quadrature Method to Structural and Vibration Analysis | 186 |

72 Differential Quadrature Analysis of Beams | 188 |

722 Numerical Discretization | 189 |

723 Implementation of Boundary Conditions | 190 |

7232 Modification of Weighting Coefficient Matrices | 191 |

Governing Equations | 194 |

Free Vibration Analysis of A Uniform Beam | 196 |

73 Differential Quadrature Analysis of Thin Plates | 197 |

732 Numerical Discretization | 199 |

733 Implementation of Boundary Conditions | 200 |

7333 Direct Substitution of Boundary Conditions into Discrete Governing Equations | 202 |

7334 General Approach | 205 |

Free Vibration Analysis of Square Plates | 207 |

74 Differential Quadrature Analysis of Shells | 209 |

742 Numerical Discretization | 218 |

743 Implementation of Boundary Conditions | 219 |

Free Vibration Analysis of A Composite Laminated Conical Shell | 222 |

Miscellaneous Applications of Differential Quadrature Method | 224 |

83 Application to Chemical Reactor | 228 |

84 Application to Lubrication Problems | 232 |

85 Application to Waveguide Analysis | 235 |

86 Solution of the Helmholtz Equation | 239 |

87 Effect of Mesh Point Distribution on Accuracy of DQ Results | 242 |

Application of Differential Quadrature to Complex Problems | 245 |

921 Topology of Interface | 246 |

9212 Overlapped Interface | 248 |

922 Multidomain DQ Application in Fluid Mechanics | 249 |

923 Multidomain DQ Application in Solid Mechanics | 251 |

924 Multidomain DQ Application in Waveguide Analysis | 252 |

93 DQ Application in Curvilinear Coordinate System | 254 |

932 DO Simulation of Incompressible Flows in Irregular Domains | 256 |

933 DQ Vibration Analysis of Irregular Plates | 260 |

9332 Complete Transformation | 261 |

9333 Implementation of Boundary Conditions | 262 |

9334 Sample Application | 264 |

94 Differential Cubature Method for Complex Problems | 266 |

Generalized Integral Quadrature GIQ and Its Application to Solve Boundary Layer Equations | 267 |

102 Generalized Integral Quadrature GIQ | 268 |

1022 Error Analysis | 271 |

1023 Extension to Multidimensional Cases | 272 |

1024 Sample Application of GIQ Method | 273 |

103 DQGIQ Algorithm to Solve Boundary Layer Equations | 275 |

1032 Primitive Variables as Dependent Variables | 277 |

104 Steady Boundary Layer Solutions | 279 |

1042 Twodimensional Case | 280 |

1043 Threedimensional Case | 281 |

105 Unsteady Boundary Layer Solutions | 285 |

A Fortran Program for Simulation of Natural Convection in A Square Cavity | 288 |

A Fortran Program for Vibration Analysis of Rectangular Plates | 305 |

A Fortran Program for LShaped Waveguide Analysis by Multidomain DQ Method | 315 |

References | 324 |

336 | |

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### Common terms and phrases

accuracy applied approach base polynomials base vectors Bellman Bert CW boundary conditions boundary layer boundary points computational domain compute the weighting continuity equation convergence curvilinear coordinate derivative condition differential quadrature method DIMENSION direction Dirichlet condition DQ discretization DQ method DQ results DQ weighting coefficient DQ-GIQ expansion-based differential quadrature finite difference scheme Fourier series Fourier series expansion functional values given by Equation grid point distribution Helmholtz equation interior points iterative methods linear vector space LU decomposition Malik matrix mesh point distribution multi-domain DQ Navier-Stokes equations Neumann condition number of grid numerical results obtained one-dimensional order finite difference ordinary differential equations PDQ and FDQ polynomial approximation pressure correction problem satisfied second order derivative set of base skew centrosymmetric solve stream function Striz subdomains SUBROUTINE Substituting Equation Taylor series technique test functions transformation two-dimensional velocity vorticity Wang waveguide wavenumbers weighting coefficient matrices zero

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Page iv - Engineering - National University of Singapore 10 Kent Ridge Crescent...

Page 329 - An Experimental and Theoretical Study of Natural Convection in the Annulus between Horizontal Concentric Cylinders, J.

Page 329 - Application of Differential Quadrature to Static Analysis of Structural Components,