Differential Quadrature and Its Application in Engineering

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Springer Science & Business Media, Jan 14, 2000 - Mathematics - 340 pages
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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
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
Index
336
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