Advanced differential quadrature methods:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2009
|
| Series: | Chapman & Hall/CRC applied mathematics and nonlinear science series
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 327-337) and index |
| Physical Description: | 339 S. graph. Darst. |
| ISBN: | 1420082485 9781420082487 |
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MARC
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Record in the Search Index
| _version_ | 1804140697819807744 |
|---|---|
| adam_text | Advanced
Differential
Quadrature
Methods
The original direct differential quadrature (DQ) method has been known to fail for
problems with strong nonlinearity and material discontinuity as well as for problems
involving singularity, irregularity, and multiple scales. But now researchers in applied
mathematics, computational mechanics, and engineering have developed a range of
innovative DQ-based methods to overcome these shortcomings. Advanced Differential
Quadrature Methods explores new DQ methods and uses these methods to solve
problems beyond the capabilities of the direct DQ method.
After a basic introduction to the direct DQ method, the book presents a number of
DQ methods, including complex DQ, triangular DQ. multi-scale DQ. variable order DQ.
multi-domain DQ, and localized DQ. It also provides a mathematical compendium that
summarizes Gauss elimination, the Runge-Kutta method, complex analysis, and more.
The final chapter contains three codes written in the FORTRAN language, enabling
readers to quickly acquire hands-on experience with DQ methods.
Focusing on leading-edge DQ methods, this book helps readers understand the
majority of journal papers on the subject. In addition to gaining insight into the dynamic
changes that have recently occurred in the field, readers will quickly master the use of
DQ methods to solve complex problems.
Features
•
Discusses the limitations of direct DQ methods and common strategies to remove
these limitations
•
Presents formulations of several new DQ methods
•
Applies novel DQ methods to solve problems that are beyond the capabilities of
the direct DQ method
•
Includes results that represent the latest important developments of DQ methods
•
Provides FORTRAN code for readers to quickly acquire hands-on experience with
DQ methods
Contents
Approximation and Differential
Quadrature
1
1.1 Approximation and best
approximation
............ 1
1.2
Interpolating bases
....................... 5
1.2.1
Polynomial bases
..................... 5
1.2.2
Fourier expansion basis
................. 10
1.3
Differential quadrature
..................... 12
1.4
Direct differential quadrature method
............. 17
1.5
Block marching in time with DQ discretization
........ 20
1.6
Implementation of boundary conditions
........... 31
1.7
Conclusions
............................ 40
Complex Differential Quadrature Method
43
2.1
Differential quadrature in the complex plane
......... 44
2.2
Complex DQ method for potential problems
......... 49
2.3
Complex DQ method for plane linear elastic problems
.... 58
2.3.1
Revisit of two-dimensional problems in the theory of
linear elasticity in an
isotropie
medium
........ 58
2.3.2
Complex DQ method for plane linear elasticity
.... 60
2.4
Conformai
mapping aided complex differential quadrature
. 71
2.4.1
Conformai
mapping aided by
Lagrange
interpolation
72
2.4.2
Conformai
mapping aided CDQ method
........ 80
2.5
Conclusions
............................ 87
Triangular Differential Quadrature Method
89
3.1
TDQ method in standard triangle
............... 89
3.2
TDQ method in curvilinear triangle
.............. 95
3.3
Geometric transformation
.................... 98
3.4
Governing equations of Reissner-Mindlin plates on Pasternak
foundation
............................ 100
3.4.1
Governing equations in physical coordinate system
. . 100
3.4.2
Governing equations in the standard triangle
..... 101
3.4.3
Boundary conditions
................... 102
3.4.4
Symmetry consideration
................. 102
3.5
Conclusions
............................ 106
Multiple
Scale
Differential
Quadrature Method
115
4.1
Multi-scale DQ method for potential problems
........ 116
4.2
Solutions of potential problems
................. 118
4.2.1
Gauss elimination
.................... 118
4.2.2
One-dimensional band storage
............. 120
4.2.3
Successive over-relaxation
(SOR)
method
....... 121
4.3
SOR-based multi-scale DQ method
.............. 122
4.4
Asymptotic multi-scale DQ method
.............. 128
4.4.1
Basic three-dimensional equations
........... 129
4.4.2
Nondimensionalization
.................. 132
4.4.3
Asymptotic expansion
.................. 133
4.4.4
Successive integration and CST
............. 134
4.4.5
Edge conditions
...................... 137
4.5
DQ solution to multi-scale poroelastic problems
....... 151
4.5.1
The multi-scale approach
................ 151
4.5.2
Governing equations of poroelasticity
.......... 154
4.5.3
DQ solution of poroelastic governing equations
.... 157
4.6
Conclusions
............................ 163
Variable Order DQ Method
169
5.1
Direct DQ discretization and dynamic numerical instability
. 170
5.2
Variable order approach
..................... 172
5.2.1
Boundary effect
..................... 172
5.2.2
Variable order DQ method
............... 175
5.2.3
Numerical examples
................... 179
5.2.4
Discussion
........................ 186
5.3
Improvement of temporal integration
............. 187
5.3.1
The first-order scheme
.................. 188
5.3.2
The second-order scheme
................ 189
5.3.3
The particular solution
................. 192
5.3.4
Numerical examples
................... 195
5.4
Conclusions
............................ 200
Multi-Domain Differential Quadrature Method
201
6.1
Linear plane elastic problems with material discontinuity
. . 201
6.2
A multi-domain approach for numerical treatment of material
discontinuity
........................... 205
6.2.1
Numerical examples
................... 209
6.3
Multi-domain DQ method for irregular domain
....... 214
6.3.1
Coordinate transformation for irregular domain
.... 217
6.3.2
Quadratic sub-domain
.................. 219
6.3.3
Cubic sub-domain
.................... 221
6.4
Multi-domain DQ formulation of plane elastic problems
. . 223
6.4.1
Multi-domain DQ for plane elastic heterogeneous solids
223
6.4.2
Governing equations and its DQ discretization
.... 224
6.4.3
Compatibility conditions
................. 228
6.4.4
Numerical examples
................... 229
6.5
Conclusions
........................... 238
7
Localized Differential Quadrature (LDQ) Method
241
7.1
DQ and its spatial discretization of the wave equation
. . . 242
7.2
Stability analysis
........................ 243
7.3
Coordinate-based localized DQ
................ 248
7.3.1
DQ localization in one dimension
............ 248
7.3.2
DQ localization in two dimensions
........... 249
7.3.3
Numerical examples
................... 250
7.4
Spline-based localized DQ method
.............. 257
7.4.1
Cardinal cubic spline interpolation
........... 258
7.4.2
Weighting coefficients for spline-based DQ
....... 261
7.4.3
Application of spline-based DQ method to initial-value
problems
......................... 261
7.4.4
Stability of spline-based DQ
............... 263
7.4.5
Examples
......................... 264
7.4.6
Discussions
........................ 266
7.5
Conclusions
........................... 267
8
Mathematical Compendium
273
8.1
Gauss elimination
........................ 273
8.2
Successive over-relaxation
(SOR)
method
........... 276
8.3
One-dimensional band storage
................. 279
8.4
Runge-Kutta method (constant time step)
.......... 279
8.5
Complex analysis
........................ 281
8.5.1
Complex variable
.................... 281
8.5.2
Holomorphic functions
.................. 283
8.5.3
Meromorphic functions
................. 283
8.6
QR algorithm
.......................... 284
8.6.1
QR algorithm
....................... 284
8.6.2
QR decomposition
.................... 285
8.6.3
Connection to a determinant or a product of eigenvalues
291
9
Codes
293
9.1
DQ for numerical evaluation of function cos(x)
........ 293
9.2
CDQ for harmonic problem
................... 295
9.3
Localized DQ method
...................... 318
References
327
Index
338
|
| any_adam_object | 1 |
| author | Zong, Zhi |
| author_facet | Zong, Zhi |
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| ctrlnum | (OCoLC)276274690 (DE-599)HBZHT015821533 |
| dewey-full | 518/.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518/.6 |
| dewey-search | 518/.6 |
| dewey-sort | 3518 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T22:04:06Z |
| institution | BVB |
| isbn | 1420082485 9781420082487 |
| language | English |
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| physical | 339 S. graph. Darst. |
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| spelling | Zong, Zhi Verfasser aut Advanced differential quadrature methods Zhi Zong ; Yingyan Zhang Boca Raton [u.a.] CRC Press 2009 339 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC applied mathematics and nonlinear science series Includes bibliographical references (p. 327-337) and index Differential equations Numerical solutions Numerical integration Numerische Integration (DE-588)4172168-8 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Numerische Integration (DE-588)4172168-8 s DE-604 Zhang, Yingyan Sonstige oth Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628903&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628903&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zong, Zhi Advanced differential quadrature methods Differential equations Numerical solutions Numerical integration Numerische Integration (DE-588)4172168-8 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4172168-8 (DE-588)4012249-9 |
| title | Advanced differential quadrature methods |
| title_auth | Advanced differential quadrature methods |
| title_exact_search | Advanced differential quadrature methods |
| title_full | Advanced differential quadrature methods Zhi Zong ; Yingyan Zhang |
| title_fullStr | Advanced differential quadrature methods Zhi Zong ; Yingyan Zhang |
| title_full_unstemmed | Advanced differential quadrature methods Zhi Zong ; Yingyan Zhang |
| title_short | Advanced differential quadrature methods |
| title_sort | advanced differential quadrature methods |
| topic | Differential equations Numerical solutions Numerical integration Numerische Integration (DE-588)4172168-8 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Differential equations Numerical solutions Numerical integration Numerische Integration Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628903&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628903&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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