商品簡介
Modern Tools to Perform Numerical DifferentiationThe 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.
目次
Approximation and Differential Quadrature
Approximation and best approximation
Interpolating bases
Differential quadrature (DQ)
Direct DQ method
Block marching in time with DQ discretization
Implementation of boundary conditions
Conclusions
Complex Differential Quadrature Method
DQ in the complex plane
Complex DQ method for potential problems
Complex DQ method for plane linear elastic problems
Conformal mapping-aided complex DQ
Conclusions
Triangular Differential Quadrature Method
Triangular DQ method in standard triangle
Triangular DQ method in curvilinear triangle
Geometric transformation
Governing equations of Reissner–Mindlin plates on Pasternak foundation
Conclusions
Multiple Scale Differential Quadrature Method
Multi-scale DQ method for potential problems
Solutions of potential problems
Successive over-relaxation (SOR)-based multi-scale DQ method
Asymptotic multi-scale DQ method
DQ solution to multi-scale poroelastic problems
Conclusions
Variable Order Differential Quadrature Method
Direct DQ discretization and dynamic numerical instability
Variable order approach
Improvement of temporal integration
Conclusions
Multi-Domain Differential Quadrature Method
Linear plane elastic problems with material discontinuity
A multi-domain approach for numerical treatment of material discontinuity
Multi-domain DQ method for irregular domain
Multi-domain DQ formulation of plane elastic problems
Conclusions
Localized Differential Quadrature Method
DQ and its spatial discretization of the wave equation
Stability analysis
Coordinate-based localized DQ
Spline-based localized DQ method
Conclusions
Mathematical Compendium
Gauss elimination
SOR method
One-dimensional band storage
Runge–Kutta method (constant time step)
Complex analysis
QR algorithm
Codes
DQ for numerical evaluation of function cos(x)
Complex DQ for harmonic problem
Localized DQ method
References
Index
