Modern Tools to Perform Numerical Differentiation
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.
Dalian University of Technology, Dalian, China National University of Singapore, Singapore Texas A&M University, College Station, USA University of Surrey, UK
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