Some modified Adams-Bashforth methods based upon the weighted Hermite quadrature rules

In this paper, we first introduce a modification of linear multistep methods, which contain, in particular, the modified Adams-Bashforth methods for solving initial-value problems. The improved method is achieved by applying the Hermite quadrature rule instead of the Newton-Cotes quadrature formulas with equidistant nodes. The related coefficients of the method are then represented explicitly, the local error is given, and the order of the method is determined. If a numerical method is consistent and stable, then it is necessarily convergent. Moreover, a weighted type of the new method is introduced and proposed for solving a special case of the Cauchy problem for singular differential equations. Finally, several numerical examples and graphical representations are also given and compared.     

A weak form quadrature element method for plane elasticity problems

A weak form quadrature element method is proposed and applied to analysis of plane elasticity problems. A variational formulation of plane elasticity problems is established and the differential quadrature analog of the derivatives in the functional is introduced. Several typical plane elasticity problems are studied to verify the proposed method. Results show that the method is highly efficient and promising. It is applied to the analysis of nearly incompressible materials and shown to be robust against volumetric locking. Similarities and dissimilarities, advantages and disadvantages as compared with other numerical methods, typically the p-version finite element method are discussed.     

The Asymptotic Behavior for Numerical Solution of a Volterra Equation

Long-time asymptotic stability and convergence properties for the numerical solution of a Volterra equation of parabolic type are studied.The methods are based on the first-second order backward difference methods.The memory term is approximated by the comvolution quadrature and the interpolant quadrature.Discretization of the spatial partial differential operators by the finite element method is also considered.     

The differential quadrature element method irregular element torsion analysis model

The differential quadrature element method (DQEM) has been proposed. The element weighting coefficient matrices are generated by the differential quadrature (DQ) or generic differential quadrature (GDQ). By using the DQ or GDQ technique and the mapping procedure the governing differential or partial differential equations, the transition conditions of two adjacent elements and the boundary conditions can be discretized. A global algebraic equation system can be obtained by assembling all of the discretized equations. This method can convert a generic engineering or scientific problem having an arbitrary domain configuration into a computer algorithm. The DQEM irregular element torsion analysis model is developed.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Gaussian Quadrature Rules with Simple Node-Weight Relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.     

Accurate vibration analysis of skew plates by the new version of the differential quadrature method

An accurate free vibration analysis of skew plates is presented by using the new version of the differential quadrature method (DQM). Eight combinations of simply supported (S), clamped (C) and free (F) boundary conditions are considered. Detailed solution procedures are given and key points for success by using the DQM are emphasized. A way to simplifying the programming in using the DQM is proposed. Convergence study is made for the simply supported skew plate with a large skew angle. Good convergence of frequencies is observed. The DQ results agree very well with the existing first known accurate upper bound solutions, obtained by using Ritz method taking into considerations of the bending stress singularities occurred at corners having obtuse angles. Since slight discrepancy between the DQ data and the known accurate solutions is observed for plates with large skew angles, the DQ results are also compared with data obtained by using finite element method with very fine meshes to verify their accuracy.     

A GENERALIZED GAUSS-TYPE QUADRATURE FORMULA AND ITS APPLICATIONS TO PSEUDOSPECTRAL METHOD

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Localized triangular differential quadrature

A localized triangular differential quadrature method is introduced in this article. Not only is the existing limitation on the approximation order in the triangular differential quadrature eliminated but also the convergent rate is enhanced in the new method. As an example to validate the new method, elastic torsion of prismatic shaft with regular polygonal cross section is studied and excellent agreement with available theoretical and analytic solutions is reached. It is believed that the present work further widens the applicability of the triangular differential quadrature technique. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 682–692, 2003     

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions. AMS subject classification (2000) 65D32, 65T20, 65Y20     

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

In this study, both the dual reciprocity boundary element method and the differential quadrature method are used to discretize spatially, initial and boundary value problems defined by single and system of nonlinear reaction–diffusion equations. The aim is to compare boundary only and a domain discretization method in terms of accuracy of solutions and computational cost. As the time integration scheme, the finite element method is used achieving solution in terms of time block with considerably large time steps. The comparison between the dual reciprocity boundary element method and the differential quadrature method solutions are made on some test problems. The results show that both methods achieve almost the same accuracy when they are combined with finite element method time discretization. However, as a method providing very good accuracy with considerably small number of grid points differential quadrature method is preferrable.     

矩形平板自由和强迫振动以及屈曲的有限元-微分求积混合法(FE-DQM)

首次提出有限元法(FEM)和微分求积法(DQM)的组合应用,分析矩形平板的振动和屈曲问题.混合法综合了FEM几何适应性强,以及DQM的高精度和高效率.与已有文献的计算结果比较,验证了该方法的正确性.研究表明,使用少量的有限单元和不多的DQM样本点,就可以得到高精度的结果.由于该方法简单且具备进一步发展的潜力,被认为适用于这类问题的求解.     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Chebyshev series method for computing weighted quadrature formulas

In this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions. The main goal is to evaluate accurately a definite integral, whose mass is highly concentrated near some points. The numerical implementation of this approach is based on the calculation of Chebyshev series and some integration formulas which are exact for polynomials. In terms of accuracy, the proposed method can be compared with rational Gauss quadrature formula.     

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Adaptive Quadrature—Revisited

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.     

High-order Gauss–Lobatto formulae

Currently, the method of choice for computing the (n+2)-point Gauss–Lobatto quadrature rule for any measure of integration is to first generate the Jacobi matrix of order n+2 for the measure at hand, then modify the three elements at the right lower corner of the matrix in a manner proposed in 1973 by Golub, and finally compute the eigenvalues and first components of the respective eigenvectors to produce the nodes and weights of the quadrature rule. In general, this works quite well, but when n becomes large, underflow problems cause the method to fail, at least in the software implementation provided by us in 1994. The reason is the singularity (caused by underflow) of the 2×2 system of linear equations that is used to compute the modified matrix elements. It is shown here that in the case of arbitrary Jacobi measures, these elements can be computed directly, without solving a linear system, thus allowing the method to function for essentially unrestricted values of n. In addition, it is shown that all weights of the quadrature rule can also be computed explicitly, which not only obviates the need to compute eigenvectors, but also provides better accuracy. Numerical comparisons are made to illustrate the effectiveness of this new implementation of the method.     

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.