High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

The Timoshenko beam model of the differential quadrature element method

A new numerical approach for solving Timoshenko beam problems is proposed. The approach uses the differential quadrature method (DQM) to discretize the Timoshenko beam equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of Timoshenko beam structures. The resulting overall discrete equation can be solved by using a solver of the linear algebra. Numerical results of the DQEM Timoshenko beam model are presented. They demonstrate the DQEM numerical method.     

DQEM for free vibration analysis of Timoshenko beams on elastic foundations

 A differential quadrature element method (DQEM) based on first order shear deformation theory is developed for free vibration analysis of non-uniform beams on elastic foundations. By decomposing the system into a series of sub-domains or elements, any discontinuity in loading, geometry, material properties, and even elastic foundations can be considered conveniently. Using this method, the vibration analysis of general beam-like structures is to be studied. The governing equations of each element, natural compatibility conditions at the interface of two adjacent elements and the external boundary conditions are developed in a systematic manner, using Hamilton's principle. The present DQEM is to be implemented to Timoshenko beams resting on partially supported elastic foundations with various types of boundary conditions under the action of axial loading. The general versality, accuracy, and efficiency of the presented DQEM are demonstrated having solved different examples and compared to the exact or other numerical procedure solutions. Received: 11 October 2002/Accepted: 26 November 2002     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method

A new version of the differential quadrature method is presented in this paper to overcome the difficulty existing in the ordinary differential quadrature method for applying multi‐boundary conditions in two‐dimensional problems. Since the weighting coefficients of the first derivative are the same as for the ordinary differential quadrature method even with the introduction of multi‐degree‐of‐freedom at the boundary points, the method is easier to extend to two‐ or three‐dimensional problems. A new version of the differential quadrature plate element has been established for demonstration. The essential difference from the existing old version of the differential quadrature plate element is the way the weighting coefficients are determined. The methodology is worked out in detail and some numerical examples are given to show the efficiency of the present method. Copyright © 2004 John Wiley & Sons, Ltd.     

A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field. Fundamental equations of plane electromagnetic, temperature and elastic fields are formulated. To the best of the authors’ knowledge, this is the first attempt to apply the DQM to magneto-thermo-elasticity and the first attempt to analyze a finite two-dimensional magneto-thermo-elastic problem. The fundamental equations and the inhomogeneous time-dependent boundary conditions are discretized in spatial and temporal domain by differential quadrature (DQ) rules. The unknowns satisfying the governing equations, the boundary and initial conditions simultaneously are computed in the entire domain by means of DQM with high efficiency and accuracy using dramatically less grid points in both spatial and time domain. Solutions of magnetic field, eddy current, temperature change and dynamic solutions of stresses and deformations are illustrated graphically.     

Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Semi‐analytical analysis for multi‐directional functionally graded plates: 3‐D elasticity solutions

Semi‐analytical 3‐D elasticity solutions are presented for orthotropic multi‐directional functionally graded plates using the differential quadrature method (DQM) based on the state‐space formalism. Material properties are assumed to vary not only through the thickness but also in the in‐plane directions following an exponential law. The graded in‐plane domain is solved numerically via the DQM, while exact solutions are sought for the thickness domain using the state‐space method. Convergence studies are performed, and the present hybrid semi‐analytical method is validated by comparing numerical results with the exact solutions for a conventional unidirectional functionally graded plate. Finally, effects of material gradient indices on the displacement and stress fields of the plates are investigated and discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media

Based on the Porous Media Theory presented by de Boer, the governing differential equations for a layered space-axisymmetrical fluid-saturated porous elastic body are firstly established, in which the suitable interface conditions between layers are presented. Then, a differential quadrature element method (DQEM) is developed, and the DQEM and the second-order backward difference scheme are applied to discretize the governing differential equations of the problem in the spatial and temporal domain, respectively. In order to show the validity of the present analysis, the dynamic response of a fluid-saturated porous medium is analyzed, and the obtained numerical results are directly compared with the existing analytical results. The effects of the numbers of the elements and grid points on the convergence of the numerical results are considered. Finally, the dynamic characteristics of a layered fluid-saturated elastic soil cylinder subjected to a water pressure or a dynamic loading are studied, and the effects of material parameters are considered in detail. From the above numerical results, it can be found that the DQEM has advantages, such as little amount in computation, good stability and convergence as well as high accuracy, so it is a very efficient method for solving the problems in soil mechanics, especially such problems with discontinuities.     

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

The two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of DQM as an effective simulation tool for buckling response of delaminated composite plates

The application of differential quadrature method (DQM), as an effective and robust numerical method, for the analysis of buckling of delaminated composite plates is introduced. The analysis investigated the response of laminated composite plates hosting a circular or an elliptical delamination. The delaminations were assumed to be fairly thinner than the plate hosting them, and thus, they could be treated as plates with clamped edges. Several case studies were used to verify the integrity of DQM in predicting the buckling strain of the plates. The investigation included the examination of several parameters influencing the buckling strength. The results obtained from DQM were compared with those obtained by the Rayleigh–Ritz and finite element solutions of other workers.     

3-D Free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM)

In this paper, the free vibration of functionally graded annular plates on elastic foundations, based on the three-dimensional theory of elasticity, using the differential quadrature method for different boundary conditions including simply supported–clamped, clamped–clamped and free–clamped ends is investigated. The foundation is described by the Pasternak or two-parameter model. A semi-analytical approach composed of differential quadrature method (DQM) and series solution are adopted to solve the equations of motions. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The new results can be used as benchmark solutions for future researches.     

The generalized differential quadrature rule for fourth‐order differential equations

The generalized differential quadrature rule (GDQR) proposed here is aimed at solving high‐order differential equations. The improved approach is completely exempted from the use of the existing δ‐point technique by applying multiple conditions in a rigorous manner. The GDQR is used here to static and dynamic analyses of Bernoulli–Euler beams and classical rectangular plates. Numerical error analysis caused by the method itself is carried out in the beam analysis. Independent variables for the plate are first defined. The explicit weighting coefficients are derived for a fourth‐order differential equation with two conditions at two different points. It is quite evident that the GDQR expressions and weighting coefficients for two‐dimensional problems are not a direct application of those for one‐dimensional problems. The GDQR are implemented through a number of examples. Good results are obtained in this work. Copyright © 2001 John Wiley & Sons, Ltd.     

A robust methodology for the simulation of postbuckling response of composite plates

The application of a robust numerical method, namely the differential quadrature method (DQM) for the analysis of buckling and postbuckling of laminated composite plates is introduced. The method is combined with an arc-length strategy to solve the resulting system of nonlinear equations. The treatment accounts for the effect of large deformation by including the von Karman strains. Imperfections in the form of global deflection, conforming to the preferred buckling modes are introduced throughout the plate. Case studies are used to evaluate the geometrically nonlinear response of composite plates under the set conditions, and the results are compared with those obtained by finite element solution and the results obtained from a published literature.     

Nonlocal free vibration of orthotropic non-prismatic skew nanoplates

The free vibration of orthotropic non-prismatic skew nanoplate based on the first-order shear deformation theory (FSDT) in conjunction with Eringen’s nonlocal elasticity theory is presented. As a simple, accurate and low computational effort numerical method, the differential quadrature method (DQM) is employed to solve the related differential equations. For this purpose, after deriving the equations of motion and the related boundary conditions, they are transformed from skewed physical domain to rectangular computational domain of DQM and accordingly discretized. After validating the formulation and method of solution, the effects of nonlocal parameter in combination with geometrical parameters and boundary conditions on the natural frequencies of the orthotropic skew nanoplates are investigated.     

A new crack tip element for the phantom‐node method with arbitrary cohesive cracks

We have developed a new crack tip element for the phantom‐node method. In this method, a crack tip can be placed inside an element. Therefore, cracks can propagate almost independent of the finite element mesh. We developed two different formulations for the three‐node triangular element and four‐node quadrilateral element, respectively. Although this method is well suited for the one‐point quadrature scheme, it can be used with other general quadrature schemes. We provide some numerical examples for some static and dynamic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

DQE方法及其在平板振动分析中的应用

介绍一种新的数值方法－DQE方法用以分析具有不连续几何形状和具有集中质量平板的自由振动,其基本思想是根据平板的具体情况将求解区域分割为几个单元,在每个单元内部用DQ方法将其控制方程离散,各单元之间的连接点上应用平衡条件和连续条件,再加上边界约束条件可得到问题的特征方程。作为数值算例,计算了具有集中质量平板和L型平板的固有频率,并与已有数值结果作了比较。     

Elasticity solution of functionally graded circular and annular plates integrated with sensor and actuator layers using differential quadrature

Based on three-dimensional theory of elasticity axisymmetric static analysis of functionally graded circular and annular plates imbedded in piezoelectric layers is investigated using differential quadrature method (DQM). The plate has various edges boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. This method can give an analytical solution along the graded direction using the state space method (SSM) and an effective approximate solution along the radial direction using the one-dimensional DQM. The method is validated by comparing numerical results with the results obtained in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers on the mechanical behavior of plate is studied. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the static behavior of FG circular and annular plates are investigated.     

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.     

Differential quadrature Trefftz method for irregular plate problems

Differential quadrature Trefftz method (DQTM) is developed to deal with plate problems defined in irregular domains. DQTM divides the solution into two parts, a particular solution for inhomogeneous biharmonic equation and the general solution for homogeneous biharmonic equation. For the former, differential quadrature method based on the interpolation of the highest derivative (DQIHD) is involved. For the latter, polynomial basis functions are adopted instead of fundamental solutions. We will show that DQTM not only keeps the advantages of traditional differential quadrature method (DQM), high efficiency and accuracy, but also has no difficulties to deal with geometrically irregular domains.