Differential Quadrature Solution Research Articles

3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders

The paper proposes a comparison between a three-dimensional (3D) exact solution and several two-dimensional (2D) numerical solutions. Numerical methods include classical 2D finite elements (FEs), and classical and refined 2D generalized differential quadrature (GDQ) solutions. The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configuration considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by means of the 3D exact solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach where the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D numerical solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

Combining surface effects and non‐local two variable refined plate theories on the shear/biaxial buckling and vibration of silver nanoplates

In this reported work, surface effects and non-local two variable refined plate theories are combined on the shear/biaxial buckling and vibration of rectangular nanoplates. A silver sheet is selected as the case study to investigate the numerical results. Surface effects are considered by Gurtin–Murdoch's theory. The differential quadrature method is used to solve the governing equations. Differential quadrature solutions are verified by Navier's method. The influences of the non-local parameter on the surface effects of shear/biaxial buckling and vibration are investigated for various boundary conditions. Results show that by increasing the non-local parameter, the effects of surface on the buckling and vibration increase. This result is in contrast with the works of other researchers in the field. Moreover, the non-local effects on the shear buckling and vibration are more important than that of biaxial, whereas the surface effects on the biaxial buckling and vibration are more notable than that of shear.

A Three-Dimensional Layerwise-Differential Quadrature Free Vibration of Thick Skew Laminated Composite Plates

Using the layerwise theory in conjunction with the differential quadrature method (DQM), an accurate and efficient solution procedure for the free vibration analysis of thick skew laminated composite plates is introduced. By employing Hamilton's principle and the layerwise theory, the discretized form of the equations of motion in the thickness direction is obtained. Then, the DQM is used to solve the resulting system of differential equations. The fast rate of convergence and high accuracy of the method is established. Then, some new results for skew laminated plates are presented, which can be used as a benchmark solution for future works.

Free vibration analysis of thick CGFR annular sector plates resting on elastic foundations

This paper deals with free vibration analysis of continuous grading fiber reinforced (CGFR) and bi-directional FG annular sector plates on two-parameter elastic foundations under various boundary conditions, based on the three-dimensional theory of elasticity. The plates with simply supported radial edges and arbitrary boundary conditions on their circular edges are considered. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. 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. Results indicate that the non-dimensional natural frequency parameter of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. It results that the CGFR plate attains natural frequency higher than those of traditional discretely laminated composite ones and this can be a benefit when higher stiffness of the plate is the goal and that is due to the reduction in spatial mismatch of material properties. Moreover, it is shown that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional one-dimensional functionally graded material. The multidirectional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material. The new results can be used as benchmark solutions for future researches.

Decoupling the nonlocal elasticity equations for thermo-mechanical vibration of circular graphene sheets including surface effects

This article deals with the axisymmetric vibration properties of circular single-layered graphene sheet embedded in a polymer matrix under thermo-mechanical loading. Both surface and nonlocal effects are taken into account. To this end, Gurtin–Murdoch continuum elasticity in conjunction with the nonlocal elasticity theory is used to develop a modified continuum plate model for free vibration analysis of the nanoplates. The governing equations are derived by decoupling the nonlocal constitutive equations of Eringen theory in the polar coordinate. Galerkin׳s method is used to obtain the vibration frequencies. To verify the accuracy of the Galerkin results, a differential quadrature (DQ) solution is also developed. Galerkin results are successfully verified with those of the DQ method. A good agreement is also found between the present results and experimental data. Further, in comparison to the available molecular dynamics simulation results, the present formulation with appropriate values of surface and nonlocal parameters provides more accurate results than those by the classical plate model.

Two-Dimensional Differential Quadrature Solution for Vibration Characteristics of Two-Dimensional Functionally Graded Metal/Ceramic Open Cylindrical Shells

In this article, semi-analytical 3D elasticity solutions are presented to study the vibration characteristics of two-dimensional functionally graded (2D FGM) metal/ceramic open cylindrical shells under various boundary conditions. In the present formulation, the shell has a smooth variation of volume fractions of metal and ceramic in the radial and axial directions with power law functions. The study is carried out based on the three-dimensional, linear and small strain elasticity theory. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The two-dimensional generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. Results indicate the advantages of using functionally graded open cylindrical shells with graded volume fractions in two directions to a more flexible design than the conventional one-dimensional FGM. The effects of different boundary conditions, various geometrical parameters and volume fraction indices in radial and axial directions on the vibration characteristics of 2D FGM open cylindrical shells are investigated. Finally, frequency parameters of 2D FGM open cylindrical shells are compared with radially and axially conventional FGM.

Differential quadrature solution of heat- and mass-transfer equations

The heat- and mass-transfer equations have an important role in various thermal and diffusion processes. These equations are nonlinear, due to the solution dependent diffusion coefficient and the source term. In this study, one- and two-dimensional nonlinear heat- and mass-transfer equations are solved numerically. To this end, the differential quadrature method is used to discretize the problem spatially and the resulting nonlinear system of ordinary differential equations in time are solved using the Runge–Kutta method. The solution is improved in time iteratively by solving considerably small sized linear system of resulting equations. To demonstrate its usefulness and accuracy, the proposed method is applied to four test problems, involving different nonlinearities.

General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature in the evaluation of the stress resultants three different approaches, specifically Qatu approach, Toorani–Lakis approach and Reissner–Mindlin approach, are considered and compared. An improvement of the Classical Reissner–Mindlin Theory (CRMT) using a different kinematical model is considered. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Four different anisotropic shell theories, namely General First-order Shear Deformation Theory by Qatu (GFSDTQ), General First-order Shear Deformation Theory by Toorani–Lakis (GFSDTTL), General First-order Shear Deformation Theory by Reissner–Mindlin (GFSDTRM) and Classical Reissner–Mindlin Theory (CRMT), are compared in order to show the differences and the accuracy of these theories. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Results are obtained taking the meridian and circumferential coordinates into account, without using the Fourier modal expansion methodology. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.

Differential quadrature solution of nonlinear Klein–Gordon and sine-Gordon equations

Differential quadrature method (DQM) is proposed to solve the one-dimensional quadratic and cubic Klein–Gordon equations, and two-dimensional sine-Gordon equation. We apply DQM in space direction and also blockwise in time direction. Initial and derivative boundary conditions are also approximated by DQM. DQM provides one to obtain numerical results with very good accuracy using considerably small number of grid points. Numerical solutions are obtained by using Gauss–Chebyshev–Lobatto (GCL) grid points in space intervals, and GCL grid points in each equally divided time blocks.

Differential Quadrature Solution of Hyperbolic Telegraph Equation

Differential quadrature method (DQM) is proposed for the numerical solution of one‐ and two‐space dimensional hyperbolic telegraph equation subject to appropriate initial and boundary conditions. Both polynomial‐based differential quadrature (PDQ) and Fourier‐based differential quadrature (FDQ) are used in space directions while PDQ is made use of in time direction. Numerical solution is obtained by using Gauss‐Chebyshev‐Lobatto grid points in space intervals and equally spaced and/or GCL grid points for the time interval. DQM in time direction gives the solution directly at a required time level or steady state without the need of iteration. DQM also has the advantage of giving quite good accuracy with considerably small number of discretization points both in space and time direction.

Differential Quadrature Solution for One-Dimensional Aquifer Flow

Differential Quadrature Method (DQM) has been applied to the solution of aquifer flow problems. Three examples from of each of the three one-dimensional aquifer flow equation problems, a confined aquifer flow with time dependent boundary conditions, a composite confined aquifer and an unconfined aquifer with seepage, were examined. The results of DQM solution were then compared with the results obtained from analytical solution, the Explicit Finite Differences Method and Implicit Finite Differences Method. Based on the comparison results, it was concluded that the DQM provides similar results but with relatively faster calculation speed, less nodes and memory usage.

Differential Quadrature Solutions of the Generalized Burgers–Fisher Equation with a Strong Stability Preserving High-Order Time Integration

Numerical solutions of the generalized Burgers-Fisher equation are presented based on a polynomial-based differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a third-order strong stability preserving Runge-Kutta scheme in time have been used. The proposed technique successfully worked to give reliable results in the form of numerical approximation converging very rapidly. The computed results have been compared with the exact solution to show the required accuracy of the method. The approximate solutions to the nonlinear equations were obtained. The approach is seen to be a very reliable alternative to the rival techniques for realistic problems.

Homotopy analysis and differential quadrature solution of the problem of free-convective magnetohydrodynamic flow over a stretching sheet with the Hall effect and mass transfer taken into account

This paper presents an analytical solution of the problem of free-convective magnetohydrodynamic flow over a stretched sheet with the Hall effect and mass transfer taken into account. A similarity transform reduces the Navier-Stokes, energy, Ohm law, and mass-transfer equations to a system of nonlinear ordinary differential equations. The governing equations are solved analytically using an analytical method for solving nonlinear problems, namely, the homotopy analysis method. The results are compared with the results of a promising numerical method of differential quadrature developed by the authors. It is shown that there is very good agreement between analytical results and those obtained by the differential quadrature method. The differential quadrature method was validated, and the effects of non-dimensional parameters on the velocity, temperature and concentration profiles were studied.

The differential quadrature solution of nonlinear reaction-diffusion and wave equations using several time-integration schemes

Three different time-integration schemes, namely the finite difference method (FDM) with a relaxation parameter, the least-squares method (LSM) and the finite element method (FEM), are applied to the differential quadrature (DQM) solution of one-dimensional nonlinear reaction–diffusion and wave equations. In the solution procedure, the space derivatives are discretized using DQM, which may also be used without the need of boundary conditions. The aim of the paper is to find computationally more efficient time-integration algorithm by comparison. For the nonlinear reaction–diffusion equation, which is a parabolic type, FEM is found to be the method of choice in terms of accuracy and computational cost. For the nonlinear wave equation, it is seen that both the FDM and LSM are better than FEM. Therefore, LSM can be preferred as a direct method for the nonlinear wave equation. Copyright © 2009 John Wiley & Sons, Ltd.

Elastoplastic buckling analysis of thick rectangular plates by using the differential quadrature method

The paper investigates the elastoplastic buckling behavior of thick rectangular plates by using the Differential Quadrature (DQ) method. Mindlin plate theory is adopted to take the transverse shear effect into considerations. Both incremental theory and deformation theory are employed. Due to the material non-linearity, iteration processes are involved for obtaining solutions. Detailed methodology and procedures are derived. The elastoplastic buckling behavior of thick rectangular plates with ten combinations of boundary conditions and under various loadings is studied. To verify the DQ solution procedures, DQ results are compared with existing analytical solutions for plates with two boundaries simply supported and the others simply supported, clamped, or free. Then the DQ method is used to obtain solutions of rectangular thick plates with other combinations of boundary conditions. Since no analytical solutions for such cases are available, the buckling loads obtained by the DQ method could serve as a reference. The phenomenon reported in the literature, namely, the deformation theory generally gives consistently lower buckling loads than the incremental theory and large discrepancy in predictions between the two theories exists with increasing of plate thickness. E / σ 0 , and c in the Ramberg–Osgood relations, is also observed for the cases studied herein. Apart from the phenomenon reported earlier by Durban that deformation theory predicted a progressively lower in-plane shear modulus as the level of plasticity increased, thus predicted lower buckling loads, another reason is given herein to explain the large discrepancy in predictions for thicker plates with the deformation theory and incremental theory.

2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.

DQM in-plane free vibration of laminated moderately thick circular deep arches

A differential quadrature (DQ) solution for moderately thick laminated circular arches with general boundary conditions is presented. The governing equations are based on the Reissner-Naghdi type shell theory, which include the effects of transverse shear deformation and rotary inertia. Explicit discrete forms of the equations of motion and the boundary conditions are presented. It is shown that using only few grid points, accurate results are obtained which demonstrate the efficiency and convenience of the DQ method (DQM) for the problem under consideration. The effects of different types of boundary conditions, opening angle, thickness-to-length and orthotropy ratios on the frequencies of antisymmetric cross-ply laminated arches are studied.

Differential quadrature solution of nonlinear reaction–diffusion equation with relaxation-type time integration

This paper presents the combined application of differential quadrature method (DQM) and finite-difference method (FDM) with a relaxation parameter to nonlinear reaction–diffusion equation in one and two dimensions. The polynomial-based DQM is employed to discretize the spatial partial derivatives by using Gauss–Chebyshev–Lobatto points. The resulting system of ordinary differential equations is solved, discretizating the time derivative by an explicit FDM. A relaxation parameter is used to position the solution from the two time levels, aiming to increase the convergence rate with a moderate time step to the steady state and also to obtain stable solution. Numerical experiments are given to illustrate the scheme for one-dimensional Fisher-type problems and also for two-dimensional reaction–diffusion boundary-value problems. The agreement of the solution with the exact solution is very good in two-dimensional case while some other numerical schemes may result in some unwanted oscillations in the computed solution. Optimal value of the relaxation parameter is obtained numerically to prevent the use of very small time steps and to achieve stable solutions. The DQM with a relaxation-type finite-difference time integration scheme exhibits superior accuracy at large time values for the problems tending towards a steady state.

A Differential Quadrature Solution of MHD Natural Convection in an Inclined Enclosure With a Partition

Magnetohydrodynamics natural convection in an inclined enclosure with a partition is studied numerically using a differential quadrature method. Governing equations for the fluid flow and heat transfer are solved for the Rayleigh number varying from 104 to 106, the Prandtl numbers (0.1, 1, and 10), four different Hartmann numbers (0, 25, 50, and 100), the inclination angle ranging from 0degto90deg, and the magnetic field with the x and y directions. The results show that the convective flow weakens considerably with increasing magnetic field strength, and the x-directional magnetic field is more effective in reducing the convection intensity. As the inclination angle increases, multicellular flows begin to develop on both sides of the enclosure for higher values of the Hartmann number if the enclosure is under the x-directional magnetic field. The vorticity generation intensity increases with increase of Rayleigh number. On the other hand, increasing Hartmann number has a negative effect on vorticity generation. With an increase in the inclination angle, the intensity of vorticity generation is observed to shift to top left corners and bottom right corners. Vorticity generation loops in each region of enclosure form due to multicelluar flow for an x-directional magnetic field when the inclination angle is increased further. In addition, depending on the boundary layer developed, the vorticity value on the hot wall increases first sharply with increasing y and then begins to decrease gradually. For the high Rayleigh numbers, the average Nusselt number shows an increasing trend as the inclination angle increases and a peak value is detected. Beyond the peak point, the foregoing trend reverses to decrease with the further increase of the inclination angle. The results also show that the Prandtl number has only a marginal effect on the flow and heat transfer.

A Differential Quadrature Solution of Natural Convection in an Enclosure with a Partial Partition

Natural convection in a partially divided enclosure has been examined numerically using a differential quadrature method. Governing equations for the flow and heat transfer have been constructed by the vorticity-stream function formulation and computational results have been obtained for the Rayleigh numbers, 104, 105, 106, the locations of the partition, 0.2, 0.4, 0.6, 0.8, and the partition ratios, 0.25, 0.50, 0.75, 1.0. The results show that circulation strength, and therefore heat transfer, decreases considerably with increasing partition height especially for the higher values of the Rayleigh number. As the distance of the partition from the hot wall increases, first a decrease and then an increase is observed in the average Nusselt number for the low Rayleigh numbers and first an increase and then a decrease turns out for the high Rayleigh numbers. As for the average Nusselt number on the partition, first it shows a decreasing trend and then an increasing trend as the partition is distanced from the hot wall towards the cold wall. The increase in the partition height brings about significant increase in the average Nusselt number on the partition.