Polynomial-based Differential Quadrature Research Articles

Thermal buckling analysis of a functionally graded microshell based on higher-order shear deformation and modified couple stress theories

In this study, thermal buckling responses of a functionally graded (FG) cylindrical microshell are investigated. Material properties of the FG microshell are considered to be graded through the thickness based on a simple power-law distribution. The governing equations and associated boundary conditions are derived based on classical higher-order shear deformation theory. Size dependence of the FG microshell is properly characterized based on the modified couple stress theory (MCST). Two types of differential quadrature method (DQM) – polynomial based differential quadrature (PDQ) and Fourier expansion based differential quadrature (FDQ) – are adopted to transform the differential equations into algebraic equations. The standard eigenvalue equation in the whole computational domain is subsequently established to obtain critical buckling temperature. The present numerical technique shows very fast convergence speed and is validated by comparison with available data in the open literature. Parametric studies are provided to demonstrate the influences of the temperature distribution profile, edge boundary conditions, length scale parameter, length-to-radius ratio, FG power-law index on the critical buckling temperature. In addition, the values of critical thermal buckling loads predicted by classical and MCST models are compared.

A polynomial differential quadrature-based numerical scheme to simulate the nerve pulse propagation in the spatial Fitzhugh-Nagumo model

Nonlinear dynamics connect the neurons that form the brain, and thus, the complex information is produced and transported. The function of the neurons and the problem of understanding the dynamics of the brain has been the research area of mathematical neuroscience. In this study, the modelling and simulation of the propagation of the electric field based Action Potential (AP) on the Two Dimensional (2-D) field of axon network, whose matrix consists of 128×128 electrically coupled neurons were done using nonlinear Spatial FitzHugh Nagumo (SFN) equations. SFN equations are a particular class of Partial Differential Equation’s (PDE’s) exhibiting travelling wave behaviour occurred in neuron systems. The motivation of this paper is to evaluate the SFN equation, which is a special kind of the time-dependent nonlinear reaction-diffusion problem governing neuron dynamics numerically in 2-D space addressed by investigating the Polynomial-based Differential Quadrature Method (PDQM) having Chebyshev-Gauss-Lobatto quadrature points. The solution occurs as elliptical spiral waves induced by electrical stimulation. Thus, the neuronal system behaviour and the interaction with the specific type of Boundary Conditions (BC’s) are predicted. The space derivatives are discretised through PDQM. In this way, the problem is reduced into a system of first-order non-linear differential equations. Hereafter the time derivatives of the SFN equation are solved through the Finite Difference Method (FDM). The various dynamical behaviour that governs the travelling wave pattern regarding the Initial Condition’s (IC’s), BC’s and way of stimulation of the neuron model examined in details. Numerical results indicated that the proposed PDQM provide reliable, fast, and efficient solutions.

Thermoelastic Analysis of Rotating Functionally Graded Truncated Conical Shell by the Methods of Polynomial Based Differential Quadrature and Fourier Expansion-Based Differential Quadrature

This paper focuses on the three-dimensional (3D) asymmetric problem of functionally graded (FG) truncated conical shell subjected to thermal field and inertia force due to the rotating part. The FG properties are assumed to be varied along the thickness according to power law distribution, whereas Poisson’s ratio is assumed to be constant. On the basis of 3D Green-Lagrange theory in general curvilinear coordinate, the fundamental equations are formulated and then two versions of differential quadrature method (DQM) including polynomial based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) are applied to discretize the resulting differential equations. The reliability of the present approach is validated by comparing with known literature where good agreement is reached using considerably few grid points. The effects of different mechanical boundary conditions, temperature fields, rotating angular speed, and shell thickness on the distributions of stress components and displacement in thickness direction for both axisymmetric and asymmetric cases are graphically depicted.

Vibrational responses of antisymmetric angle-ply laminated conical shell by the methods of polynomial based differential quadrature and Fourier expansion based differential quadrature

Vibrational response of antisymmetric angle-ply laminated conical shell based on transverse shear deformation theory is analyzed in this research. The equilibrium and corresponding boundary equations including the extension-coupling terms are derived in terms of displacement and rotational components. Two types of differential quadrature methods (DQM) including polynomial based differential quadrature (PDQ) and Fourier expansion based differential quadrature (FDQ) are applied to convert the existing differential equations to algebraic equations. After discretization of existing differential equations, the eigenvalue equation system for the functional value in the whole domain is established and solved. A series of numerical tests are carried out to validate the present approach that highly accurate results are obtained only few grid points. Finally, the parametric studies are performed to illustrate the effects of different boundary conditions, fiber angle and number of layers and also various aspect ratio of the shell including thickness to radius ratio length to radius on the frequency parameter.

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

In this article, we introduce a new, simple, and accurate computational technique for one‐dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017

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.

Application of the Generalized Differential Quadrature Method in Solving Burgers' Equations

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge—Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with the exact solutions. The method can compete against the methods applied in the literature.

Numerical simulation of natural convection in a partitioned enclosure using PDQ method

PurposeThis paper seeks to investigate the effect of a heat conducting vertical partition in an enclosure on natural convection heat transfer and fluid flow using the polynomial‐based differential quadrature (PDQ) method.Design/methodology/approachThe PDQ method with the non‐uniform Chebyshev‐Gauss‐Lobatto grid point distribution given below is used to transform the governing equations into a set of algebraic equations. After numerical discretization, the resulting algebraic equations are solved by the successive over‐relaxation iteration method.FindingsIt is found that the average Nusselt number decreases towards a constant value as the partition is distanced from the hot wall towards the middle of the enclosure. Furthermore, with decreasing thermal conductivity ratio, the average Nusselt number first increases and passes a peak point and then begins to decrease. The average heat transfer rate exhibits little dependence on the width of the partition in the range taken into consideration in this study for the thickness of the partition.Originality/valueThis study offers more knowledge on natural convection in partitioned enclosures.

Approximations of Sturm–Liouville eigenvalues using differential quadrature (DQ) method

The polynomial-based differential quadrature (PDQ) and the Fourier expansion-based differential quadrature (FDQ) methods are applied in this work to compute eigenvalues of the Sturm–Liouville problem. It is demonstrated through some examples that accurate results for the first k th eigenvalues of the problem, where k = 1 , 2 , 3 , … , can be obtained by using (at least) 2 k mesh points. The results of this work are compared with other published results in the literature.

RADIAL BASIS FUNCTION-ENHANCED DOMAIN-FREE DISCRETIZATION METHOD AND ITS APPLICATIONS

The recently proposed domain-free discretization (DFD) method is based on the Lagrange interpolation and polynomial-based differential quadrature (PDQ) method. In this article, the radial basis function (RBF) approximation is used in the DFD method as the interpolation scheme for function approximation, and the RBF-DQ method is applied to derivative approximation. The new variant of DFD method is then applied to simulation of natural convection in both concentric and eccentric annuli between inner circular and outer square cylinders. It is found from the numerical examples that the usage of radial basis function can greatly enhance the DFD method in terms of the computational stability and convergence speed.

AN EFFICIENT APPROACH TO SIMULATE NATURAL CONVECTION IN ARBITRARILY ECCENTRIC ANNULI BY VORTICITY-STREAM FUNCTION FORMULATION

The global method of polynomial-based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) is applied in this work to simulate the natural convection in an annulus between two arbitrarily eccentric cylinders. The vorticity-stream function formulation in the curvilinear coordinate system is taken as the governing equation, and the pressure single value condition is converted to an explicit formulation to update the stream function value on the inner cylinder wall. The present approach is very efficient, which combines the high efficiency and accuracy of the differential quadrature (DQ) method with simple implementation of pressure single value condition. When the present approach is applied to the concentric case, it was found that the computed stream function on the inner cylinder is almost zero and the flow field is symmetric. The computed average equivalent conductivity for the concentric case also agrees very well with available data in the literature. For the eccentric case, it was found that the computed stream function on the inner cylinder is not zero and there is a global circulation. The present result confirms the findings by Guj and Stella (Numer. Heat Transfer, vol. 27, pp. 89-105, 1995).

Application of differential quadrature method to simulate natural convection in a concentric annulus

In this paper, the Fourier expansion-based differential quadrature (FDQ) and the polynomial-based differential quadrature (PDQ) methods are applied to simulate the natural convection in a concentric annulus with a horizontal axis. The comparison and grid independence of PDQ and FDQ results are studied in detail. It was found that both PDQ and FDQ can obtain accurate numerical solutions using just a few grid points and requiring very small computational resources. It was demonstrated in the paper that the FDQ method can be applied to a periodic problem or a non-periodic problem. When FDQ is applied to a non-periodic problem (half of annulus), it can achieve the same order of accuracy as the PDQ method. And when FDQ is applied to the periodic problem (whole annulus), it is very efficient for low Rayleigh numbers. However, its efficiency is greatly reduced for the high Rayleigh numbers. The benchmark solution for Ra=102, 103, 3×103, 6×103, 104, 5×104 are also presented in the paper. Copyright © 1999 John Wiley & Sons, Ltd.

Solution of Helmholtz equation by differential quadrature method

The polynomial-based differential quadrature (PDQ) and the Fourier expansion-based differential quadrature (FDQ) methods are applied in this work to solve the two-dimensional Helmholtz equation. It is demonstrated through some sample examples that accurate numerical solution can be obtained by using only 2 to 3 mesh points per wavelength. It was found that the FDQ approach is more suitable for solving the Helmholtz equation.