Chebyshev Differentiation Matrices Research Articles

Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section

The present paper investigates the transverse vibration of a non-uniform axially functionally graded Timoshenko beam with cross-sectional and material properties varying in the beam length direction. The Chebyshev collocation method is used to spatially discretize the governing partial differential equations of motion of the beam into time-dependent ordinary differential equations in terms of Chebyshev differentiation matrices. An algebraic eigenvalue equation in matrix form is then formed to study the free vibration behavior of non-uniform axially functionally graded Timoshenko beams. Several results of natural frequencies of the beams are evaluated and compared with those in the published literature to assure the accuracy of the proposed model. The effects of taper ratio, material graded index, slenderness ratio, material compositions and restraint types on the natural frequencies of tapered axially functionally graded Timoshenko beams are examined.

Vibration Analysis of Bidirectional Functionally Graded Timoshenko Beams Using Chebyshev Collocation Method

This paper studies the vibration behaviors of bidirectional functionally graded (BDFG) Timoshenko beams based on the Chebyshev collocation method. The material properties of the beam are assumed to vary simultaneously in the beam length and thickness directions. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem for free vibration analysis. To validate the accuracy of the proposed model, some calculated results are compared with those obtained by other investigators. Good agreement has been achieved. Then the effects of slenderness ratios, material distribution types, gradient indexes, and restraint types on the natural frequency of BDFG beams are examined. Through the parametric study, the influences of the various geometric and material parameters on the vibration characteristics of BDFG beams are evaluated.

Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams Based on Chebyshev Collocation Method

The bending vibration behavior of a non-uniform axially functionally graded Euler-Bernoulli beam is investigated based on the Chebyshev collocation method. The cross-sectional and material properties of the beam are assumed to vary continuously across the axial direction. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem associated with the free vibration. Some calculated results are compared with numerical results in the published literature to validate the accuracy of the present model. A good agreement is observed. The effects of the taper ratio, volume fraction index, and restraint types on the natural frequency of axially functionally graded beams with non-uniform cross section are examined.

Nonlocal Timoshenko representation and analysis of multi-layered functionally graded nanobeams

The vibration properties of functionally graded multiple nanobeams is studied using Eringen’s nonlocal elasticity theory. Beam layers are considered to be continuously connected by layer of linear springs and the nonlocal Timoshenko beam theory is used to model each layer of beam which applies the size dependent effects in FG beam. The material behaviors of FG nanobeams are assumed to vary over the thickness based to the power law. The Hamilton’s principle is used to derive the governing differential equations of motion according to Eringen nonlocal theory and a Chebyshev spectral collocation method is employed to convert the coupled equations of motion into algebraic equations. The discretized boundary conditions are applied to adjust the Chebyshev differentiation matrices, and the system of equations is then expressed in the matrix–vector form. Next, the coupled natural frequencies and corresponding mode shapes are obtained by solving the standard eigenvalue problem. The model is confirmed by comparing the obtained results with benchmark results existing in the literature. Next, a parametric study is carried out to determine the influences of the material gradation, length scale, and stiffness parameters on the vibration properties of multiple functionally graded nanobeams. It is demonstrated that these parameters are vital in examination of the free vibration of a multi-layered FG nanobeam.

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

Numerical Solution of Two Dimensional Laplace’s Equation on a Regular Domain Using Chebyshev Differentiation Matrices

This work presents an efficient procedure based on Chebychev spectral collocation method for computing the 2D Laplace’s equation on a rectangular domain. The numerical results and comparison of finite difference and finite element methods are presented. We obtained a satisfactory result when compared with other numerical solutions.

Application of reaction diffusion model in Turing pattern and numerical simulation

Turing proposed a model for the development of patterns found in nature in 1952. Turing instability is known as diffusion-driven instability, which states that a stable spatially homogeneous equilibrium may lose its stability due to the unequal spatial diffusion coefficients. The Gierer-Mainhardt model is an activator and inhibitor system to model the generating mechanism of biological patterns. The reaction-diffusion system is often used to describe the pattern formation model arising in biology. In this paper, the mechanism of the pattern formation of the Gierer-Meinhardt model is deduced from the reactive diffusion model. It is explained that the steady equilibrium state of the nonlinear ordinary differential equation system will be unstable after adding of the diffusion term and produce the Turing pattern. The parameters of the Turing pattern are obtained by calculating the model. There are a variety of numerical methods including finite difference method and finite element method. Compared with the finite difference method and finite element method, which have low order precision, the spectral method can achieve the convergence of the exponential order with only a small number of nodes and the discretization of the suitable orthogonal polynomials. In the present work, an efficient high-precision numerical scheme is used in the numerical simulation of the reaction-diffusion equations. In spatial discretization, we construct Chebyshev differentiation matrices based on the Chebyshev points and use these matrices to differentiate the second derivative in the reaction-diffusion equation. After the spatial discretization, we obtain the nonlinear ordinary differential equations. Since the spectral differential matrix obtained by the spectral collocation method is full and cannot use the fast solution of algebraic linear equations, we choose the compact implicit integration factor method to solve the nonlinear ordinary differential equations. By introducing a compact representation for the spectral differential matrix, the compact implicit integration factor method uses matrix exponential operations sequentially in every spatial direction. As a result, exponential matrices which are calculated and stored have small sizes, as those in the one-dimensional problem. This method decouples the exact evaluation of the linear part from the implicit treatment of the nonlinear reaction terms. We only solve a local nonlinear system at each spatial grid point. This method combines with the advantages of the spectral method and the compact implicit integration factor method, i.e., high precision, good stability, and small storage and so on. Numerical simulations show that it can have a great influence on the generation of patterns that the system control parameters take different values under otherwise identical conditions. The numerical results verify the theoretical results.

Meshless methods for one-dimensional oscillatory Fredholm integral equations

In this paper, efficient and simple algorithms based on Levin’s quadrature theory and our earlier work involving local radial basis function (RBF) and Chebyshev differentiation matrices, are adopted for numerical solution of one-dimensional highly oscillatory Fredholm integral equations. This work is focused on the comparative performance of local RBF meshless and pseudospectral procedures. We have tested the proposed methods on phase functions with and without stationary phase point(s), both on uniform and Chebyshev grid points. The proposed procedures are shown accurate and efficient, and therefore provide a reliable platform for the numerical solution of integral equations. From the numerical results, we draw some conclusions about accuracy, efficiency and robustness of the proposed approaches.

Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations

This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.

Chebyshev Collocation Solutions for Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions

This paper applies the Chebyshev collocation method to finding accurate solutions of natural frequencies for circular cylindrical shells. The shells with different boundary conditions are considered in the parametric study. By using the method to solve the coupled differential equations of motion governing the vibration of the shell, numerical results are obtained from the algebraic eigenvalue equation using the Chebyshev differentiation matrices. And the results satisfy both the geometric and force boundary conditions. Based on the numerical examples, the proposed method shows its capacity and reliability in predicting accurate frequency results for circular cylindrical shells with various boundary conditions as compared to some exact solutions available in the literature.

Efficient modified Chebyshev differentiation matrices for fractional differential equations

This paper compares several fractional operational matrices for solving a system of linear fractional differential equations (FDEs) of commensurate or incommensurate order. For this purpose, three fractional collocation differentiation matrices (FCDMs) based on finite differences are first proposed and compared with Podlubny’s matrix previously used in the literature, after which two new efficient FCDMs based on Chebyshev collocation are proposed. It is shown via an error analysis that the use of the well-known property of fractional differentiation of polynomial bases applied to these methods results in a limitation in the size of the obtained Chebyshev-based FCDMs. To compensate for this limitation, a new fast spectrally accurate FCDM for fractional differentiation which does not require the use of the gamma function is proposed. Then, the Schur-Pade and Schur decomposition methods are implemented to enhance and improve numerical stability. Therefore, this method overcomes the previous limitation regarding the size limitation. In several illustrative examples, the convergence and computation time of the proposed FCDMs are compared and their advantages and disadvantages are outlined.

Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, an efficient technique based on the Chebyshev spectral collocation method for computing the eigenvalues of fourth-order Sturm–Liouville boundary value problems is proposed. The excellent performance of this scheme is illustrated through four examples. Numerical results and comparison with other methods are presented.

Spectral method for modeling waves in cylindrical poroelastic structures.

The modeling of propagating modes in poroelastic fluid‐saturated cylindrical structures is important for the interpretation of acoustic and seismic measurements in hydrocarbon wells as well as laboratory measurements on core samples. In order to obtain the velocities of different modes the roots of the appropriate dispersion equation have to be found. This is a very difficult task especially when some layers are represented by poroelastic materials with frequency‐dependent attenuation. A new technique is presented for modeling those modes, which adopts a spectral method to discretize the underlying partial differential equations using Chebyshev differentiation matrices. The corresponding system of linear equations can then be solved as a generalized eigenvalue problem. This means that, for a given frequency, the eigenvalues correspond to the slownesses of different modes. This approach is implemented for an arbitrary number of cylindrical fluid and solid layers as well as poroelastic layers. In addition to the dispersion, the attenuation curves are computed as well as radial profiles of displacements and stresses for various frequencies. The resulting approach can be used to model wave propagations in open and completed boreholes surrounded by permeable formations. This can pave the way to invert for formation and completion permeability from acoustic measurements.