Zeros Of Chebyshev Polynomials Research Articles

Rarefied Poiseuille Flow in a Circular Tube

An isothermal rarefied gas flow through a long circular tube due to longitudinal pressure gradient (a three-dimensional Poiseuille problem) was studied using the linearized Bhatnagar-Gross-Krook model kinetic equation over the whole range of the Knudsen numbers covering both free molecular and hydrodynamic regimes. The solution of the model kinetic equation with the diffuse boundary condition is obtained by the collocation method. This approach is based on the Chebyshev polynomials and rational Chebyshev functions. Choosing the zeros of Chebyshev polynomials in the multivariate range of integration for the collocation points, we reduce this problem to a set of algebraic equations. Based on the proposed approach, we have calculated the mass and the heat fluxes through the tube. The obtained results have also been compared with other studies. The developed approach may also be applied to a more general class of problems of rarefied gas flows in microand nanotubes.

Fast factorization of rectangular Vandermonde matrices with Chebyshev nodes

The polynomial interpolation problem with distinct interpolation points and the polynomial represented in the power basis gives rise to a linear system of equations with a Vandermonde matrix. This system can be solved efficiently by exploiting the structure of the Vandermonde matrix with the aid of the Bjorck–Peyrera algorithm. We are concerned with polynomial least-squares approximation at the zeros of Chebyshev polynomials. This gives rise to a rectangular Vandermonde matrix. We describe fast algorithms for the factorization of these matrices. Both QR and QR-like factorizations are discussed. The situations when the nodes are extreme points of Chebyshev polynomials or zeros of some classical orthogonal polynomial also are considered.

A new sequential sampling method for constructing the high-order polynomial surrogate models

PurposeThis paper aims to study the sampling methods (or design of experiments) which have a large influence on the performance of the surrogate model. To improve the adaptability of modelling, a new sequential sampling method termed as sequential Chebyshev sampling method (SCSM) is proposed in this study.Design/methodology/approachThe high-order polynomials are used to construct the global surrogated model, which retains the advantages of the traditional low-order polynomial models while overcoming their disadvantage in accuracy. First, the zeros of Chebyshev polynomials with the highest allowable order will be used as sampling candidates to improve the stability and accuracy of the high-order polynomial model. In the second step, some initial sampling points will be selected from the candidates by using a coordinate alternation algorithm, which keeps the initial sampling set uniformly distributed. Third, a fast sequential sampling scheme based on the space-filling principle is developed to collect more samples from the candidates, and the order of polynomial model is also updated in this procedure. The final surrogate model will be determined as the polynomial that has the largest adjusted R-square after the sequential sampling is terminated.FindingsThe SCSM has better performance in efficiency, accuracy and stability compared with several popular sequential sampling methods, e.g. LOLA-Voronoi algorithm and global Monte Carlo method from the SED toolbox, and the Halton sequence.Originality/valueThe SCSM has good performance in building the high-order surrogate model, including the high stability and accuracy, which may save a large amount of cost in solving complicated engineering design or optimisation problems.

Estimates for the Remainders of Certain Quadrature Formulas

Estimates of the remainders of certain quadrature formulas are obtained, in particular, of quadrature formulas with nodes that are the zeros of Chebyshev polynomials of the first kind in classes of differentiable functions characterized by a generalized modulus of continuity.

Incremental modeling of a new high-order polynomial surrogate model

This study will develop a new high-order polynomial surrogate model (HOPSM) to overcome routines of expensive computer simulations in engineering. The proposed HOPSM is expected to keep advantages of the traditional low-order polynomial models in efficiency, transparency and simplicity, while avoid their disadvantage in accuracy. The zeros of Chebyshev polynomials having the highest allowable order will be utilized as the sampling candidates to improve stability and accuracy of the approximation. In the numerical process, a space-filling scheme is used to generate the initial set of samples, and then an incremental method based on the maximin principle is established to select more samples from all candidates. At the same time, the order of HOPSM is sequentially updated by using an order incremental scheme, to adaptively increase the polynomial order along with the increase of the sample size. After the order increment, the polynomial with the largest adjusted R-square is determined as the final HOPSM. Several typical test functions and two engineering applications are used to demonstrate the effectiveness of the proposed surrogate modeling method.

On radially symmetric vibrations of circular sandwich plates of non-uniform thickness

In this paper, the free axisymmetric vibrations of circular sandwich plates with relatively stiff core of parabolically varying thickness have been presented on the basis of first order shear deformation theory. For the present model, the plate is assumed to be symmetric with respect to the middle surface. The facings are taken of the same material and of the same thickness. Due to parabolic variation in thickness of the core, facings take the shape of paraboloid of revolution and membrane forces of facings contribute to the transverse shear and bending of the core of sandwich plate. The equations of motion have been derived by Hamilton׳s energy principle. The first three natural frequencies for clamped, simply supported and free edge conditions have been obtained using a differential quadrature method, taking the grid points as zeros of Chebyshev polynomials. The effect of various plate parameters such as taper parameter, facing thickness and core thickness on the frequency parameter has been investigated. Comparison of results for some special cases with published results obtained from other approximate methods has been presented which shows an excellent agreement. Mode shapes for specified plates have been plotted.

A numerical scheme for solving differential equations with space and time-fractional coordinate derivatives

In this paper, we apply a numerical scheme for solving fractional differential equations. Our approach is based on an operational matrix of fractional Riemann-Liouville integration with Legendre basis and zeros of Chebyshev polynomials. In this framework the fractional Burgers equation, modified fractional Korteweg-de Vries equation and fractional wave equation are considered as prototype examples. The results reveal that the present method is very effective and accurate.

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.

An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels

This paper proposes a new design optimization framework for suspension systems considering the kinematic characteristics, such as the camber angle, caster angle, kingpin inclination angle, and toe angle in the presence of uncertainties. The coordinates of rear inner hardpoints of upper control arm and lower control arm of double wishbone suspension are considered as the design variables, as well as the uncertain parameters. In this way, the actual values of the design variables will vary surrounding their nominal values. The variations result in uncertainties that are described as interval variables with lower and upper bounds. The kinematic model of the suspension is developed in software ADAMS. A high-order response surface model using the zeros of Chebyshev polynomials as sampling points is established, termed as Chebyshev metamodel, to approximate the kinematic model. The Chebyshev meta-model is expected to provide higher approximation accuracy. Interval uncertain optimization problems usually involve a nested computationally expensive double-loop optimization process, in which the inner loop optimization is to calculate the bounds of the interval design functions, while the outer loop is to search the optimum for the deterministic optimization problem. To reduce the computational cost, the interval arithmetic is introduced in the inner loop to improve computational efficiency without compromising numerical accuracy. The numerical results show the effectiveness of the proposed design method.

Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation

In this paper, the asymptotic convergence of cubic Hermite collocation method in continuous time for the parabolic partial differential equation is established of order Oh2. The linear combination of cubic Hermite basis taken as approximating function is evaluated using the zeros of Chebyshev polynomials as collocation points. The theoretical results are verified for two test problems.

THE SIMULTANEOUS APPROXIMATION OF QUASI-HERMITE INTERPOLATION ON THE WEIGHTED MEAN NORM

We research the simultaneous approximation problem of the quasi-Hermite interpolation that based on the zeros of Chebyshev polynomials under weighted Lp-norm where our estimation is sharp.

Vibration analysis of non-homogeneous circular plate of nonlinear thickness variation by differential quadrature method

Free axisymmetric vibrations of non-homogeneous isotropic circular plates of nonlinear thickness variation have been analyzed on the basis of classical plate theory employing the differential quadrature (DQ) method. The non-homogeneity is assumed to arise due to the variation in Young's modulus and also the density of the plate material. The first three natural frequencies have been obtained for clamped, simply supported and free edge conditions, taking grid points as zeros of Chebyshev polynomials. The effect of non-homogeneity and thickness variation on natural frequencies of vibration has been investigated for the first three modes of vibration. The results for linear as well as parabolic thickness variations have been obtained as special cases. Comparison studies have been carried out to establish the accuracy and versatility of the present DQ method.

On the rate of weightedL p convergence by the Grünwald interpolatory operators

This paper gives the weighted Lp convergence rate estimations of the Grunwald interpolatory polynomials based on the zeros of Chebyshev polynomials of the first kind, and proves that the order of the estimations is optimal for p≥1.

Maximum entropy method and equilibrium charge state distributions

The maximum entropy method is implemented in order to describe equilibrium distributions arising in beam-foil spectroscopy. Since there are very few charge states involved, the usual moment conditions, based on simple powers xi, give rise to severe numerical difficulties already for three moments and cannot be applied in these systems. Earlier devised methods, based on Lagrange interpolation polynomials Qi(x) with abscissae chosen as zeros of Chebyshev polynomials in the interval being studied, are adapted and implemented for the present problems of charge state distributions. A reduced variable x=(q-(q))/(q), where q is the charge and (q) the mean charge, is chosen. Using the method described above calculations of equilibrium charge state distributions for copper ions (exit energy range 0.559-2.304 MeV u-1) colliding with carbon foils are carried out in order to exhibit the usability of the method. The new moment conditions associated to the Qi(x) provide a framework for a systematic analysis of equilibrium distributions.

Zeros of Chebyshev polynomials in Markov systems

We prove, under certain additional assumptions, that a Markov system has dense span if and only if the zeros of the associated Chebyshev polynomials are dense.

On the approximation of the continuous functions by some Hermite-Fejer interpolation polynomials

The object of this note is to improve Some wellknown results, which are related with the approximation problems of the continuous functions by Hermite-Fejer interpolation which based on the zeros of Chebyshev polynomials of the first or second kind.