A univariate Chebyshev polynomials method for structural systems with interval uncertainty

This paper presents an effective univariate Chebyshev polynomials method (UCM) for interval bounds estimation of uncertain structures with unknown-but-bounded parameters. The interpolation points required by the conventional collocation methods to generate the surrogate model are the tensor product of each one-dimensional (1D) interpolating point. Therefore, the computational cost is expensive for uncertain structures containing more interval parameters. To deal with this issue, the univariate decomposition is derived through the higher-order Taylor expansion. The structural system is decomposed into a sum of several univariate subsystems, where each subsystem only involves one uncertain parameter and replaces the other parameters with their midpoint value. Then the Chebyshev polynomials are utilized to fit the subsystems, in which the coefficients of these subsystems are confirmed only using the linear combination of 1D interpolation points. Next, a surrogate model of the actual structural system composed of explicit univariate Chebyshev functions is established. Finally, the extremum of each univariate function that is obtained by the scanning method is substituted into the surrogate model to determine the interval ranges of the uncertain structures. Numerical analysis is conducted to validate the accuracy and effectiveness of the proposed method.     

Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials

The effects of damaged boundaries on natural frequencies and critical loads of beams and columns of variable cross section with conservative and non-conservative loads are investigated. The shifted Chebyshev polynomials are used to solve the one-dimensional transverse vibration problem, in which the ordinary differential equation is reduced to an algebraic eigenvalue problem. The advantages of this method are that it is easily employed in a symbolic form and that the number of polynomials may be adjusted to attain convergence. In the present study, the damaged boundary is modeled by linear translational and torsional springs, and the effects of the damage severity on the natural frequencies are studied. It is shown that as the amount of damage increases the natural frequencies decrease at rates which vary with the mode number. The method is applied to the instability problems of both uniform and uniformly tapered beams with and without follower forces, and the results for the undamaged cases show agreement when compared with results available in the literature. Convergence studies are carried out to determine the number of Chebyshev polynomials that should be used in the proposed method.     

Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials

This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay‐differential equations (DDEs) with time‐periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the ‘infinite‐dimensional Floquet transition matrix U’. Two different formulas for the computation of the approximate U, whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second‐order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time‐periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance‐modulated turning. The results indicate that this method is an effective way to study the stability of time‐periodic DDEs. Copyright © 2004 John Wiley & Sons, Ltd.     

分数阶电报方程的Chebyshev多项式数值解法研究

分数阶电报方程作为通信工程中的一类重要方程,在实际应用中往往很难求得解析解,因而对其进行数值求解就显得至关重要．为了求得分数阶电报方程的数值解,本文借助Chebyshev多项式函数构造相应的微分算子矩阵,并结合Tau方法将待求方程转化为非线性代数方程组,然后对该方程组进行数值离散求解,最后给出的数值算例也验证了该方法的可行性及有效性．     

一类带变系数的空间分数阶偏微分方程的Chebyshev拟谱分法

分数阶微分方程已经广泛地应用于工程等各个领域。在本文中,我们针对一类带变系数的空间分数阶偏微分方程,提出了一种Chebyshev拟谱的数值方法,其中分数阶导数是由Caputo分数阶导数定义。该方法能将空间分数阶偏微分方程转化为一个常微分方程,然后在时间上用有限差分方法离散。数值实验表明该方法是有效的。     

A new variable sampling control scheme at fixed times for monitoring the process dispersion

The variable sampling rate (VSR) schemes for detecting the shift in process mean have been extensively analyzed; however, adding the VSR feature to the control charts for monitoring process dispersion has not been thoroughly investigated. In this research, a novel VSR control scheme, sequential exponentially weighted moving average inverse normal transformation (EWMA INT) at fixed times chart (called (SEIFT) chart), which integrates the sequential EWMA scheme at fix times with the INT statistic, is proposed to detect both the increase and decrease in process dispersion. Moreover, the sample size at each sampling time is also allowed to vary. The Markov chain method is used to evaluate the performance of this new control chart. Numerical analysis reveals that this SEIFT chart gives significant improvement on detection ability than the fixed sampling rate schemes. Compared with other control schemes, the good properties of the INT statistic makes this SEIFT chart easy to design and convenient to implement. Copyright © 2009 John Wiley & Sons, Ltd.     

基于第二类切比雪夫多项式的语音线性谱对参数的计算

对线性谱对(LSP)参数的计算方法提出改进算法，该算法利用第二类切比雪夫多项式的迭代性质对初始函数降阶。理论分析表明，改进算法可以获得更简洁的数学表达式。实验结果显示，改进算法中基本消除了乘法运算，同时随着线性预测分析阶数的增加可以进一步降低算法复杂度。     

A new algorithm for the normal distribution function

In this paper new algorithms for the rapid, efficient and accurate evaluation of the standard normal integral and its tail are developed. It is shown how the accuracy of the computation can easily be improved so as to achieve machine accuracy for the particular computer being used.     

A Chebyshev collocation based sequential matrix exponential method for the generalized density evolution equation

In perspective of global approximation, this study presents a numerical method for the generalized density evolution equation (GDEE) based on spectral collocation. A sequential matrix exponential solution based on the Chebyshev collocation points is derived in consideration of the coefficient or velocity term of GDEE being constant in each time step, then the numerical procedure could be successively implemented without implicit or explicit difference schemes. The results of three numerical examples indicate that the solutions in terms of the sequential matrix exponential method for GDEE have good agreement with the analytical results or Monte Carlo simulations. For sufficiently smooth cases, there need no more than one hundred representative points to achieve a satisfied solution by the proposed method, whereas for the case in presence of severe discontinuity a few more sampling points are required to keep numerical stability and accuracy.     

A Chebyshev collocation method for computing the eigenvalues of the Laplacian

Chebyshev collocation techniques are developed in this paper to compute the eigenvalues of the Laplacian based on a boundary integral formulation for two-dimensional domains with piecewise smooth boundaries. Unlike the traditional domain methods (for example, the finite element method) which discretizes the eigenfunctions on the two-dimensional domain, only a one-dimensional function defined on the boundary is discretized. Global expansions in terms of Chebyshev polynomials are used in each smooth piece of the boundary to solve the integral equation. Comparing with the boundary element method, this method obtains higher accuracy for a smaller discretized matrix. Finally, an efficient algorithm for generating the discretized matrix (say, n × n) is developed that requires only O(n² log n) operations.     

Automated sampling decision scheme for the AVM system

Reducing the sampling rate to as low as possible is a high priority for many factories to reduce production cost. Automatic virtual metrology based intelligent sampling decision (ISD) scheme had been previously developed for reducing the sampling rate and sustaining the virtual metrology (VM) accuracy. However, the desired sampling rate of the ISD scheme is fixed and set manually. Hence, whenever the VM accuracy gets worse, it cannot adaptively increase the default sampling rate in the ISD scheme. As a consequence, it would take more time to collect enough samples for improving the VM accuracy. Moreover, when the VM accuracy performs well all the time, it cannot automatically decrease the default sampling rate in ISD, which may result in unnecessary waste. Accordingly, this paper proposes an automated sampling decision (ASD) scheme to adaptively and automatically modify the sampling rate online and in real time for continuous improvement. The ASD scheme can monitor the VM accuracy online as well as update the VM models in real time for maintaining the VM accuracy when the VM accuracy becomes poor. Also, the ASD scheme can automatically reduce the sampling rate while the VM accuracy performs well.     

An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field

We establish a formula for the number of irreducible polynomialsf(x) over the binary fieldF ₂ of given degreen 2 for which the coefficient ofx ^n-1 and ofx is equal to 1. This formula shows that the number of such polynomials is positive for alln 2 withn 3. These polynomials can be applied in a construction of irreducible self-reciprocal polynomials overF ₂ of arbitrarily large degrees.     

A new efficient factorization algorithm for polynomials over small finite fields

We present a new deterministic factorization algorithm for polynomials over a finite prime fieldF _p . As in other factorization algorithms for polynomials over finite fields such as the Berlekamp algorithm, the key step is the linearization of the factorization problem, i.e., the reduction of the problem to a system of linear equations. The theoretical justification for our algorithm is based on a study of the differential equationy ^(p –1)+y ^p =0 of orderp–1 in the rational function fieldF _p(x). In the casep=2 the new algorithm is more efficient than the Berlekamp algorithm since there is no set-up cost for the coefficient matrix of the system of linear equations.     

A new plate bending element based on orthogonal polynomials expansion of the curvature field

A new approach to derive finite elements for the thin plate model is presented. The proposed method approximates compatible Kirchhoff formulations by means of orthogonal polynomials expansion of the curvature field depending only on the element boundary traces. With respect to the compatible formulation the proposed method produces elements that beneficially underestimate the deformation energy. A simple triangular element is developed and investigated from both the theoretical and the numerical point of view and numerically compared with other two well‐known elements. Copyright © 2007 John Wiley & Sons, Ltd.     

A resubmitted sampling scheme by variables inspection for controlling lot fraction nonconforming

The paper attempts to develop a resubmitted sampling scheme by variables inspection for controlling lot fraction nonconforming when the quality characteristic follows a normal distribution and has two-sided specification limits. In this paper, the plan parameters are determined by the classical two-point condition on the operating characteristic curve, which will satisfy the quality requirements and allowable risks by the producer and the consumer simultaneously. The behaviour of the proposed sampling plan is discussed and compared with the conventional single sampling plan by variables. The proposed plan requires smaller sample size for inspection with the same protection to the producer and the consumer especially when the quality of the submitted lot is good enough. Tables of the plan parameters under several selected quality requirements and risks are provided for practical applications, and the operating procedure is also presented and illustrated with an example.     

Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

This article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.     

A boundary meshless method using Chebyshev interpolation and trigonometric basis function for solving heat conduction problems

A boundary meshless method has been developed to solve the heat conduction equations through the use of a newly established two‐stage approximation scheme and a trigonometric series expansion scheme to approximate the particular solution and fundamental solution, respectively. As a result, no fundamental solution is required and the closed form of approximate particular solution is easy to obtain. The effectiveness of the proposed computational scheme is demonstrated by several examples in 2D and 3D. We also compare our proposed method with the finite‐difference method and the other meshless method showed in ?arler and Vertnik (Comput. Math. Appl. 2006; 51 :1269–1282). Excellent numerical results have been observed. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of the two-dimensional wave equation by using wave polynomials

The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.     

实验模态分析中一种改进的傅氏域离散正交多项式

针对在借助有理分式多项式构建结构频响函数的分析模型的过程中由于负频率引入的虚拟测点将导致方程式在多项式阶次较高时出现病态的问题,对实域离散点列上的正交多项式进行了推广,得到傅氏域离散点列上的正交多项式.该多项式不仅可避免由负频率引入的冗余计算,而且亦使方程式得到解耦,从而使该方法更为高效.