Study on the elastic‐plastic buckling strength of plate structures

Abstract

Shifted Legendre polynomials are applied to solve the state equations of linear system. The computation procedure is greatly simplified by introducing the operational matrix for the integration of shifted Legendre vectors whose elements are shifted Legendre polynomials. The key of the method is that the state and forcing functions are expressed in terms of a series of shifted Legendre polynomials with expansion coefficients. Ordinary differential equations of state system are transformed into a series of algebraic equations of the shifted Legendre expansion coefficients and then are solved by employing the technique of matrix inverse. The methods of the computational algorithms are also investigated in order to simplify the calculation procedure and make the calculation convergent.     

Second-order statistics of integrated intensities and detected photons,the exact analysis I. Theory

Abstract

We express the exact probability density distribution function as the product of a gamma distribution and a series of associated Laguerre polynomials, with the expansion coefficients determined by moments of the integrated intensity. Orthogonal polynomials with respect to the exact probability distribution function are then expanded in similar fashion. These polynomials are then used to construct an expansion of the joint probability distribution function in the second-order photoelectron statistics. Since the polynomials are identical with the corresponding Laguerre polynomials when the exact probability distribution function is the gamma distribution, the new polynomials are generalized versions of the associated Laguerre polynomials. The joint photoelectron statistics may be studied with these new polynomials.     

Solutions of population balance equations by double Legendre polynomial transformation

Abstract

First‐order partial differential equations of population balance are solved by employing the Legendre polynomials. The key of the method is that the dependent variable of the population density function is assumed to be expressed by a double series of Legendre polynomials with respect to time and space variables. The approach algorithm is that a series of ordinary differential equations are obtained by making the Legendre transformation with respect to the space coordinate. The series of time‐function ordinary differential equations are further transformed into algebraic equations of expansion coefficients with respect to time. The expansion coefficients of the Legendre polynomials are obtained by solving matrix equations which represent the series of algebraic equations. Illustrative examples are given, and the computational results are compared with those of other numerical values given in the literature. Satisfactory agreements are obtained.     

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.     

A Method of Measuring Current–Voltage Characteristics Based on the Hammerstein–Chebyshev Model

A method of measuring the current–voltage characteristics of nonlinear components is proposed, which involves applying a sinusoidal voltage to the object being measured, finding the Fourier spectrum to read out the output current, and a calculation using the spectrum of the coefficients of the expansion of the measured characteristic in Chebyshev polynomials. The properties of the Hammerstein–Chebyshev model are considered, a block diagram of a measuring instrument is presented, and recommendations are made for minimizing the systematic errors due to approximating their characteristics by Chebyshev polynomials.     

A higher-order theory for functionally graded beams based on Legendre's polynomial expansion

In this article a higher-order theory for functionally graded beams based on the expansion of the two-dimensional (2D) equations of elasticity for functionally graded materials into Fourier series in terms of Legendre's polynomials is presented. Starting from the 2D equations of elasticity, the stress and strain tensors, displacement, traction, and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way, the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients are obtained. In particular, the first- and second-order approximations of the exact infinite dimensional beam theory are considered in more detail. The obtained boundary-value problems are solved by the finite element method with MATHEMATICA, MATLAB, and COMSOL multiphysics software. Numerical results are presented and discussed.     

The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

A simple and rapid numerical quadrature is developed for the evaluation of the diffraction-based optical transfer function for on-axis systems, using a Tchebycheff polynomial expansion of the pupil function. The integration of the autocorrelation integral of the pupil function is replaced by once and for all evaluations of the cross-correlation of respective polynomials. However, the expansion coefficients themselves of the Tchebycheff series are linear sums of the sampled pupil function and thus a series of coefficients can be generated that weight the pupil function at various points to give the resultant OTF. The coefficients for a tenth-order Tchebycheff expansion are included in the paper, and a set of tables of OTF values calculated with these coefficients and 64 2 64 Gaussian quadrature for a diffraction-limited system, and one with one wavelength of primary spherical aberration.     

On a functional approximation for inversion of Laplace transforms

Abstract

A functional representation for inversion of the Laplace transform of a function is considered. The function is given in Laguerre polynomials expansion. The coefficients of the polynomials are in terms of weighted moments which are directly determined from the Laplace transform. The applications to rational and irrational Laplace transforms are presented to illustrate the satisfactory results that the method provides.     

Method for surface reconstruction from slope or curvature measurements of rectangular areas

The problem of surface reconstruction from slope or curvature measurements of rectangular areas has been examined by use of the orthogonality property of Legendre polynomials. A relation is given between the integrals involving slope or curvature data and the coefficients used in the expansion of the surface with such polynomials. It is shown that those coefficients are retrieved independently of each other. The efficiency and stability of the method have been tested with numerical examples.     

通信管道静摩擦系数标准器检定装置计量标准研究

介绍了通信管道静摩擦系数标准器检定装置计量标准的量传关系、组成、测量能力、主要技术参数、工作原理和检定方法,对可能影响检定装置测量结果准确性的因素进行了研究,确定了检定装置所用的标准试棒、采用的拉伸速度和对测量不确定度影响的因素。结果表明:检定装置所用的标准试棒与标准试件组成的摩擦副的静摩擦系数为0.120±0.005（拉伸速度为1.000mm/s）;温度和湿度对静摩擦系数的影响可以忽略;拉伸速度、重力加速度、标准试棒纵向中心线与推拉力计测头中心线重合度对测量结果均会造成影响,应作为B类不确定度分量加以评定。     

Expressing the coefficients of the expansion of a scattering indicatrix through Mie coefficients

Equations are given expressingthe coefficients of the expansion of a spherical-particle scattering indicatrix by Legendre polynomials directly through Mie coefficients. The equations are converted to a form suitable for use in a computer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 4, pp. 648–650, October, 1978.     

基于正交展开方法的Duffing振子随机最优控制

基于多维Hermite多项式的经典均相混沌展开,考察了Duffing振子随机最优多项式控制的正交展开方法,阐明了多项式系数演化与振子系统反应、最优控制力概率特性之间的联系.系统输入采用Karhunen-Loève展开表现的随机地震动.为降低混求解规模,引入位移-速度范数准则,发展了自适应混沌多项式展开策略.同时,基于Lyapunov稳定条件设计控制器的控制增益参数.数值算例分析表明,受控后系统位移和加速度的均方特征得到改善、振子系统的非线性程度减小,基于混沌多项式展开的最优控制方法能明显降低系统的随机涨落和显著改善系统的非线性反应性态.     

A Note on the Spherical Harmonic Expansion of the Mie Scattering Kernel

Abstract

Three methods of obtaining the Mie scattering function are considered and compared. Legendre polynomials and Clebsh-Gordon coefficients are employed and an exact expression for the expansion of the Mie kernel in spherical harmonics is given.     

Wavefront expansion basis functions and their relationships

Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.     

An adaptive strategy for <Emphasis Type="Italic">hp</Emphasis>-FEM based on testing for analyticity

We present an hp-adaptive strategy that is based on estimating the decay of the expansion coefficients when a function is expanded in L ²-orthogonal polynomials on a triangle or a tetrahedron. We justify this approach by showing that the decay of the coefficients is exponential if and only if the function is analytic. Numerical examples illustrate the performance of this approach, and we compare it with two other hp-adaptive strategies.     

A new view on the Peřina-Mišta form of the Glauber-Sudershan P function

Abstract

The ‘regularized’ form of the Glauber-Sudershan P function in terms of a series of Laguerre polynomials proposed by Perina and Mista is reconsidered. It is shown that the corresponding expansion coefficients result from averaging sampling functions well known from optical homodyne tomography with respect to the quadrature distribution of the signal field. An illustrative example of a nonclassical state is considered.     

Calibration of Heat Stress Monitor and its Measurement Uncertainty

Wet-bulb globe temperature (WBGT) equation is a heat stress index that gives information for the workers in the industrial areas. WBGT equation is described in ISO Standard 7243 (ISO 7243 in Hot environments—estimation of the heat stress on working man, based on the WBGT index, ISO, Geneva, 1982). WBGT is the result of the combined quantitative effects of the natural wet-bulb temperature, dry-bulb temperature, and air temperature. WBGT is a calculated parameter. WBGT uses input estimates, and heat stress monitor measures these quantities. In this study, the calibration method of a heat stress monitor is described, and the model function for measurement uncertainty is given. Sensitivity coefficients were derived according to GUM. Two-pressure humidity generators were used to generate a controlled environment. Heat stress monitor was calibrated inside of the generator. Two-pressure humidity generator, which is located in Turkish Standard Institution, was used as the reference device. This device is traceable to national standards. Two-pressure humidity generator includes reference temperature Pt-100 sensors. The reference sensor was sheltered with a wet wick for the calibration of natural wet-bulb thermometer. The reference sensor was centred into a black globe that has got 150 mm diameter for the calibration of the black globe thermometer.     

Comparison among three boundary element methods for torsion problems: CPM,CVBEM, LEM

This paper provides solutions for De Saint-Venant torsion problem on a beam with arbitrary and uniform cross-section. In particular three methods framed into complex analysis have been considered: Complex Polynomial Method (CPM), Complex Variable Boundary Element Method (CVBEM) and Line Element-less Method (LEM), recently proposed. CPM involves the expansion of a complex potential in Taylor series, computing the unknown coefficients by means of collocation points on the boundary. CVBEM takes advantage of Cauchy's integral formula that returns the solution of Laplace equation when mixed boundary conditions on both real and imaginary parts of the complex potential are known. LEM introduces the expansion in the double-ended Laurent series involving harmonic polynomials, proposing an element-free weak form procedure, by imposing that the square of the net flux of the shear stress across the border is minimized with respect to the series coefficients. These methods have been compared with respect to numerical efficiency and accuracy. Numerical results have been correlated with analytical and approximate solutions that can be already found in literature.     

射频功率器件宽带多层TRL校准算法研究

多层TRL校准是微波毫米波大功率器件测试中常用的一种校准方法。针对经典TRL校准方法在大功率器件测试中易出现的误差系数相位跳变问题,提出了一种结合先验知识和动态最小化误差逼近的宽带TRL校准优化算法。基于国产AV3672矢量网络分析仪和负载牵引测试系统在3.8GHz及三次谐波对该算法的有效性进行了验证。实验结果证明,该算法有效地修正了误差系数相位跳变的问题,对器件大信号工作状态最优阻抗点的分析更为准确,并且算法复杂度没有大幅增加,可以应用为经典TRL算法的后续修正步骤,具备极强的实用性和通用性。     

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.