谱表示法模拟风场的误差分析

研究了原型谱表示法模拟的非各态历经性多变量风场的统计矩的时域估计值和目标值之间误差的概率描述。基于原型谱表示法的模拟公式，以三变量风场为例，导出了模拟结果的均值、相关函数、功率谱密度函数和根方差等四项统计特征的单样本时域估计表达式，它们是随机变量或随机过程。运用概率论的计算方法，推导出了上述随机变量或过程的前二阶矩的解析表达式，得到了模拟风场的统计特征时域估计的偏度误差和随机误差。将三变量过程的结果加以推广，给出了误差计算的通式。通过算例中统计误差值和理论误差值的对比，验证解析解的正确性。探讨了可能的降低随机误差的方法。求得的误差闭合解将有利于结合误差传播理论进行可靠性分析。     

Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function. I. Theory

We approach the perturbative solution to the diffusion equation for the case of absorbing inclusions embedded in a heterogeneous scattering medium by using general properties of the radiative transfer equation and the solution of the Fredholm equation of the second kind given by the Neumann series. The terms of the Neumann series are used to obtain the expression of the moments of the generalized temporal point-spread function derived in transport theory. The moments are calculated independently by using Monte Carlo simulations for validation of the theory. While the mixed moments are correctly derived from the theory by using the solution of the diffusion equation in the geometry of interest, in order to obtain the self moments we should reframe the problem in transport theory and use a suitable solution of the radiative transfer equation for the calculation of the multiple integrals of the corresponding Neumann series. Since the rigorous theory leads to impractical formulas, in order to simplify and speed up the calculation of the self moments, we propose a heuristic method based on the calculation of only a single integral and some scaling parameters. We also propose simple quadrature rules for the calculation of the mixed moments for speeding up the computation of perturbations due to multiple defects. The theory can be developed in the continuous-wave domain, the time domain, and the frequency domain. In a companion paper [J. Opt. Soc. Am. A23, 2119 (2006)] we discuss the conditions of applicability of the theory in practical cases found in diffuse optical imaging of biological tissues.     

On the existence of maximum entropy distributions with four and more assigned moments

It is known that the probability distribution satisfy the Maximum Entropy Principle (MEP) if the available data consist in four moments of probability density function. Two problems are typically associated with use of MEP: the definition of the range of acceptable values for the moments M_i; the evaluation of the coefficients a_j. Both problems have already been accurately resolved by analytical procedures when the first two moments of the distribution are known.

In this work, the analytical solution in the case of four known moments is provided and a criterion for confronting the general case (whatever the number of known moments) is expounded. The first four moments are expressed in nondimensional form through the expectation and the coefficients of variation, skewness and kurtosis. The range of their acceptable values is obtained from the analytical properties of the differential equations which govern the problem and from the Schwarz inequality.     

A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data

A method is presented to estimate the process capability index (PCI) for a set of non‐normal data from its first four moments. It is assumed that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties. The probability density function of non‐normal data is expressed in Chebyshev–Hermite polynomials up to tenth order from the first four moments. An effective range, defined as the value for which a pre‐determined percentage of data falls within the range, is solved numerically from the derived cumulative distribution function. The PCI with a specified limit is hence obtained from the effective range. Compared with some other existing methods, the present method gives a more accurate PCI estimation and shows less sensitivity to sample size. A simple algebraic equation for the effective range, derived from the least‐square fitting to the numerically solved results, is also proposed for PCI estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

Improved beam propagation method equations

Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff-Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens.     

Exact spread function for a pulsed collimated beam in a medium with small-angle scattering

A solution has been obtained for the spatial and temporal distribution function for a pulsed fully collimated beam propagating through a homogeneous medium with Gaussian small-angle scattering. The solution was obtained first by separation of the general problem into two plane problems, which results in a partial differential equation in three variables. A Fourier transform on two projected variables (one angular and one spatial) and a Laplace transform on the projected temporal variable yielded a set of nonlinear differential equations, which were solved. A recursion relation for the moments of the distribution function was also obtained, and the software MATHEMATICA was used to evaluate these moments to high orders. The contractions on certain variables are also presented; they correspond to the solutions of less-general problems contained in the main problem. A change in the definition of the time-delay produces a remarkable change in the structure of the equations. These solutions should be quite useful for lidar studies in atmospheric and oceanic optics, x-ray and radio-wave scattering in the atmosphere and interstellar medium, and in medical physics.     

An orthonormal basis functions method for moment problems

In this paper, a new computational method is developed to recover an unknown function from its moments with respect to general kernel functions. By using the Gram–Schmidt orthonormalization technique, our method is shown to be efficient and can be interpreted as a generalization of the Talenti method. Convergence and error estimates are also discussed. For the purposes of verification and application, the method is applied to solve both Cauchy problem for Laplace equation and a Fredholm integral equation of the first kind.     

Cell renormalized FPK equation for stochastic non-linear systems

The Fokker–Planck–Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.     

Moment-matching method for extraction of asymmetric beam jitters and bore sight errors in simulations and experiments with actively illuminated satellites of small physical cross section

Optical returns from remote resident space-based objects such as satellites suffer from pointing and tracking errors. In a previously reported paper [Appl. Opt.46, 5608 (2007)APOPAI0003-693510.1364/AO.46.005608], we developed a moment-matching technique that used the statistics of time series of these optical returns to extract information about bore sight and symmetric beam jitter errors (symmetric here implies that the standard deviations of the jitter measured along two orthogonal axes, perpendicular to the line of sight, are equal). In this paper, we extend that method to cover the case of asymmetric beam jitter and bore sight. The asymmetric beam jitter may be due to the combination of symmetric atmospheric turbulence beam jitter and optical beam train jitter. In addition, if a tracking control system is operating, even the residual atmospheric tracking jitter could be asymmetric because the power spectrum is different for the slewing direction compared to the cross-track direction. Analysis of the problem has produced a set of nonlinear equations that can be reduced to a single but much higher-order nonlinear equation in terms of one of the jitter variances. After solving for that jitter, all the equations can be solved to extract all jitter and bore sight errors. The method has been verified by using simulations and then tested on experimental data. In order to develop this method, we derived analytical expressions for the probability density function and the moments of the received total intensity. The results reported here are valid for satellites of small physical cross section, or else those with retroreflectors that dominate the signal return. The results are, in general, applicable to the theory of noncircular Gaussian speckle with a coherent background.     

Solution of the Pontriagin–Vitt equation for the moments of time to first passage of the randomly accelerated particle by the finite element method

The Pontriagin–Vitt equation governing the mean of the time of first passage of a randomly accelerated particles has been studied extensively by Franklin and Rodemich.¹ In their paper is presented the analytic solution for the two-sided barrier problem and solutions by several finite difference procedures. This note demonstrates solution of the problem by a Petrov–Galerkin finite element method using upstream weighting functions,² shown to give rapidly convergent results. In addition, the equation is generalized to include higher statistical moments, and solutions for the first few ordinary moments are reported.     

Recognition of motion-blurred images by use of the method of moments

Image motion causes a blur that changes features of objects and therefore complicates the task of automatic recognition. In this work we develop two recognition methods for motion-blurred images. For the first method we assume that the motion function and direction during the exposure are given. We develop the relation between the blurred-image moments and the original-image moments based on the motion function only. The recognition is carried out by comparing the moments of the restored image against the moments of the image database. In the second method the motion function is not known. In this case image moments that are invariant with respect to the motion blur are identified, and only these moments are used for recognition. The advantage of the suggested methods is that no time-consuming image restoration is required prior to recognition.     

Nonlinear vibration and postbuckling of unsymmetrically laminated imperfect shallow cylindrical panels with mixed boundary conditions resting on elastic foundation

This study is analytically concerned with the titled problem. The rotational stiffness variation is assumed to be identical along opposite edges. The panel is subjected to inplane edge forces but not tangential boundary forces. A unified approximate solution is formulated on the basis of the dynamic Marguerre-type equations. The edge condition for the rotational stiffness variation is satisfied by expansion of the edge bending moments and the varying rotational edgerestraint coefficients into generalized Fourier series. These moments are also replaced by an equivalent lateral pressure near these edges. The Galerkin procedure furnishes an equation for the time function which is solved by the method of perturbation. In the postbuckling case the equation reduces to a relation between the postbuckling load and the maximum deflection. Numerical results for nonlinear vibration and postbuckling behavior of orthotropic and unsymmetrically laminated angle-ply cylindrical panels are presented graphically for different parameters and compared with available data.     

Improving mass conservation in simulation of incompressible flows

We propose a technique for improving mass‐conservation features of fractional step schemes applied to incompressible flows. The method is illustrated by using a Lagrangian fluid formulation, where the mass loss effects are particularly apparent. However, the methodology is general and could be used for fixed grid approaches. The idea consists in reflecting the incompressibility condition already in the intermediate velocity. This is achieved by predicting the end‐of‐step pressure and using this prediction in the fractional momentum equation. The resulting intermediate velocity field is thus much closer to the final incompressible one than that of the standard fractional step scheme. In turn, the predicted pressure can be used as the boundary condition necessary for the solution of the pressure Poisson equation in case a continuous Laplacian matrix is employed. Using this approximation of the end‐of‐step incompressible pressure as the essential boundary condition considerably improves the conservation of mass, specially for the free surface flows of fluids with low viscosity. The pressure prediction does not require the resolution of any additional equations system. The efficiency of the method is shown in two examples. The first one shows the performance of the method with respect to mass conservation. The second one tests the method in a challenging fluid–structure interaction benchmark, which can be naturally resolved by using the presented approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.     

Chloride diffusion coefficient: A comparison between impedance spectroscopy and electrokinetic tests

This paper proposes to compare several experimental ways of obtaining the chloride diffusion coefficient through saturated porous materials. The first method is based on the application of the Nernst-Einstein equation for which the conductivity of the saturated sample is measured by impedance spectroscopy for two kinds of materials: inert samples of TiO₂, and concretes based on type I and type V cements. The second method is a migration test in which the flux of chloride measured upstream allows calculating the diffusion coefficient by means of the Nernst-Planck equation. In the third case, the diffusion coefficient is calculated by current measurements in an equivalent configuration as the second method. It is shown that the formation factor does not vary neither with the ionic strength of the saturation solution, nor with the change in the pore solution constituents when the material is without mineral additions. The CEM-V concrete exhibits a specific behavior with a strong influence of the addition of chloride in its pore solution on the formation factor. The diffusion coefficients calculated with the three methods are in good agreement provided the metrology of the experiments is carefully controlled.     

On the convergence of iterative solutions of the integral magnetic field equation

The solution of the integral magnetic field equation by numerical iteration is discussed. Using a simple linear example, it is shown rigorously that relaxation techniques are required to obtain convergence. The range of permissible relaxation parameters is examined and that particular value which yields most rapid convergence is determined. An iterative solution to a simple nonlinear problem is shown to converge rapidly if the relaxation parameter is adjusted appropriately at each step in the iteration. For the general case of a saturable media of complex geometric shape, a relaxation matrix method is proposed in order to achieve rapid convergence.     

Beam moments by a general finite difference formulation in cable-assisted bridge analysis

This paper presents a general finite difference equation for the determination of beam bending moments. The scope of its application to the solution of cable-assisted bridge problems is discussed. Its derivation and relation to previous difference formulations is explained. It is found that some previous formulations can be conveniently represented by the proposed general difference equation through the variation of a general difference coefficient. The accuracy of the proposed equation is illustrated with numerical examples.     

Recent advances in frequency domain techniques for electromagneticscattering problems

Some recent applications of numerical techniques for the analysis of electromagnetic scattering problems involving several general classes of geometries are reviewed. The solution procedures are all formulated directly in the frequency domain, using the method of moments or the finite-element method, in terms of an integral equation and/or a partial differential equation. Geometrical configurations discussed include three-dimensional conducting scatterers and two- and three-dimensional homogeneous and inhomogeneous dielectric scatterers     

Padé approximants for the numerical solution of singular integral equations

Summary A method for accelerating the convergence of the numerical solution of a singular integral equation, based on Padé Approximants, is given in this paper. At first the general form of the Padé Table and of the “epsilon” algorithm are presented. Taking into consideration the classical quadrature method, based on the Gauss-Jacobi quadrature rule, an approximate formula is derived for the unknown density function of the Cauchy-type singular integral equation or of the equivalent Fredholm integral equation. In this formula applying the “epsilon” algorithm to the solution for the stress intensity factors, the convergence is achieved after a few operations. The number of numerical operations required for the determination of stress intensity factors is considerable reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method.     

Estimating Standard Deviations for Inventory Control

This article investigates two alternative methods for estimating standard deviations of demand in inventory control systems. One method requires only the average demand rate (in units), while the second is based on the average number of units requested per customer order and the average customer order rate. For a sample of 489 products with 48 months of demand data, the second method pro vides more accurate estimates of standard deviations than does the first, with the largest improvement in the low-and medium-volume categories. For one numerical example, the improvement offers an expected reduction in total inventory-associated costs of 3.94 percent.