高斯流场中单自由度线性体系非高斯振动响应试验研究

对于风湍流等高斯分布流速场中的线性结构体系,当考虑荷载中脉动流速二次项的影响时,理论上其振动响应将呈现非高斯分布特性。基于调试得到的不同粗糙工况高斯流场,开展了单自由度线性体系顺流向振动响应测试,研究了单自由度线性体系加速度响应的非高斯分布特性,分析了粗糙度对响应非高斯成分的影响,讨论了三种常见非高斯概率密度逼近方法对响应的拟合效果。试验结果表明:试验高斯流场中单自由度线性体系的顺流向加速度响应主要呈现出尖峰非高斯分布特征,且随着紊流度的提高,响应非高斯性有增强的趋势;响应的非高斯概率密度宜采用高斯混合模型方法进行拟合。     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.     

Multivariate non-Gaussian process simulation based on HPM-JTM hybrid model

For the simulation of wind field or wind load, the Gaussian assumption is not applicable in some situations. Hence the simulation of non-Gaussian process becomes significant. In order to avoid the iteration and maintain a wide application range, a HPM-JTM hybrid model is proposed based on translation process. During the simulation, the correlation function of the related Gaussian process can be solved by analytical or numerical expression. Through a numerical example, the application of the model is presented. Results show that the model provides a reasonable estimation for the target case, which can be regarded as an appropriate candidate for the simulation of multivariate non-Gaussian process.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.     

A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process

Some widely used methodologies for simulation of non-Gaussian processes rely on translation process theory which imposes certain compatibility conditions between the non-Gaussian power spectral density function (PSDF) and the non-Gaussian probability density function (PDF) of the process. In many practical applications, the non-Gaussian PSDF and PDF are assigned arbitrarily; therefore, in general they can be incompatible. Several techniques to approximate such incompatible non-Gaussian PSDF/PDF pairs with a compatible pair have been proposed that involve either some iterative scheme on simulated sample functions or some general optimization approach. Although some of these techniques produce satisfactory results, they can be time consuming because of their nature. In this paper, a new iterative methodology is developed that estimates a non-Gaussian PSDF that: (a) is compatible with the prescribed non-Gaussian PDF, and (b) closely approximates the prescribed incompatible non-Gaussian PSDF. The corresponding underlying Gaussian PSDF is also determined. The basic idea is to iteratively upgrade the underlying Gaussian PSDF using the directly computed (through translation process theory) non-Gaussian PSDF at each iteration, rather than through expensive ensemble averaging of PSDFs computed from generated non-Gaussian sample functions. The proposed iterative scheme possesses two major advantages: it is conceptually very simple and it converges extremely fast with minimal computational effort. Once the underlying Gaussian PSDF is determined, generation of non-Gaussian sample functions is straightforward without any need for iterations. Numerical examples are provided demonstrating the capabilities of the methodology.     

Non-Gaussian models for stochastic mechanics

Memoryless transformations of Gaussian processes and transformations with memory of the Brownian and Lévy processes are used to represent general non-Gaussian processes. The transformations with memory are solutions of stochastic differential equations driven by Gaussian and Lévy white noises. The processes obtained by these transformations are referred to as non-Gaussian models. Methods are developed for calibrating these models to records or partial probabilistic characteristics of non-Gaussian processes. The solution of the model calibration problem is not unique. There are different non-Gaussian models that are equivalent in the sense that they are consistent with the available information on a non-Gaussian process. The response analysis of linear and non-linear oscillators subjected to equivalent non-Gaussian models shows that some response statistics are sensitive to the particular equivalent non-Gaussian model used to represent the input. This observation is relevant for applications because the choice of a particular non-Gaussian input model can result in inaccurate predictions of system performance.     

Non-Gaussian solution for random rocking of slender rigid block

This paper adopts a random vibration approach to study the response of the slender rigid block to seismic action. The problem is strongly non-linear because of (i) the restoring term and (ii) the quadratic dissipation of energy due to the inelastic impacts, modeled as an impulsive process. The excitation process is firstly assumed to be a Gaussian white noise; secondly, a non-stationary filtered Gaussian white noise is assumed to simulate seismic shaking more accurately. The solution of the associated Fokker-Planck equation in terms of moments of the response is obtained by means of a non-Gaussian closure technique, that enables the complete statistical definition of the approximated transient response process to be achieved. The mean upcrossing rates and the response spectra in terms of displacement are evaluated. The reliability of the solutions derived is assessed by comparing them with Monte Carlo simulations.     

Distribution-free multivariate process control based on log-linear modeling

This paper considers Statistical Process Control (SPC) when the process measurement is multivariate. In the literature, most existing multivariate SPC procedures assume that the in-control distribution of the multivariate process measurement is known and it is a Gaussian distribution. In applications, however, the measurement distribution is usually unknown and it needs to be estimated from data. Furthermore, multivariate measurements often do not follow a Gaussian distribution (e.g., cases when some measurement components are discrete). We demonstrate that results from conventional multivariate SPC procedures are usually unreliable when the data are non-Gaussian. Existing statistical tools for describing multivariate non-Gaussian data, or transforming the multivariate non-Gaussian data to multivariate Gaussian data, are limited, making appropriate multivariate SPC difficult in such cases. In this paper, we suggest a methodology for estimating the in-control multivariate measurement distribution when a set of in-control data is available, which is based on log-linear modeling and which takes into account the association structure among the measurement components. Based on this estimated in-control distribution, a multivariate CUSUM procedure for detecting shifts in the location parameter vector of the measurement distribution is also suggested for Phase II SPC. This procedure does not depend on the Gaussian distribution assumption; thus, it is appropriate to use for most multivariate SPC problems.     

Probability distributions of peaks and troughs of non-Gaussian random processes

This paper deals with the development of probability density functions applicable for peaks, troughs and peak-to-trough excursions of a non-Gaussian random process where the response of a non-linear system is represented in the form of Volterra's second-order functional series. The density functions of peaks and troughs are derived in closed form and presented separately. It is found that the probability density function applicable to peaks (and troughs) is equivalent to the density function of the envelope of a random process consisting of the sum of a narrow-band Gaussian process and sine wave having the same frequency. Furthermore, for a non-Gaussian random process for which the skewness of the distribution is less than 1.2, the density function of peaks (and troughs) can be approximately presented in the form of a Rayleigh distribution. The parameter of the Rayleigh distribution is given as a function of parameters representing the non-Gaussian characteristics. The results of comparisons between newly derived density functions and histograms of peaks, troughs and peak-to-trough excursions constructed from data with strong non-linear characteristics show that the distributions well represent the histograms for all cases.     

一种非白非高斯数据的数值仿真方法

使用复合抽样法,可以产生具有指定概率密度形式的加性分布非高斯序列。通过在极零图上直接指定数对极零点,可以实现定性色化低阶自回归滤波器设计。把非高斯激励序列通过自回归滤波器,即可得到非高斯信号处理仿真研究中频繁使用的非高斯有色序列。结合一组混合高斯有色数据数值仿真实例,演示了这一由复合抽样法加定性色化构成的非白非高斯数据快捷数值仿真方法的有效性。     

Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.     

Entanglement distillation from Gaussian input states by coherent photon addition

The entanglement between Gaussian entangled states can be increased by non-Gaussian operations. We design a new scheme, named coherent photon addition, which can coherently add one photon generated by a spontaneous parametric down-conversation process to Gaussian quadrature-entangled light pulses created by a non-degenerate optical parametric amplifier. This operation can increase the entanglement of input two-mode Gaussian states as an entanglement distillation, and provides us with a new method of non-Gaussian operation. This scheme can also help us to study the decoherence of adding one- to two-mode Gaussian states from coherent photon addition to normal photon addition.     

Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations

The analysis of many systems in optical communications and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical model of Gaussian beam propagation. We present a full vector analysis of Gaussian beam propagation by using the well-known method of the angular spectrum of plane waves. A Gaussian beam is assumed to traverse a charge-free, homogeneous, isotropic, linear, and nonmagnetic dielectric medium. The angular spectrum representation, in its vector form, is applied to a problem with a Gaussian intensity boundary condition. After some mathematical manipulation, each nonzero propagating electric field component is expressed in terms of a power-series expansion. Previous analytical work derived a power series for the transverse field, where the first term (zero order) in the expansion corresponds to the usual scalar paraxial approximation. We confirm this result and derive a corresponding longitudinal power series. We show that the leading longitudinal term is comparable in magnitude with the first transverse term above the scalar paraxial term, thus indicating that a full vector theory is required when going beyond the scalar paraxial approximation. In spite of the advantages of a compact analytical formalism, enabling rapid and accurate modeling of Gaussian beam systems, this approach has a notable drawback. The higher-order terms diverge at locations that are sufficiently far from the initial boundary, yielding unphysical results. Hence any meaningful use of the expansion approach calls for a careful study of its range of applicability. By considering the transition of a Gaussian wave from the paraxial to the spherical regime, we are able to derive a simple expression for the range within which the series produce numerically satisfying answers.     

基于高斯混合模型的非高斯随机振动幅值概率密度函数

针对非高斯振动信号的幅值概率密度函数难以用数学模型表述的问题,提出了基于高斯混合模型的非高斯概率密度函数表示方法。首先,基于时域样本信号得到非高斯振动信号的高阶矩估计值。其次,基于高斯随机过程偶次高阶矩之间的定量关系,结合二阶高斯混合模型建立方程组,求解得到混合模型中每个高斯分量的方差和权值。然后,将各高斯分量的权值和方差代入高斯混合模型,得到适用于对称非高斯振动信号的幅值概率密度函数。最后,通过仿真信号和实测振动信号,验证了该方法的有效性和适用性。     

Efficient stationary multivariate non-Gaussian simulation based on a Hermite PDF model

An efficient stationary multivariate non-Gaussian simulation method is developed using spectral representation and third order Hermite polynomial translation. An approximate closed form relationship is employed to identify the Hermite translation parameters based on target skewness and kurtosis. This preserves a high degree of accuracy over the entire admissible range of the Hermite translation, and eliminates the need for iterative solution of the translation parameters. The Hermite PDF model is suitable for a wide range of strongly non-Gaussian stochastic process. In addition, an explicit bidirectional relationship between the target non-Gaussian and Gaussian correlation is developed to eliminate the need for iteration or numerical integration to identify the underlying Gaussian correlation. Examples apply the simulation method to both theoretical targets and experimental wind pressure data.     

On the accuracy of the polynomial chaos approximation

Polynomial chaos representations for non-Gaussian random variables and stochastic processes are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. Finite truncations of these series are referred to as polynomial chaos (PC) approximations. This paper explores features and limitations of PC approximations. Metrics are developed to assess the accuracy of the PC approximation. A collection of simple, but relevant examples is examined in this paper. The number of terms in the PC approximations used in the examples exceeds the number of terms retained in most current applications. For the examples considered, it is demonstrated that (1) the accuracy of the PC approximation improves in some metrics as additional terms are retained, but does not exhibit this behavior in all metrics considered in the paper, (2) PC approximations for strictly stationary, non-Gaussian stochastic processes are initially nonstationary and gradually may approach weak stationarity as the number of terms retained increases, and (3) the development of PC approximations for certain processes may become computationally demanding, or even prohibitive, because of the large number of coefficients that need to be calculated. However, there have been many applications in which PC approximations have been successful.     

Simulation of wind velocity time histories on long span structures modeled as non-Gaussian stochastic waves

A methodology is proposed for efficient and accurate modeling and simulation of correlated non-Gaussian wind velocity time histories along long-span structures at an arbitrarily large number of points. Currently, the most common approach is to model wind velocities as discrete components of a stochastic vector process, characterized by a Cross-Spectral Density Matrix (CSDM). To generate sample functions of the vector process, the Spectral Representation Method is one of the most commonly used, involving a Cholesky decomposition of the CSDM. However, it is a well-documented problem that as the length of the structure – and consequently the size of the vector process – increases, this Cholesky decomposition breaks down numerically. This paper extends a methodology introduced by the second and fourth authors to model wind velocities as a Gaussian stochastic wave (continuous in both space and time) by considering the stochastic wave to be non-Gaussian. The non-Gaussian wave is characterized by its frequency–wavenumber (FK) spectrum and marginal probability density function (PDF). This allows the non-Gaussian wind velocities to be modeled at a virtually infinite number of points along the length of the structure. The compatibility of the FK spectrum and marginal PDF according to translation process theory is secured using an extension of the Iterative Translation Approximation Method introduced by the second and third authors, where the underlying Gaussian FK spectrum is upgraded iteratively using the directly computed (through translation process theory) non-Gaussian FK spectrum. After a small number of computationally extremely efficient iterations, the underlying Gaussian FK spectrum is established and generation of non-Gaussian sample functions of the stochastic wave is straightforward without the need of iterations. Numerical examples are provided demonstrating that the simulated non-Gaussian wave samples exhibit the desired spectral and marginal PDF characteristics.     

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.