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.     

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.     

环境激励下基于奇异谱互熵的结构损伤诊断方法

环境激励下的结构响应是一个随机过程,结构发生破损时其响应将随之变化,因而可将描述随机过程特性的参数作为评判结构状况的指标。熵是测量随机过程不确定性的一个比较方便的方法,能够用于高斯及非高斯分布的情况。在相空间重构与奇异值分解的基础上建立了奇异谱互熵的概念,提出了一种环境激励下用奇异谱互熵诊断结构损伤的方法。以ASCE学会提出的基准结构为对象进行研究,利用NExT技术获得响应,采用伪邻近法确定相空间的嵌入维数,讨论了不同工况及噪声对诊断结果的影响,分析结果验证了该方法的有效性和鲁棒性。     

A non-stationary stochastic model for periodic excitation with random phase modulation

The paper presents a non-stationary stochastic model for periodic excitation with random phase modulation, where the phase modulation is modeled as a modulated stationary. Gaussian process. Applications of the model are demonstrated by analysis of response of a single-degree-of-freedom (SDOF) system under such an excitation. The response is, in general, non-Gaussian. Cases of step, rectangular, and exponential envelopes are considered in the present study. The nonstationary second and fourth order moments are calculated by numerically solving the transient moment equations. Non-Gaussianity of the response is studied in terms of the non-stationary excess factor. Some numerical results are presented. The influences of system parameters, build-up and decay rates as well as duration of random phase modulation on the moment response of the SDOF system are discussed.     

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.     

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.     

The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus

This paper deals with the study of the Green’s functions of a layered soil with random characteristics. The dynamic shear modulus of the soil is modelled as a non-Gaussian random process that varies in the vertical direction and is characterized by a marginal probability density function and a correlation function. The stochastic finite element method is applied to a hybrid thin layer — direct stiffness formulation in order to obtain the stochastic system equations, which are solved by means of a Monte Carlo simulation. The influence of the variations of the dynamic shear modulus on the Green’s functions is illustrated for different excitation frequencies and receiver positions.     

A Monte Carlo simulation model for stationary non-Gaussian processes

A class of stationary non-Gaussian processes, referred to as the class of mixtures of translation processes, is defined by their finite dimensional distributions consisting of mixtures of finite dimensional distributions of translation processes. The class of mixtures of translation processes includes translation processes and is useful for both Monte Carlo simulation and analytical studies. As for translation processes, the mixture of translation processes can have a wide range of marginal distributions and correlation functions. Moreover, these processes can match a broader range of second order correlation functions than translation processes. The paper also develops an algorithm for generating samples of any non-Gaussian process in the class of mixtures of translation processes. The algorithm is based on the sampling representation theorem for stochastic processes and properties of the conditional distributions. Examples are presented to illustrate the proposed Monte Carlo algorithm and compare features of translation processes and mixture of translation processes.     

Non-Gaussian random processes in ocean engineering

This paper presents a state-of-the-art review on stochastic analysis and probabilistic prediction of non-Gaussian random processes in ocean engineering. The derivation of probability density functions which constitute the basis for stochastic analysis of non-Gaussian processes is discussed in detail, and then the probability distributions of peaks and troughs of non-Gaussian random process is discussed to provide information necessary for engineering design. As an example of application of these probability distribution functions, the procedure for predicting responses of an offshore structure which has substantial non-linear characteristics in random seas is presented.     

Generation of non-Gaussian stochastic processes using nonlinear filters

Non-Gaussian stochastic processes are generated using nonlinear filters in terms of Itô differential equations. In generating the stochastic processes, two most important characteristics, the spectral density and the probability density, are taken into consideration. The drift coefficients in the Itô differential equations can be adjusted to match the spectral density, while the diffusion coefficients are chosen according to the probability density. The method is capable to generate a stochastic process with a spectral density of one peak or multiple peaks. The locations of the peaks and the band widths can be tuned by adjusting model parameters. For a low-pass process with the spectrum peak at zero frequency, the nonlinear filter can match any probability distribution, defined either in an infinite interval, a semi-infinite interval, or a finite interval. For a process with a spectrum peak at a non-zero frequency or with multiple peaks, the nonlinear filter model also offers a variety of profiles for probability distributions. The non-Gaussian stochastic processes generated by the nonlinear filters can be used for analysis, as well as Monte Carlo simulation.     

斜拉索在轴向白噪声随机激励下的响应

建立了在考虑垂度以及拉索张力沿索长变化时拉索的参激随机微分方程,给出了在白噪声随机激励下预测拉索响应的近似理论解--用统计矩截断法求解矩方程,获得高斯闭合解和一阶非高斯闭合解.并采用数值模拟方法进行验证.随后以南京长江二桥A20拉索为研究对象,采用MonteCarlo数值方法对近似理论解进行验证.最后分析了白噪声激励强度、拉索阻尼及拉索张力对响应的影响.结果表明,斜拉索在零均值随机位移激励下的响应为非零均值;在高斯白噪声激励下的一阶位移响应随激励强度的增大而增大.随拉索阻尼和张力增大而减小.     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

Dynamic analysis of bridge with non-Gaussian uncertainties under a moving vehicle

An analysis method on the bridge-vehicle interaction problem with uncertainties is proposed. The bridge is modeled as a simply supported Euler-Bernoulli beam with non-Gaussian material parameters with a vehicle moving on top modeled by a deterministic four degrees-of-freedom mass-spring system. The non-Gaussian uncertainty in bridge is modeled by the Spectral Stochastic Finite Element Method (Ghanem and Spanos (1991) [17]), and the mathematical model of the coupled bridge-vehicle system, with the road surface roughness assumed as a Gaussian random process, will be solved by the Newmark-β method. The proposed model is verified by the Monte Carlo Simulation with numerical examples. Different levels of uncertainties in both the excitation and system parameters are investigated. Criteria on the selection of both the order of Polynomial Chaos and the threshold for truncation in the Karhunen-Loève expansion are provided. Results show that the proposed algorithm is promising for the dynamic analysis of the bridge-vehicle interaction problem even with a high level of system and excitation uncertainties.     

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.     

Equations for probability density of response of dynamic systems to a class of non-Poisson random impulse process excitations

The excitation considered in the present paper is a random train of impulses driven by two classes of non-Poisson counting processes. The impulse processes are obtained by selecting impulses from a Poisson and from an Erlang-driven trains of impulses with the aid of an additional, purely jump stochastic process, assumed as an auxiliary state variable. The variable introduced for the first class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Poisson processes, with different parameters, and is tantamount to a two-state Markov chain. The variable introduced for the second class of non-Poisson processes is governed by the stochastic differential equation driven by two independent Erlang processes, with different parameters. As each Erlang process is tantamount to a number of Markov states, the Markov chain for the whole problem is constructed. The equations governing the joint probability density-distribution function of the state vector of the dynamic system and of the Markov states are derived from the general integro-differential forward Chapman–Kolmogorov equation. The necessary jump probability intensity functions are evaluated for both classes of impulse processes and for purely external as well as parametric excitations. Parametric excitation multiplicative to the displacement and to the velocity state variable is considered. The resulting set of coupled integro-partial differential equations is obtained.     

Global sensitivity analysis for stochastic processes with independent increments

This paper is a first attempt to develop a numerical technique to analyze the sensitivity and the propagation of uncertainty through a system with stochastic processes having independent increments as input. Similar to Sobol’ indices for random variables, a meta-model based on Chaos expansions is used and it is shown to be well suited to address such problems. New global sensitivity indices are also introduced to tackle the specificity of stochastic processes. The accuracy and the efficiency of the proposed method is demonstrated on an analytical example with three different input stochastic processes: a Wiener process; an Ornstein–Uhlenbeck process and a Brownian bridge process. The considered output, which is function of these three processes, is a non-Gaussian process. Then, we apply the same ideas on an example without known analytical solution.     

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.     

泊松白噪声激励下斜拉索的面内随机振动

目前针对斜拉索非线性随机振动的研究已广泛开展,但仅限于高斯随机激励情形。然而,现实中大部分的随机扰动都是非高斯的。若使用高斯激励模型将产生较大误差。假设拉索所受非高斯激励为泊松白噪声,研究了泊松白噪声激励下斜拉索面内随机振动。推导了受泊松白噪声激励的斜拉索面内振动的随机微分方程,建立了支配系统平稳响应概率密度函数的广义FPK方程。提出迭代加权残值法求解了四阶广义FPK方程,得到了系统响应概率密度函数的近似稳态闭合解。考察了垂跨比、阻尼系数以及脉冲到达率对拉索面内随机振动响应的影响。结果表明:拉索的响应随着垂跨比的增大,响应呈现不对称现象愈加明显;随阻尼比增加,系统响应得到显著抑制;当脉冲到达率增大,拉索的响应也随之增大,并逐渐接近于高斯白噪声激励的情形。另外,获得的理论结果与蒙特卡罗模拟的结果吻合地非常好。     

Stochastic and chaotic sub- and superharmonic response of shallow cables due to chord elongations

The paper deals with the non-linear response of shallow cables driven by stochastically varying chord elongations caused by random vibrations of the supported structure. The chord elongation introduces parametric excitation in the linear stiffness terms of the modal coordinate equations, which are responsible for significant internal subharmonic and superharmonic resonances. Under harmonically varying support motions coupled ordered or chaotic in-plane and out-of-plane subharmonic and superharmonic periodic motions may take place. If the harmonically varying chord elongation is replaced by a zero-mean, stationary narrow-band random excitation with the same standard deviation and center frequency, qualitatively and quantitatively completely different modes of vibration are registered no matter how small the bandwidth of the excitation process is. Additionally, the stochastic excitation process tends to enhance chaotic behavior. Based on Monte Carlo simulation on a reduced non-linear two-degree-of freedom system the indicated effects have been investigated for stochastic subharmonic resonance of order 2:1, and stochastic superharmonic resonances of orders 1:2 and 2:3. By analyzing the responses for two chord elongation processes with almost identical auto-spectral density function, but completely different amplitudes, it is shown that the indicated qualitative and quantitative changes of the subharmonic resonance primarily are caused by the slowly varying phase of the stochastic excitation. The superharmonic stochastic responses are dominated by random jumps between a single mode in-plane and a coupled mode attractor, which are caused by the variation of the amplitude of the random excitation. Such jumps do not occur in the subharmonic response, because the single mode in-plane attractor is unstable.     

New probabilistic approach to estimate vehicle failure trajectories in curve driving

The vehicle trajectories analysis on dangerous bends is an important task to improve road safety. This paper proposes a new methodology to predict failure trajectories of light vehicles in curve driving. It consists to use a stochastic modelling and reliability analysis in order to estimate the failure probability of vehicle trajectories.Firstly, we build probabilistic models able to describe real trajectories in a given bend. The models are transforms of scalar normalized second order stochastic processes which are stationary, ergodic and non-Gaussian. The process is characterized by its probability density function and its power spectral density estimated starting from the experimental trajectories. The probability density is approximated by using a development on the basis of Hermite polynomials.The second part is devoted to apply a reliability strategy intended to associate a risk level to each class of trajectories. Based on the joint use of probabilistic methods for modelling uncertainties, reliability analysis for assessing risk levels and statistics for classifying the trajectories, this approach provides a realistic answer to the tackled problem. Experiments show the relevance and effectiveness of this method.