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.     

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.     

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.     

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.     

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.     

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。     

高斯/非高斯混合随机风压场的模拟方法

针对作用于屋盖结构随机风压场样本的统计特性要求,基于零记忆非线性转化法的理论,给出了随机风压场的具体模拟过程。其中,解决了两个关键问题：（1）推导了服从对数正态分布和韦布尔分布的多点非高斯随机过程向量的标准化协方差,与相应高斯随机过程向量的标准化协方差的函数转化关系;（2）提出了分解谱密度函数修正法,解决利用谐波合成法模拟多点高斯随机过程向量时,功率谱密度函数矩阵在某些频率点出现负定的问题。经过具体算例表明,所提出的方法能生成合乎风洞实验数据统计特性要求的随机风压场样本。     

非高斯风压峰值因子估计:基于矩的转换过程法的对比研究

建筑围护结构抗风设计需要准确估计非高斯风压极值或者峰值因子。对于非高斯风压峰值因子估计,常用的基于矩的转换过程法有Hermite多项式模型（HPM）、Johnson转换模型（JTM）及平移广义对数正态分布（SGLD）模型。极值通常由母本概率密度函数（PDF）的尾部决定,现阶段对于三种模型基于相同前四阶矩预测的非高斯母本PDF尾部的差别尚不清楚,自然,对于这三种模型预测的极值或者峰值因子的差别尚无答案。为了探明三种模型的异同,从而提供一定的选取原则,该文就三种方法对非高斯风压峰值因子估计效果进行了系统的对比研究。首先从理论上对比了三种方法预测得到的母本PDF的差异和估计的峰值因子差别;其次,选用长时距风洞试验风压数据检验了三种方法对非高斯风压峰值因子的估计效果。结果表明在三种模型都适用的偏度和峰度组合范围内,HPM对非高斯风压峰值因子估计结果相比SGLD模型和JTM模型估计结果更准确。     

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 non-Gaussian field applied to wind pressure fluctuations

A simulation algorithm to generate non-Gaussian wind pressure fields is proposed. This algorithm uses the correlation–distortion method based on translation vector processes. Conditions on the matrix of cross-covariance functions are given to assure the applicability of the model. The proposed method does not require iterative procedures and it is well suited when experimental data are available. In particular it requires cross-covariance functions and marginal distribution that can be directly estimated from data. To illustrate the procedure, the model is calibrated on experimental results obtained from wind tunnel tests on a tall building. The efficiency of the proposed methodology for reproducing the non-Gaussian nature of pressure fluctuations on separated flow regions is demonstrated.     

非高斯脉动风压的模拟研究

对于风荷载数值模拟,大多数研究都是假定风荷载为平稳高斯随机过程。然而,在分离流作用的一些局部重要区域,例如建筑物屋盖边缘、屋面转角等,风荷载表现出强烈的非高斯特性。为了能够有效地模拟非高斯脉动风压,本文提出一种模拟非高斯脉动风压的框架。首先,根据风速与风压间的关系,严格推导出脉动风压功率谱密度函数。然后,基于导出的功率谱密度函数、Johnson转换系统和数字滤波理论,提出一种能够快速而有效的生成指定偏度、峰度和功率谱非高斯脉动风压的方法。最后,通过一个一维单变量非高斯脉动风压的模拟算例对该方法的可行性和正确性进行验证。数值结果表明：模拟生成的单样本非高斯脉动风压的统计参数例如偏度和峰度与目标偏度和峰度非常吻合,而且模拟功率谱与目标功率谱两者也吻合得很好。     

Conditional simulation of non-Gaussian wind velocity profiles: Application to buffeting response of Vincent Thomas suspension bridge

For a proper prediction of wind-induced vibrations of a long suspension bridge, it is necessary to use a representative wind velocity profile at the bridge site. However, as the simultaneous collection of wind velocity data at closely spaced locations of the entire bridge is not a viable option, simulation has become a powerful tool for this purpose. In this study, a new iterative approach of non-Gaussian conditional simulation is proposed to conditionally simulate the wind velocity profiles by utilizing the measured wind velocities at a few locations. The approach utilizes the well-known spectral representation technique in conjunction with nonlinear Gaussian to non-Gaussian mapping technique. Focusing on Vincent Thomas suspension bridge on which three anemometers have been installed recently at three strategic locations, this study compares buffeting responses evaluated using different simulation schemes. It has been found that the bridge response evaluated using the non-Gaussian simulation scheme may be higher when compared with other simulation schemes.     

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.     

Crossing rate statistics of quadratic transformations of Gaussian processes

The main focus of this paper is the development of a numerical procedure for calculating the average crossing rates of a stochastic process that can be expressed as a sum of a linear and a nonlinear, quadratic transformation of a Gaussian process. Such a representation applies for instance to the motion response of a linear structure subjected to wind loading, when the loading model is proportional to the square of a Gaussian wind velocity process. It is also the standard model for expressing the total wave forces or horizontal excursion responses of a moored floating offshore platform in a random sea way. Knowledge of the crossing rate is a key to many important quantities related to response statistics and reliability applications. It is demonstrated how the proposed numerical procedure can be used for calculating the average crossing rate of the type of response processes considered.     

Advanced Scientific Computing in BASIC With Applications in Chemistry,Biology, and Pharmacology

Tuning and calibration are processes for improving the representativeness of a computer simulation code to a physical phenomenon. This article introduces a statistical methodology for simultaneously determining tuning and calibration parameters in settings where data are available from a computer code and the associated physical experiment. Tuning parameters are set by minimizing a discrepancy measure while the distribution of the calibration parameters are determined based on a hierarchical Bayesian model. The proposed Bayesian model views the output as a realization of a Gaussian stochastic process with hyper-priors. Draws from the resulting posterior distribution are obtained by the Markov chain Monte Carlo simulation. Our methodology is compared with an alternative approach in examples and is illustrated in a biomechanical engineering application. Supplemental materials, including the software and a user manual, are available online and can be requested from the first author.     

A new model for non-Gaussian random excitations

Very often one is called upon to model time series data which are clearly non-Gaussian, but which retain some aspects of a Gaussian process. In the present paper, a novel methodology which helps in modelling such data is presented. The method is essentially to express the process as a series with finite number of terms, wherein the first term is a Gaussian process with zero mean and unit standard deviation. Non-Gaussian higher order correction terms are added to this such that each succeeding term is orthogonal or uncorrelated with all the previous terms. The unknown coefficients in the series representation can be expressed in terms of the estimated moments of the data. Further the autocorrelation or PSD of the data can be exactly reproduced by the non-Gaussian model. The use of the proposed model is illustrated by considering the unevenness data of railway tracks. Application to response of systems under non-Gaussian excitation is also briefly discussed.     

大跨屋盖脉动风压的非高斯特性研究

在大跨屋盖表面局部区域,特别是迎风边缘区域和屋盖拐角区,风荷载会表现出明显的非高斯特性,如果仍采用高斯模型来描述,往往会产生较大误差。基于五种典型大跨度屋盖结构的风洞试验,对屋盖表面局部风压的高斯和非高斯特性进行了研究。首先通过对第三阶、第四阶矩统计量归纳分析,给出划分高斯区和非高斯区的标准并对大跨度屋盖进行分区;同时,运用基于k-s检验的曲线拟合方法也得到风压非高斯分区,利用分区结果,得到保证率为99.38%的峰值因子取值。将两种方法相对比,发现得出的分区结果相似:非高斯区域往往集中在来流前缘、后部尾流区及高点角区附近。此外,分析结果表明,对于大跨屋盖结构,应适当提高我国荷载规范中的峰值因子并按结构分区取值。     

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。     

复杂体型屋盖表面风压的高阶统计量与非高斯峰值因子

该文研究了某站台结构刚性屋盖风洞实验中风压统计量对于数据时长的敏感性。采用不同时长的数据计算长时距脉动风压数据的偏度和峰度,结果表明高阶统计量具有非平稳特性,峰度的值受数据时长的变化影响显著。根据随机数据的峰度是否大于3,可以将其划分为软响应过程和硬响应过程。通过分析此建筑屋盖表面风压数据,发现屋盖表面存在峰度小于3的硬响应测点。而现有的峰值因子计算方法都没有具体探讨其对于这种峰度小于3的硬响应测点的适用性,该文将不同计算方法得到的软响应和硬响应峰值因子结果与标准统计方法计算值进行对比,进而判断各种方法的优劣性。结果表明非高斯峰值因子计算当中不宜引入峰度这个参数,TPP方法对于计算硬响应过程和软响应过程的非高斯峰值因子都有很好的适用性。     

New formulation of Cholesky decomposition and applications in stochastic simulation

Monte Carlo simulation plays a significant role in the mechanical and structural analysis due to its versatility and accuracy. Classical spectral representation method is based on the direct decomposition of the power spectral density (PSD) or evolutionary power spectral density (EPSD) matrix through Cholesky decomposition. This direct decomposition of complex matrix usually results in large computational time and storage memory.In this study, a new formulation of the Cholesky decomposition for the EPSD/PSD matrix and corresponding simulation scheme are presented. The key idea to this approach is to separate the phase from the complex EPSD/PSD matrix. The derived real modulus matrix evidently expedites decomposition compared to the direct Cholesky decomposition of the complex EPSD/PSD matrix. In the proposed simulation scheme, the separated phase can be easily assembled. The modulus of EPSD/PSD matrix could be further decomposed into the modulus of coherence matrix (or lagged coherence matrix), which describes the basic coherence structure of stochastic process. The lagged coherence matrix is independence of time and thus remarkably improves the Cholesky decomposition efficiency.The application of the proposed schemes to Gaussian stochastic simulations is presented. Firstly, the previous closed-form wind speed simulation algorithm for equally-spaced locations is extended to a more general situation. Secondly, the proposed approach facilitates the application of interpolation technique in stochastic simulation. The application of interpolation techniques in the wind field simulation is studied as an example.