Simulation of Highly Skewed Non-Gaussian Stochastic Processes

In this paper, a simulation methodology is proposed to generate sample functions of a stationary, non-Gaussian stochastic process with prescribed spectral density function and prescribed marginal probability distribution. The proposed methodology is a modified version of the Yamazaki and Shinozuka iterative algorithm that has certain difficulties matching the prescribed marginal probability distribution. Although these difficulties are usually sufficiently small when simulating non-Gaussian stochastic processes with slightly skewed marginal probability distributions, they become more pronounced for highly skewed probability distributions (especially at the tails of such distributions). Two major modifications are introduced in the original Yamazaki and Shinozuka iterative algorithm to ensure a practically perfect match of the prescribed marginal probability distribution regardless of the skewness of the distribution considered. First, since the underlying “Gaussian” stochastic process from which the desired non-Gaussian process is obtained as a translation process becomes non-Gaussian after the first iteration, the empirical (non-Gaussian) marginal probability distribution of the underlying stochastic process is calculated at each iteration. This empirical non-Gaussian distribution is then used instead of the Gaussian to perform the nonlinear mapping of the underlying stochastic process to the desired non-Gaussian process. This modification ensures that at the end of the iterative scheme every generated non-Gaussian sample function will have the exact prescribed non-Gaussian marginal probability distribution. Second, before the start of the iterative scheme, a procedure named “spectral preconditioning” is carried out to check the compatibility between the prescribed spectral density function and prescribed marginal probability distribution. If these two quantities are found to be incompatible, then the spectral density function can be slightly modified to make it compatible with the prescribed marginal probability distribution. Finally, numerical examples (including a stochastic process with a highly skewed marginal probability distribution) are provided to demonstrate the capabilities of the proposed algorithm.     

Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation

The first-passage probability, describing the probability that a scalar process exceeds a prescribed threshold during an interval of time, is of great engineering interest. This probability is essential for estimating the reliability of a structural component whose response is a stochastic process. When considering the reliability of an engineering system composed of several interdependent components, the probability that two or more response processes exceed their respective safe thresholds during the operation time of the system is an equally essential quantity. This paper proposes simple and accurate formulas for approximating this joint first-passage probability of a vector process. The nth order joint first-passage probability is obtained from a recursive formula involving lower order joint first-passage probabilities and the out-crossing probability of the vector process over a safe domain. Interdependence between the crossings is approximately accounted for by considering the clumping of these events. The accuracy of the proposed formulas is examined by comparing analytical estimates with those obtained from Monte Carlo simulations for stationary Gaussian processes. As an example application, the reliability of a system of interconnected equipment items subjected to a stochastic earthquake excitation is estimated by linear programming bounds employing marginal and joint component fragilities obtained by the proposed formulas.     

Evolutionary Spectra Estimation Using Wavelets

A wavelets-based method is developed to estimate the evolutionary power spectral density (EPSD) of nonstationary stochastic processes. The method relies on the property that the continuous wavelet transform of a nonstationary process can be treated as a stochastic process with EPSD given in terms of the EPSD of the process in a closed form. This yields an equation in the frequency domain relating the instantaneous mean-square value of the wavelet transform to the EPSD of the process. A number of these equations are considered, each related to a certain scale of the wavelet transform, in conjunction with representing the target EPSD as a sum of time-independent shape functions modulated by time-dependent coefficients; the squared moduli of the Fourier transforms of the wavelets associated with the selected scales are taken as shape functions. This leads to a linear system of equations which is solved to determine the unknown time-dependent coefficients; the same system matrix applies for all time instances. Numerical examples demonstrate the accuracy and computational efficiency of the proposed method.     

Multivariate Extreme Value Distributions for Random Vibration Applications

The problem of determining the joint probability distribution of extreme values associated with a vector of stationary Gaussian random processes is considered. A solution to this problem is developed by approximating the multivariate counting processes associated with the number of level crossings as a multivariate Poisson random process. This, in turn, leads to approximations to the multivariate probability distributions for the first passage times and extreme values over a given duration. It is shown that the multivariate extreme value distribution has Gumbel marginal and the first passage time has exponential marginal. The acceptability of the solutions developed is examined by performing simulation studies on bivariate Gaussian random processes. Illustrative examples include a discussion on the response analysis of a two span bridge subjected to spatially varying random earthquake support motions.     

Peak Response of a Nonlinear Beam

A model for estimating the peak dynamic response distribution of a nonlinear beam, based on a special class of non-Gaussian stochastic processes, is proposed in this paper. It is shown that the stochastic response of a cantilever beam with geometrically nonlinear behavior can be accurately calibrated with translation processes. Different models to describe the significant bimodal features in the marginal probability density functions of the response time histories are proposed. Finally, two of these models are used to estimate the response peak value distributions and the results are compared. This comparison demonstrates the effects of inaccurate models for the parent response processes on the peaks estimation.     

Simulation of Nonstationary Stochastic Processes by Spectral Representation

This paper presents a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes integrating Priestly’s evolutionary spectral representation theory. Applying this formula, sample functions can be generated with great computational efficiency. The simulated stochastic process is asymptotically Gaussian as the number of terms tends to infinity. This paper shows that (1) these sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number of terms in the cosine series is large, i.e., the ensemble averaged evolutionary power spectral density function (PSDF) or autocorrelation function approaches the corresponding target function as the sample size increases, and (2) the simulation formula, under certain conditions, can be reduced to that for nonstationary white noise process or Shinozuka’s spectral representation of stationary process. In addition to derivation of simulation formula, three methods are developed in this paper to estimate the evolutionary PSDF of a given time-history data by means of the short-time Fourier transform (STFT), the wavelet transform (WT), and the Hilbert-Huang transform (HHT). A comparison of the PSDF of the well-known El Centro earthquake record estimated by these methods shows that the STFT and the WT give similar results, whereas the HHT gives more concentrated energy at certain frequencies. Effectiveness of the proposed simulation formula for nonstationary sample functions is demonstrated by simulating time histories from the estimated evolutionary PSDFs. Mean acceleration spectrum obtained by averaging the spectra of generated time histories are then presented and compared with the target spectrum to demonstrate the usefulness of this method.     

Stochastic Response of a Random Mass Structure

This paper proposes a polynomial approximation approach for the estimation of the stochastic response of a random mass structure subjected to an evolutionary random excitation. A bounded, monopeak, and symmetrically distributed probability density function, called λ-PDF, together with the Gegenbauer polynomial approximation, is introduced to deal with stochastic dynamic response problems of the random mass structures. The λ-PDF model is used to describe the random parameters in the engineering random structures. And then the Gegenbauer polynomial approximation method is used to reduce the random structures into its deterministic equivalent one. The numerical example shows the effectiveness of the proposed method to explore dynamic phenomena in random structures.     

Simulation of Stochastic Wind Velocity Field on Long-Span Bridges

An improved algorithm is introduced in this paper for digital simulation of the stochastic wind velocity field on long-span bridges, when the cross-spectral density matrix of the field is given. The target wind velocity field is assumed to be a one-dimensional, multivariate, homogeneous stochastic process. The basic method of simulation used is the spectral representation method. It is improved by explicitly expressing Cholesky's decomposition of the cross-spectral density matrix in the form of algebraic formulas, then cutting off as many as possible of the cosine terms, so long as the accuracy of results is not affected. The fast Fourier transform technique is used to enhance the efficiency of computation. A numerical example of simulation for buffeting analysis is included in this paper to illustrate the improved method introduced. It is demonstrated that deviations between the simulated correlation functions and the target are sufficiently small and that the simulated power spectra are close to the target.     

Error Assessment for Spectral Representation Method in Wind Velocity Field Simulation

Stochastic wind velocity fields are usually simulated as N-variate stationary Gaussian processes by using the spectral representation method (SRM). However, the temporal statistics estimated from one SRM-simulated sample wind process cannot coincide with the target characteristics; the disagreements can be described by the bias and stochastic errors. For controlling the errors efficiently, this paper assesses the errors produced by both the Cholesky decomposition-based SRM (CSRM) and the eigendecomposition-based SRM (ESRM). The SRM is revisited first, followed by computing the temporal mean value, correlation function, power spectral density (PSD), and standard deviation of the SRM-simulated wind process. It is shown that the temporal correlation function and standard deviation are Gaussian, while the temporal PSD is non-Gaussian. Further, as mathematical expectations and standard deviations of the corresponding temporal estimations, the bias errors and stochastic errors of all the first- and second-order statistics are obtained in closed form for both the CSRM and the ESRM; the closed-form solutions are then verified in the numerical example. More importantly, this example is employed for taking a clear look at and making a comparison between the stochastic errors produced by the CSRM and by the ESRM; observations suggest that (1) in sum, the ESRM produces smaller stochastic errors than the CSRM and (2) if the ESRM is employed, stochastic errors will be distributed to each component of the wind process in a more uniform pattern. Moreover, some practical approaches are proposed to control the stochastic errors.     

Probability Density and Excursions of Structural Response to Imperfectly Periodic Excitation

A single-degree-of-freedom system under periodic excitation with random phase modulation is considered. Probability density functions (PDF) of the response are obtained numerically using the path integration method. Basic results are presented in the form of expected number of excursions over a given displacement level. They clearly illustrate the transformation of the response PDF with increasing excitation/system bandwidth ratio from one corresponding to the sinusoid with random phase at small values to asymptotically Gaussian PDF at high values of the above ratio. While this qualitative trend is known from previous analysis of the response excess factor by the method of moments, the present quantitative results may be of direct use for reliability predictions. An analytical study is also made for reduced stochastic differential equations of motion, as obtained by stochastic averaging, by the method of moments resulting in a simple explicit expression for the mean square amplitude.     

Lognormal Distribution Provides an Optimum Representation of the Concrete Delivery and Placement Process

The process of concrete supply and delivery exhibits traits of random nature, which render control arduous. This random nature allows concrete placement operations to be considered as a stochastic system and therefore cannot be analyzed by deterministic techniques. In modeling the process realistically, it may be necessary to recreate the variability through fitting a sufficiently flexible theoretical probability distribution to observed data. There is a significant body of work relating to the probability distributions which represent general construction activities. However, there is a lack of literature aimed at determining the optimum representative of the concrete placement process. Therefore, the aim of this paper is to identify a sufficiently flexible representative of this process. To this end, construction data were collected from a number of concrete pours in Scotland, U.K. To identify a suitable distribution in this context, a computer package, PDFit, is presented. It is founded upon the production of probability density functions of select theoretical probability distributions plotted against the histogram of the input data. The principal method of assessment is a visual comparison of the shape of the probability density function and the input data histogram. This paper presents the lognormal distribution as a sufficiently flexible theoretical probability distribution to represent the concrete placing process.     

Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction

A procedure is presented for the probabilistic analysis of the seismic soil-structure interaction problem. The procedure accounts for uncertainty in both the free-field input motion as well as in local site conditions, and structural parameters. Uncertain parameters are modeled using a probabilistic framework as stochastic processes. The site amplification effects are accounted for via a randomized relationship between the soil shear modulus and damping on the one hand, and the shear strain of the subgrade on the other hand, as well as by modeling the shear modulus at low strain level as randomly fluctuating with depth. The various random processes are represented by their respective Karhunen-Loève expansions, and the solution processes, consisting of the accelerations and generalized forces in the structure, are represented by their coordinates with respect to the polynomial chaos basis. These coordinates are then evaluated by a combination of weighted residuals and stratified sampling schemes. The expansion can be used to carry out very efficiently, extensive Monte Carlo simulations. The procedure is applied to the seismic analysis of a nuclear reactor facility.     

Linear Stochastic Dynamical System under Uncertain Load: Inverse Reliability Analysis

The reliability of a linear dynamical system driven by a partially known Gaussian load process, specified only through its total average energy, is studied. A simple dynamic parallel and series system reliability network is investigated for the failure analysis using the crossing theory of stochastic processes. The critical input power spectral density of the load process which maximizes the mean crossing rate of a parallel or series system network emerges to be fairly narrow banded and hence fails to represent the erratic nature of the random input realistically. Consequently, a tradeoff curve between the maximum mean crossing rate of the reliability network and the disorder in the input, quantitatively measured through its entropy rate, is generated. Mathematically, the generation of the tradeoff curve of nondominated solutions, known as the Pareto optimal set, leads to a nonlinear, nonconvex, and multicriteria optimization problem with conflicting objectives. A recently developed Pareto optimization technique, implemented through genetic algorithm, has been successfully exploited to solve the optimization problem. A suitable exploitation of stochastic process theory circumvents the task of handling an apparent robust optimization problem (i.e., optimization under uncertainty) and at the same time, reduces the dimensionality of the multiobjective optimization scheme in view of both the multiobjectivity and constraints in the aforementioned methodology. The optimally disordered inputs which simulate the worst performance of the system of a spring-supported coupled rod assembly, has been studied as a numerical illustration of the mathematical formulation.     

Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.     

Probability Density Estimation Using Entropy Maximization

We propose a method for estimating probability density functions and conditional density functions by training on data produced by such distributions. The algorithm employs new stochastic variables that amount to coding of the input, using a principle of entropy maximization. It is shown to be closely related to the maximum likelihood approach. The encoding step of the algorithm provides an estimate of the probability distribution. The decoding step serves as a generative mode, producing an ensemble of data with the desired distribution. The algorithm is readily implemented by neural networks, using stochastic gradient ascent to achieve entropy maximization.     

Stochastic Averaging of Preisach Hysteretic Systems

The sustained progress in the study of the hysteretic behavior of structural and mechanical systems has led to the adoption of increasingly sophisticated and reliable mathematical representations. Models based on the distributed elements (hysterons) appear to be quite versatile. Among these, the Preisach hysteretic model has received considerable attention in the field of engineering mechanics. In this paper, the stochastic response of a Preisach hysteretic system driven by a white noise process is investigated. In this regard, the method of stochastic averaging is modified to be applicable for the determination of the probability density of the stationary system response envelope. Remarkably, this probability density expression in conjunction with the response of an auxiliary linear system can also be used to determine the power spectrum of the system response. The approximate theoretical solutions are validated by data derived by a pertinent Monte Carlo study.     

Peak Non-Gaussian Wind Effects for Database-Assisted Low-Rise Building Design

Current procedures for estimating the peaks of the stochastic response of tall buildings to wind are based on the assumption that the response is Gaussian. Those procedures are therefore inapplicable to low-rise buildings, in which time histories of wind-induced internal forces are generally non-Gaussian. In this paper, an automated procedure is developed for obtaining from such time histories sample statistics of internal force peaks for low-rise building design and codification. The procedure is designed for use in software for calculating internal force time series by the database-assisted design approach. A preliminary step in the development of the procedure is the identification of the appropriate marginal probability distribution of the time series using the probability plot correlation coefficient method. The result obtained is that the gamma distribution and a normal distribution are appropriate for estimating the peaks corresponding, respectively, to the longer and shorter tail of the time series’ histograms. The distribution of the peaks is then estimated by using the standard translation processes approach. It is found that the peak distribution can be represented by the Extreme Value Type I (Gumbel) distribution. Because estimates obtained from this approach are based on the entire information contained in the time series, they are more stable than estimates based on observed peaks. The procedure can be used to establish minimum acceptable requirements with respect to the duration and sampling rate of the time series of interest, so that the software used for database-assisted design will be both efficient and accurate.     

Statistical Moments of Polynomial Dimensional Decomposition

This technical note presents explicit formulas for calculating the response moments of stochastic systems by polynomial dimensional decomposition entailing independent random input with arbitrary probability measures. The numerical results indicate that the formulas provide accurate, convergent, and computationally efficient estimates of the second-moment properties.