Simulation of Multivariate Stationary Gaussian Stochastic Processes: Hybrid Spectral Representation and Proper Orthogonal Decomposition Approach

By observing that the optimal basis for the proper orthogonal decomposition (POD) can be obtained from the cross power spectral density (XPSD) matrix of a multivariate stationary Gaussian stochastic process, the computational efficiency, in both time and memory consumption, of simulations of this process is improved by using a hybrid spectral representation and POD approach with negligible loss of accuracy. This hybrid approach actually simulates another multivariate process with many fewer variables in an optimal subspace obtained by the POD. This approach is straightforward, effective, and does not place any conditions on the XPSD matrices. Furthermore, the error induced by the reduction of variables is predictable and controllable prior to the simulation procedure. The spectral representation method (SRM) is discussed in a heuristic way. In this paper, a specific POD theorem is formally stated, proved, and related to XPSD matrices. A numerical example is given to demonstrate the effectiveness of this hybrid approach. This approach may also have potential applications for simulations of nonstationary non-Gaussian processes.     

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.     

Polynomial Chaos Decomposition for the Simulation of Non-Gaussian Nonstationary Stochastic Processes

A method is developed for representing and synthesizing random processes that have been specified by their two-point correlation function and their nonstationary marginal probability density functions. The target process is represented as a polynomial transformation of an appropriate Gaussian process. The target correlation structure is decomposed according to the Karhunen–Loève expansion of the underlying Gaussian process. A sequence of polynomial transformations in this process is then used to match the one-point marginal probability density functions. The method results in a representation of a stochastic process that is particularly well suited for implementation with the spectral stochastic finite element method as well as for general purpose simulation of realizations of these processes.     

Amplitude Variability in Simulated Incoherent Seismic Ground Motions

This note compares in detail four commonly used schemes for the simulation of spatially variable ground motions. Emphasis is placed not only on the conformity of the simulations with the power and cross spectral density of the random field but, also, on the examination of the consistency of the simulations with the homogeneity condition, and the (Fourier) amplitude variability of the simulations. It is shown that, whereas three techniques that simulate ground motions in parallel satisfy the homogeneity requirement, produce simulations with random amplitudes, and amplitude and phase variability consistent with that of recorded data, one technique that simulates motions in sequence does not.     

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.     

Approximate Reliability-Based Optimization Using a Three-Step Approach Based on Subset Simulation

A novel three-step approach is proposed to solve reliability-based optimization (RBO) problems. The new approach is based on a novel approach previously developed by the authors for estimating failure probability functions. The major advantage of the new approach is that it is applicable to RBO problems with high-dimensional uncertainties and with arbitrary system complexities. The basic idea is to transform the reliability constraint in the target RBO problem into nonprobabilistic one by first estimating the failure probability function and the confidence intervals using minimal amount of computation, in fact, using just a single subset simulation (SubSim) run for each reliability constraint. Samples of the failure probability function are then drawn from the confidence intervals. In the second step, candidate solutions of the RBO problems are found based on the samples, and in the third step, the final design solution is screened out of the candidates to ensure that the failure probability of the final design meets the target, which also only costs a single SubSim run. Four numerical examples are investigated to verify the proposed novel approach. The results show that the approach is capable of finding approximate solutions that are usually close to the actual solution of the target RBO problem.     

Identification of Statistically Homogeneous Soil Layers Using Modified Bartlett Statistics

Stationarity or statistical homogeneity is an important prerequisite for subsequent statistical analysis on a given section of a soil profile to be valid. The estimation of important soil statistics such as the variance is likely to be biased if the profile is not properly demarcated into stationary sections. Existing classical statistical tests are inadequate even for simple identification of stationarity in the variance because the spatial variations of soil properties are generally correlated with each other. In this paper, a modified Bartlett statistical test is proposed to provide a more rational basis for rejecting the null hypothesis of stationarity in the correlated case. The accompanying rejection criteria are determined from simulated correlated sample functions and summarized into a convenient form for practical use. A statistical-based soil boundary identification procedure is then developed using the modified Bartlett test statistic. Based on the analysis of a piezocone sounding record, two advantages can be discerned. First, the proposed procedure provides a useful supplement to existing empirical soil classification charts, especially in situations where inherent variability tends to complicate interpretation of soil layers. Second, various key assumptions in geostatistical analysis such as stationarity and choice of trend function can be verified more rigorously using the framework of hypothesis testing.     

Spectral Analysis and Parametric Study of Stochastic Pavement Loads

This paper investigates the effects of vehicle parameters, speed, and surface roughness on the power spectral density (PSD) of stochastic pavement loads. Pavement surface roughness is modeled as a zero-mean stationary random field. A quarter-vehicle model is established to simulate the vibrations of heavy and passenger vehicles with typical parameters. Tire damping is also included in the consideration of stochastic pavement loads; this was assumed to be zero in many previous investigations. The PSD roughness proposed by the ISO is adopted in the simulation of the loads. An important indicator of the stochastic loads, the so-called energy cumulative distribution function, is introduced to describe the frequency distribution of load energy. The results show that passenger vehicles produce more high-frequency loads than heavy vehicles, while more of the loads generated by heavy vehicle are primarily distributed in the low-frequency region. It is also found that the effect of tire damping on stochastic pavement loads is not negligible especially if the loads of interest are concentrated in the high-frequency region. The results of the study may be useful in optimum design of vehicle suspensions and prediction of dynamic pavement response.     

Quick Seismic Response Estimation of Prestressed Concrete Bridges Using Artificial Neural Networks

Seismic early warning has been very important and has become feasible in Taiwan. Perhaps because of the lack of quick and reliable estimations of the induced structural response, however, the triggering criteria of almost all of the existing earthquake protection or early warning systems in the world are merely based on the collected or estimated data of the ground motion, without any information regarding the structural response. This paper presents a methodology of generating quick seismic response estimations of a prestressed concrete (PC) bridge using artificial neural networks (ANNs), which may be incorporated in a seismic early warning system for the bridge. In the methodology ANNs were applied to model the critical structural response of a PC bridge subjected to earthquake excitation of various magnitudes along various directions. The objective was to implement a well-trained network that is capable of providing a quick prediction for the critical response of the target bridge. The well-known multilayer perception (MLP) networks with back propagation algorithm were employed. A simple augmented form of MLP that can be quantitatively determined was proposed. These networks were trained and tested based on the analytical data obtained from the nonlinear dynamic finite fiber element analyses of the target PC bridge. The augmented MLPs were found to be much more efficient than the MLPs in modeling the critical bending moments of the piers and girder of the PC bridge.     

Stochastic Streamflow Simulation Using SAMS-2003

SAMS is a specialized software that has been developed for analyzing, modeling, and generating synthetic samples of hydrologic and water resources time series such as monthly streamflows. The 2003 version of SAMS provides enhanced technical capabilities from the earlier versions of the software. The graphical user interface and the mechanisms for handling the data have been entirely rewritten in MS Visual C++. As a result SAMS-2003 is easier to use and easier to update and maintain. In addition, substantial changes and restructuring have been made to enhance the modeling and data generation capabilities. The package provides many menu option windows that focus on three primary application modules—statistical analysis of data, fitting of a stochastic model (including parameter estimation and testing), and generating synthetic series. SAMS has the capability of analyzing and modeling single site and multisite annual and seasonal data such as monthly and weekly streamflows based on a number of single site and multisite stochastic models, and aggregation and disaggregation modeling schemes. The models are then utilized for generating synthetic data. Results from the various computations, e.g., the generated samples, can be presented in graphical and tabular forms and, if desired, saved to an output file. Some illustrations are provided to demonstrate the improved technical capabilities of the program using flow data of the Colorado River system.     

Stochastic Analysis of a Single-Degree-of-Freedom Nonlinear Experimental Moored System Using an Independent-Flow-Field Model

Stochastic characteristics of the surge response of a nonlinear single-degree-of-freedom moored structure subjected to random wave excitations are examined in this paper. Sources of nonlinearity of the system include a complex geometric configuration and wave-induced quadratic drag. A Morison-type model with an independent-flow-field formulation and a three-term-polynomial approximation of the nonlinear restoring force is employed for its proven excellent prediction capability for the experimental results investigated. Wave excitations considered in this study include nearly periodic waves, which take into account the presence of tank noise, noisy periodic waves that have predominant periodic components with designed additive random perturbations, and narrow-band random waves. A unified wave excitation model is used to describe all the wave conditions. A modulating factor governing the degree of randomness in the wave excitations is introduced. The corresponding Fokker–Planck formulation is applied and numerically solved for the response probability density functions (PDFs). Experimental results and simulations are compared in detail via the PDFs in phase space. The PDFs portray coexisting multiple response attractors and indicate their relative strengths, and experimental response behaviors, including transitions and interactions, are accordingly interpreted from the ensemble perspective. Using time-averaged probability density functions as an invariant measure, probability distributions of large excursions in experimental and simulated responses to various random wave excitations are demonstrated and compared. Asymptotic long-term behaviors of the experimental responses are then inferred.     

Investigation of Turbulence Effects on Torsional Divergence of Long-Span Bridges by Using Dynamic Finite-Element Method

Aerostatic stability of super long-span bridges is a much concerned issue during the design stage. Typical aerostatic instability is the so-called torsional divergence which may lead to abrupt structural failure. The iterative static-based FEM, which generally entails the assumption of smooth oncoming flow, has been widely used to evaluate the aerostatic stability of the bridge concerned. However, the wind in atmospheric boundary layer is naturally turbulent and the effect of turbulence on bridge torsional divergence should be therefore considered, and that is the main concern of the present study. To take into account the effects of turbulence on torsional divergence, a dynamic-based time domain finite-element (FE) procedure for predicting bridge aerostatic stability is introduced first. Then the quasi-steady wind loads expressions are presented and discussed, into which the aerodynamic torsional stiffness, which is indispensable for the evaluation of aerostatic stability, has been demonstrated to be incorporated indirectly by a frequency-domain-based approach. Finally, the aerostatic performances of the longest suspension bridge in China are investigated, of which the torsional divergence is the primary concern. Numerical results show that the torsional divergence pattern in turbulent flow differs considerably from that in smooth flow. The primary difference is, while the torsional instability in smooth flow manifests as an abrupt mounting up of the twist deformation of the main girder with the increasing of the wind velocity, that in turbulent flow manifests as an unstable stochastic vibration with large peak values. Another difference is that the wind velocity for divergence in turbulent flow is obviously lower than that in smooth wind and there does not present an obvious wind velocity threshold for divergence, which is distinguished from the torsional divergence in smooth flow characterized by a clear threshold. Based on the presented time domain FE analysis procedure, the influence of turbulence intensity and gusts spatial correlation upon torsional divergence is also investigated and shown to play an important role on the aerostatic stability.     

Updating Properties of Nonlinear Dynamical Systems with Uncertain Input

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to quantify the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is first presented for single-degree-of-freedom systems and then for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator, an elastoplastic system and a four-story inelastic structure are presented to illustrate the proposed approach.     

Performance of Wavelet Transform and Empirical Mode Decomposition in Extracting Signals Embedded in Noise

Time-frequency transformations have gained increasing attention for the characterization of nonstationary signals in a broad spectrum of science and engineering applications. This study evaluates the performance of two popular transformations, the continuous wavelet transform and empirical mode decomposition with Hilbert transform (EMD+HT), in estimating instantaneous frequency (IF) in the presence of noise. The findings demonstrate that under these conditions wavelets seeking harmonic similitude at various scales produce lower variance IF estimates than EMD+HT. The shortcomings of the latter approach are attributed to its empirical, envelope-dependent nature, leading to bases that are themselves derived from noise.     

Stochastic Time-Cost-Resource Utilization Optimization Using Nondominated Sorting Genetic Algorithm and Discrete Fuzzy Sets

In a construction project, the cost and duration of activities could change due to different uncertain variables such as weather, resource availability, etc. Resource leveling and allocation strategies also influence total time and costs of projects. In this paper, two concepts of time-cost trade-off and resource leveling and allocation have been embedded in a stochastic multiobjective optimization model which minimizes the total project time, cost, and resource moments. In the proposed time-cost-resource utilization optimization (TCRO) model, time and cost variables are considered to be fuzzy, to increase the flexibility for decision makers when using the model outputs. Application of fuzzy set theory in this study helps managers/planners to take these uncertainties into account and provide an optimal balance of time, cost, and resource utilization during the project execution. The fuzzy variables are discretized to represent different options for each activity. Nondominated sorting genetic algorithm (NSGA-II) has been used to solve the optimization problem. Results of the TCRO model for two different case studies of construction projects are presented in the paper. Total time and costs of the two case studies in the Pareto front solutions of the TCRO model cover more than 85% of the ranges of total time and costs of solutions of the biobjective time-cost optimization (TCO) model. The results show that adding the resource leveling capability to the previously developed TCO models provides more practical solutions in terms of resource allocation and utilization, which makes this research relevant to both industry practitioners and researchers.     

Efficacy of Hilbert and Wavelet Transforms for Time-Frequency Analysis

Two independently emerging time-frequency transformations in Civil Engineering, namely, the wavelet transform and empirical mode decomposition with Hilbert transform (EMD+HT), are discussed in this study. Their application to a variety of nonstationary and nonlinear signals has achieved mixed results, with some comparative studies casting significant doubt on the wavelet’s suitability for such analyses. Therefore, this study shall revisit a number of applications of EMD+HT in the published literature, offering a different perspective to these commentaries and highlighting situations where the two approaches perform comparably and others where one offers an advantage. As this study demonstrates, much of the differing performance previously observed is attributable to EMD+HT representing nonlinear characteristics solely through the instantaneous frequency, with the wavelet relying on both this measure and the instantaneous bandwidth. Further, the resolutions utilized by the two approaches present a secondary factor influencing performance.     

CTAN for Risk Assessments Using Multilevel Stochastic Networks

Measuring projects’ cost and schedule risks in an integrated framework using simulation has several modeling challenges that have yet to be addressed by researchers. This paper presents a multilevel network modeling approach that aims to integrate a combination of different networks in one framework, and presents a computer simulation implementation to the cost and time risk assessment network (CTAN). The CTAN is an integrated network that includes uncertainties in the realization of the schedule logic, in activities durations, in project scope, and in cost. The simulation model is a decision support simulation system (DSSS) that currently consists of three modules: the CTAN, the stochastic decision trees, and the stochastic shortest/longest rout network. The CTAN-DSSS may be used in cost and schedule risk assessment. It completely integrates with other DSSS networks and deals with risks associated with cost, time, and scope at equal importance. The DSSS was verified by conducting several tests and validated by its extensive use for both undergraduate and graduate courses in Civil Engineering at the University of Calgary over the last three years.