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.     

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.     

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.     

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.     

Hilbert-Huang Transform Analysis of Dynamic and Earthquake Motion Recordings

This study examines the rationale of Hilbert-Huang transform (HHT) for analyzing dynamic and earthquake motion recordings in studies of seismology and engineering. In particular, this paper first provides the fundamentals of the HHT method, which consist of the empirical mode decomposition (EMD) and the Hilbert spectral analysis. It then uses the HHT to analyze recordings of hypothetical and real wave motion, the results of which are compared with the results obtained by the Fourier data processing technique. The analysis of the two recordings indicates that the HHT method is able to extract some motion characteristics useful in studies of seismology and engineering, which might not be exposed effectively and efficiently by Fourier data processing technique. Specifically, the study indicates that the decomposed components in EMD of HHT, namely, the intrinsic mode function (IMF) components, contain observable, physical information inherent to the original data. It also shows that the grouped IMF components, namely, the EMD-based low- and high-frequency components, can faithfully capture low-frequency pulse-like as well as high-frequency wave signals. Finally, the study illustrates that the HHT-based Hilbert spectra are able to reveal the temporal-frequency energy distribution for motion recordings precisely and clearly.     

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.     

Effect of Asynchronous Earthquake Motion on Complex Bridges. II: Results and Implications on Assessment

Nonuniform seismic excitation has been shown through previous analytical studies to adversely affect the response of long-span bridge structures. To further understand this phenomenon, this study investigates the response of complex straight and curved long-span bridges under the effect of parametrically varying asynchronous motion. The generation process and modeling procedures are presented in a companion paper. A wide-ranging parametric study is performed aimed at isolating the effect of both bridge curvature and the two main sources of asynchronous strong motion: geometric incoherence and the wave-passage effect. Results from this study indicate that response for the 344?m study structure is amplified significantly by nonsynchronous excitation, with displacement amplification factors between 1.6 and 3.4 for all levels of incoherence. This amplification was not constant or easily predicable, demonstrating the importance of inelastic dynamic analysis using asynchronous motion for assessment and design of this class of structure. Additionally, deck stiffness is shown to significantly affect response amplification, through response comparison between the curved and an equivalent straight bridge. Study results are used to suggest an appropriate domain for consideration of asynchronous excitation, as well as an efficient methodology for analysis.     

Effect of Asynchronous Earthquake Motion on Complex Bridges. I: Methodology and Input Motion

Based on observed damage patterns from previous earthquakes and a rich history of analytical studies, asynchronous input motion has been identified as a major source of unfavorable response for long-span structures, such as bridges. This study is aimed at quantifying the effect of geometric incoherence and wave arrival delay on complex straight and curved bridges using state-of-the-art methodologies and tools. Using fully parametrized computer codes combining expert geotechnical and earthquake structural engineering knowledge, suites of asynchronous accelerograms are produced for use in inelastic dynamic analysis of the bridge model. Two multi-degree-of-freedom analytical models are analyzed using 2,000 unique synthetic accelerograms with results showing significant response amplification due to asynchronous input motion, demonstrating the importance of considering asynchronous seismic input in complex, irregular bridge design. The paper, Part 1 of a two-paper investigation, presents the development of the input motion sets and the modeling and analysis approach employed, concluding with sample results. Detailed results and implications on seismic assessment are presented in the companion paper: Effect of Asynchronous Motion on Complex Bridges. Part II: Results and Implications on Assessment.     

基于MATLAB的双馈调速系统矢量控制的仿真

双馈调速具有能同时调节定子有功、无功等优点,是一种非常具有竞争力的调速方案。分析了基于定子磁场定向的矢量控制的基本原理及其算法,并利用MATLAB/SIMULINK软件对双馈调速系统的矢量变换部分进行了仿真且给出了仿真波形。     

Response Spectrum Analysis of Suspension Bridges for Random Ground Motion

The response spectrum method of analysis for suspension bridges subjected to multicomponent, partially correlated stationary ground motion is presented. The analysis is based on the relationship between the power spectral density function and the response spectrum of the input ground motion and fundamentals of the frequency domain spectral analysis. The analysis duly takes into account the spatial correlation of ground motions between the supports, the quasi-static component of the response, and the modal correlation between different modes of vibration. A suspension bridge is analyzed under a set of important parametric variations in order to (1) compare between the responses obtained by the response spectrum method of analysis and the frequency domain spectral analysis; and (2) investigate the behavior of suspension bridges under seismic excitation. The parameters include the spatial correlation of ground motion, the angle of incidence of the earthquake, the ratio between the three components of ground motion, the number and nature of modes considered in the analysis, and the nature of the power spectral density function of ground motion. It is shown that the response spectrum method of analysis provides a fair estimate of responses under parametric variations considered in the study.     

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.     

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.     

Subset Simulation and its Application to Seismic Risk Based on Dynamic Analysis

A method is presented for efficiently computing small failure probabilities encountered in seismic risk problems involving dynamic analysis. It is based on a procedure recently developed by the writers called Subset Simulation in which the central idea is that a small failure probability can be expressed as a product of larger conditional failure probabilities, thereby turning the problem of simulating a rare failure event into several problems that involve the conditional simulation of more frequent events. Markov chain Monte Carlo simulation is used to efficiently generate the conditional samples, which is otherwise a nontrivial task. The original version of Subset Simulation is improved by allowing greater flexibility for incorporating prior information about the reliability problem so as to increase the efficiency of the method. The method is an effective simulation procedure for seismic performance assessment of structures in the context of modern performance-based design. This application is illustrated by considering the failure of linear and nonlinear hysteretic structures subjected to uncertain earthquake ground motions. Failure analysis is also carried out using the Markov chain samples generated during Subset Simulation to yield information about the probable scenarios that may occur when the structure fails.     

Formation Processes and Configuration of Channel-Flow Dominated Alluvial Deltas by Numerical Simulation

A horizontal two-dimensional mobile bed model for simulating the formation of river-dominated deltas in the river mouth or reservoir is presented, which is composed of shallow water equations, sediment transport formula, and a sediment continuity equation. Geometry similarity of river deltas during the processes of formation is discussed. Stability analysis and sensitivity analysis of parameters in the model are analyzed, which indicates that bed configuration is sensitive to the incipient-motion criteria of bed–load particles. The effect of gravity component on the initiation of sediment movement, therefore, is recommended to be considered in the modeling. The bed configuration including the reverse slope in the longitudinal profile and concave in the transverse profiles are correctly simulated with help from the correction of incipient-motion criteria. Simulation results are verified with a series of experiments and are consistent with series geometric functions and dimensionless profiles inducted from experimental data. This reflects the great reliability of the model. Historical topographical records of two typical in-land deltas depicting their earlier developmental stages are discussed to show the usefulness of this study.     

基于MATLAB／SimMechanics的离心式飞剪机的建模与仿真

飞剪是连续轧钢机组中的重要设备之一,其结构参数的设计直接影响产品质量和生产效率。简要介绍离心式飞剪的工作原理,运用SimMechanics工具箱建立了机构仿真模型,模拟了飞剪的运动过程,得到了飞剪的运动规律。仿真结果表明该飞剪结构参数满足实际生产要求。而且,该建模方法和仿真结果对于深入分析剪切过程、进行参数优化以及设计同类飞剪具有参考价值。     

Application Framework for Mapping and Simulation of Waste Handling Processes in Construction

This research is focused on modeling waste-handling processes in construction, with particular emphasis on how to map out and simulate on-site waste sorting processes. The research proposes an application framework for (1) guiding the development of process mapping models and simulation models; and (2) further assessing the cost effectiveness of on-site waste sorting efforts under practical site constraints (such as labor resource availability, time control on refuse chute usage, and limited working area space in a building site). The connection has been established between the mapping and simulation techniques in the context of modeling waste handling processes in construction sites, such that the process flowchart resulting from the mapping technique can serve as convenient model input to facilitate the creation of a “dynamic” operations simulation model. A case study of the on-site waste sorting method with one refuse chute for waste classification is presented to demonstrate the complete application framework spanning (1) process mapping; (2) mapping-to-simulation model conversion; and (3) method optimization based on valid simulations.     

Numerical Simulation of Turbulent Flows,Heat and Mass Transfer in Metallurgical Induction Processes

Comprehensive knowledge of the turbulent flows, heat and mass transfer processes in the melt of induction applications is required to realize efficient metallurgical processes. Experimental and numerical studies of the melt flow in induction furnaces show that the flow pattern, which comprises several vortices of the mean flow, and the temperature distribution in the melt are significantly influenced by low‐frequency large‐scale flow oscillations. Two‐ and three‐dimensional hydrodynamic calculations of the melt flow, using two‐equation turbulence models based on Reynolds Averaged Navier‐Stokes approach, do not predict the large‐scale periodic flow instabilities obtained from the experimental data. That is why the Large Eddy Simulation (LES) numerical technique was approved to be an alternative for the various k‐? model modifications. The results of the transient 3D LES simulation of the turbulent melt flow revealed the large‐scale periodic flow instabilities and the temperature distribution in the melt, which both are in good agreement with the expectations based on the data from the experiments. The studies, presented in this paper, demonstrate the possibility of using the three‐dimensional transient LES approach for successful simulation of heat and mass transfer processes in metallurgical applications.     

Proper Orthogonal Decomposition-Based Modeling, Analysis, and Simulation of Dynamic Wind Load Effects on Structures

Multicorrelated stationary random processes/fields can be decomposed into a set of subprocesses by diagonalizing their covariance or cross power spectral density (XPSD) matrices through the eigenvector/modal decomposition. This proper orthogonal decomposition (POD) technique offers physically meaningful insight into the process as each eigenmode may be characterized on the basis of its spatial distribution. It also facilitates characterization and compression of a large number of multicorrelated random processes by ignoring some of the higher eigenmodes associated with smaller eigenvalues. In this paper, the theoretical background of the POD technique based on the decomposition of the covariance and XPSD matrices is presented. A physically meaningful linkage between the wind loads and the attendant background and resonant response of structures in the POD framework is established. This helps in better understanding how structures respond to the spatiotemporally varying dynamic loads. Utilizing the POD-based modal representation, schemes for simulation and state-space modeling of random fields are presented. Finally, the accuracy and effectiveness of the reduced-order modeling in representing local and global wind loads and their effects on a wind-excited building are investigated.