Spectral Representation Method Research Articles

Dimension-Reduction Spectral Representation of Soil Spatial Variability and Its Application in the Efficient Reliability Analysis of Seismic Response in Tunnels

Reliability analysis of tunnels considering soil spatial variability was traditionally carried out using Monte Carlo simulations, which are computationally expensive. This study proposes a dimension-reduction spectral representation method (SRM) for simulating parameter stochastic fields to capture uncertainties in soil properties, which can be coupled with the probability density evolution method (PDEM). Additionally, the PDEM is employed to conduct seismic reliability assessments of tunnels in a more efficient manner. By integrating a non-intrusive dynamic random finite element method (FEM), this study explores the horizontal drift angle and vertical relative settlement of tunnel dynamic response under different coefficient of variations (COVs) and horizontal autocorrelation lengths of elastic modulus (E). Results show the changes in COVs have more pronounced effects on the mean and standard deviation of horizontal drift angle and vertical relative settlement of tunnel than alterations in horizontal autocorrelation lengths. A limit state pertaining to the horizontal drift angle and vertical relative settlement is established through the use of a probability density function (PDF) and cumulative distribution function (CDF). The results suggest that the combination of soil spatial variability and PDEM can serve as an effective approach for practical tunnel design.

Structural reliability analysis under stochastic seismic excitations and multidimensional limit state based on gamma mixture model and copula function

A novel method for analyzing the reliability of structures under non-stationary stochastic seismic excitations, considering the combined effect of multiple structural demand extreme values, is proposed. The spectral representation method is employed to establish a non-stationary stochastic seismic excitation model, and based on the theory of first-passage probability, multiple integral formulas for seismic reliability under multidimensional limit states are derived. The extreme value distribution is established using the Gamma mixture model (GMM). The equations for estimating the model parameters are derived based on both fractional moments and moment-generating functions, while the determination of the number of gamma distribution components is guided by the probability distribution and statistical characteristics of the samples. The joint probability density function (JPDF) for multiple demand extreme values is established by incorporating copula functions to account for correlation, and the fitting accuracy of different copula functions is assessed. The proposed method is illustrated using reinforced concrete (RC) frame structures. The results demonstrate that the fitting accuracy of extreme value distribution can be enhanced by adjusting the number of gamma distribution components in the GMM, which exhibits high accuracy in fitting both the main and tail regions. The presence of correlation can induce variations in the JPDF, thereby exerting an influence on the failure probability. Consequently, disregarding correlation is not conducive to reliability analysis.

Higher-order spectral representation method: New algorithmic framework for simulating multi-dimensional non-Gaussian random physical fields

This study derives a novel higher-order spectral representation method (HOSRM) to represent and simulate multi-dimensional fourth-order non-Gaussian random physical fields. The method primarily extends the traditional second-order spectral representation method (SRM) for simulating non-Gaussian random physical fields by introducing higher-order cumulant function tensors and trispectrum tensors, thereby accomplishing the modeling of fourth-order non-Gaussian random physical fields (symmetric nonlinear physical fields) from a frequency domain perspective. In an endeavor to enhance the simulation efficiency of this theoretical framework, the Fast Fourier Transform (FFT) algorithm is astutely amalgamated into the simulation. This integration contributes significantly to the augmentation of computational efficiency. Furthermore, exhaustive derivations and proofs are presented for the first-order, second-order, and fourth-order ensemble properties of simulated fourth-order non-Gaussian random physical fields. Subsequently, the reliability and accuracy of the proposed algorithm framework are validated through numerical simulations of two two-dimensional and two three-dimensional non-Gaussian random physical fields. The findings demonstrate that the simulated sample function effectively captures the probability characteristics of the random field, including mean, variance, and kurtosis. Finally, an in-depth analysis of numerical simulation results highlights the difference between the proposed method and the traditional second-order spectral representation method, which further underscores the distinctive attributes and potential superiority of the proposed method.

Conditional simulation of stationary non-Gaussian processes based on unified hermite polynomial model

The conditional simulation of non-Gaussian excitations utilizing records from the monitoring system is of great significance for hazard mitigation. To this end, this paper proposes a novel conditional non-Gaussian simulation method. In this method, the Unified Hermite Polynomial Model (UHPM) is used to describe the transformation relationship between recorded and unrecorded non-Gaussian processes and their underlying Gaussian counterparts. Meanwhile, an explicit transformation model between their correlation functions is also provided. Then, the covariance matrix of Fourier coefficients of the underlying Gaussian processes is constructed. Based on this covariance matrix, the conditional samples of Fourier coefficients are generated and substituted into the Spectral Representation Method (SRM) to perform the conditional simulation of the underlying Gaussian processes. Finally, the conditionally simulated samples of the underlying Gaussian processes are transformed into the non-Gaussian samples by the UHPM. To showcase the precision and efficacy of the proposed method, two numerical examples involving the conditional simulations of non-Gaussian ground motions and non-Gaussian wind pressure coefficients are provided.

Seismic analyses of single-layer dome structures with random geometrical imperfections under stochastic ground motions

In this paper, seismic analyses of single-layer dome structures are carried out, with a particular emphasis on the incorporation of randomness in both geometrical imperfections and ground motions. First, the paper introduces models for random geometrical imperfections and stochastic ground motions, employing the Stochastic Imperfection Mode Superposition Method (SIMSM) for imperfections and utilizing the Spectral Representation Method (SRM) for ground motions. These models are designed to comprehensively account for the inherent uncertainties in real-world structural behavior. Subsequently, the Probability Density Evolution Method (PDEM) is applied to investigate the probabilistic response and seismic reliability of dome structures, which allows for a detailed examination of key performance indicators, such as displacement, plastic ratio, and strength damage, under the influence of random factors. Both deterministic and stochastic analyses are conducted to gain insights into the impact of randomness in geometrical imperfections and ground motions. The computational results unveil a crucial finding: an unfavorable distribution of imperfections can significantly compromise the dynamic performance and safety of single-layer dome structures. Furthermore, the paper delves into parametric analyses, with a specific focus on imperfection amplitude, the rise-span ratio and roof load as pivotal parameters. These analyses provide valuable insights into the sensitivity of the structural response to variations in these critical design factors.

An efficient methodology for simulating multivariate non-Gaussian stochastic processes

Simulation of multivariate non-Gaussian stochastic processes is still a challenging task, where the efficiency and accuracy may not be balanced. In this paper, an efficient methodology is presented to address this challenge. This methodology makes use of a probability weighted moments-based Hermite polynomial model and a wavenumber–frequency spectrum density-based spectral representation approach. First, a probability weighted moments-based Hermite polynomial model is employed to establish a transformation relationship between Gaussian and non-Gaussian samples. Importantly, this model only requires the solution of a solvable system of equations. Second, the target non-Gaussian wavenumber–frequency spectrum density can be promptly transformed into the Gaussian one, where a fast root matching strategy and fast Fourier transform are utilized. Finally, the underlying Gaussian samples are generated by the spectral representation method, and the non-Gaussian samples are obtained via the proposed transformation model. The computational results from numerical examples demonstrate that the proposed methodology is considered to be an accurate, applicable, and efficient approach for simulating multivariate non-Gaussian stochastic processes.

Stochastic Optimization of Electromagnetic Inertial Mass Dampers in Nonlinear Hybrid Base Isolation Systems Under Seismic Excitations

Recently developed inerter-based dampers have been installed at the base layer to reduce base displacement and simultaneously improve the superstructure’s performance. However, in the existing research on optimizing damper parameters, the nonlinear property of base isolation is simplified and stochastic linearization technique is introduced to approximate such nonlinearity, which would inevitably introduce errors. In this paper, an improved method is proposed for stochastic optimization of electromagnetic inertial mass dampers (EIMD) in nonlinear hybrid isolation systems subject to seismic excitations. First, a stochastic seismic excitation is represented by one elementary random variable adopting the dimension-reduction spectral representation method. Then, statistical moments of nonlinear base-isolated structural responses are estimated using two-point estimation method (2PEM). Finally, the EIMD optimal parameters are obtained via the modified directional bat algorithm (MDBA) recently proposed by the authors through minimizing base layer response moments. This proposed improved method could fully consider the base isolation nonlinearity compared with the previous stochastic linearization technique. Moreover, the proposed stochastic optimization of the EIMD only requires two dynamic analyses to achieve its efficiency. A seven-story base-isolated frame building equipped with an EIMD is investigated in the numerical simulation, and results indicate that EIMD optimized by this proposed method obviously outperforms the results by the stochastic linearization technique and the conventional viscous damper (VD) in the base layer displacement reduction.

A frequency-dependent uniform discretization scheme for simulating fluctuating wind field based on a frequency-wavenumber spectrum

The frequency-wavenumber spectrum-based spectral representation method (FWS-based SRM) serves as an effective and broadly utilized technique for simulating multivariate fluctuating wind fields. Despite its usefulness, it struggles with accuracy in low frequency range during simulations involving structures of considerable length when employing conventional uniform discretization in both frequency and wavenumber domains. To overcome this challenge, this study introduces a frequency-dependent uniform discretization scheme. This strategy first applies uniform discretization to the frequency domain, and then ascertains the cut-off wavenumbers based on the truncation error, and finally implements uniformly discretization in the wavenumber domain. Subsequently, the uniformly discrete wavenumbers and frequencies are integrated into the FWS-based SRM to generate wind speed time histories, in combination with the FFT technique. The numerical example serves to validate the exceptional accuracy exhibited by the frequency-dependent uniform discretization scheme in low frequency range.

Stochastic simulation of earthquake ground motions based on improved finite-fault model

In the seismic reliability analysis of engineering structures, it is often necessary to adopt stochastic ground motions as inputs. However, the current methods used to generate stochastic samples, such as the spectral representation method, the spectral transfer method, the dynamic source method, the physical source method, and the comprehensive method, etc., are not very convincing since they do not contain the complete information of seismic occurrence and propagation. In this study, an improved finite-fault model for stochastic simulation of ground motions is proposed. Compared to existing stochastic finite-fault model, the improved model considers the propagation law of phase spectrum, local site effect, and the impact of wave velocity on the arrival time of seismic waves. Besides, identification and statistics of model parameters are carried out based on ground motion records. Reasonable samples of stochastic ground motions are then obtained. In addition, parameter sensitivity analysis is conducted to determine critical random variables, which reduces the complexity of stochastic simulation. Simulation results are compared with recorded accelerograms from the 1999 Chi-Chi earthquake. It is shown that the improved stochastic finite-fault model is reliable, and the acceleration response spectra of simulated samples are consistent with those of station records collected at various site classes and the design spectra.

Seismic performance assessment of hydraulic tunnels considering oblique incoming nonstationary stochastic SV waves based on the generalized PDEM

An important aspect of a seismic performance assessment framework is to adequately consider the stochastic nature of SGMs. This requires an appropriate method for synthesizing random SGMs, a crucial step in conducting structural response analysis. Additionally, affected by topographic and geological conditions, the incident angle of SGMs is also uncertain and random, which may induce unexpected damage to structures. To date, a framework is presented to assess the seismic performance of hydraulic tunnels under stochastic obliquely incident SV waves. The framework incorporates the spectral representation method of SGMs and obliquely incident SV waves into stochastic dynamic analysis. To create the appropriate PDFs and CDFs for numerous performance indices, the generalized PDEM and equivalent extreme-value event are introduced. Subsequently, stochastic dynamic analysis and seismic performance assessment of the hydraulic tunnel are performed. Finally, the disparity between the hydraulic tunnel fragility curves generated using conventional linear regression and generalized PDEM is examined. The difference between vertically incident and obliquely incident SV waves is discussed. The results indicated that (1) different performance indices exhibit great differences in the stochastic dynamic response of structures and the failure probability of seismic performance evaluation; (2) compared with the traditional linear regression method, the probability of damage of hydraulic tunnels under strong SGMs may be overestimated; and (3) nonstationary stochastic obliquely incident SV waves can cause more seismic demand than nonstationary stochastic vertically incident SV waves under low/medium earthquake intensity levels for hydraulic tunnel.

Neural network-aided simulation of non-Gaussian stochastic processes

Simulation of non-Gaussian stochastic processes is of paramount importance to dynamic reliability assessment of structures driven by non-Gaussian loads. In this paper, a novel translation model based on neural network is proposed to convert the target non-Gaussian power spectrum to the underlying Gaussian power spectrum for simulating non-Gaussian stochastic processes. First, a valid neural network model, whose structure is similar to the transformation from a non-Gaussian power spectrum to its underlying Gaussian power spectrum, is established. Then, starting with an initial underlying Gaussian power spectrum, the non-Gaussian power spectrum corresponding to the target non-Gaussian distribution is obtained through the Wiener-Khintchine transform and translation process. After that, the discrete data of non-Gaussian power spectrum is employed as the input layer in the neural network above, while the discrete data of Gaussian power spectrum serves as the output layer. With the trained neural network, the required underlying Gaussian power spectrum can be directly obtained by inputting the target non-Gaussian power spectrum. Finally, Gaussian samples can be generated by using the spectral representation method, which are then converted into non-Gaussian samples via the translation process. The effectiveness, accuracy, and applicability of the proposed method are demonstrated through two numerical examples.

Efficient simulation of fully nonstationary wind velocity field by an enhanced numerical truncation method

The spectral representation method (SRM) is widely utilized to simulate the fully nonstationary wind velocity field with time-varying coherence. The simulation samples are generated by double summation of harmonic functions. Despite the fast Fourier transform (FFT) technique has been employed to accelerate the summation in the frequency dimension, the computational efficiency is still constrained by the number of simulation points and samples due to the time-consuming summation operation in the spatial dimension. In this study, a numerical truncation method is presented to further quicken the SRM-based simulation of a fully nonstationary wind velocity field. The fundamental idea is that the harmonic functions with tiny amplitude are inconsequential and can be omitted. A novel diagonal decoupling approach is first developed to rearrange the double summation. Further, the frequency truncation scheme is established to simplify the summation in the spatial dimension. Meanwhile, the FFT operations to calculate the summation in the frequency dimension are also considerably reduced. Finally, the superior performance of the proposed method on precision and efficiency is demonstrated by numerical examples.

An efficient method for simulating fluctuating wind speed fields in two-spatial dimensions based on a frequency-dependent acceptance-rejection scheme

The simulation of fluctuating wind speed fields holds paramount importance in the wind-resistant design of significant structures such as high-rise buildings, long-span bridges, and wind turbines. To address the computational challenges posed by the original spectral representation method (SRM), a novel approach has emerged. This approach, rooted in the frequency-wavenumber spectrum (FWS), offers a solution to the complexities associated with the original SRM. An enhanced non-uniform discrete technique based on the acceptance-rejection (A-R) criterion was introduced to improve the efficiency of the FWS-based SRM. Nonetheless, this technique currently faces certain constrains, including its demand for a substantial allocation of random access memory (RAM) and its inability to be seamlessly integrated with the fast Fourier transform (FFT) algorithm. Given this scenario, a notably more effective strategy for selecting representative points emerges -the frequency-dependent acceptance-rejection (A-R) scheme. This innovative scheme holds the advantage of diminishing RAM utilization, transforming wind field modeling from a supercomputers-exclusive task to one feasible on personal computers. Moreover, its seamless integration with FFT technology bolsters simulation efficiency. The provided numerical simulation instances of two-spatial dimensional fluctuating wind speed fields underscore the proficiency of this proposed frequency-dependent A-R scheme. The outcomes demonstrated the combined A-R and FFT technique’s efficiency and precision, in simulating two-spatial dimensions fluctuating wind speed fields.

Uncertainty Quantification and Simulation of Wind-Tunnel-Informed Stochastic Wind Loads

The simulation of stochastic wind loads is necessary for many applications in wind engineering. The proper-orthogonal-decomposition-(POD)-based spectral representation method is a popular approach used for this purpose, due to its computational efficiency. For general wind directions and building configurations, the data-informed POD-based stochastic model is an alternative that uses wind-tunnel-smoothed auto- and cross-spectral density as input, to calibrate the eigenvalues and eigenvectors of the target load process. Even though this method is straightforward and presents advantages, compared to using empirical target auto- and cross-spectral density, the limitations and errors associated with this model have not been investigated. To this end, an extensive experimental study on a rectangular building model considering multiple wind directions and configurations was conducted, to allow the quantification of uncertainty related to the use of short-duration wind tunnel records for calibration and validation of the data-informed POD-based stochastic model. The results demonstrate that the data-informed model can efficiently simulate stochastic wind loads with negligible model errors, while the errors associated with calibration to short-duration wind tunnel data can be important.

Probabilistic analysis of crown settlement in high-speed railway tunnel constructed by sequential excavation method considering soil spatial variability

High-speed railways (HSRs) are extensively constructed in China, with the help of tunnels to cross through complex geological conditions. The sequential excavation method (SEM) is vastly adopted in the construction of loess tunnel. This study performs a probabilistic analysis of crown settlement, taking into account the spatial variability of the surrounding soil through Monte Carlo (MC) simulations. To illustrate the influence of horizontal and vertical scale fluctuations (SOFs) on crown settlement, an anisotropic random field of loess elastic modulus is simulated using the spectral representation method (SRM) and unified Hermite polynomial model (UHPM) for the random finite difference method (RFDM). The mean crown settlements obtained through stochastic analysis are consistently larger than the deterministic results in the 1st, 6th, and 9th construction steps. The maximum probabilities of stochastic crown settlement exceeding the deterministic results are 77.3% and 78.2% with varying horizontal and vertical SOFs, respectively. Amplification factors of 1.417 and 1.306 are necessary to ensure that the stochastic crown settlement of the HSR tunnel, constructed in spatially variable loess, satisfies a 5% and 10% exceeding probability. This study provides valuable guidance for the construction of HSR tunnels.

Wavelet packets-based simulation of non-stationary multivariate ground motions

This study proposes a novel approach to simulate non-stationary multivariate earthquake accelerograms for the seismic analysis of extended structures. This approach is based on an extension of the traditional stochastic decomposition technique applied to the wavelet packet theory using the spectral representation method. Using the advantages of a wavelet packet to identify the time and frequency characteristics of ground motion effectively, the generated recordings successfully reflected the non-stationarity in both the time and frequency domains. Accordingly, the spatial correlation trend was consistent with the selected coherence function. Compared to the ground motions observed using the traditional method, the proposed approach was exemplified by simulating ensembles of ground motions at three points spaced 200 m apart. The multivariate ground motions simulated by different methods were applied as multi-support excitations for the dynamic time–history analysis of a 190 m continuous rigid-frame box girder bridge. Notable variations in structural response were observed when considering the non-stationarity of the frequency domain, contrasting with the cases where it was not taken into account. This study provides an effective approach for generating non-stationary multivariate earthquake accelerograms, which is of great significance for structural response analysis in engineering applications.

Simulating Multivariate Multidimensional Homogenous Non-Gaussian Field Based on Unified Hermite Polynomial Model

Reasonable simulation of stochastic fields is a key prerequisite for addressing stochastic mechanics problems with Monte Carlo simulation solver. This paper presents a novel method to simulate m-variate and n-dimensional (mV-nD) homogenous non-Gaussian fields according to the prescribed power spectral density matrix (PSDM) and first four moments. In the proposed method, the unified Hermite polynomial model (UHPM) is first extended to mV-nD homogenous non-Gaussian fields. Then, based on the extended UHPM, a complete transformation model from the normalized non-Gaussian correlation function matrix (CFM) into the underlying Gaussian CFM with its applicable range is derived. In addition, two types of potential incompatibility of the transformation arising from the prescribed PSDM and first four moments are defined and neatly solved. Finally, a unified simulation framework for mV-nD homogenous non-Gaussian fields is presented, where the fast Fourier transform technique can be embedded in both spectral representation method and Wiener-Khintchine transformation to speed up the simulation progress. Two numerical examples including the simulations of non-Gaussian wind fields and non-Gaussian fields of material properties are presented to demonstrate the effectiveness of the proposed method.

Estimation of extreme wind speed based on upcrossing rate of mean wind speeds with Weibull distribution

The estimation of the extreme mean wind speed has a pivotal role in the structural wind-resistance design. For yearly maximum mean wind speed samples usually not enough, it is necessary to estimate the extreme mean wind speed from short-term mean wind speed records, such as about a few years. The upcrossing rate method, which relies on the distribution of the mean wind speed, is one of the popular extreme estimation methods. It is widely recognized that the mean wind speed follows the Weibull distribution usually and the Rayleigh distribution at some areas. The upcrossing rate method for mean wind speed time series following Rayleigh distribution was proposed by Harris. The upcrossing rate method for mean wind speed time series following the Weibull distribution is studied in the present paper, by translating the Weibull process to the Rayleigh process based on the theory of cumulative distribution function mapping. The uncertainty of the extreme mean wind speed estimated by the proposed method is investigated to consider the sampling error from short-term mean wind speed records. The effectiveness of the proposed upcrossing rate method and uncertainty analysis are verified by the numerically generated mean wind speeds based on the spectral representation method with Von der Hoven wind speed spectrum.

Simulation of turbulent wind field in multi-spatial dimensions using a novel non-uniform FFT enhanced stochastic wave-based spectral representation method

Simulation of stochastic wind field is necessary for wind-induced vibration analysis of wind-sensitive structures. Stochastic wave-based spectral representation method (SWSRM) is one of the most widely used methods for wind field simulation. However, it may face challenges to simulate turbulent wind field in multi-spatial dimensions due to its large memory consumption and low simulation efficiency. In this study, a novel non-uniform FFT (NUFFT) enhanced SWSRM is proposed. The approach for selecting sampling points of wave number is also established based on adaptive integral method. Numerical experiments are further designed to demonstrate the validity and effectiveness of the enhanced SWSRM. Results show that the proposed method can significantly improve the simulation efficiency and reduce the memory consumption in multi-spatial dimensions. In addition, the simulation points can be arbitrarily selected using NUFFT enhanced SWSRM. More importantly, the proposed method can improve the turbulent spectral accuracy in low-frequency region.

Simulating Temporally and Spatially Correlated Wind Speed Time Series by Spectral Representation Method

In this paper, it aims to model wind speed time series at multiple sites. The five-parameter Johnson distribution is deployed to relate the wind speed at each site to a Gaussian time series, and the resultant m- dimensional Gaussian stochastic vector process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> is employed to model the temporal-spatial correlation of wind speeds at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> different sites. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ln$</tex> general, it is computationally tedious to obtain the autocorrelation functions (ACFs) and cross-correlation functions (CCFs) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> , which are different to those of wind speed times series. In order to circumvent this correlation distortion problem, the rank ACF and rank CCF are introduced to characterize the temporal-spatial correlation of wind speeds, whereby the ACFs and CCFs of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> can be analytically obtained. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{Then}$</tex> , Fourier transformation is implemented to establish the cross-spectral density matrix of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{Z}(t)$</tex> , and an analytical approach is proposed to generate samples of wind speeds at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> different sites. Finally, simulation experiments are performed to check the proposed methods, and the results verify that the five-parameter Johnson distribution can accurately match distribution functions of wind speeds, and the spectral representation method can well reproduce the temporal-spatial correlation of wind speeds.