non-Gaussian Random Processes Research Articles

Non-Gaussian buffeting analysis of large structures by means of a Proper Orthogonal Decomposition

Proper Orthogonal Decomposition is implemented in a Bispectral Stochastic Analysis of large MDOF structures. Although Higher Order Stochastic Analysis has been introduced few decades ago, its establishment has found place only in a theoretical way, or applied to very small systems. Its heavy computational cost as well as resource consumption burden has let it aside for practical civil engineering applications. Nevertheless, in wind engineering, the community is increasingly considering the importance of non-Gaussian nature of wind induced vibrations. This is of interest for the extreme value analysis, for which many theoretical and empirical models exist for non-Gaussian random processes. In this context, a (bi-)spectral approach is realized in a 2-D frequency space where the bispectrum is computed then the 3rd statistical moment is obtained by means of a twofold integration. These operations are quite heavy as soon as the dimension of the problem increases. A novel algorithm implementation is proposed. It hinges on (i) the use of Proper Orthogonal Decomposition and the (ii) development of an optimized numerical method. The proposed algorithmic arrangement minimizes the number of operations, and consequently saves time and memory, while conserving precision in estimating 3rd order statistical moments of structural responses.

Assessment of train running safety on railway bridges based on velocity-related indices under random near-fault ground motions

Random analysis has emerged as the dominant approach for analyzing train-bridge coupled (TBC) systems, replacing deterministic analysis. This shift is driven by considerations of system economy and randomness. However, the seismic train running safety assessment with multiple non-Gaussian distribution parameters faces the double dilemma of low computational efficiency and inaccuracy for TBC systems. In this study, an approach is applied by the combination of the new point estimate method and moment expansion approximation (NPEM-MEA) to address this problem. Furthermore, the application of this approach is illustrated by the random dynamic responses of a three-dimensional TBC system subjected to random near-fault ground motions. Validation of the NPEM-MEA’s feasibility is conducted using the Monte Carlo method, and comparative analysis confirms the accuracy and efficiency of the method. Additionally, the influences of random near-fault ground motions are discussed based on the random dynamic responses and train running safety indices in the TBC system. The simulation results underscore the necessity of simultaneously considering the randomness of multiple parameters in seismic train running safety assessment. This methodology holds potential for safety assessments involving non-Gaussian random processes in complex systems of a similar nature.

A spectral method for accurate evaluation of fatigue damage induced by wide-band non-Gaussian random processes

Evaluating the fatigue damage of marine structures subjected to random environmental loadings is of great practical significance at the design stage. In many situations, the structural responses of marine structures exhibit wide-band non-Gaussian properties. This study presents the fatigue damage assessment of wide-band softening non-Gaussian random processes by a spectral method proposed by Benasciutti and Tovo (2005b). Following this spectral method, the rainflow fatigue damage is approximated by a linear combination of its upper and lower bounds. A refined formulation of the weighting factor is proposed, which depends on the bandwidth parameters, slope parameter of the S‒N curve, and skewness and kurtosis of non-Gaussian processes. By introducing a suitable transformation function, the simulated non-Gaussian process is generated from its underlying Gaussian process. Through case studies, the effectiveness and accuracy of the present method for wide-band non-Gaussian processes are verified by comparisons with the results from the time-domain rainflow counting method and the other existing spectral methods.

First Passage Dynamic Reliability Analysis of Non-Stationary and Non-Gaussian Buffeting Random Dynamic Responses of Long-Span Suspension Bridge

In the complex mountain wind environment, especially in the strong wind, fluctuating wind speed is a non-stationary and non-Gaussian random process. With the increase of suspension bridges’ span, they are more sensitive to wind-induced vibration response. In this paper, a non-stationary and non-Gaussian buffeting reliability analysis method for long-span bridges is presented. Firstly, the non-stationary wind speed is simulated by the modulation function based on the stationary wind speed model. Secondly, the load is obtained by simulated wind speed, the obtained loads are loaded on the finite element model by ANSYS and the whole bridge time-domain analysis is performed, the time history of displacement response is obtained. Thirdly, based on the first excursion failure criterion of random vibration, the samples of the non-stationary and non-Gaussian displacement time history are transformed into a standard Gaussian process through a modified Fleishman approximation method, and then the non-stationary Poisson distribution method is used for structural reliability analysis. Finally, a mountainous long-span suspension bridge is used as the engineering background, and the proposed reliability analysis method is applied to analyze the reliability of the bridge. The influence of different wind attack angles and modulation functions on the dynamic reliability of the bridge is further studied. The results indicate that the displacement response obtained by the transformation of non-stationary and non-Gaussian random process is less reliable than that obtained by traditional analysis method. Its dynamic reliability is maximum at 1[Formula: see text] wind attack angle, the reliability of negative wind attack angle is lower than that of positive wind attack angle, and decreases with the increase of wind attack angle. The reliability obtained by using different modulation functions is lower than that of traditional method. If the traditional analysis method is still used for reliability analysis, it will produce unsafe consequences and reduce the engineering safety reserve.

Simulation of 4th-order non-Gaussian random processes by higher-order spectral representation method

In this study, a new higher-order spectral representation method (HOSRM) is derived to simulate fourth-order (kurtosis-order) non-Gaussian random processes. Applying the HOSRM allows fourth-order non-Gaussian random process (symmetric non-Gaussian) sample functions to be generated accurately. As the number of terms in the cosine function used in the HOSRM increases, the simulated non-Gaussian random process sample functions can more accurately reflect the second-order (variance) and fourth-order (kurtosis) statistical properties of the objective sample functions. In particular, if the number of terms tends to infinity, then the simulated sample function characteristics will be the same as the target value. Meanwhile, a reasonable three-dimensional Hermite interpolation method is proposed to improve the simulation efficiency of stochastic processes. Finally, two examples are provided to demonstrate the accuracy and effectiveness of the proposed HOSRM for generating fourth-order non-Gaussian random processes. The results show that the statistical properties (power spectral density, trispectrum, and kurtosis) of the simulated non-Gaussian random process are consistent with those of the target random process.

An innovative method for space-time-dependent reliability analysis

The spatiotemporal variability exists in almost all engineering problems, since the environmental actions and material properties may change both spatially and temporally in a stochastic manner. Therefore, reliability analysis is inherently space–time-dependent. In this study, an innovative space–time-dependent outcrossing rate (STOR) method is proposed for dealing with such reliability problems. The failure probability of a structure is determined based on outcrossing rate which is theoretically derived for the first time, considering the spatiotemporal variability of all basic variables. Then, an efficient numerical algorithm for computing the failure probability is proposed based on Gauss-Legendre quadrature. The application of STOR method is investigated through three examples, including non-Gaussian random processes, non-Gaussian random fields, strong nonlinear limit state functions and small failure probability problems. The proposed STOR method is found to be efficient for space–time-dependent reliability analysis with enough accuracy.

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.

Probabilistic physics-informed machine learning for dynamic systems

This paper develops a physics-informed machine learning approach for response prediction in dynamic systems, by augmenting a physics-based model with a machine learning model for model error; both models are probabilistic. The physics-based dynamic system model has discrepancy in the predicted output (due to incomplete physics or model form error in the physics-based model) when compared against observation data. The model form error is quantified using Bayesian state estimation with observation data, and a probabilistic machine learning model is trained to predict the output discrepancy for untrained inputs. Two different computational options are developed to implement this overall approach, with different levels of sophistication and computational effort. The methods are developed for problems with stationary and non-stationary, and Gaussian and non-Gaussian random process inputs. Strategies for computational effort reduction at different stages, such as model training, model error quantification, and reliability prediction, are discussed. The proposed approach is demonstrated using two numerical illustrations: (a) a deep beam subjected to dynamic loading (a single physics system), and (b) hypersonic flow over a flexible aircraft panel (a multi-physics system).

Simulation of non-stationary and non-Gaussian random processes by 3rd-order Spectral Representation Method: Theory and POD implementation

This paper introduces the 3rd-order Spectral Representation Method for simulation of non-stationary and non-Gaussian stochastic processes. The proposed method extends the classical 2nd-order Spectral Representation Method to expand the stochastic process from an evolutionary bispectrum and an evolutionary power spectrum, thus matching the process completely up to third-order. A Proper Orthogonal Decomposition (POD) approach is further proposed to enable an efficient FFT-based implementation that reduces computational cost significantly. Two examples are presented, including the simulation of a fully non-stationary seismic ground motion process, highlighting the accuracy and efficacy of the proposed method.

Detection and Analysis of Bionic Motion Pose of Single Leg and Hip Joint Based on Random Process.

Based on the spectral representation method of random function and combined with memoryless nonlinear translation theory, this paper analyzes the transformation relationship between potential Gaussian random process and non-Gaussian random process, and successfully generates a stationary non-Gaussian random process that conforms to the target non-Gaussian random process. For the non-stationary non-Gaussian random process simulation, on the basis of the stationary Gaussian random process, the intensity non-stationary uniform modulation model is used to modulate it, and combined with the nonlinear translation theory, the non-stationary non-Gaussian random process conforming to the target non-Gaussian random process is obtained. Aiming at the single-leg bouncing model based on the flexible rotary hip joint, the stability of its bouncing motion under passive motion is studied, and the influence of the flexible hip rotary joint on the motion stability is analyzed by comparing the single-leg bouncing motion characteristics of the free rotary hip joint. Based on the inverse dynamic control of the air phase, the fixed point distribution of the single-leg bounce of the flexible rotary hip joint was improved, and the function of the flexible rotary hip joint in the energy conversion of the bouncing motion was studied by establishing the energy consumption evaluation function. The kinematic performance verification, dynamic performance verification, dynamic parameter identification verification, and modal experiment simulation analysis were carried out for the built experimental platform, and the comparison and analysis with its theoretical model were carried out. The results show that the theoretical motion trajectory of the test mobile platform is basically consistent with the actual motion trajectory in the X and Y directions, and there is a small error in the Z-axis direction, and the error is within an acceptable range, indicating that the experimental platform system can be used to simulate the human hip joint. There is a large error between the theoretical value of the driving torque calculated by the theoretical value of the dynamic parameters and the measured value, and the dynamic theoretical model cannot accurately predict the driving torque. The predicted value of the driving torque calculated by using the identification value of the dynamic parameters is in good agreement with the measured torque, and its confidence is increased by 10–16%, indicating that the dynamic parameter identification method in this paper has a high degree of confidence.

Nonlinear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method and a second-order projected differentiation (PD2) method. The proposed approach is called AV2–PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes following the traditional translation process model. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first- and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal cumulative distribution function (CDF) of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOMs as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach are demonstrated using a nonlinear Duffing oscillator problem under a quadratic random excitation and a nonlinear half-car suspension example.

Variance-based sensitivity analysis of dynamic systems with both input and model uncertainty

This paper develops a methodology to compute variance-based sensitivity indices for dynamic systems with time series inputs and outputs, while accounting for both aleatory and epistemic uncertainty sources, and both random process and random variable inputs. We present semi-analytical methods for computing sensitivity indices for linear systems with Gaussian random process inputs, and for the general case of nonlinear systems with non-Gaussian random process inputs. The novel elements in this approach are the treatment of model form error, quantifying the cumulative effects of uncertainty sources over time, and evaluating sensitivity indices for multi-physics models. Bayesian state and parameter estimation methods are incorporated to quantify the model uncertainty arising from unknown model parameters and model form errors, and sensitivity indices are computed before and after model updating. The proposed methods are illustrated for (a) a linear Timoshenko beam erroneously modeled as an Euler-Bernoulli beam, and (b) hypersonic flow behavior of a flexible panel represented by a coupled multi-physics nonlinear model.

Fatigue life prediction of ship and offshore structures under wide-banded non-Gaussian random loadings: Part II: extension to wide-banded non-Gaussian random processes

Ships and offshore structures are exposed to various cyclic loads such as winds, waves and currents in their lifetime. These loads can result in fatigue failures at hot spots, and to prevent such failures, accurate estimates of the fatigue life of hot spots should be performed in the design stage. The cycle distribution indicative of the probability distribution of cycles is an important factor in estimation of fatigue damage to mechanical systems in frequency-domain methods. It depends on not only the counting method but also the statistical properties of random processes. Many studies deriving both theoretical and approximate models of rainflow-counted cycle distributions in wide-banded non-Gaussian processes have been conducted. However, some of the existing models require multi-dimensional integrations with much computational time or yield inaccurate representations of rainflow-cycle distributions. Therefore, a new model providing accurate cycle distributions is needed for fatigue analysis of offshore structures under wide-banded non-Gaussian random loadings.This paper consists of two parts: Part I is devoted to the development of a new approximation model for the joint probability distribution of mean and amplitude of rainflow-counted cycles in wide-band Gaussian random processes. This probabilistic model derived in Gaussian random processes could be transformed to the corresponding non-Gaussian processes through non-linear transformation technique. Part II discusses the detailed procedure and accuracy of the proposed model in fatigue damage assessment of offshore structures under non-Gaussian random loading. Rainflow cycle distribution in wide-band non-Gaussian processes is derived using the Hermite transformation function. The accuracy of the proposed model is verified through two numerical examples, and compared to other existing models.

Varied approaches to loading assessment in fatigue studies

In this paper, the authors want to draw readers’ attention to one long-standing question: which approach is preferable for estimation longevities in fatigue problem, the time domain approach (Rainflow) or the frequency domain one (Dirlik and others)? This question is important in engineering problems, particularly in problems of prolongation of the guaranteed service life. The discussion here is restricted by the longevity evaluation only at the post-processing stage of unidirectional loaded machine parts. It means the realizations might be recorded.Some experimental and speculative evidence of preferable use the Rainflow method is shown. Taking into account the huge computer's power nowadays, the question of the irrelevance of appellation to the calculation accelerating using the spectral methods is specially discussed.There are areas, where the spectral methods are really necessary. There is only a need to recommend the restriction of their application scope to these special situations. It seems there is no need in inventing new spectral complicated algorithms only to stress out at the end, that their result coincides with the Rainflow outcome. That might be confusing for the practicing engineers.Currently, the main attention among supporters of spectral methods is focused on non-stationary and non-Gaussian random processes. (to estimate the spectral density is impossible for non-stationary processes, according to definition). These researchers seem to have forgotten, that even for these complicated situations the decision has already existed: that is the Rainflow and its analogues.The paper shows extensive laboratory experiment results of random fatigue testing of aluminum flat specimens under regular (to build the fatigue curve) and irregular (random) loading. The fatigue life curve (Gassner curve) has been built. These results allowed to compare the existing computation methods of longevity estimation. In the particular situation of narrow-band process, the methods seem to provide comparative results.Considered are methodological issues related to the assessment of the necessary and sufficient realization length, the influence of RMS, cycle counting methods and some possibilities of computing resources saving when using the Rainflow method. The stability of the Rainflow estimates is confirmed. Some problems with the choice of parameters during the longevity assessment by the Dirlik method were noted.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads Using a Sensitivity-Based Propagation of Moments

Abstract The paper proposes a new methodology for time-dependent reliability analysis of vibratory systems using a combination of a first-order, four-moment (FOFM) method and a non-Gaussian Karhunen–Loeve (NG-KL) expansion. The approach can also be used for random vibrations studies. The vibratory system is nonlinear and is excited by stationary non-Gaussian input random processes which are characterized by their first four marginal moments and autocorrelation function. The NG-KL expansion expresses each input non-Gaussian process as a linear combination of uncorrelated, non-Gaussian random variables and computes their first four moments. The FOFM method then uses the moments of the NG-KL variables to calculate the moments and autocorrelation function of the output processes based on a first-order Taylor expansion (linearization) of the system equations of motion. Using the output moments and autocorrelation function, another NG-KL expansion expresses the output processes in terms of uncorrelated non-Gaussian variables in the time domain, allowing the generation of output trajectories. The latter are used to estimate the time-dependent probability of failure using Monte Carlo simulation (MCS). The computational cost of the proposed approach is proportional to the number of NG-KL random variables and is significantly lower than that of other recently developed methodologies which are based on sampling. The accuracy and efficiency of the proposed methodology is demonstrated using a two-degree-of-freedom nonlinear vibratory system with random coefficients excited by a stationary non-Gaussian random process.

Numerical Stochastic Model of Non-stationary Time Series of the Wind Chill Index

A numerical stochastic model of the high-resolution time series of the wind chill index is considered. The model is constructed under the assumption that time series of the wind chill index are non-stationary non-Gaussian random processes with time-dependent one-dimensional distributions. This assumption makes possible to take into account both daily and seasonal variations of real meteorological processes. Data of the long-term real observations at weather stations were used for estimating the model parameters and for the verification of the model. Based on the simulated trajectories, some statistical properties of rare and adverse weather events, like long periods of time with a low wind chill index, are studied. The model is also used to study the dependence of the statistical properties of the wind chill index time series on a climate change.

Extrema Prediction and Fatigue Assessment of Highly Non-Gaussian Random Processes Using a Homogenous Reproduction Method

AbstractGiven a limited number of time histories of a non-Gaussian random process, it is of interest to reproduce a much greater number of samples while retaining homogenous statistical characteris...

Simulation of stationary non-Gaussian multivariate wind pressures using moment-based piecewise Hermite polynomial model

The wind pressure at the flow separation region of building surfaces generally exhibits non-Gaussian characteristics and is modeled as a non-Gaussian random process. To evaluate the structural response under this non-Gaussian excitation, the time-domain response analysis is commonly used, especially in order to take into account the structural and aerodynamics nonlinearity. This makes the simulation of stationary non-Gaussian multivariate random processes becomes an essential task. The traditional correlation distortion simulation method based on the Hermite polynomial model (HPM) is widely used because it has relatively satisfactory simulation efficiency and is easy for the use. Nonetheless, its application range and simulation accuracy are restricted by the effective zone and modeling accuracy of the moment-based HPM. In this study, a novel correlation distortion simulation method using moment-based piecewise HPM (PHPM) is proposed, where the formulas of the PHPM corresponding to four cases, correlation distortion relationship and simulation scheme are developed. In comparison with the method using HPM, the proposed method has not only wider application range, but also higher simulation accuracy. To demonstrate the effectiveness of the proposed method, the simulations of measured univariate and multivariate non-Gaussian wind pressure data are conducted. Results show that for the example of wind pressures out of the monotonic region, which cannot be simulated by the method using HPM, the PDF and PSD of the simulated samples by the proposed method agree with the targets well. For the other examples, the simulation accuracy of the proposed method is overall higher than that of the simulation method using the moment-based HPM. These verify that the proposed method has larger applicability and higher accuracy than the simulation method using the HPM. Note that the proposed method can also be applied to simulate other non-Gaussian excitations such as the wind speed in the complex area.

A modified spectral representation method to simulate non-Gaussian random vector process considering wave-passage effect

Wave-passage effect is always of great importance when performing the dynamic analysis of lager scale structures. This effect can be easily considered in the simulation of Gaussian vector process, but there is not an efficient way to consider it in simulating non-Gaussian vector process. In this paper, a method is proposed to generate stationary non-Gaussian vector process considering wave-passage effect. In the simulation of vector process, wave-passage effect leads to a complex-defined cross spectral density matrix (CSDM). To separate the phase part from the CSDM, an improved form of spectral representation method (SRM) is proposed, in which the phase part representing the wave-passage effect can be directly expressed in an explicit form in the simulation formula. Then the characteristic of phase part representing the wave-passage effect during the nonlinear translation (translating underlying Gaussian vector process to the prescribed non-Gaussian vector process) is investigated. Interestingly, it is found that, the phase part is not changed. Based on this good feature, the modified SRM and the iterative scheme can be combined to simulate non-Gaussian vector process considering wave-passage effect. Two examples, involving the simulation of seismic ground motions and wind velocity time histories, are presented to show the characteristics of the simulated vector process and to verify the capabilities of the proposed method.

Methods of mathematical modeling of non-Gaussian random variables quantities and processes

Mathematical methods allowing to model non-Gaussian random variables and processes are considered. The models and description of non-Gaussian correlated processes in the form of generated Gaussian noise are analyzed, as well as the methods of formation of stationary random processes defined by the one-dimensional density distribution of Vero -abilities and the autocorrelation function. Examples of formation of non-Gaussian random variables and processes are given.