Probability Density Evolution Research Articles

An extended two-scale random field model for stochastic response analysis and its application to RC Short-leg shear wall structure

For the stochastic response analysis of concrete structures, it is necessary to model the heterogeneous concrete with significant spatial variability by random fields. In practical applications, multiple material parameters are typically required to characterize the nonlinear constitutive relationship of concrete. As a result, the material parameters of concrete will be represented by a multivariate random field in the nonlinear stochastic response analysis. Nevertheless, a vast number of random variables are frequently involved in representing the multivariate random field, and it is difficult to quantify the probabilistic dependence among the material parameters. To address the preceding issues, this paper proposes an extended two-scale random field model (ETSRF) to quantify the spatial variability of the constitutive relationship of concrete. The model can accurately depict the fluctuation of material properties at the micro-meso scale and the spatial correlation at the macro scale. Besides, it is proved that there is a linear relationship between the mesoscopic random field and ETSRF. Furthermore, a new framework of stochastic response analysis is proposed, where the probability density evolution method (PDEM) is applied to obtain the probabilistic information of the structural responses, and ETSRF is adopted to describe the spatial variability of the material properties. Finally, the proposed model is validated by two illustrative examples, one of which simulates the random fields of compressive and tensile strength of a plain concrete slab, and the other of which deals with the stochastic response analysis of a RC short-leg shear wall structure. The results indicate that the proposed model can depict the spatial variability of the nonlinear constitutive relationship of concrete effectively.

Retracted: Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

Retracted: Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

Data-driven probability density forecast for stochastic dynamical systems

In this paper, a data-driven non-parametric approach is presented for forecasting the probability density evolution of stochastic dynamical systems. The method is based on stochastic Koopman operator and extended dynamic mode decomposition (EDMD). To approximate the finite-dimensional eigendecomposition of the stochastic Koopman operator, EDMD is applied to the training data set sampled from the stationary distribution of the underlying stochastic dynamical system. The family of the Koopman operators form a semigroup, which is generated by the infinitesimal generator of the stochastic dynamical system. A significant connection between the generator and Fokker-Planck operator is leveraged to construct an orthonormal basis of a weighted Hilbert space, enabling a spectral decomposition of the probability density function to be performed in this space. This approach is a data-driven method and used to predict the probability density evolution and real-time moment estimation. In the limit of the large number of snapshots and observables, the data-driven probability density approximation converges to the Galerkin projection of the semigroup solution of Fokker-Planck equation on a basis adapted to an invariant measure. This method shares similarities with diffusion forecasting but is capable of producing more accurate probability density forecasts than diffusion forecasting. A few numerical examples are presented to illustrate the performance of the proposed data-driven probability density forecast.

Stochastic analysis of galloping piezoelectric energy harvesters under turbulent flow conditions based on the probability density evolution method

Harvesting wind energy to power wireless sensors is a very promising technology. Galloping piezoelectric energy harvesters (GPEHs) are considered to be an excellent solution that can be miniaturized but high efficiency. In order to better understand the working mechanism and determine the design method, it is very necessary to develop accurate theoretical model of GPEHs. In conventional method, the wind was usually simplified as uniform flow with constant wind speed. In practice, however, the wind is turbulent flow that consists of uniform and fluctuating components. The influence of fluctuating component on GPEHs is usually neglected in existing literatures, which will lead to significant errors. To address this issue, this work proposes a stochastic method for GPEHs based on the probability density evolution method (PDEM). Aerodynamic coefficients are regarded as random variables to describe the random excitation induced by turbulent flow. The confidence bands of the output power and the cut-in wind speed of GPEHs under turbulent flow condition are predicted, which satisfy specific probability density distribution. The probabilistic concentrated phenomenon of the output power at low wind speed and far from the optimal resistance is observed. Its mechanism is explored in detail. In addition, a parametric study is conducted to highlight the influence of the variability of the aerodynamic coefficients and the probability distributions of the aerodynamic coefficients on the output power. The proposed method is verified by Monte Carlo Simulation (MCS) and experiment.

Dynamic reliability analysis of three-dimensional slopes considering the spatial variability in soil parameters

Slopes are commonly a significant aspect of geotechnical engineering projects. The incorrect evaluation and prediction of slope stability can cause serious disasters when it experiences seismic activity, and reliability is the key index a slope’s dynamic state. Spatial variability is an inherent soil property and has an essential influence on the dynamic response of a slope body. However, in previous studies, soil was often described deterministically, and the interaction between the uncertainty and spatial dimensions was ignored. To fill this gap, in this paper, practical expressions for the cohesion and friction angle in each working condition are given to simulate spatial variability by a lognormal random field. Then, the finite element method and effective dynamic reliability calculation method are used to obtain the safety factor of a 3D slope. Finally, the generalized probability density evolution method is performed to obtained the probability density information of the dynamic slope response and judge the influence of the soil spatial variability probabilistically. Due to the spatial variability in soil, the probability density distribution under each condition is distributed in a large range and thus may not provide results that are much more dangerous than those of the deterministic simulation. Meanwhile, the friction angle is more sensitive than cohesion and shows stronger discreteness in the spatial variability. The change in the cross-correlation coefficient has a greater influence on reliability than the autocorrelation length. Consequently, the simplification of model slopes in dimension and variability could deviate the reliability results from actual dynamic responses, either overestimating or underestimating, resulting in incorrect designs and evaluations.

Random Dynamic Analysis and Running Safety Assessment of Long-Span Cable-Stayed Bridge Under High-Speed Trains Subjected to Earthquake Excitations and Track Irregularities

Running trains on a long-span bridge under earthquake is an inevitable problem for the high-speed railway in seismic regions. This paper aims to investigate the seismically random dynamic behavior of train–cable stayed bridges interaction system using the probability density evolution method (PDEM). The system simultaneously involves the multi-effect of random multi-point earthquake excitations with traveling waves and random track irregularity. The multi-point earthquake excitations, composed of lateral and vertical positions, are generated by stochastic harmful functions (SHFs) and uniformly modulated into nonstationary random processes. The motion equation of such a system is established by coupling the 38-degree vehicles and bridges through the refined random wheel/rail contact relationship and accounting for the phase-lags of seismic travelling waves between pier excitations. PDEM is proven to be applicable to such time-dependent systems and is then used to transform the random excitations into a series of deterministic representative excitations with initial probability. By solving for the corresponding deterministic probability responses, various nonstationary random responses, including the time-dependent probability density evolution functions of random responses, mean values curves and standard deviations, can be obtained easily. A case study based on the cable-stayed bridge is then presented, and the results show the effectiveness and accuracy of the proposed method by comparison with Monte Carlo Simulation. Additionally, the influence of train speed and seismic intensity on the safety and reliability of trains is discussed.

Structural design optimization under dynamic reliability constraints based on probability density evolution method and quantum-inspired optimization algorithm

Dynamic-reliability-based design optimization (DRBDO) has been a promising approach for designing structures under dynamic excitations in the presence of uncertainties. This paper proposes an effective scheme for solving a class of DRBDO problems. The proposed scheme is based on the quantum particle swarm optimization (QPSO) algorithm, a quantum-inspired algorithm that utilizes quantum mechanisms to achieve better exploration and exploitation. During the optimization process, the probability density evolution method (PDEM) combined with the extreme value distribution strategy is employed to evaluate the structural dynamic reliability. Due to the high efficiency of the PDEM, the computational cost associated with reliability assessments can be considerably reduced. Numerical examples involving linear and nonlinear structures with different types of design variables are presented to demonstrate the effectiveness and efficiency of the proposed scheme.

Reliability assessment and performance improvement of long-span roof structure subjected to coupled wind-snow multi-hazard

Sensitivity to wind and snow loads has been the focus of reliability analysis and safety design of long-span roof structures in cold regions. In this study, the reliability assessment and performance improvement of a long-span roof structure subjected to coupled wind-snow multi-hazard are addressed. To this end, the average wind pressure coefficients of roof surface are first derived from wind tunnel tests, and the two-dimensional wind velocity field simulation is carried out by using the refined spectral representation scheme based on joint wavenumber-frequency power spectrum and stochastic harmonic function. The snow pressure coefficients of structural roof considering wind-induced snow drift effects are obtained by conducting computational fluid dynamics simulation. Then the global reliability of the structure subjected to coupled wind-snow multi-hazard with 100-yr return period is assessed using the probability density evolution method (PDEM). To improve structural performance, a control scheme of distributed tuned mass dampers (d-TMDs) is designed according to the structural dynamics of their deployment positions. Numerical results show that the fluctuating wind is a critical factor resulting in deformation failure of the long-span roof structure, and the proposed control scheme has sound effectiveness and robustness to mitigate the fluctuating wind effect and thus improve the structural performance.

Dynamic Reliability Analysis of Running Safety and Stability of a High-Speed Maglev Train on a Guideway Bridge

The dynamic performance of high-speed maglev trains, a next-generation rapid transit system, has received continuous attention in recent years. In this study, a dynamic reliability analysis method for a high-speed maglev train–guideway coupled system was proposed. First, a refined model of the maglev vehicle–bridge interaction system was established, where the vehicle subsystem was simulated as a rigid body-spring-damper model with 101 degrees of freedom. The guideway subsystem was simulated as a finite element model, and these two subsystems were coupled as an entire system through a magnet–rail interaction model with a proportional-derivative (PD) controller. Second, a dimension-reduction method for the simulation of representative samples of track irregularities was developed, and thus the number of random variables in the system was reduced to four. Finally, an efficient method for the calculation of the dynamic reliability of a maglev train–guideway coupled system was proposed using the probability density evolution method-based equivalent extreme value principle. With numerical examples, the accuracy of the maglev train–guideway interaction model was verified by comparing it with field measurement data from the Shanghai high-speed maglev line. The accuracy of the proposed dynamic reliability analysis method was confirmed by comparing three types of results, that is, the mean value time-history curve, the probability density function, and the cumulative distribution function of extreme values, obtained by the Monte Carlo method. Finally, the dynamic reliability of the running safety and stability of the maglev vehicle at a speed of 430 km/h and the variation laws of the dynamic reliabilities with train speed were examined in detail.

PDEM-based reliability assessment of RC frames against progressive collapse considering initial local failure

With the increased frequency of progressive collapse events, an accurate and efficient reliability-based method for preventing potential structural collapse has become a research focus. Several methods for structural reliability against progressive collapse have recently been proposed; however, they ignore the effect of initial local failure, which may have a significant influence on risk management in protecting structures against progressive collapse. Therefore, this study developed a computation framework for assessing the overall reliability of reinforced concrete (RC) frames against progressive collapse that considers the impact of the initial local failure. The proposed framework quantifies the overall reliability of the structure using the full probability method, which comprises three levels of determination of the initial local failure location, initial local failure assessment, and structural progressive collapse analysis. The efficient probability density evolution method (PDEM) was then employed to calculate the initial local failure probability and structural progressive collapse probability, which are required for achieving a trade-off between computational efficiency and accuracy. Thereafter, the developed framework was applied to a prototype RC frame, in which the structural resistance was captured using 3-D macro-based numerical modeling, which was initially validated by experimental data. Finally, the overall reliability of the RC frame was evaluated according to three different failure criteria, and based on reliability results, various structural strengthening strategies were discussed. Results show that the developed framework can effectively quantify the impact of initial local failure on the reliability of RC frames against progressive collapse. The overall reliability of the structure increases as the input energy decreases. When the input energy greater than or equal to 132 kJ, the overall reliability of the structure investigated is 0.933 for the ultimate damage criterion.

An efficient time-variant reliability analysis strategy embedding the NARX neural network of response characteristics prediction into probability density evolution method

Stochastic response analysis and first-passage reliability evaluation of multi-degree of freedom nonlinear systems subject to non-stationary seismic excitation have been a critical issue in the field of compound random vibrations, yet remain unsolved. This study extends the time-varying reliability analysis method based on the probability density evolution method, making it applicable to the problem of stochastic response analysis and first-passage reliability evaluation of complex engineering structures. To efficiently represent the response of representative samples, a nonlinear autoregressive with exogenous inputs neural network model is introduced in this study. After comparing three different optimization methods, the neural network model based on the Bayesian regularization algorithm is selected as the predictor for the response of complex nonlinear systems, enabling efficient stochastic response analysis and time-varying reliability evaluation. To this end, leveraging the high efficiency of the probability density evolution method, the analysis of time-varying reliability of large-scale structures has been successfully conducted. Furthermore, the proposed method can be easily expanded to dynamic reliability evaluation problems represented by extreme value theory. Three numerical examples are quoted to demonstrate the applicability and advantages of the proposed method in the field of first-passage and extreme-value problems of systems, which are also compared with PCE, SVR, and Kriging. The results from these numerical examples indicate that the proposed method can effectively reduce the required calculational cost for the reliability analysis of complex structures while maintaining high analytical accuracy.

Verification of a stochastic dynamic analysis method by shaking table tests of a slope retaining system

Various sources of uncertainty can greatly affect the seismic responses and seismic reliability of slope retaining systems. Many studies have observed and highlighted that ignoring the effect of uncertainties, especially the randomness of ground motions, would cause an inaccurate evaluation of seismic performance of slope retaining system. It is thus of great significance to employ the stochastic dynamic analysis method to quantify these uncertainties and corresponding effect. A recently developed stochastic dynamic analysis method, namely probability density evolution method (PDEM), have been proposed for the stochastic seismic responses analysis of geotechnical engineering. Although its efficiency and accuracy have already been proved by numerical cases, its feasibility and accuracy has not been verified from the experimental perspective, and its application in the stochastic seismic responses analysis using the field monitor data and experiment results in geotechnical engineering has not yet been studied. In the present study, a hard-strength rock slope model reinforced by pile-anchor structures is firstly constructed for the repeating shaking under a series of ground motions. A set of fully non-stationary stochastic ground motions are generated to quantify the randomness of input ground motions. Then, the stochastic seismic responses of the slope retaining system, including acceleration responses, amplification effect and earth pressure along pile-anchor structures, are analyzed and compared by the PDEM and various assumed probabilistic distribution models to illustrate the feasibility and accuracy of PDEM from the experimental perspective. The results demonstrate that the PDEM can obtain relatively more accurate stochastic seismic response results, and can also be used to analyze the experimental data from the probabilistic perspective.

Uncertainty propagation of turbulence parameters for typhoon and Non-typhoon winds in buffeting analysis of Long-span bridges

Wind turbulence-induced buffeting vibration affects the serviceability of long-span bridges and lead to fatigue problems. The randomness of turbulence parameters in the power spectrum density function is always disregarded, failing to well reproduce the dispersion of structural buffeting responses and achieve the extreme vibration amplitude. This study proposes a probabilistic buffeting analysis approach based on multi-point estimation, dimension reduction techniques, and the principle of maximum entropy accounting for the uncertainty of turbulence parameters. The uncertainties in turbulence power spectral density functions are quantified based on the long-term real observational wind data captured by the sensors installed at Xihoumen Bridge. These uncertainties are propagated into the buffeting response of the bridge using the proposed method. The results are compared with those of Monte Carlo simulation, suggesting a good agreement with substantially lower computational costs can be attained by the proposed method. Then, the probability density evolution of buffeting response versus wind speed is calculated. Results reveal that typhoon-induced buffeting response is greater than non-typhoon’s under low mean wind speeds (e.g., 25 m/s), whereas it is the opposite at high wind speeds (e.g., 30 m/s). The predicted probability density evolution of buffeting response exhibits a great consistency with real observations. By introducing the mean wind speed hazard curves for both typhoon and non-typhoon winds, the exceeding probability of vibration within a given period can be easily attained. It allows the estimation of return periods of structural responses and development of the uniform-risk approach for structure design rather than the conventional uniform-hazard design method.

Uncertainty quantification for granular materials with a stochastic discrete element method

Significant uncertainty exists in granular materials, as demonstrated by experimental and simulation studies. Quantifying this uncertainty by integrating refined discrete element analysis with direct stochastic simulation is challenging due to computational cost constraints. To this end, the probability density evolution method (PDEM) is introduced to propose the methodology of stochastic discrete element analysis. The objective of this paper is to develop a formulation of the stochastic discrete element method (DEM) on uncertainty quantification for granular materials. First, the uncertainty characterization of inter-particle friction coefficients (μ) is performed via experimental data, resulting in the identification of an optimal probability distribution. Second, the 100 representative deterministic points of basic random variables are selected for subsequent DEM shearing simulation until the critical state. Third, the PDEM is developed to obtain the probability density function (PDF) surface representing the system’s performance, thus offering a comprehensive prediction of uncertainty for granular materials. The results show that the uncertainty of μ propagates to the behavior of granular materials. Utilizing the stochastic DEM methodology clarifies which behavior parameters exhibit significant randomness and which possess negligible randomness. Notable uncertainties are observed in large strain macro-scale behavior and the proportion of inactive particles. It can be inferred that conventional methods that neglect probability distributions might produce spurious results. Moreover, the influence of uncertainties on granular material behavior varies depending on the its shearing state and related soil parameters. In contrast, limited uncertainties are identified for the randomness of small strain stiffness, and the randomness on the particle scale stress transmission shape is also limited. This study marks the initial application of the PDEM in geotechnical engineering, suggesting that considering probability density evolution could enhance geotechnical engineering design. The current study highlights the necessity of using statistical methods to assess key geotechnical parameters.

Reliability Evaluation of a Nonlinear Frame Structure under Explosive Ground Motions Generated by Dimension-Reduction Method

In the present study, a stochastic model of explosive ground motions applying the dimension-reduction method is proposed, and the reliability evaluation of a nonlinear frame structure under such excitations is realized by means of the probability density evolution method and an equivalent extreme-value-based reliability evaluation strategy. Firstly, the evolutionary power spectrum density function of the explosive ground motions is modeled by respectively identifying the normalized total energy distribution function and the frequency total energy distribution function on the basis of the measured motion records. In addition, an exponential model is constructed to forecast the seismic characteristics of the explosive ground motions based on the given distance to the explosive source and the charge quantity. Then, the representative samples of the explosive ground motions are simulated using the dimension-reduction method. The simulation results show that the generated acceleration samples have significant seismic characteristics of the explosive ground motions, and the accuracy is verified by comparing the second-order statistics with the sample set and the corresponding targets. Due to the fact that the probabilities of the representative samples simulated by the dimension-reduction method can compose a comprehensive probability set, it contributes to the refined dynamic response analysis and reliability evaluation of complex structures combining with the probability density evolution method. The accurate dynamic response analysis and reliability evaluation of a nonlinear frame structure illustrates the effectiveness of the proposed model and the dimension-reduction method for simulating the explosive ground motions. The numerical results demonstrate that the explosive ground motions have a substantial effect on the nonlinear behavior and the security of engineering structures.

Uncertainty quantification of mechanism motion based on coupled mechanism—motor dynamic model for ammunition delivery system

In this paper, a dynamic modeling method of motor driven electromechanical system is presented, and the uncertainty quantification of mechanism motion is investigated based on this method. The main contribution is to propose a novel mechanism-motor coupling dynamic modeling method, in which the relationship between mechanism motion and motor rotation is established according to the geometric coordination of the system. The advantages of this include establishing intuitive coupling between the mechanism and motor, facilitating the discussion for the influence of both mechanical and electrical parameters on the mechanism, and enabling dynamic simulation with controller to take the randomness of the electric load into account. Dynamic simulation considering feedback control of ammunition delivery system is carried out, and the feasibility of the model is verified experimentally. Based on probability density evolution theory, we comprehensively discuss the effects of system parameters on mechanism motion from the perspective of uncertainty quantization. Our work can not only provide guidance for engineering design of ammunition delivery mechanism, but also provide theoretical support for modeling and uncertainty quantification research of mechatronics system.

A practical approach to train-bridge system performance evaluation with consideration of random uncertainty and weighted evaluation indexes

In this paper, a practical methodology for evaluating the dynamic performance of train-bridge systems is presented, which incorporates both random uncertainty and the weighting of evaluation indexes. Firstly, a train-bridge stochastic model is established by combining an existing train-bridge deterministic model with the Generalized F-discrepancy (GF-discrepancy) minimized point selection strategy and the probability density evolution method (PDEM). The model enables the accurate and efficient quantification of the variability of any dynamic response quantity of interest in the train-bridge system. Building on this foundation, an assessment model based on the multi-layer fuzzy comprehensive evaluation method is developed to determine the possibility of the train-bridge stochastic system in different performance states with consideration of the weighted evaluation indexes. In the assessment model, the main criteria of the train-bridge system are collected, classified, and further structured a logical hierarchical framework, and the weightings of the criteria are solved by analytic hierarchy process method and entropy weighting method. As demonstrated by a numerical example, the proposed method can properly consider system uncertainties, appraisal hierarchy, and the weightings of evaluation indexes, which helps identify critical evaluation indexes and improve evaluation reliability. The proposed approach can provide a useful tool for the performance evaluation of train-bridge systems.

Probabilistic analysis of seismic mitigation of base‐isolated structure with sliding hydromagnetic bearings based on finite element simulations

AbstractBase isolation is one of the most efficient methods for seismic mitigation of building and bridge structures in high‐intensity areas. However, the base‐isolated structure designed according to current codes based on the local seismic hazard and a few station records might not guarantee the performance of the base isolation system and ensure the safety of the structure. Carrying out a probabilistic analysis of seismic mitigation of base‐isolated structures, considering the randomness associated with earthquake ground motions, is thus necessary. To this end, the stochastic seismic response and reliability analysis of base‐isolated structures subjected to stochastic ground motions are conducted in this study, in conjunction with the probability density evolution method and the principle of equivalent extreme value event. The recently developed sliding hydromagnetic bearing is employed to constitute the base isolation layer of the base‐isolated structure. A refined finite element analysis is carried out to accurately reveal the dynamic characteristics of the base‐isolated structure. It is shown that the base‐isolated structure can be well simulated by the finite element model, which is verified through a comparative study with the results of shaking‐table tests. The base‐isolated structure gains a sound seismic mitigation compared to the base‐fixed structure in a probabilistic sense, and can ensure the safety of the structure by deploying a moderately designed base isolation system.

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

Seismic response analysis of subway station structure under random excitation based on probability density evolution method

The subway station structure mainly faces the uncertainty of ground motion and surrounding soil mass. Comprehensive analysis of the seismic response of subway station structures under random excitation is the key to ensuring the safety of urban people's lives and property. In this paper, the random shear wave velocity profile model and the random ground motion model characterizing uncertainty are established. Using a high-dimensional space optimization point selection strategy significantly reduces the number of sample points and computational costs of nonlinear time history analysis (NLTHA). Combining the probability density evolution method (PDEM) with finite element analysis, the stochastic seismic response analysis of the subway station structure under dual random excitations is realized. In addition, statistical methods are used to analyze the distribution characteristics of the most unfavorable seismic responses of the subway station structure. The research results show that when random excitation is not considered, the adverse seismic damage risk of subway station structures will be ignored. The evolution of the seismic dynamic response of subway station structures is mainly controlled by the amplitude of random ground motion acceleration. Additional random shear wave velocities can increase the lateral deformation range of the subway station structure, leading to a decrease in the dynamic reliability of the structure. PDEM can conduct stochastic seismic response analysis of the subway station structure from both time and response dimensions. The proposed method provides strong support for a comprehensive analysis of the seismic response of underground structures.