Probability Density Evolution Research Articles

Slope displacement reliability analysis considering rock parameters spatial variability subjected to stochastic mainshock-aftershock earthquake

A new and efficient framework for analyzing the stochastic dynamic response and reliability of the rock slope was constructed based on the generalized probability density evolution method (GPDEM). The left bank slope of Jinping dam was taken as an illustrative example and the effects of mainshock-aftershock sequence randomness, spatial variability of geotechnical materials and coupled uncertainties on the slope dynamic response and reliability were compared and analyzed for the first time. Firstly, a series of mainshock-aftershock sequences and parameter random fields were generated as stochastic factors inputs. Then, the effectiveness of GPDEM was validated by comparing with Monte Carlo Simulation (MCS). The Newmark slip displacement method was combined with GPDEM to analyze the slope dynamic response and reliability subjected to single and coupled uncertainties from a probabilistic viewpoint. Finally, the fragility curves of three risk levels were established based on the reliabilities with different seismic intensities. The result suggests that parameter spatial variability affects the dynamic response and reliability of the slope to some extent, but this effect will be overshadowed by the uncertainty of mainshock-aftershock sequences when it is considered simultaneously. Besides, aftershocks will cause the decrease of slope reliability and the impact of aftershock cannot be overlooked.

Multi-output multi-physics-informed neural network for learning dimension-reduced probability density evolution equation with unknown spatio-temporal-dependent coefficients

The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients.

Velocity divergence—Generalized adaptive probability density evolution method

AbstractThis study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads.

Research on the influence of random stiffness of the rubber tie plate on the coupled vehicle-turnout system based on the probability density evolution method

Track supporting stiffness plays a crucial role in the running quality of a vehicle crossing through a turnout. This paper mainly focuses on the influence of random stiffness of the rubber tie plate on the vehicle-turnout coupling system. The relevant random field is constructed via the number theoretical method (NTM). The coupled vehicle-turnout model with flexible rails is built via the modal superposition method. The probability density function (PDF) is obtained via the probability density evolution method (PDEM). Cases of normal distributions with various mean values of stiffness are compared to contrast the relevant impact on the dynamic wheel–rail interaction. Research results indicate that an increase of the stiffness of the rubber tie plate can abate the transverse sway of the vehicle and low down the wheel number. The target stiffness should be taken as 35 kN/mm and corresponding normal distribution with a confidence interval of plus and minus 5 kN/mm can be accepted. The discrepancy between the mean value and the relevant PDF contour clarify the importance and necessity for analysis of randomness. Research results of this paper can serve as the theoretical fundament of the track stiffness homogenisation of a turnout in the future.

Multiscale stochastic optimal control of hysteretic structures based on wavelet transform and probability density evolution method

PurposeContemporary stochastic optimal control by synergy of the probability density evolution method (PDEM) and conventional optimal controller exhibits less capability to guarantee economical energy consumption versus control efficacy when non-stationary stochastic excitations drive hysteretic structures. In this regard, a novel multiscale stochastic optimal controller is invented based on the wavelet transform and the PDEM.Design/methodology/approachFor a representative point, a conventional control law is decomposed into sub-control laws by deploying the multiresolution analysis. Then, the sub-control laws are classified into two generic control laws using resonant and non-resonant bands. Both frequency bands are established by employing actual natural frequency(ies) of structure, making computed efforts depend on actual structural properties and time-frequency effect of non-stationary stochastic excitations. Gain matrices in both bands are then acquired by a probabilistic criterion pertaining to system second-order statistics assessment. A multi-degree-of-freedom hysteretic structure driven by non-stationary and non-Gaussian stochastic ground accelerations is numerically studied, in which three distortion scenarios describing uncertainties in structural properties are considered.FindingsTime-frequency-dependent gain matrices sophisticatedly address non-stationary stochastic excitations, providing efficient ways to independently suppress vibrations between resonant and non-resonant bands. Wavelet level, natural frequency(ies), and ratio of control forces in both bands influence the scheme’s outcomes. Presented approach outperforms existing approach in ensuring trade-off under uncertainty and randomness in system and excitations.Originality/valuePresented control law generates control efforts relying upon resonant and non-resonant bands, and deploys actual structural properties. Cost-function weights and probabilistic criterion are promisingly developed, achieving cost-effectiveness of energy demand versus controlled structural performance.

Efficient reliability analysis of stochastic dynamic first-passage problems by probability density evolution analysis with subset supported point selection

This study introduces a novel point selection procedure for the Probability Density Evolution Method (PDEM) to estimate time-dependent reliability and failure probabilities in dynamic systems under first-passage failure conditions. The method integrates and modifies features of the Subset simulation procedure to adaptively generate dependent sample sets suitable for a reliability analysis by a direct probability integration approach levering PDEM. Performance function assessments are used as weighting factors in the Subset supported Point Selection (S-PS), enhancing the ultimate failure probability estimation accuracy. The presented approach effectively identifies samples in the failure region, particularly benefiting for dynamic systems under stochastic excitation, tested with random dimensions up to 60. It also offers a computationally efficient structural reliability estimation procedure by analyzing full time–history responses. The proposed method provides deeper insights into rare failure events and mechanisms through the visualization of intermediate results. This research presents an advanced framework for estimating structural reliability and understanding critical events in dynamic systems.

Response prediction and probabilistic analysis of the vehicle-ballasted track system considering track irregularity based on long-short term memory neural network

It is necessary to efficiently and accurately predict the dynamic behavior of the train-track system subjected to massive random track geometric excitations for system evaluation and maintenance. This paper proposed an engineering practical approach for the train-track system behavior prediction by integrating the deep learning surrogate model-based long-short term memory (LSTM) and physical model of the train-ballasted track interaction (TBTI) system, and then the generalized probability density evolution method (GPDEM) is introduced to achieve the probabilistic assessment of the TBTI system under limited track irregularities, where the track irregularity probability model (TIPM) coupled with generalized-F discrepancy points sets is employed to compress the track irregularity sets to reduce the number of system responses (as training target) but without loss of the responses completeness, therefore, it can promote the efficiency and accuracy of training model. The prediction performance of the surrogate model is illustrated by comparing the time-frequency domain information, mean and standard deviation of the responses obtained by the physical model and surrogate model. The results indicated the proposed method can accurately predict the system response, especially the low and medium frequency response below 50 Hz, and improve the efficiency of stochastic analysis for the TBTI system by approximately 20 times. Furthermore, the applicability of the proposed method is further examined by investigating the influence of probability levels of track irregularity and train speeds on train performance. Its extended model is also implemented to predict the response of systems induced by any measured track irregularity in real-time, which promotes the generalization capability of the proposed surrogate model.

Novel study on active reliable PID controller design based on probability density evolution method and interval-oriented sequential optimization strategy

Vibration active control is a significant problem in practical engineering, but the influence of hybrid uncertainties in practical engineering cannot be ignored. For the uncertainties with adequate data, probability theory can be used while for the uncertainties with insufficient data, probability theory is limited and interval quantification can be used. Taking the widely used proportional-integral-differential (PID) control as an example, a novel study on controller design based on probability density evolution method and interval-oriented sequential optimization strategy considering interval and random uncertainties is proposed. Firstly, under a certain sample point of interval uncertainties, probability density evolution equation is utilized to calculate the probability density function of uncertain response considering only random uncertainties. In this way the probabilistic reliability corresponding to the sample point can be obtained. Then the hybrid reliability is defined as the average probability within the interval uncertainty and the adaptive dichotomy method is utilized to choose proper sample points of interval uncertainties, in which the reliability curve and the interval uncertainty axis is regarded as the judgement of convergence. Next the hybrid reliability is introduced to the optimization as a constraint and the PID gains reliable design optimization is established to balance the energy-consuming and safety. Finally, the sequential optimization strategy is adopted to improve the efficiency of optimization. In every iteration step of the optimization, the constraint of deterministic optimal design is adjusted according to the static reliability at the moment before the passage or reaching the maximum value. Three numerical examples are presented to demonstrate the accuracy and practicability of the proposed framework.

An attention-based deep learning method for safety of uncertain vehicle-bridge system with random near fault earthquakes

In this paper, a novel approach, based on the principle of the deep learning method, is proposed to study the stochastic responses of vehicle-bridge system (VBS) subjected to near fault earthquakes (NFEs), which also considers the effects of uncertain parameters. To generate the training data as the input of the proposed deep learning model, the dynamic formulas of the VBS are deduced by Newmark-β method. The proposed analysis model comprises three modules: the CNN module for seismic data feature extraction, the Attention Mechanism module for enhancing the selection for information between time series to improve the accuracy and efficiency of the final prediction, and the Bidirectional Gated Recurrent Unit (BiGRU) for predicting VBS responses. The mapping connection between earthquake action and the system response is established. The BiGRU model is capable of conveying both the excitation's randomness and the system's uncertain parameters. An actual railway cable-stayed bridge subjected to the running railway vehicle and NFEs is utilized to verify the proposed model. The uncertain train weight, bridge damping ratio and the randomness of NFEs are incorporated into the dynamic responses analysis of VBS. As a result, the time-varying responses obtained by the proposed model show significant agreement with results from a validated dynamic VBS framework. The mean value and standard deviation (STD) of the responses obtained by the proposed method are also compared with those by the Monte Carlo method and probability density evolution method. Therefore, both the individual sample of the dynamic response and the statistical data from diverse stochastic responses are chosen to validate the model's accuracy and efficiency in the VBS analysis under NFEs. In addition, the effects of the stochastic characteristics on the system's random vibrations are also explored through the time-histories of statistical data and the probability density function of the absolute maximum of responses.

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.

An enhanced fatigue damage model based on Weibull strength distribution

AbstractConsidering the inherent characteristics of strength degradation, a residual strength model is improved reasonably through the incorporation of the two‐parameter Weibull distribution. Building upon this foundation, an enhanced nonlinear fatigue damage accumulation model that is closer to the actual situation is proposed in view of the randomness of fatigue damage. The model takes into account the effect of load action on fatigue damage. To ascertain the feasibility of the proposed model, the time‐varying reliability of the model is evaluated by applying the probability density evolution method. The model's precision in predicting residual life is corroborated through five experimental data sets under bilevel and multilevel loading conditions. The results demonstrate that the proposed model exhibits excellent applicability and remarkable precision, thereby confirming its potential value in practical applications.

Random dynamic response analysis of an asphalt concrete core wall dam on deep overburden with double random factors

The spatial variability of material parameters and the randomness of ground motion are factors that influence the seismic response of a dam. To obtain more reliable seismic response results, it is crucial to study the influence of various uncertain factors on the response to earthquakes. In this paper, the K-L series expansion method and LH sampling technique are combined to characterize the spatial correlation of material parameters using a Gaussian autocorrelation function. This allows for the simulation of random fields of material parameters. By improved the Clough-Penzien power spectrum model and incorporating the concept of a random function, it becomes possible to generate nonstationary random ground motion that accurately reflects the site conditions. Taking an asphalt concrete core dam on deep overburden as an example, a random dynamic response analysis was conducted, considering various random factors. Statistical analysis and distribution testing were conducted on the horizontal acceleration and vertical permanent deformation responses. The process of the evolution of the probability density of the horizontal acceleration at the top of the dam was determined using the generalized probability density evolution method. This method helps to reveal the influence of various random factors on the response of the dam body. The research results show that among the spatial variability of material parameters and the randomness of ground motions, the randomness of ground motions is the main influencing factor of the random response results of the dam body. The mean response of two random factors is significantly larger than that of a single random factor. When considering dual random factors simultaneously, the complexity of the random factors increases, resulting in statistical properties that differ from those of single random factors. Therefore, to obtain more reliable random seismic response results, it is necessary to consider both the spatial variability of material parameters and the randomness of ground motions.

Global sensitivity analysis of the maximum live load and its applications

The design live loads are determined by the probability distribution of the maximum live load, which is influenced by the amplitudes and time intervals of various sustained and extraordinary loads. If the relative impact of different input variables on the maximum can be clarified, more targeted load surveys and modeling can be achieved. However, there is currently no global sensitivity analysis that simultaneously considers all input variables. In this study, the probability density function of the maximum live load is determined using the load coincidence principle and probability density evolution method. The relative entropy is employed as a measure for conducting a global sensitivity analysis across five common building occupancy types. The results indicate a significant imbalance in the impact of different input variables. The load amplitudes have a much greater effect than the time intervals. Among various load amplitudes, those related to the extraordinary loads often have the most significant impact. Regarding the time intervals, the occurrence intervals corresponding to the extraordinary loads caused by furniture stacking and normal crowding consistently have the least influence. For the time intervals with minimal impact, it is suggested to treat them as deterministic values in the live load modeling. This treatment has a negligible impact (not exceeding 10%) on the mean and upper fractile of the maximum, which are generally used for design in load codes.

A numerical critical shear crack model and its application to post‐peak behavior assessment of RC and SFRC beams

AbstractIn this paper, a numerical critical shear crack model, in which the shear‐flexural coupling effect is considered and the full process of stress and strain is captured, is proposed to evaluate the post‐peak behavior of reinforced concrete (RC) and steel fiber reinforced concrete (SFRC) beams. The shape of critical shear crack is identified by combining the modified compression field theory considering the bridge effect of steel fibers and the section analysis method. Based on the shape of critical shear crack, the shear capacity of beam members is provided by the shear tension zone and the shear compression zone. The shear capacity of the shear tension zone is calculated by the modified compression field theory and that of the shear compression zone is determined by multiaxial strength criterion of concrete. The vertical displacements caused by the flexure deformation and shear deformation are deduced by the moment area method and integration of shear strain. To verify the proposed numerical approach, a test database of 486 RC beams and 313 SFRC beams was established to predict the shear strength, and the force‐displacement relationships of twelve beam members are used to validate the feasibility for full process analysis. The stochastic analysis of beams with different failure modes is conducted via GF‐discrepancy‐based point selection method and probability density evolution method. The limitation between different failure modes is defined according to the degradation percent of shear capacity and it is taken as threshold value of failure domain, and the failure probability analysis indicates that the designed flexure beam member suffered severe degradation of shear capacity, resulting in a significant decline in safety probability.

Probability density evolution method based stochastic simulation of near-fault pulse-like ground motions

Quantifying the near-fault effect and establishing a reasonable model of near-fault pulse-like ground motions are particularly important for seismic design of structures in near-fault regions. Given the pronounced randomness associated with earthquakes, this study first proposes a novel stochastic model of near-fault pulse-like ground motions by combining the improved finite-fault model (IFFM) and the multivariate copula-based velocity-pulse model (CVPM). Further, a probability density evolution method (PDEM) based stochastic simulation method is developed, by which the model parameters can be determined in a unified probability space so as to ensure the consistency of two independent models. For illustrative purposes, the observed records collected from the 1999 Chi-Chi earthquake are used to generate new stochastic ground motions set. Two ground motions sets based on classical stochastic simulation methods are also presented for comparison. Numerical results show that the proposed method for stochastic simulation of near-fault pulse-like ground motions is reliable; the statistics of peak ground accelerations and spectral characteristics of simulated samples are consistent with station records. Besides, the proposed method accommodates the noteworthy randomness and proportion consistency of components associated with near-fault pulse-like ground motions, making it suitable for the stochastic response and reliability analysis of seismic structures in near-fault regions. This superiority is challenging to classical stochastic simulation methods that lack reasonable consideration of randomness and correlation associated with model parameters.

Wind-resistant reliability assessment of a super-high tower crane considering the randomness and correlation of turbulent spectral parameters

The need for wind-sensitive super-high tower cranes during the construction of high-rise structures has been growing, which makes it critical to assess the wind-resistant reliability of cranes. The accuracy of assessment is dominated by the refined simulation of real stochastic wind fields, which is usually based on the deterministic turbulent spectra including fixed spectral parameters. However, it might cause significant errors to estimate extreme responses of structures. Therefore, a simulation method for stochastic wind fields considering the randomness and correlation of turbulent spectral parameters is proposed in this study. The random functions of correlated spectral parameters are constructed based on several independent elementary random variables. A novel simulation method for stochastic wind fields involving the established random turbulent spectral functions is utilized to generate fluctuating wind samples at the Ma'anshan Yangtze River (MYR) Bridge site. In combination with the probability density evolution method, the stochastic response analysis and reliability assessment of wind-induced vibration of the crane are addressed. Results prove the effectiveness of the proposed approach in depicting probabilistic properties of real wind fields. Meanwhile, the wind-resistant comfort reliability of the tower crane is higher when considering the randomness and correlation of turbulent spectral parameters.

Evolutionary probability density reconstruction of stochastic dynamic responses based on physics-aided deep learning

Probability density evolution is the vital probabilistic information for stochastic dynamic system. However, it may face big challenges when using numerical methods to solve the generalized probability density evolution equation (GDEE) of complicated real-world system under complex excitations. Recently, physics-informed deep learning has gained popularity in solving partial differential equations due to the superior approximation and generalization capabilities, which opens a promising intelligent way to solve the GDEE. In this work, a mesh-free learning model, i.e., Phy-EPDNN, is proposed for evolutionary probability density (EPD) reconstruction, where a normalized GDEE is derived and embedded in the developed network as the prior physical knowledge. The computational domain can be customized by users on demand, in which the GDEE is not required to be solved in the whole computational domain as that in numerical methods. There is no need of boundary and initial conditions for the proposed model. A data augmentation method is also proposed to obtain sufficient collection data for supervised learning. Three specially selected analytical models and two practical scenarios verify the potential applicability of the proposed general framework for EPD reconstruction. Parametric analysis is performed to discuss the influence of the major network parameters on reconstruction performance.

Revaluation of occupancy duration for live load using big data of enterprise credit information

The occupancy duration is an essential variable in establishing the time-varying model of the live load. The manual questioning and inquiries, which are the common survey methods for occupancy duration, suffer from being expensive and time-consuming. This has hindered the updating of the data basis and progress in live load research. This study proposes a novel big data survey method, which utilizes 210 million enterprise credit information to collect occupancy duration samples. Along with the big data survey method, an address-oriented mining method is presented to efficiently extract duration samples of the target type from big data. Based on the actual distribution of the extracted duration data, a more general time-varying model, called the compound renewal process, is suggested to describe the live load. The stochastic harmonic function representation method and probability density evolution method are then adopted to access the maximum load distribution of the live load process. The maximum load distributions of the sustained and combined loads are calculated and compared with previous studies to evaluate the impact of the new survey results. Comparison shows that the occupancy duration of office buildings is significantly shorter, resulting in an increase in the maximum load. The proposed data survey method enables continuous updating of duration samples, and the compound renewal process allows better compatibility for new survey data. The up-to-date duration data and a realistic time-varying model provide new ideas for determining the design live load.

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.

Seismic safety assessment of dam slopes considering rockfill softening characteristics, shear strength uncertainties, and stochastic ground motion

The stability of the dam slope is one of the crucial indicators for maintaining the overall stability and reliability of the dam. The seismic stability response analysis of the dam slope involves a significant number of factors that introduce uncertainty, primarily including the stochastic ground motion and the uncertainty of the shear strength parameters of rockfill. Furthermore, the rockfill material is susceptible to strain-softening behavior at low confining pressures when subjected to ground motion, which exacerbates its progressive failure. Therefore, the assessment of seismic safety for dam slopes should give due consideration to this characteristic of rockfill material. This study extensively explores the influence of rockfill softening, stochastic ground motion, and the variability of rockfill shear strength parameters on the stability of the dam slope. Conducting probabilistic analysis in conjunction with the generalized probability density evolution method (GPDEM), the findings suggest that it is imperative to evaluate the safety characteristics of the dam slope by considering the coupled effects of stochastic ground motion, the variability of shear strength parameters, and the effect of rockfill material softening. Based on this, a preliminary dam slope failure classification was proposed, and a framework for dam slope seismic safety assessment was established.