Probability Density Evolution Equation Research Articles

An improved solution of the probability density evolution equation for analyzing stochastic structural responses with adaptive time steps and initial conditions

The probability density evolution method (PDEM) offers an innovative approach for analyzing the stochastic vibration responses of structures. Given the complexities inherent in real-world challenges, the probability density evolution equations (PDEE) often require numerical methods for effective resolution. Consequently, improving computational efficiency and solution accuracy is of paramount importance. Currently, the impulse function discretization method (IFDM) faces several challenges, including limited differentiability in the spatial direction for initial value conditions. Furthermore, the probability density function produced by this method frequently deviates significantly from the true value, undermining the accuracy of the PDEE. Additionally, when addressing vibration responses characterized by rapid evolutionary velocities, the use of a globally fixed uniform time step size (FUTSS) results in a marked decrease in computational efficiency. To tackle these challenges, this study introduces an improved probability density evolution method (IPDEM), which integrates the Gaussian kernel density estimation discretization method (GKDEDM) and an adaptive non-uniform time step size (ANUTSS). The advantages of the IPDEM are illustrated through two numerical simulation examples. The findings indicate that the GKDEDM not only enhances spatial differentiability but also provides a more accurate estimation of the probability density at the initial moment, thereby improving the accuracy of the PDEE. Furthermore, the ANUTSS significantly reduces the number of iterations required while preserving precision, leading to a substantial optimization of computation time.

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.

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.

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.

A modified Chebyshev collocation method for the generalized probability density evolution equation

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time . If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density . The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location and the amount of time spent at the origin. The continuum limit involves rescaling by a factor , where is the lattice spacing. In the limit , the rescaled functional becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

Refined probabilistic response and seismic reliability evaluation of high-rise reinforced concrete structures via physically driven dimension-reduced probability density evolution equation

Refined probabilistic response and seismic reliability evaluation of high-rise reinforced concrete structures via physically driven dimension-reduced probability density evolution equation

Comparison of PDEM and MCS: Accuracy and efficiency

The accuracy and efficiency of two methods for stochastic analysis, the probability density evolution method (PDEM) and the Monte Carlo simulation (MCS) method, are compared in terms of how well they reflect the physical properties of stochastic systems. The basic principle and the numerical implementation details of PDEM and MCS are revisited. The analytical solutions of generalized probability density evolution equation (GDEE) for three typical stochastic systems are given and are to be used as the basis for comparing the two methods. It is verified that, with the rational partition of the probability space, the PDEM provides a continuous and complete reflection of physical properties over the whole probability space. Meanwhile, with the help of the numerical solution of GDEE, PDEM is efficient and accurate to describe the process of the probability density evolution of stochastic systems. In contrast, the random samples in the MCS may not reflect the physical properties of a stochastic system adequately, and the local cluster of sample points may cause redundant calculation, which leads to lower computational efficiency. Through three typical numerical examples, the paper compares the accuracy and efficiency of PDEM and MCS specifically. It is shown that, as the numerical approaches for the stochastic response of a system, the PDEM could get much higher numerical accuracy than MCS with the same number of samples. To achieve the same level of calculation accuracy, MCS needs a much higher number of samples than PDEM.

Flutter stability analysis of aeroelastic systems with consideration of hybrid uncertain parameters

Uncertainties are usually ubiquitous and of different sources, and especially hybrid uncertainties widely exist in the aeroelastic system including external loads and material properties. In this context, a novel numerical method for flutter stability analysis of aeroelastic systems considering hybrid uncertain parameters is proposed. Uncertain parameters in aeroelastic systems are characterized by stochastic and interval model, and the flutter analysis model with hybrid uncertain parameters is established. The interval-based probability density evolution method is then developed to capture the probability density function of the lower and upper bound of generalized eigenvalues. By introducing the virtual time parameter, the probability density evolution equation can be transformed into a standard form. Using the finite difference method and the total variation diminishing (TVD) scheme, the probability density function as well as the second-order statistical quantities of the lower and upper bound of the maximum real part of generalized eigenvalues are predicted efficiently. Thus, the stability of aeroelastic systems with hybrid uncertain parameters can be analyzed based on the probability distribution of the interval bound of the maximum real part of generalized eigenvalues. Two numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method and improve the computational efficiency.

Dynamic Response and Fatigue-Life Analysis of High-Speed Railway Tunnel Base with Defects

Based on the repeated “tensile-compressive-tensile-compressive” stress characteristics of tunnel-base concrete structure, double scalar damage variables were defined using the concrete stochastic damage mechanics. The elastoplastic Helmholtz free energy was corrected by introducing the hardening parameter. A stochastic dynamic damage constitutive model with the nonlinear and strain-rate effect behavior of concrete was derived via dynamic expansion of damage energy based on the principle of energy dissipation. An “elastic prediction-plastic correction” numerical analysis algorithm was developed based on the solution of the probability density evolution equation. Secondary development of the algorithm was achieved using the Universal Distinct Element Code (UDEC) for numerical calculation of the proposed constitutive model. A comparison was made between the calculation result and the result of laboratory rapid uniaxial compression test to verify the model. Dynamic response and damage features of the tunnel-base structure in three working conditions, i.e., filled layer without seam, filled layer with seam and ground water, and filled layer with seam but without water, were analyzed based on engineering practice. According to the analysis, structural vibration response was intensified in the presence of a seam. With and without groundwater, the vertical dynamic stress attenuation at both sides of the seam was 41.07% and 47.13%, respectively; vertical vibration acceleration was attenuated by 91.17% and 91.73%, respectively; and the acceleration amplitude at the upper structure of the seam increased by 724.67% and 765.02%, respectively. Groundwater in the seam would aggravate damage accumulation. It could be seen from the analysis that the current design parameters satisfied the antifatigue requirements within the design reference period at a train speed of 300 km/h when there was no seam in the tunnel-base concrete structure with IV-class surrounding rocks. When there was a seam in the tunnel-base concrete structure, however, antifatigue life was 56 years in the presence of groundwater and 62 years without groundwater, which suggested that current design parameters failed to satisfy the antifatigue requirements within the design reference period.

A reliability analysis method of dynamic irregularity for track–viaduct system with low stiffness

To evaluate the reliability of the dynamic irregularity of a track structure in satisfying the safety requirements, a stochastic virtual track inspection (SVTI) system that can be used to predict the time-varying probability distribution of dynamic irregularity was developed by combining virtual track inspection (VTI) system and the generalized probability density evolution method (GPDEM) in this study. A train–track–bridge dynamic interaction model (TTBDIM) was built using the Universal Mechanism platform. The inertial reference method (IRM) was simulated by finding the frequency-domain quadratic integration of the axle box acceleration and filtering it, based on which the deterministic VTI system was established. For the sake of probability analysis, the track geometry irregularities (TGI) was considered as a stochastic variable. A number-theory-based uniform experimental design was used to determine the representative points in the high-dimensional probability space. SVTI was established by coupling the deterministic VTI system and the generalized probability density evolution equation. The probability density evolution function of the dynamic irregularity was solved for using the finite difference method with the total variation diminishing (TVD) format. In this study, floating slab tracks on a U-shaped beam viaduct were taken as an example to analyze the influence of track stiffness irregularity (TSI) on the basic signal characteristics of the dynamic irregularity of the tracks using the deterministic VTI system. Furthermore, the SVTI system was used to analyze the influence of TSI on the time-varying probability distribution and determine whether the dynamic inspection requirements of the dynamic irregularity were satisfied. The research findings suggested that the TSI of the track–viaduct system with a low stiffness had a significant influence on the time-varying characteristics of the dynamic irregularity probability distribution. As the stiffness of the track–viaduct system decreased, the mean probability distribution of the dynamic irregularity extrema shifted to the higher-level area with an increasing standard deviation. When rubber spring floating slab tracks (RSFSTs) and steel spring floating slab tracks (SSFSTs) were laid on the 35-m-span U-shaped beam, the probabilities of dynamic irregularity exceeding the limit were 16.13% and 30.02%, respectively. The deterministic VTI system might not be able to detect irregularities exceeding the limit within the limited simulation mileage, which could lead to misjudgments and track regularity overestimation. In conclusion, the proposed SVTI system will provide direction for the design of track–viaduct systems with low stiffnesses in urban rail transit construction.

Dynamics of contentment

This work formulates a mathematical model aimed at prediction of temporal evolution of joint probability density of wealth and contentment in a society. A continuous variable changing between 0 and 1 is introduced to characterise contentment, or satisfaction with life, of an individual and an equation governing its evolution is postulated from analysis of several factors likely to affect the contentment. As contentment is strongly affected by material well-being, a similar equation is formulated for wealth of an individual. Subsequently, an evolution equation for the joint probability density of individuals’ wealth and contentment within a society is derived from these two equations and an integral representation of marriage effects. As an illustration of this model capabilities, effects of the wealth tax rate over a long period of time are simulated for a society with an initially low variation of wealth and contentment: the model predicts that a higher taxation in the longer run may lead to a wealthier and more content society. It is also shown that lower rates of the wealth tax lead to pronounced stratification of the society in terms of both wealth and contentment and that there is no direct relationship between the average values of the latter two variables thus providing an explanation to the Easterlin paradox.

An RKPM-based formulation of the generalized probability density evolution equation for stochastic dynamic systems

In the stochastic dynamic analysis, the probability density evolution method (PDEM) provides an optional way to capture the complete probability distribution of the stochastic response of general nonlinear systems. In the PDEM, the key point is to solve the generalized probability density evolution equation (GDEE), which governs the evolution of the joint probability density function (PDF) of the response and the randomness. In this paper, a new numerical method based on the reproducing kernel particle method (RKPM) is proposed. The GDEE can be approximated through the RKPM. By some particles in the response domain, the instantaneous PDF and its partial derivative with respect to response are smoothly expressed. Then, the approximated GDEE can be discretized directly at the collocation points in the response domain. At the same time, discretization in the time domain is achieved by the difference scheme. Therefore, the RKPM-based formulation to obtain the numerical solution of GDEE is formed. The implementation procedure of the proposed method is given in detail. The accuracy and efficiency of this method are illustrated with some numerical examples. Some details of parameter analysis are also discussed.

The PDEM-based time-varying dynamic reliability analysis method for a concrete dam subjected to earthquake

A time-varying dynamic reliability analysis method based on the generalized probability density evolution method (GPDEM) is proposed for a concrete gravity dam under seismic loading. Characterized by considering both randomness and time-variability, the method could be applied to predict the time-varying seismic performance of concrete dams from the perspective of probability. Two probability density evolution equations (PDEEs) are established in the method: the extreme value distribution-based PDEE.1 considering the randomness of material parameters under seismic loading and the PDEE.2 considering the deterioration of material properties during the service life. To solve the PDEE.2, a novel initial condition is derived from the PDEE.1 for the first time. The numerical implementation is illustrated by a concrete gravity dam. The results show that the abundant and continuous probability evolution information of the time-varying seismic performance could be captured. Then, the continuous time-varying reliabilities are obtained through the specified thresholds. The accuracy and effectiveness of the newly proposed method are compared with those of the MCS. Finally, the proposed method is verified to be efficient and be suitable for complex nonlinear structures under seismic loading, which could provide a potential tool for the life-cycle seismic design.

Flutter stability analysis of stochastic aeroelastic systems via the generalized eigenvalue-based probability density evolution method

This paper proposes a generalized eigenvalue-based probability density evolution method for aeroelastic systems considering stochastic uncertainties in structural and aerodynamic parameters. Uncertain parameters in aeroelastic systems are characterized by stochastic model, and the flutter analysis model with stochastic parameters is established. The generalized eigenvalue-based probability density evolution method is then developed to capture the probability density function of the maximum real part of generalized eigenvalues. By introducing the virtual time parameter, the probability density evolution equation can be transformed into a standard form. Using the finite difference method and the total variation diminishing (TVD) scheme, the probability density function as well as the second-order statistical quantities of the maximum real part of generalized eigenvalues are predicted efficiently. Thus, the flutter stability of aeroelastic systems with stochastic parameters can be analyzed based on the probability distribution of the maximum real part of generalized eigenvalues. Two numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method and improves the computational efficiency.

On the combination of nonlinear load effects of structures

A new method for combination of nonlinear load effects of structures is suggested in the paper. At first, the concepts of load combination and load effect combination are discussed, respectively. A load combination method based on probability space partition is introduced, through which the load effects of structures could be accurately estimated. Afterwards, the principle of nonlinear load effect combination of structures under multiple loads is presented. In order to reflect the nonlinear performances of structures, the physical equations of solid mechanics with a reasonable elastoplastic damage constitutive model are applied to derive structural deterministic nonlinear responses. The probability density evolution equation (PDEE) is used to acquire the stochastic response characteristics of structures under multiple random loads. Hence, the combination of nonlinear load effects of structures is well realized. Furthermore, the reliability of structures under multiple loads is evaluated by using the physical synthesis method. Finally, in order to illustrate the proposed method, the nonlinear load effect combination analyses for a reinforced concrete (RC) plane frame under gravity load and earthquake and a concrete bridge under long-term vehicular loads and earthquakes are carried out. The analysis results demonstrate the applicability and superiority of the proposed method.

A PDEM-based perspective to engineering reliability: From structures to lifeline networks

Research of reliability of engineering structures has experienced a developing history for more than 90 years. However, the problem of how to resolve the global reliability of structural systems still remains open, especially the problem of the combinatorial explosion and the challenge of correlation between failure modes. Benefiting from the research of probability density evolution theory in recent years, the physics-based system reliability researches open a new way for bypassing this dilemma. The present paper introduces the theoretical foundation of probability density evolution method in view of a broad background, whereby a probability density evolution equation for probability dissipative system is deduced. In conjunction of physical equations and structural failure criteria, a general engineering reliability analysis frame is then presented. For illustrative purposes, several cases are studied which prove the value of the proposed engineering reliability analysis method.

Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method

This paper proposes a novel numerical method for predicting the probability density function of generalized eigenvalues in the mechanical vibration system with consideration of uncertainties in structural parameters. The eigenproblem of structural vibration is presented by first and the sensitivity of generalized eigenvalues with respect to structural parameters can be derived. The probability density evolution method is then developed to capture the probability density function of generalized eigenvalues considering uncertain material properties. Within the proposed method, the probability density evolution equation for the generalized eigenvalue problem is established accounting for the sensitivity of generalized eigenvalues with respect to structural parameters. A new variable which connects generalized eigenvalues to structural parameters is then introduced to simplify the original probability density evolution equation. Next, the simplified probability density evolution equation is solved by using the finite difference method with total variation diminishing schemes. Finally, the probability density function as well as the second-order statistical quantities of generalized eigenvalues can be predicted. Numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method within significantly less computation time and the coefficients of variation of uncertain parameters as well as the total number of them have remarkable effects on stochastic characteristics of generalized eigenvalues.

Fatigue reliability analysis of wind turbine tower under random wind load

It is important to understand the vibration response and fatigue characteristics of the wind turbine structure. The fatigue reliability of the tower flange and bolt is studied in this paper. Firstly, by using the orthogonal expansion method and the Numerical-Theoretical method of random dynamic action, the random fluctuating wind field is expanded into a set of scattered points to calculate the wind load. The numerical calculation of the dynamic response of the wind turbine is verified with the measured data. Then, a fatigue probability calculation method based on the probability density evolution method is proposed. It is an effective method to substitute the fatigue damage D obtained by a rain flow counting method into the probability density evolution equation to solve the probability density of fatigue damage. In the end, a 1.5 MW wind turbine tower was taken as an example for analyzing the fatigue damage of its flange and bolt which considers the wind speed and wind direction distribution. By using the probability density evolution method, the evolution probability density curve clusters of the structural responses can be given accurately and quantitatively; thus, the reliability of the structure under random load can be conveniently obtained.