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.     

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.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

包含突变过程的结构时变可靠度的概率密度演化方法及其应用

结构的局部破坏或加固均会引起性能突变,导致结构功能函数严重不连续,从而增加可靠度分析的难度。为此,该文拟在概率密度演化理论的框架内建立突变结构的时变可靠度分析方法。首先,引入Heaviside函数建立了突变结构时变功能函数的统一表达式;其次,基于此表达式推导了突变结构承载力裕量的广义密度演化方程,本质上该方程为包含无穷系数的分段偏微分方程,数值求解困难;再次,针对该方程的形式解析解引入Dirac#x003b4;序列算法,为承载力裕量概率密度函数的获取提供了可行的方法;然后,给出了突变结构时变可靠度分析的一维积分公式,建立了包含突变过程的时变可靠度分析的概率密度演化方法;最后,将其应用于改造加固结构的时变可靠度分析,并以一个简单的悬臂梁破坏-加固算例验证了建议算法的可行性,且通过与MonteCarlo法的对比验证了建议方法的高效性和准确性。     

A Chebyshev collocation based sequential matrix exponential method for the generalized density evolution equation

In perspective of global approximation, this study presents a numerical method for the generalized density evolution equation (GDEE) based on spectral collocation. A sequential matrix exponential solution based on the Chebyshev collocation points is derived in consideration of the coefficient or velocity term of GDEE being constant in each time step, then the numerical procedure could be successively implemented without implicit or explicit difference schemes. The results of three numerical examples indicate that the solutions in terms of the sequential matrix exponential method for GDEE have good agreement with the analytical results or Monte Carlo simulations. For sufficiently smooth cases, there need no more than one hundred representative points to achieve a satisfied solution by the proposed method, whereas for the case in presence of severe discontinuity a few more sampling points are required to keep numerical stability and accuracy.     

Solution of generalized density evolution equation via a family of <Emphasis Type="Italic">δ</Emphasis> sequences

The traditional probability density evolution equations of stochastic systems are usually in high dimensions. It is very hard to obtain the solutions. Recently the development of a family of generalized density evolution equation (GDEE) provides a new possibility of tackling nonlinear stochastic systems. In the present paper, a numerical method different from the finite difference method is developed for the solution of the GDEE. In the proposed method, the formal solution is firstly obtained through the method of characteristics. Then the solution is approximated by introducing the asymptotic sequences of the Dirac δ function combined with the smart selection of representative point sets in the random parameters space. The implementation procedure of the proposed method is elaborated. Some details of the computation including the selection of the parameters are discussed. The rationality and effectiveness of the proposed method is verified by some examples. Some features of the numerical results are observed.     

随机结构动力反应的极值分布

提出了求解随机结构动力反应极值分布的概率密度演化方法。基于随机结构动力反应概率密度演化分析的基本思想，可构造一个与随机结构动力反应极值有关的具有“虚拟时间参数”的随机过程及其导数过程，导出了一维概率密度演化方程。结合结构动力反应的时程分析方法与有限差分方法，可求解该随机过程的一维概率密度函数。当虚拟时间参数为1时，即得到随机结构动力反应的极值分布。这一方法可用来求解一般的随机抽样和随机过程的极值分布。与随机抽样极大值分布的理论结果比较表明，本文建议方法具有良好的精度。在此基础上，分析了八层框架结构随机动力反应极值分布的若干特征。     

Strategy for selecting representative points via tangent spheres in the probability density evolution method

A strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non‐linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non‐overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables. Copyright © 2007 John Wiley & Sons, Ltd.     

Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures

Based on a partition of probability-assigned space, a strategy for determining the representative point set and the associated weights for use in the probability density evolution method (PDEM) is developed. The PDEM, which is capable of capturing the instantaneous probability density function of responses of linear and nonlinear stochastic systems, was developed in the past few years. The determination of the representative point set and the assigned probabilities is of paramount importance in this approach. In the present paper, a partition of probability-assigned space related to the representative points and the assigned probabilities are first examined. The error in the resulting probability density function of the stochastic responses is then analyzed, leading to two criteria on strategies for determining the representative points and a set of indices in terms of discrepancy of the point sets. A two-step algorithm is proposed, in which an initial uniformly scattered point set is mapped to an optimal set. The implementation of the algorithm is elaborated. Two methods for generating the initial point set are outlined. These are the lattice point sets and the Number-Theoretical nets. A density-related transformation yielding the final point set is then analyzed. Numerical examples are investigated, where the results are compared to those obtained from the standard Monte Carlo simulation and the Latin hyper-cube sampling, demonstrating the accuracy and efficiency of the proposed approach.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

随机结构复合随机振动分析的概率密度演化方法

提出了随机结构复合随机振动分析的概率密度演化方法。通过引入扩展状态向量,构造具有随机初始条件的状态方程,导出了复合随机振动反应的概率密度演化方程。结合精细时程积分方法和Lax-Wendroff差分格式对概率密度演化方程提出了数值求解方法。进行了八层层间剪切框架结构复合随机振动反应的概率密度演化分析,证明提出的方法具有计算高效、收敛性稳定与精度高的特点。研究表明随着时间的增长,复合随机振动反应概率密度曲线趋于复杂,基于正态分布假定的二阶矩分析方法可能造成可靠度分析结果的显著偏差。与仅考虑结构参数随机性和仅考虑输入随机性时的结构反应相比,复合随机振动反应概率密度曲线峰值降低、分布变宽,且随机涨落显著增强。     

随机结构反应概率密度演化分析的切球选点法

发展了随机结构反应概率密度演化分析中随机参数空间的切球选点法。密度演化方法是一类直接获取随机结构动力反应概率密度函数及其演化过程的有效方法。在多个随机变量时，随机变量空间中的离散代表点选点规则直接关系到密度演化方法的精度和效率。本文构造了平面内等半径相切圆圆心分布定位的算法，以此为基础，建立了三维空间中等半径相切球球心坐标定位的计算公式。从而给出随机变量空间中的离散代表点及其赋得概率。计算表明，基于空间切球法的选点规则具有良好的精度和效率，在2个和3个随机变量情况下是较为理想的选点方法。     

Lifecycle operational reliability assessment of water distribution networks based on the probability density evolution method

In this study, a lifecycle operational reliability assessment framework for water distribution networks (WDNs) is proposed on the basis of the probability density evolution method (PDEM). The occurrence models of daily accidents are fitted using the maintenance data provided by a local water administration sector. For a given accident, two types of accidents (e.g., leaks and bursts) are distinguished in different occurrence probabilities and simulated in various ways. The pipe deterioration process in the lifecycle is reflected by incorporating the time-dependent pipe roughness model. Considering various randomness in the model, PDEM, a newly proposed and developed method for a stochastic system, is used to evaluate the lifecycle operational reliability of WDNs. The framework is demonstrated using an actual WDN, and the nodal reliabilities in the lifecycle are obtained. Comparisons of the operational reliabilities of all nodes calculated via the PDEM and Monte Carlo simulations prove that PDEM is an accurate and highly efficient method.     

Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method

The development of high-speed railway has made it important to clarify the influence of random system parameters (i.e. vehicle load, elastic modulus, damping ratio, and mass density of bridge) on train-bridge dynamic interactions. The probability density evolution method (PDEM), a newly developed theory which is applicable to train-bridge systems, can capture instantaneous probability density functions of dynamic responses. In this study, PDEM is employed to implement random dynamic analysis of a 3D train-bridge system subjected to random system parameters. The number theory method (NTM) is employed to choose the representative point sets of random parameters, whose initial probability distribution is divided by Voronoi cells., MATLAB® software is prepared for calculation, the Newmark-β integration method and the bilateral difference method of TVD (total variation diminishing) are adopted for solution. A case study is presented in which the train travels on a three-span simply supported high-speed railway bridge. The calculation accuracy and computational efficiency of the PDEM has been verified and some conclusions are provided. Furthermore, the influence of train speed under various combinations of random parameters is beyond discuss.     

Probability density evolution analysis of engineering structures via cubature points

The probability density evolution method (PDEM) is a new approach for stochastic dynamics whereby the dynamic response and reliability evaluation of multi-degree-of-freedom nonlinear systems could be carried out. The apparent similarity and subtle distinction between the ordinary cubature and PDEM are explored with the aid of the concept of the rank of an integral. It is demonstrated that the ordinary cubature are rank-1 integrals, whereas an rank-?? integral is involved in PDEM. This interprets the puzzling phenomenon that some cubature formulae doing well in ordinary high-dimensional integration may fail in PDEM. A criterion that the stability index does not exceed unity is then put forward. This distinguishes the cubature formulae by their applicability to higher-rank integrals and the adaptability to PDEM. Several kinds of cubature formulae are discussed and tested based on the criterion. The analysis is verified by numerical examples, demonstrating that some strategies, e.g. the quasi-symmetric point method, are preferred in different scenarios. Problems to be further studied are pointed out.     

基于模型结构振动台试验的概率密度演化方法验证

以广义概率密度演化方程为核心的概率密度演化方法可应用于一般随机动力系统的反应分析与可靠度评价。该文基于随机地震动作用下模型结构振动台试验实测数据,将试验模型典型动力响应的样本集合直接统计结果与概率密度演化分析结果进行对比,以从试验角度验证概率密度演化方法的正确性。研究结果表明,概率密度演化分析结果,无论从均值与标准差过程,还是典型时刻的概率分布上,均分别与样本统计结果吻合良好,从而证明了概率密度演化方法在随机动力系统分析中的精确性与可靠性。     

Response and reliability analysis of a high-dimensional stochastic system

The efficiency of the probability density evolution method (PDEM) is improved in this paper by embedding the Kullback–Leibler (K–L) relative sensitivity in the response analysis of a stochastic dynamic system. The response reliability obtained and the probability density function of the response peaks are used for ranking to get a reduced set of random variables for the PDEM analysis. The need of complicated point selection technique with the high-dimensional uncertain variables is therefore alleviated. The proposed method is illustrated with the response analysis of a random crowd-structure system where the load randomness is considered. The acceleration response induced by the presence of the crowd is evaluated with the proposed method. Results obtained highlight the significant improvements in the computation efficiency of the probabilistic response analysis of a high-dimensional dynamic system.     

近海风力发电高塔波浪随机动力响应分析

基于拟层流风波生成机制建立的随机Fourier海浪模型,采用概率密度演化理论研究了近海风力发电高塔在随机波浪作用下的动力响应问题,给出了结构响应概率密度函数的时间演化过程、概率密度等值线图及其均值和标准差.其中随机波浪力由线性波浪理论和M0rison公式计算.结果表明,概率密度演化方法可以获得结构波浪动力响应的时变概率密度函数和等概率密度响应轨迹.据此计算的均值及标准差与Monte Carlo计算结果吻合较好.