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.     

An ensemble evolution numerical method for solving generalized density evolution equation

The key issue of the probability density evolution method (PDEM) is to solve a generalized density evolution equation (GDEE). Previously, the GDEE was solved in the framework of the point evolution method which is essentially a zero-order ensemble evolution method. In this paper, a first-order ensemble evolution method is proposed aiming at increasing the accuracy and robustness of the PDEM. The main idea of the proposed method is to incorporate information of standard deviation of each probability subdomain into the probability density evolution equation (PDEE) by introducing an ensemble velocity term. Compared with the point evolution method, the proposed method can truly reflect the fluctuation of a stochastic dynamic system. In order to estimate the ensemble velocity term accurately, a piecewise quadratic polynomial fitting method is also proposed. In addition, a GF-discrepancy based point selection method and a finite difference scheme that is total variation diminishing are adopted to solve the new PDEE. A single-degree-of-freedom oscillator, a Riccati equation and a 2-span 8-storey frame structure are investigated in detail to demonstrate the advantage of the proposed method over the original one.     

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.     

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。     

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。     

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。     

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.     

动力可靠度约束下基于概率测度变换-全局收敛移动渐近线法的结构优化设计EI北大核心CSCD

基于动力可靠度的结构优化是实现随机动力系统优化设计的重要途径。针对设计变量为系统中部分随机变量分布均值的情形,提出了一种基于动力可靠度的结构优化设计方法。在该方法中,通过概率密度演化理论实现了结构动力可靠度的高效分析。在此基础上,结合概率测度变换,可以在不增加任何确定性结构分析的前提下,实现动力可靠度对设计变量的灵敏度分析。进而,通过将上述概率密度演化-测度变换方法嵌入全局收敛移动渐近线法,实现了基于动力可靠度的结构优化设计问题的高效求解。数值算例的结果表明,所提方法可以显著降低结构分析次数,具有较高的效率与稳健性。     

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.     

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.     

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.     

The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems

Dynamic response analysis of nonlinear structures involving random parameters has for a long time been an important and challenging problem. In recent years, the probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) of the dynamic response and its evolution, has been proposed and developed for nonlinear stochastic dynamical systems. In the probability density evolution method, the strategy of selecting representative points is of critical importance to the efficiency especially when the number of random parameters is large. Enlightened by Cantor’s set theory, a strategy of dimension-reduction via mapping is proposed in the present paper. In the strategy, a two-dimensional domain is firstly considered and discretized such that the grid points are assigned with probabilities associated to the joint PDF. These points are then sorted and set on a virtual line according to a certain principle. Partitioning the sorted points on the virtual line into a certain number of intervals and selecting one single point in each interval, the two random variables can be transformed to a single comprehensive random variable. The associated probability of each point is simultaneously transformed accordingly. In the case of multiple random parameters, the above dimension-reduction procedure from two to one could be used recursively such that the random vector is finally transformed to one single comprehensive random variable. Numerical examples are investigated, showing that the proposed method is of high efficiency and fair accuracy.     

A global sensitivity index based on Fréchet derivative and its efficient numerical analysis

Sensitivity analysis plays an important role in reliability evaluation, structural optimization and structural design, etc. The local sensitivity, i.e., the partial derivative of the quantity of interest in terms of parameters or basic variables, is inadequate when the basic variables are random in nature. Therefore, global sensitivity such as the Sobol’ indices based on the decomposition of variance and the moment-independent importance measure, among others, have been extensively studied. However, these indices are usually computationally expensive, and the information provided by them has some limitations for decision making. Specifically, all these indices are positive, and therefore they cannot reveal whether the effects of a basic variable on the quantity of interest are positive or adverse. In the present paper, a novel global sensitivity index is proposed when randomness is involved in structural parameters. Specifically, a functional perspective is firstly advocated, where the probability density function (PDF) of the output quantity of interest is regarded as the output of an operator on the PDF of the source basic random variables. The Fréchet derivative is then naturally taken as a measure for the global sensitivity. In some sense such functional perspective provides a unified perspective on the concepts of global sensitivity and local sensitivity. In the case the change of the PDF of a basic random variable is due to the change of parameters of the PDF of the basic random variable, the computation of the Fréchet-derivative-based global sensitivity index can be implemented with high efficiency by incorporating the probability density evolution method (PDEM) and change of probability measure (COM). The numerical algorithms are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method.     

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.     

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.     

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.     

Dynamic response and reliability analysis of non-linear stochastic structures

A new probability density evolution method is proposed for dynamic response analysis and reliability assessment of non-linear stochastic structures. In the method, a completely uncoupled one-dimensional governing partial differential equation is derived first with regard to evolutionary probability density function (PDF) of the stochastic structural responses. This equation holds for any response or index of the structure. The solution will put out the instantaneous PDF. From the standpoint of the probability transition process, the reliability of the structure is evaluated in a straightforward way by imposing an absorbing boundary condition on the governing PDF equation. However, this does not induce additional computational efforts compared with the dynamic response analysis. The computational algorithm to solve the PDF equation is studied. A deterministic dynamic response analysis procedure is embedded to compute coefficient of the evolutionary PDF equation, which is then numerically solved by the finite difference method with total variation diminishing scheme. It is found that the proposed hybrid algorithm may deal with non-linear stochastic response analysis problem with high accuracy. Numerical examples are investigated. Parts of the results are illustrated. Some features of the probabilistic information of the response and the reliability are observed and discussed. The comparisons with the Monte Carlo simulations demonstrate the accuracy and efficiency of the proposed method.