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。     

The Number Theoretical Method in Response Analysis of Nonlinear Stochastic Structures

A strategy of determining representative point sets through the number theoretical method (NTM) in analysis of nonlinear stochastic structures is proposed. The newly developed probability density evolution method, applicable to general nonlinear structures involving random parameters, is capable of capturing instantaneous probability density function. In the present paper, the NTM is employed to pick out the representative point sets in a hypercube, i.e., the multi-dimensional random parameters space. Further, a hyper-ball is imposed on the point sets to greatly reduce the number of the finally selected points. The accuracy of the proposed method is ensured in that he error estimate is proved. Numerical examples are studied to verify and validate the proposed method. The investigations indicate that the proposed method is of fair accuracy and efficiency, with the computational efforts of a problem involving multiple random parameters reduced to the level of that involving only one single random parameter.     

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.     

考虑轮轨非线性接触的车辆-轨道-桥梁垂向耦合系统随机振动分析

基于列车-轨道-桥梁耦合动力学理论,考虑轮轨接触非线性,采用广义概率密度演化理论建立了列车-轨道-桥梁垂向耦合系统非线性随机振动方程。采用数论选点法结合谱表示-随机函数法生成轨道随机不平顺样本,实现了用两个随机变量和少量样本较精确地反映轨道不平顺功率谱的随机特性。以高速列车-简支梁桥上CRTSⅠ型板式无砟轨道为例,从概率及可靠度角度,考虑非线性轮轨关系中跳轨现象以及轨道随机平顺影响,对比分析了线性与非线性轮轨对车辆运行安全性的影响;计算了不同轨道谱、车辆运行速度下轮重减载率概率密度演化规律及其对车辆运行安全性影响。结果表明,建议的方法可通过较少的随机变量和样本计算得到车辆-轨道-桥梁耦合系统非线性随机动力响应及其概率分布,可为车辆运行安全性评价提供更好的指导。     

多向不规则波浪模拟的降维方法

基于多向不规则波的3种传统Monte Carlo模拟方法,提出了相应的时空随机场源谱表达,通过引入随机函数的降维思想,推导了3种多向不规则波模拟的降维方法,实现了仅用2个～4个基本随机变量即可精确表达多向不规则波随机场。数值算例表明,采用三种降维方法模拟所得代表性样本集的有效波高均符合规范要求。同时,代表性样本集的二阶统计量(模拟值)与目标值拟合一致,且总体上比传统Monte Carlo方法精度更高,验证了降维方法的有效性。此外,通过对比分析三种降维方法的模拟精度和效率,建议了最适用于工程应用的多向不规则波模拟方法,这为结合概率密度演化理论进行海洋工程结构在随机海浪作用下的动力可靠度精细化分析提供了有效途径。     

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.     

Uncertainty quantification of multidimensional dynamical systems based on adaptive numerical solutions of the Liouville equation

Propagation of uncertainty in multidimensional dynamical systems, in the presence of parametric uncertainties, can be quantified by the solution of the underlying Liouville equation that governs the evolution of a multivariate joint probability density function of random variables associated with states and parameters. In this paper we propose an efficient numerical solution of the Liouville equation that involves (a) sampling at Gauss-quadrature nodes of random variables corresponding to uncertain parameters and (b) evolution of the associated conditional probability density functions using a finite difference method with time-adaptive computational domains in multiple dimensions. The proposed approach is designed to accurately predict long-time statistics of random variables corresponding to system states, including moments and probability density function, for dynamical systems of moderate dimension. The proposed approach is applied to four different dynamical systems, including (i) single spring–mass system, (ii) Van der Pol oscillator, (iii) double spring–mass system and (iv) a typical section nonlinear aeroelastic model. When compared to a conventional finite difference based numerical solution on a fixed grid, the solution obtained from the proposed adaptive grid based approach involves a considerable reduction in the required number of grid points for equivalent accuracy. For the single spring–mass system, for which an analytical solution is found, comparison with Monte Carlo simulation results indicates that the proposed adaptive numerical solution approach is generally one to two orders of magnitude more computationally efficient for a given level of accuracy.     

基于分解法的最大熵随机有限元法

摘要：提出了一种新的基于分解法的最大熵随机有限元方法,利用单变量分解将多维随机响应函数表述为单维随机响应函数的组合形式,从而将求解随机结构响应统计矩的多维积分表达式转化为单维积分式,对单维积分采用高斯-埃尔米特积分格式求解。在获得结构响应的统计矩之后,利用最大熵原理求得结构响应的概率密度函数解析表达式。该法不涉及求导运算,对于非线性随机问题非常适用。算例结果表明,本文方法具有较好的精度与计算效率。
     

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.     

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.     

Generalized F-discrepancy-based point selection strategy for dependent random variables in uncertainty quantification of nonlinear structures

In the performance evaluation of structures under disastrous actions, for example, earthquakes, it is important to take into account the randomness of structural parameters. Generally, these random parameters are treated either as independent or perfectly dependent, but practically they are partly dependent. This article aims at developing a point selection strategy for uncertainty quantification of nonlinear structures involving probabilistically dependent random parameters characterized by copula function. For this purpose, the point selection strategy for structures involving independent basic variables is first revisited. As an improvement, a generalized F-discrepancy diminishing oriented iterative screening algorithm is proposed. Then, combining with the conditional sampling method, a conditional point set rearrangement method and a conditional iterative screening-rearrangement method are proposed for probabilistically dependent variables. These new point selection strategies are readily incorporated into the probability density evolution method for uncertainty quantification of nonlinear structures involving dependent random parameters, which is characterized by copula function. The proposed methods are illustrated by two examples including a shear frame with hysteretic restoring forces and a reinforced concrete frame structure with the damage constitutive model of concrete, where the material parameters are probabilistically dependent. The results demonstrate the effectiveness of the proposed method. Problems to be studied are discussed.     

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.     

Approximate solution of the Fokker–Planck–Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.     

Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

Efficient path integration methods for nonlinear dynamic systems

New approaches for numerical implementation of the path integration (PI) method are described. In essence the PI method is a stepwise calculation of the joint probability density function (PDF) of a set of state space variables describing a white noise excited nonlinear dynamic system. The basic idea behind the proposed procedure is to apply a splines interpolation method to the logarithm of the calculated PDF to obtain an accurate representation of the PDF over the whole domain and not only at the chosen grid points. This exploits the fact that the logarithm of the PDF shows a more polynomial behaviour than the PDF itself, and therefore is better adapted to a splines representation. It is demonstrated that the proposed techniques may lead to significantly improved performance in calculating the response statistics of large classes of nonlinear oscillators excited by white or coloured noise when compared to other available implementations of the PI method. An advantage of the new approaches is that they allow time-variant dynamic systems to be analysed without significant increase in computer time. Numerical results for both 2D and 3D problems are presented.     

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。     

巨子型有控结构体系中附加柱的连接方式

附加柱是巨子型有控结构体系（Mega-sub controlled structural system,即MSCSS）中的必要组成部分,附加柱的连接方式对MSCSS的动力特性有较大影响。结合概率密度演化理论及切球选点方法,通过对巨子型有控结构体系在结构参数及地震激励均为随机变量情况下进行附加柱应力响应计算,得到了附加柱应力响应的概率密度演化曲面,对不同附加柱与巨型梁连接方式下的附加柱应力进行了研究。结果表明,释放附加柱的水平约束,可以有效地降低附加柱应力响应的均值及标准差,整体提高巨子型有控结构体系的抗震能力;而附加柱应力概率密度的不均匀性表明,即使巨子型有控结构体系受到平稳随机地震作用激励,其附加柱应力响应也将表现出明显的非平稳性。     

On deriving exact probability density functions for functionally graded exponential profile orthotropic plates driven by laterally random excitation

In this article, the exact probability density function for a nonlinear exponential functionally graded orthotropic plate under lateral random excitation is investigated. After that, a study on the probability density function (PDF) is done to examine the dynamic instability and bifurcation. Using nondimensional parameters, the results are justified for a wide range of plates. The outcomes are studied with respect to the variation of both mean value of lateral loads and in-plane forces. The famous Monte Carlo simulation is employed to validate the obtained analytical PDFs. An analogical study is done between the behaviors of homogenous plates with their corresponding functionally graded material ones. Finally, it is shown how the exponential profile functionally graded orthotropic material can affect the quality of instability and bifurcation.