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 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.     

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.     

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.     

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。     

Solutions of the FPK equation for time-delayed dynamical systems with the continuous time approximation method

This paper presents a method of finite-dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay and numerical solutions of probability density functions of the systems. Solutions of probability density functions of time-delayed systems are rare in the literature. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems and to compute the probability density functions efficiently. The solutions of the FPK equation for a linear time-delayed stochastic system are presented. The effects of different spectral differentiation schemes for the FDMP method on the probability density functions are compared.     

一类受随机激励的强非线性包装振动系统的随机平均

对一类受有界噪声激励作用的强非线性包装振动系统使用随机平均法和MLP方法,给出了此类系统的随机平均方程和FPK方程.利用路径积分方法,初步得出了一些关于系统稳态响应的联合概率密度的模拟结果.     

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.     

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.     

First-passage reliability of high-dimensional nonlinear systems under additive excitation by the ensemble-evolving-based generalized density evolution equation

The reliability analysis of high-dimensional stochastic dynamical systems subjected to random excitations has long been one of the major challenges in civil and various engineering fields. Despite great efforts, no satisfactory method with high efficiency and accuracy has been available as yet for high-dimensional systems even when they are linear systems, not to mention generic nonlinear systems. In the present paper, a novel method by imposing appropriate absorbing boundary condition on the newly developed ensemble-evolving-based generalized density evolution equation (EV-GDEE) combined with a feasible numerical method is proposed to capture the time-variant first-passage reliability of high-dimensional systems enforced by additive white noise excitation. In the proposed method, the equivalent drift coefficients in EV-GDEE can be estimated by analytical expression or captured by some representative deterministic dynamic analyses. Further, imposing the absorbing boundary condition and then solving the EV-GDEE, a one-or two-dimensional partial differential equation (PDE), yield the remaining probability density of the response of interest. Consequently, by integrating the remaining probability density, the numerical solution of time-variant first-passage reliability can be obtained. Several numerical examples are illustrated to verify the efficiency and accuracy of the proposed method. Compared to the Monte Carlo simulation, the proposed method is of much higher efficiency. Problems to be further studied are finally discussed.     

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.     

A physical approach to structural stochastic optimal controls

The generalized density evolution equation proposed in recent years profoundly reveals the intrinsic connection between deterministic systems and stochastic systems by introducing physical relationships into stochastic systems. On this basis, a physical stochastic optimal control scheme of structures is developed in this paper, which extends the classical stochastic optimal control methods, and can govern the evolution details of system performance, while the classical stochastic optimal control schemes, such as the LQG control, essentially hold the system statistics since there is still a lack of efficient methods to solve the response process of the stochastic systems with strong nonlinearities in the context of classical random mechanics. It is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes. The celebrated Pontryagin’s maximum principles is employed to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation. Further, the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on a control criterion of system second-order statistics assessment. The stochastic optimal control of an active tension control system is investigated, subjected to the random ground motion represented by a physical stochastic earthquake model. The investigation reveals that the structural seismic performance is significantly improved when the optimal control strategy is applied. A comparative study, meanwhile, between the advocated method and the LQG control is carried out, indicating that the LQG control using nominal Gaussian white noise as the external excitation cannot be used to design a reasonable control system for civil engineering structures, while the advocated method can reach the desirable objective performance. The optimal control strategy is then further employed in the investigation of the stochastic optimal control of an eight-storey shear frame. Numerical examples elucidate the validity and applicability of the developed physical stochastic optimal control methodology.     

Nonlinear stochastic optimal control of Preisach hysteretic systems

The nonlinear stochastic optimal control of Preisach hysteretic systems is studied, and the control procedure is illustrated with an example of the single-degree-of-freedom Preisach system. The Preisach hysteretic system subjected to a stochastic excitation is first replaced by an equivalent non-hysteretic nonlinear stochastic system with displacement-amplitude-dependent damping and stiffness, by using the generalized harmonic balance technique. Then, the relationship between the displacement amplitude and total system energy is established, and the equivalent damping and stiffness coefficients are expressed as functions of the system energy. The averaged Itô stochastic differential equation for the system energy as one-dimensional controlled diffusion process, is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is obtained based on the stochastic dynamical programming principle, and is solved to yield the optimal control force. Finally, the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is established, and the stationary probability density of the system energy is obtained, from which the variances of the controlled system response and the optimal control force are predicted and the control efficacy is evaluated. Numerical results show that the proposed control strategy for Preisach hysteretic systems is very effective and efficient.     

横浪中船舶的随机混沌运动

采用概率密度函数和数值模拟的方法研究随机横浪中船舶的混沌运动特性和发生混沌运动的临界参数条件。综合考虑非线性阻尼、非线性恢复力矩以及白噪声横浪激励,建立了船舶的横摇非线性随机微分方程。用随机Melnikov均方准则确定混沌运动的系统参数域后,应用路径积分法求解随机微分方程得到了响应的概率密度函数。研究发现:当噪声强度大于混沌临界值时,船舶出现随机混沌运动;对于高的白噪声激励强度,系统响应有两种较大可能的状态并在这两个状态间随机跳跃,这时船舶的运动不稳定并可能发生倾覆。     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先,基于能量平衡理论,运用等效线性化方法,计算等效系统的线性阻尼及自然频率,建立统计意义下的等效线性化系统。然后,利用平均法建立随机Ito方程,得到随机响应的Markovian近似;给出描述振子振幅概率密度函数演化的Fokker-Planck方程,并得到它的稳态解。进一步,对于含有响应振幅的等效线性系统,借助由Laplace变换得到的转换函数,得到原系统的条件功率谱密度,结合振幅的稳态概率密度作为权重函数,给出原系统功率谱密度的估计,以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的,它甚至适用于Duffing振子具有强非线性回复力的情形,因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。     

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.