Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.     

Generation of non-Gaussian stochastic processes using nonlinear filters

Non-Gaussian stochastic processes are generated using nonlinear filters in terms of Itô differential equations. In generating the stochastic processes, two most important characteristics, the spectral density and the probability density, are taken into consideration. The drift coefficients in the Itô differential equations can be adjusted to match the spectral density, while the diffusion coefficients are chosen according to the probability density. The method is capable to generate a stochastic process with a spectral density of one peak or multiple peaks. The locations of the peaks and the band widths can be tuned by adjusting model parameters. For a low-pass process with the spectrum peak at zero frequency, the nonlinear filter can match any probability distribution, defined either in an infinite interval, a semi-infinite interval, or a finite interval. For a process with a spectrum peak at a non-zero frequency or with multiple peaks, the nonlinear filter model also offers a variety of profiles for probability distributions. The non-Gaussian stochastic processes generated by the nonlinear filters can be used for analysis, as well as Monte Carlo simulation.     

A semi-analytical particle filter for identification of nonlinear oscillators

Particle filters find important applications in the problems of state and parameter estimations of dynamical systems of engineering interest. Since a typical filtering algorithm involves Monte Carlo simulations of the process equations, sample variance of the estimator is inversely proportional to the number of particles. The sample variance may be reduced if one uses a Rao–Blackwell marginalization of states and performs analytical computations as much as possible. In this work, we propose a semi-analytical particle filter, requiring no Rao–Blackwell marginalization, for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. Through local linearizations of the nonlinear drift fields in the process/observation equations via explicit Ito–Taylor expansions, the given nonlinear system is transformed into an ensemble of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. This information is further exploited within the particle filter algorithm for obtaining samples from the optimal posterior density of the states. The potential of the method in state/parameter estimations is demonstrated through numerical illustrations for a few nonlinear oscillators. The proposed filter is found to yield estimates with reduced sample variance and improved accuracy vis-à-vis results from a form of sequential importance sampling filter.     

Finite element solution of Fokker–Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response – an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems.     

First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations

The first passage failure of single-degree-of-freedom (SDOF) nonlinear oscillator with lightly fractional derivative damping under real noise excitations is investigated in this paper. First, the system state is approximately represented by one-dimensional time-homogeneous diffusive Markov process of amplitude through stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established from the averaged Itô equation for Hamiltonian. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the analytical solutions are checked by those from the Monte Carlo simulation of original systems.     

Nonlinear dynamic state estimation in instrumented structures with conditionally linear Gaussian substructures

Many problems of state estimation in structural dynamics permit a partitioning of system states into nonlinear and conditionally linear substructures. This enables a part of the problem to be solved exactly, using the Kalman filter, and the remainder using Monte Carlo simulations. The present study develops an algorithm that combines sequential importance sampling based particle filtering with Kalman filtering to a fairly general form of process equations and demonstrates the application of a substructuring scheme to problems of hidden state estimation in structures with local nonlinearities, response sensitivity model updating in nonlinear systems, and characterization of residual displacements in instrumented inelastic structures. The paper also theoretically demonstrates that the sampling variance associated with the substructuring scheme used does not exceed the sampling variance corresponding to the Monte Carlo filtering without substructuring.     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

A stochastic Galerkin expansion for nonlinear random vibration analysis

An approach is developed for the numerical solution of random vibration problems. It is based on treating random variables as functions in a certain Hilbert space. Stochastic processes are described as curves defined in this space, and concepts from deterministic approximation theory are applied to represent the solution as a series involving a known basis of stochastic processes, and a set of unknown coefficients which are deterministic functions of time. Then, a Galerkin projection procedure is utilized to derive a set of ordinary differential equations which can be solved numerically to determine the coefficients in the series. The versatility of the proposed approach is demonstrated by its application to a nonlinear vibration problem involving the probability density of a non-Markovian oscillator response.     

Dynamics analysis of distributed parameter system subjected to a moving oscillator with random mass, velocity and acceleration

The problem of calculating the response of a distributed parameter system excited by a moving oscillator with random mass, velocity and acceleration is investigated. The system response is a stochastic process although its characteristics are assumed to be deterministic. In this paper, the distributed parameter system is assumed as a beam with Bernoulli–Euler type analytical behaviour. By adopting the Galerkin's method, a set of approximate governing equations of motion possessing time-dependent uncertain coefficients and forcing function is obtained. The statistical characteristics of the deflection of the beam are computed by using an improved perturbation approach with respect to mean value. The method, useful to gathering the stochastic structural effects due to the oscillator–beam interaction, is simple and leads to results very close to Monte Carlo simulation for a wide interval of variation of the uncertainties.     

Structural damage identification via time domain response and Markov Chain Monte Carlo method

The present work addresses the problem of structural damage identification built on the statistical inversion approach. Here, the damage state of the structure is continuously described by a cohesion parameter, which is spatially discretized by the finite element method. The inverse problem of damage identification is then posed as the determination of the posterior probability densities of the nodal cohesion parameters. The Markov Chain Monte Carlo method, implemented with the Metropolis–Hastings algorithm, is considered in order to approximate the posterior probabilities by drawing samples from the desired joint posterior probability density function. With this approach, prior information on the sought parameters can be used and the uncertainty concerning the known values of the material properties can be quantified in the estimation of the cohesion parameters. The assessment of the proposed approach has been performed by means of numerical simulations on a simply supported Euler–Bernoulli beam. The damage identification and assessment are performed considering time domain response data. Different damage scenarios and noise levels were addressed, demonstrating the feasibility of the proposed approach.     

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。     

Sequential Monte Carlo filters for abruptly changing state estimation

Sequential Monte Carlo techniques are evaluated for the nonlinear Bayesian filtering problem applied to systems exhibiting rapid state transitions. When systems show a large disparity between states (long periods of random diffusion about states interspersed with relatively rapid transitions), sequential Monte Carlo methods suffer from the problem known as sample impoverishment. In this paper, we introduce the maximum entropy particle filter, a new technique for avoiding this problem. We demonstrate the effectiveness of the proposed technique by applying it to highly nonlinear dynamical systems in geosciences and econometrics and comparing its performance with that of standard particle-based filters such as the sequential importance resampling method and the ensemble Kalman filter.     

Data-driven identification for approximate analytical solution of first-passage problem

The first-passage problem plays a significant role in engineering performance evaluation and design optimization. To address general stochastic dynamical systems, a data-driven method is proposed to identify approximate analytical solutions for the first-passage problem which explicitly includes parameters of the system, excitation, and those related to the initial and boundary conditions. The method consists of two successive processes. First, the probability density of the first-passage time is assumed to satisfy the modified Weibull distribution and its expansion expression is constructed by using the rule of dimensional consistency. Second, by comparing the expansion with the probability density of the first-passage time estimated from random state data, the coefficients are determined by solving a set of overdetermined linear algebraic equations. Two representative examples, including the Duffing oscillator and a 2-DOF nonlinear dynamical system, are discussed in detail to illustrate the application and efficiency of the data-driven method. The efficacies of the approximate analytical solutions for the external parameters are also verified.     

Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach

A numerical path integral approach is developed for determining the response and first-passage probability density functions (PDFs) of the softening Duffing oscillator under random excitation. Specifically, introducing a special form for the conditional response PDF and relying on a discrete version of the Chapman–Kolmogorov (C–K) equation, a rigorous study of the response amplitude process behavior is achieved. This is an approach which is novel compared to previous heuristic ones which assume response stationarity, and thus, neglect important aspects of the analysis such as the possible unbounded response behavior when the restoring force acquires negative values. Note that the softening Duffing oscillator with nonlinear damping has been widely used to model the nonlinear ship roll motion in beam seas. In this regard, the developed approach is applied for determining the capsizing probability of a ship model subject to non-white wave excitations. Comparisons with pertinent Monte Carlo simulation data demonstrate the reliability of the approach.     

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the Itô stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single-time) statistics or probability distributions.     

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.     

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.     

基于遗传算法的非线性迟滞系统参数识别

研究了非线性迟滞系统的参数识别问题。在识别过程中,将非线性迟滞系统的记忆复力用双折线模型来描述,并由此模型写出非线性迟滞系统的参数识别方程。利用相干函数,把关于模型参数非线性的参数识别方程转化线性参数识别前提下的非线性函数优化问题。     

联合分布函数蒙特卡罗模拟及结构可靠度分析

针对复杂极限状态方程可靠度计算问题,提出了基于理论联合分布函数以及2 种近似联合分布函数的结构失效概率蒙特卡罗模拟方法,并给出了计算流程图.采用2 个算例证明了所提方法的有效性.结果表明:所提的失效概率模拟方法的计算精度很高,尤其适用于复杂极限状态方程的可靠度计算问题.2 种联合分布函数近似构造方法得到的失效概率精度相当,近似方法与精确方法结果的差异随失效概率的减小而增大,而且随着变量间相关性的增加而增加.当失效概率小于10-3时,近似方法的失效概率误差较大.     

A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading

This paper presents a procedure which allows for a stochastic finite element (SFE)-based reliability analysis of large nonlinear structures under dynamic loading involving both structural and loading randomness with relatively little computational effort when compared to traditional Monte Carlo methods. The analysis is based on the identification of important random variables by means of a transformation of the vector of original random variables to the uncorrelated space and subsequent sensitivity analyses. Only few nonlinear computations using the most important identified random variables are performed to determine points on the limit state surface. Subsequently, the response surface method (RSM) is employed to estimate the reliability of the structure.