Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation

This paper presents a method for the dynamic analysis of structures with stochastic parameters to random excitation. A procedure to derive the statistical characteristics of the dynamic response for structure is proposed by using dynamic Neumann stochastic finite element method presented herein. Random equation of motion for structure is transformed into a quasi-static equilibrium equation for the solution of displacement in time domain. Neumann expansion method is developed and applied to the equation for deriving the statistical solution of the dynamic response of such a random structure system, within the framework of Monte Carlo simulation. Then, the results from Neumann dynamic stochastic finite element method are compared with those from the first- and second-order perturbation stochastic finite element methods and the direct Monte Carlo simulation with respect to accuracy, convergence and computational efficiency. Numerical examples are examined to show that the approach proposed in this paper has a very high accuracy and efficiency in the analysis of compound random vibration.     

Comparison of the approximate solutions of a noisy non-linear system based on Monte Carlo simulation

This paper is concerned with the comparison of the approximate methods in order to evaluate the values of state variables of a noisy nonlinear system by Monte Carlo simulation. The approximate methods considered here are (1) the stochastic linearisation, (2) the first-order approximation and (3) the second-order approximation. It is shown that the order of the accuracies of the covariance based on the approximate equation of Riccati type does not always coincide with the order of the practical error covariance of the original system which is obtained by using Monte Carlo simulation. Therefore the comparison of the accuracies should not be carried out by the solution obtained from the covariance equation but by the practical error covariance of the original system directly. This result is demonstrated with numerical examples. The characteristics of the random numbers which are used for Monte Carlo simulation are examined by various kinds of tests.     

蒙特卡罗仿真机及其应用

蒙特卡罗仿真机是通过对随机性问题采用Monte Carlo方法进行计算机仿真，从而得出待解问题的解。为了研究复杂的随机问题，文中提出了基于蒙特卡罗的随机模拟法的蒙特卡罗仿真机，并说明了它的基本原理。通过圆周率的计算，实践了蒙特卡罗仿真机的应用过程，从而显示出蒙特卡罗仿真法处理随机性问题的优越性和仿真普遍的适用性。     

Monte Carlo simulation-compatible stochastic field for application to expansion-based stochastic finite element method

In order to endow the expansion-based stochastic formulation with the capability of representing the characteristic behavior of stochastic systems, i.e., the non-linear dependence of the response variability on the coefficient of variation of the stochastic field, a Monte Carlo simulation-compatible stochastic field is suggested. Through a theoretical comparison of displacement vectors in the Monte Carlo method and an expansion-based scheme, it is found that the stochastic field adopted in the expansion-based scheme is not compatible with that appearing in the Monte Carlo simulation. The Monte Carlo simulation-compatible stochastic field is established by means of enforcing the compatibility between the stochastic fields in the expansion-based scheme and the Monte Carlo simulation. Employing the stochastic field suggested in this study, the response variability is reproduced with high precision even for uncertain fields with a moderately large coefficient of variation. Furthermore, the formulation proposed here can be used as an indirect Monte Carlo scheme by directly substituting the numerically simulated random fields into the covariance formula. This yields a pronounced reduction in the computation cost while resulting in virtually the same response variability as the Monte Carlo technique.     

Probabilistic solutions of some multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise

In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker–Planck–Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler–Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.     

A discontinuous Galerkin method for stochastic Cahn–Hilliard equations

In this paper, a discontinuous Galerkin method for the stochastic Cahn-Hilliard equation with additive random noise, which preserves the conservation of mass, is investigated. Numerical analysis and error estimates are carried out for the linearized stochastic Cahn-Hilliard equation. The effects of the noises on the accuracy of our scheme are also presented. Numerical examples simulated by Monte Carlo method for both linear and nonlinear stochastic Cahn-Hilliard equations are presented to illustrate the convergence rate and validate our conclusion.     

宽带随机激励下单自由度双边非对称碰撞振动系统的稳态响应

本文以一类单自由度双边非对称碰撞振动系统为研究对象,采用广义Hertz接触模型表示碰撞过程,考察系统在宽带随机激励下的稳态响应.应用基于广义谐和函数的随机平均法推导出系统在宽带随机外激励下的伊藤随机微分方程,通过求解相应的稳态FPK方程,得到系统关于幅值、能量和位移的稳态概率密度以及位移与速度的联合稳态概率密度.另外,将系统的随机响应近似为马尔可夫过程,利用广义胞映射法得到系统的近似稳态响应.最后通过与蒙特卡罗模拟结果的对比,验证了随机平均法和广义胞映射法的有效性.     

不确定性转子系统的随机有限元建模及响应分析

随机特性和随机载荷会引起转子系统动力响应的不确定性,是转子动力学分析中的重要影响因素.本文基于Timosheke梁理论,把转轴的材料和几何随机特性表示为一维随机场函数,推导出随机转轴有限元列式,建立转子系统随机动力学模型,并给出随机载荷作用下随机转子系统动力响应统计量的分析方法.分别对线性和非线性涡轮泵转子系统进行了随机动力响应分析,并同Monte Carlo仿真结果进行对比,结果表明所建立的随机有限元动力学模型和给出的随机响应分析方法是合理可行的,可以有效应用于实际转子系统随机动力学分析和设计中.     

Fault detection and isolation in non-linear stochastic systems—A combined adaptive Monte Carlo filtering and likelihood ratio approach

This paper presents the development of a new method for solving fault detection and isolation (FDI) problem in general non-linear stochastic systems. In this paper, the faults are modelled as unknown changes in system parameters and adaptive Monte Carlo filtering approach is used in deriving an FDI scheme. Essentially, a set of adaptive Monte Carlo filters are designed based on the augmented system models along with a nominal Monte Carlo filter designed based on the nominal system model. The likelihood functions of the observations are then evaluated using the particles from these (adaptive) Monte Carlo filters and FDI is eventually achieved via the likelihood ratio test. The simulation results on a highly non-linear system are provided which demonstrates the effectiveness of the proposed method.     

Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.     

Suboptimal control of linear discrete stochastic systems with linear input constraints

Optimal control of linear discrete stochastic systems with linear input constraints is considered. A lower bound on the achievable minimum of the loss function is given. A suboptimal control strategy is derived by using a truncated Taylor series to approximate the expected future loss in the Bellman equation. The performance of the suboptimal controller is studied using Monte Carlo simulation and the obtained loss is compared with the lower bound.     

半车非线性悬架的非平稳随机振动及控制

研究了1/2车非线性悬架模型在路面随机激励下的非平稳振动响应,并基于随机最优控制理论对其进行主动控制.首先利用等效线性化方法将具有非线性阻尼及迟滞刚度的非线性悬架模型线性化,然后将主动、被动悬架非平稳随机响应进行比较,结果表明非线性主动悬架的性能要优于被动悬架.最后,通过Monte-Carlo数值模拟验证了理论结果.     

Indirect estimation of stochastic differential equation models: some computational experiments

In this paper we consider the estimation of some stochastic differential equation models by an indirect estimation method proposed by Gourieroux, Monfort and Renault (1993) using discrete data. The performance of this method is analysed via Monte Carlo experiments. In particular we examine the Vasicek and the Cox, Ingersoll and Ross models used in financial economics and a system of three stochastic differential equations proposed by P.C.B. Phillips in 1972. These results show the ability of indirect estimation to remove the bias resulting from the discretisation of the continuous model.     

群决策中随机信息的贝叶斯集结与统计模拟方法

针对方案属性值和权重都为随机变量的群决策问题,结合贝叶斯理论和随机模拟,提出一种集成专家主观概率分布集结和随机多属性决策方案选优的方法.该方法首先构建一个多元正态集结模型,将多个专家估计的属性值分布集结成单一分布,然后用随机模拟算法,生成不完全权重信息,并通过计算各方案获得特定排名的可信度因子,以及反映决策者风险偏好的整体排名可信度因子,得到各方案排序.实例分析验证了方法的有效性.     

毁伤概率评估的加权蒙特卡洛方法

由于经典蒙特卡洛方法的仿真效率不高,文中利用概率论和数理统计的基本原理,可以推导得到加权蒙特卡洛方法.加权蒙特卡罗方法不但能有效的缩小了样本方差,还能提高目标事件出现概率,相当于提高了样本的抽样效率,从而提高了仿真效率.与此同时,文中分别利用经典蒙特卡洛和加权蒙特卡洛这两种方法对典型目标进行了仿真计算,仿真计算最终证明了加权蒙特卡洛方法的仿真效率明显优于经典蒙特卡洛方法,能使仿真工作量成数量级的衰减.     

高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应

应用随机平均法研究了高斯白噪声激励下含有分数阶阻尼项的Duffing-Van der Pol系统的稳态响应.首先应用基于广义谐和函数的随机平均法得到系统关于幅值的平均伊藤微分方程并建立相应的平稳FPK方程,求解该平稳FPK方程的近似理论解得到系统幅值的稳态概率密度.分析幅值、位移和速度的稳态概率密度探究分数阶阻尼项以及其它参数对系统稳态响应的影响.发现降低分数阶的阶数可以增强系统的响应而增大分数阶的系数可以减弱系统响应.最后对原系统进行Monte Carlo数值模拟验证近似理论解的有效性.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

Integrating subset simulation with probabilistic re-analysis to estimate reliability of dynamic systems

It is often expensive to estimate the failure probability of highly reliable systems by Monte Carlo simulation. Subset Simulation breaks the original problem of estimating a small probability into the estimation of a sequence of large conditional probabilities, which is more efficient. The conditional probabilities are estimated by Markov Chain simulation. Uncertainty in the power spectral density of the excitation makes it necessary to re-evaluate the reliability for many power spectral densities that are consistent with the evidence about the system excitation. Subset Simulation is more efficient than Monte Carlo simulation, but still requires a new simulation for each admissible power spectral density. This paper presents an efficient method to re-evaluate the reliability of a dynamic system under stationary Gaussian stochastic excitation for different load spectra. We accomplish that by re-weighting the results of a single Subset Simulation. This method is applicable to both linear and nonlinear systems provided that all of the spectra contain the same amount of energy. The authors are currently working on an extension of the method to nonlinear systems, even when the sampling and true power spectral density functions contain different amounts of energy.     

On time counting procedures in the DSMC method for rarefied gases

The DSMC (direct simulation Monte Carlo) method for rarefied gas dynamics is studied. The behaviour of the underlying stochastic particle system is determined by three main components – the time step between subsequent collisions, the random mechanism for the choice of the collision partners, and the random mechanism for calculating the result of the collision. The purpose of this paper is to illustrate the interplay between these various components and to propose some new modifications of the DSMC method. Different time counting procedures are derived and their influence on the other parts of the algorithm is investigated. Various modifications of the DSMC method are compared with respect to different criteria such as efficiency, systematic error, and statistical fluctuations.