A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Robust optimization of structures subjected to stochastic earthquake with limited information on system parameter uncertainty

The optimization of structures subjected to stochastic earthquake and characterized by uncertain parameters is usually posed in the form of non-linear programming with stochastic performance measures where the uncertain parameters are modelled as random variables. Such an approach, however, cannot be adopted in many real life situations when the limited information about uncertainty can be only modelled as of the uncertain but bounded (UBB) type. A robust optimization strategy for stochastic dynamic systems characterized by UBB parameters is proposed in the present study in the framework of the response surface method (RSM). In evaluating the stochastic constraints, repeated computations of the dynamic responses are avoided by applying an adaptive RSM based on the moving least squares method. Numerical results are presented to highlight the effectiveness of the proposed procedure. The effect of parameter uncertainty is also studied by comparing the results obtained from the proposed optimization approach with the conventional stochastic optimization results.     

Random fatigue crack growth in mixed mode by stochastic collocation method

Fatigue crack growth is uncertain, either for cracking rate or direction. The stochastic models proposed in the literature suffer from limited applicability or lack of physical meaning. In this paper, a new stochastic collocation method is proposed to solve mixed mode fatigue crack growth problems with uncertain parameters. This approach has the advantage of non-intrusive nature methods, such as Monte-Carlo simulations, since it allows us to decouple the stochastic and the mechanical computations. The proposed numerical implementation is very simple, as it requires only repetitive runs of deterministic finite element analysis at some specific points in the random space. The method describes a precise approximation of the mechanical response corresponding to the fatigue life, in order to assess the stochastic properties, namely the statistical moments and the probability density function of fatigue life. The performance of the stochastic collocation method for dealing with this kind of problems has been evaluated through two numerical examples, showing the high performance for practical applications. Moreover, the proposed method is extended in the last example to the failure probability assessment, with respect to the target service life.     

Decision support in early development phases—A case study from machine engineering

In the case study presented in this paper we consider early development phases of a mechanical product. We want to evaluate different concepts and decide which one(s) to pursue. A problem in early phases is that usually no test runs are available. In our case study, based on a standard, there are ways to compute the lifetime distributions of the components of the different concepts. Some parameters needed for these computations are not known precisely. Unfortunately, the lifetime distributions of the components are highly sensitive to these parameters. Our approach is to equip these parameters with distributions. These distributions would be called prior distributions in Bayesian terminology, but no update is possible since no test runs are available. Our approach implies that the distribution of the system lifetime for each concept is random, i.e. we get random elements in the space of lifetime distributions. Using Monte-Carlo simulations, we demonstrate several ways to compare the random lifetime distributions of the concepts. Some of these comparisons use stochastic orderings. We also introduce a new stochastic ordering which is particularly suitable for reliability purposes. Our case study, consisting of three scenarios, allows us to demonstrate some conclusions that can be reached.     

Stochastic reduced order models for random vectors: Application to random eigenvalue problems

An improved optimization algorithm is presented to construct accurate reduced order models for random vectors. The stochastic reduced order models (SROMs) are simple random elements that have a finite number of outcomes of unequal probabilities. The defining SROM parameters, samples and corresponding probabilities, are chosen through an optimization problem where the objective function quantifies the discrepancy between the statistics of the SROM and the random vector being modeled. The optimization algorithm proposed shows a substantial improvement in model accuracy and significantly reduces the computational time needed to form SROMs, as verified through numerical comparisons with the existing approach. SROMs formed using the new approach are applied to efficiently solve random eigenvalue problems, which arise in the modal analysis of structural systems with uncertain properties. Analytical bounds are established on the discrepancy between exact and SROM-based solutions for these problems. The ability of SROMs to approximate the natural frequencies and modes of uncertain systems as well as to estimate their dynamics in time is illustrated through comparison with Monte Carlo simulation in numerical examples.     

On the dynamics of uncertain coupled structures through a wave based method in mid- and high-frequency ranges

This paper focuses on guided wave propagation in elastic random structures. A numerical tool, referred to as the ’stochastic wave finite element method’ (SWFE) describing uncertain spectral parameters in periodic structures is presented. This approach represents an extension of the wave finite element for homogenous randomness media. The statistics of the kinematic diffusion matrix for two semi-infinite waveguides connected through an uncertain coupling element is offered. The diffusion relationships presented evaluate the statistics of reflection and transmission coefficients for semi-infinite connected waveguides subject to structural and geometrical variabilities on a coupling element. Finally, the effects of the uncertainties on kinematic and energetic parameters are investigated for two finite coupled structures based on the stochastic spectral approach. Numerical experiments show the effectiveness of the proposed formulation to predict the dynamics of periodic systems in mid- and high-frequency ranges with low CPU consumption.     

Effects of uncertain suspension parameters on dynamic responses of the railway vehicle system

Due to the manufacture error, design tolerance and time-varying factors, the suspension parameters of railway vehicles are always uncertain. This paper investigates the stochastic vibration of the railway vehicle system with uncertain suspension parameters. The energy method and Hamilton’s principle are adopted to derive the governing equations of the deterministic railway vehicle system, in which the rigid and flexible modes of the railway car body can be considered. Based on the deterministic model, the polynomial chaos expansion (PCE) method is further employed to perform the uncertain analysis of the railway vehicle system. The global sensitivity analysis of the stochastic response of the railway vehicle with uncertain parameters is further carried out based on the PCE method and Sobol indices. The accuracy of the proposed method is validated by comparing the obtained random results with those from the published literature and satisfactory agreements can be observed between them. Furthermore, the effects of uncertain suspension parameters on the stochastic vibration characteristics of the railway vehicle system are discussed, which can be used as the reference for the dynamic design of the railway vehicle system. The numerical results show that the computational efficiency of the PCE method is significantly improved compared with the Monte Carlo method.     

Identification of composite uncertain material parameters from experimental modal data

Stochastic analysis of structures using probability methods requires the statistical knowledge of uncertain material parameters. This is often quite easier to identify these statistics indirectly from structure response by solving an inverse stochastic problem. In this paper, a robust and efficient inverse stochastic method based on the non-sampling generalized polynomial chaos method is presented for identifying uncertain elastic parameters from experimental modal data. A data set on natural frequencies is collected from experimental modal analysis for sample orthotropic plates. The Pearson model is used to identify the distribution functions of the measured natural frequencies. This realization is then employed to construct the random orthogonal basis for each vibration mode. The uncertain parameters are represented by polynomial chaos expansions with unknown coefficients and the same random orthogonal basis as the vibration modes. The coefficients are identified via a stochastic inverse problem. The results show good agreement with experimental data.     

Efficient structural reliability methods considering incomplete knowledge of random variable distributions

The determination of an exact distribution function of a random phenomena is not possible using a limited number of observations. Therefore, in the present paper the stochastic properties of a random variable are assumed as uncertain quantities and instead of predefined distribution types the maximum entropy distribution is used. Efficient methods for a reliability analysis considering these uncertain stochastic parameters are presented. Based on approximation strategies this extended analysis requires no additional limit state function evaluations. Later, variance based sensitivity measures are used to evaluate the contribution of the uncertainty of each stochastic parameter to the total variation of the failure probability.     

Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics

A priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The proper generalized decomposition method is used to construct a quasi‐optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure, which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. Because the dynamic response is highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension. Copyright © 2011 John Wiley & Sons, Ltd.     

复合随机振动分析的混合PC-PEM方法

由于加工、制造等原因,实际结构系统往往所具有很多不确定性,准确评估随机系统的动力学行为不仅具有实际意义,而且是近年来结构动力学理论的一个研究热点。本文研究了同时考虑结构模型参数与所受外激励载荷具有不确定性的复合随机振动问题。结构模型参数的不确定性采用随机变量模拟,外激励载荷的不确定性采用随机过程模拟,提出了结构随机振动响应评估的混合混沌多项式-虚拟激励(PC-PEM)方法。数值算例研究了参数不确定性在21杆桁架中的传播,讨论了响应的一阶、二阶统计矩,并同蒙特卡洛方法进行对比表明提出方法的正确性和有效性。本文的工作对于考虑不确定的复杂装备与结构系统的随机振动分析具有很好的借鉴意义。     

A Bayesian approach to an adaptive preventive maintenance model

In this paper we consider a Bayesian theoretic approach to determine an optimal adaptive preventive maintenance policy with minimal repair. By incorporating minimal repair, maintenance and replacement, the mathematical formulas of the expected cost per unit time are obtained. When the failure density is Weibull with uncertain parameters, a Bayesian approach is established to formally express and update the uncertain parameters for determining an optimal adaptive preventive maintenance policy. Furthermore, various special cases of our model are discussed in detail.     

基于分类优化贝叶斯结构算法的篦冷机参数状态分析及其算法收敛性分析

针对种群算法建立贝叶斯结构存在参数多、易陷入局部最优的问题,提出一种改进贝叶斯结构学习算法。该算法将候选结构分为优劣解集,利用师生交流机制优化优解集保留精英个体,利用变异机制优化劣解集来增加结构多样性,从而加快算法收敛速度,并在准确率和运行时间上达到平衡。最后不仅利用马尔科夫链证明该算法是全局收敛的,而且通过仿真实验验证了所提出算法的性能。将该算法应用到水泥篦冷机的实际数据中,构建水泥篦冷机工艺参数的贝叶斯网络结构,并完成篦冷机参数状态分析。     

Accounting for Parameter Uncertainty in Simulation Input Modeling

We formulate and evaluate a Bayesian approach to probabilistic input modeling for simulation experiments that accounts for the parameter and stochastic uncertainties inherent in most simulations and that yields valid predictive inferences about outputs of interest. We use prior information to construct prior distributions on the parameters of the input processes driving the simulation. Using Bayes' rule, we combine this prior information with the likelihood function of sample data observed on the input processes to compute the posterior parameter distributions. In our Bayesian simulation replication algorithm, we estimate parameter uncertainty by independently sampling new values of the input-model parameters from their posterior distributions on selected simulation runs; and we estimate stochastic uncertainty by performing multiple (conditionally) independent runs with each set of parameter values. We formulate performance measures relevant to both Bayesian and frequentist input-modeling techniques, and we summarize an experimental performance evaluation demonstrating the advantages of the Bayesian approach.     

Dynamic capacitated lot sizing with random demand and dynamic safety stocks

We present a stochastic version of the single-level, multi-product dynamic lot-sizing problem subject to a capacity constraint. A production schedule has to be determined for random demand so that expected costs are minimized and a constraint based on a new backlog-oriented δ-service-level measure is met. This leads to a non-linear model that is approximated by two different linear models. In the first approximation, a scenario approach based on the random samples is used. In the second approximation model, the expected values of physical inventory and backlog as functions of the cumulated production are approximated by piecewise linear functions. Both models can be solved to determine efficient, robust and stable production schedules in the presence of uncertain and dynamic demand. They lead to dynamic safety stocks that are endogenously coordinated with the production quantities. A numerical analysis based on a set of (artificial) problem instances is used to evaluate the relative performance of the two different approximation approaches. We furthermore show under which conditions precise demand forecasts are particularly useful from a production–scheduling perspective.     

A generalized stochastic perturbation technique for plasticity problems

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations.     

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.     

A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

This work presents a data‐driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi‐physics simulation of Micro‐ElectroMechanical Systems (MEMS). The proposed framework comprises of two key steps: first, probabilistic characterization of the input uncertain parameters based on available experimental information, and second, propagation of these uncertainties through the predictive model to relevant quantities of interest. The uncertain input parameters are modeled as independent random variables, for which the distributions are estimated based on available experimental observations, using a nonparametric diffusion‐mixing‐based estimator, Botev (Nonparametric density estimation via diffusion mixing. Technical Report, 2007). The diffusion‐based estimator derives from the analogy between the kernel density estimation (KDE) procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent. The diffusion model allows for the incorporation of the prior density and leads to an improved density estimate, in comparison with the standard KDE approach, as demonstrated through several numerical examples. Following the characterization step, the uncertainties are propagated to the output variables using the stochastic collocation approach, based on sparse grid interpolation, Smolyak (Soviet Math. Dokl. 1963; 4 :240–243). The developed framework is used to study the effect of variations in Young's modulus, induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a MEMS switch. Copyright © 2010 John Wiley & Sons, Ltd.     

A Bayesian perspective on age replacement with minimal repair

In this article, a Bayesian approach is developed for determining an optimal age replacement policy with minimal repair. By incorporating minimal repair, planned replacement, and unplanned replacement, the mathematical formulas of the expected cost per unit time are obtained for two cases – the infinite-horizon case and the one-replacement-cycle case. For each case, we show that there exists a unique and finite optimal age for replacement under some reasonable conditions. When the failure density is Weibull with uncertain parameters, a Bayesian approach is established to formally express and update the uncertain parameters for determining an optimal age replacement policy. Further, various special cases are discussed in detail. Finally, a numerical example is given.     

A nonparametric model of random uncertainties for reduced matrix models in structural dynamics

Random uncertainties in finite element models in linear structural dynamics are usually modeled by using parametric models. This means that: (1) the uncertain local parameters occurring in the global mass, damping and stiffness matrices of the finite element model have to be identified; (2) appropriate probabilistic models of these uncertain parameters have to be constructed; and (3) functions mapping the domains of uncertain parameters into the global mass, damping and stiffness matrices have to be constructed. In the low-frequency range, a reduced matrix model can then be constructed using the generalized coordinates associated with the structural modes corresponding to the lowest eigenfrequencies. In this paper we propose an approach for constructing a random uncertainties model of the generalized mass, damping and stiffness matrices. This nonparametric model does not require identifying the uncertain local parameters and consequently, obviates construction of functions that map the domains of uncertain local parameters into the generalized mass, damping and stiffness matrices. This nonparametric model of random uncertainties is based on direct construction of a probabilistic model of the generalized mass, damping and stiffness matrices, which uses only the available information constituted of the mean value of the generalized mass, damping and stiffness matrices. This paper describes the explicit construction of the theory of such a nonparametric model.