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.     

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.     

The Horseracing Simulation algorithm for evaluation of small failure probabilities

Over the past decade, the civil engineering community has ever more realized the importance and perspective of reliability-based design optimization (RBDO). Since then several advanced stochastic simulation algorithms for computing small failure probabilities encountered in reliability analysis of engineering systems have been developed: Subset Simulation (Au and Beck (2001) [2]), Line Sampling (Schuëller et al. (2004) [3]), The Auxiliary Domain Method (Katafygiotis et al. (2007) [4]), ALIS (Katafygiotis and Zuev (2007) [5]), etc. In this paper we propose a novel advanced stochastic simulation algorithm for solving high-dimensional reliability problems, called Horseracing Simulation (HRS). The key idea behind HS is as follows. Although the reliability problem itself is high-dimensional, the limit-state function maps this high-dimensional parameter space into a one-dimensional real line. This mapping transforms a high-dimensional random parameter vector, which may represent the stochastic input load as well as any uncertain structural parameters, into a random variable with unknown distribution, which represents the uncertain structural response. It turns out that the corresponding cumulative distribution function (CDF) of this random variable of interest can be accurately approximated by empirical CDFs constructed from specially designed samples. The generation of samples is governed by a process of “racing” towards the failure domain, hence the name of the algorithm. The accuracy and efficiency of the new method are demonstrated with a real-life wind engineering example.     

Robust topology optimization of phononic crystals with random field uncertainty

The uncertain spatial variation of material properties can remarkably affect the band gap characteristics of phononic crystals (PnCs). It is necessary to consider this issue when designing and manufacturing PnC materials/structures. This paper investigates a robust topology optimization method for designing the microstructures of PnCs by considering random‐field material properties. Herein, the spatial distribution of the material properties is first represented by a random field and then discretized into uncorrelated stochastic variables with the expansion optimal linear estimation method; stochastic band gap analysis is then conducted with polynomial chaos expansion. Furthermore, a robust topology optimization formulation of PnCs is proposed on the basis of the relative elemental density, where a weighted objective function handles the compromise of the mean value and standard deviation of the PnC band gap. The band gap response is analyzed, employing the finite element method for each sample of polynomial chaos expansion. In this context, the sensitivities of the stochastic band gap behaviors to the design variables are also derived. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of PnCs with a relatively large width and less sensitive band gap. Additionally, the effects of the weight factors in the objective function and the variation coefficient of material properties are discussed.     

Spectral stochastic analysis of structures with uncertain parameters

We present a probabilistic analysis of a structure with uncertain parameters subject to arbitrary stochastic excitations in a frequency domain. The problem of stochastic dynamic analysis of a linear system in a frequency domain is formulated by taking into consideration the uncertainty of structural parameters. The solution is based on the idea of a random frequency response vector for stationary input excitation and a transient random frequency response vector for nonstationary one which are used in the context of spectral analysis in order to determine the influence of structural uncertainty on the random response of structure. The numerical spectral analysis of the building structure under wind and earthquake excitation is provided to demonstrate the described algorithms in the context of computer implementation.     

任意分布随机序列的产生方法

随机序列已经被广泛地使用在信号处理中。通常都是由已知概率密度得到分布函数,通过分布函数产生随机序列。然而在难以通过概率密度求得分布函数时,产生随机序列则变得十分困难。提出了一种随机数生成的新方法——基于频数的随机重构方法,即由已知的概率密度直接产生随机序列;以K分布概率密度产生随机序列为例,并用K分布的特征函数以及特征函数的标准差进行验证,通过仿真分析证明该方法的正确性。最后通过统计对比,此方法不仅简捷,而且解决了传统方法的一些局限性。     

Identification of stochastic loads applied to a non-linear dynamical system using an uncertain computational model and experimental responses

The paper is devoted to the identification of stochastic loads applied to a non-linear dynamical system for which experimental dynamical responses are available. The identification of the stochastic load is performed using a simplified computational non-linear dynamical model containing both model uncertainties and data uncertainties. Uncertainties are taken into account in the context of the probability theory. The stochastic load which has to be identified is modelled by a stationary non-Gaussian stochastic process for which the matrix-valued spectral density function is uncertain and is then modelled by a matrix-valued random function. The parameters to be identified are the mean value of the random matrix-valued spectral density function and its dispersion parameter. The identification problem is formulated as two optimization problems using the computational stochastic model and experimental responses. A validation of the theory proposed is presented in the context of tubes bundles in Pressurized Water Reactors.     

Reliability sensitivity analysis with random and interval variables

In reliability analysis and reliability‐based design, sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability. Most of the current sensitivity analysis methods are applicable for only random variables. In many engineering applications, however, some of uncertain variables are intervals. In this work, a sensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals, as well as with respect to the distribution parameters of random variables. The equations of these sensitivity indices are derived based on the first‐order reliability method (FORM). The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function calls after reliability is found. Once FORM is performed, the sensitivity information is obtained automatically. Two examples are used for demonstration. Copyright © 2009 John Wiley & Sons, Ltd.     

Response and reliability analysis of a high-dimensional stochastic system

The efficiency of the probability density evolution method (PDEM) is improved in this paper by embedding the Kullback–Leibler (K–L) relative sensitivity in the response analysis of a stochastic dynamic system. The response reliability obtained and the probability density function of the response peaks are used for ranking to get a reduced set of random variables for the PDEM analysis. The need of complicated point selection technique with the high-dimensional uncertain variables is therefore alleviated. The proposed method is illustrated with the response analysis of a random crowd-structure system where the load randomness is considered. The acceleration response induced by the presence of the crowd is evaluated with the proposed method. Results obtained highlight the significant improvements in the computation efficiency of the probabilistic response analysis of a high-dimensional dynamic system.     

论正态云模型的普适性

分布函数是分析随机现象的重要工具，正态分布是最重要的概率分布，在自然科学和社会科学中应用广泛；隶属函数是模糊集合的基石，正态隶属函数也有广泛的应用。但是，精确确定一个模糊概念的隶属函数已经成为模糊学应用的瓶颈。云模型把随机性和模糊性结合起来，用数字特征熵，揭示随机性与模糊性的关联性，并用来表示一个定性概念的粒度。正态云模型通过期望、熵和超熵构成的特定结构发生器，生成定性概念的定量转换值，体现概念的不确定性。这种特定结构不但放宽了形成正态分布的前提条件，而且把精确确定隶属函数放宽到构造正态隶属度分布的期望函数，因而更具有普遍适用性，更简单、直接地完成了定性与定量之间的相互转换过程。     

MTO生产模式的完工期预测及承诺交货期决策

在订单式生产的中小制造企业中,订单的完成过程受诸多不确定因素的影响。考虑影响完工期的确定因素及随机因素,如生产时间、插单的等待时间、机械故障等待时间、误工时间、损耗率及返工率等,根据数据统计分析假定各因素服从一定的随机分布,建立了完工期区间预测模型,通过多次仿真得到了完工期区间范围。以相关原则为依据,通过比较预测完工期及客户要求交货期,做出承诺交货期及价格决策。最后进行灵敏度分析,通过仿真得到完工期均值与各随机变量对应参数的变化关系,在此基础上提出了控制完工期影响因素的措施。     

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。     

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.     

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.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Decisions with uncertain alternatives

This paper analyses decision models with an uncertain set of alternatives defined by a deterministic objective function and constraints with uncertain coefficients. Here, in contrast to decisions under risk, the stochastic distribution of uncertain coefficients is not known. On the basis of an uncertain multiobjective decision model, we define efficient alternatives and formulate deterministic surrogate models. Uncertain decision models with recourse are introduced, and we present solution concepts that combine approaches from both multiobjective decision making and decision models with uncertain objective functions. These approaches are discussed in relation to stochastic and fuzzy programming and to models with linear partial information. In addition, ramifications of this particular approach are explored.     

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.     

刚度不确定性结构在基础随机激励下的振动响应谱分析

综合考虑动载荷和结构刚度的不确定性，建立了基础机激励下刚度不确定性结构的动力学递推方程组。将随机有限元法和随机振动理论相结合，推导了结构振动响应谱一阶、二阶变异量以及均值、方差的计算公式，建立了刚度不确定结构在基础随机激励下的振动响应谱分析方法。算例分析验证了本方法的有效性。     

Stochastic sensitivity analysis using HDMR and score function

Probabilistic sensitivities provide an important insight in reliability analysis and often crucial towards understanding the physical behaviour underlying failure and modifying the design to mitigate and manage risk. This article presents a new computational approach for calculating stochastic sensitivities of mechanical systems with respect to distribution parameters of random variables. The method involves high dimensional model representation and score functions associated with probability distribution of a random input. The proposed approach facilitates first-and second-order approximation of stochastic sensitivity measures and statistical simulation. The formulation is general such that any simulation method can be used for the computation such as Monte Carlo, importance sampling, Latin hypercube, etc. Both the probabilistic response and its sensitivities can be estimated from a single probabilistic analysis, without requiring gradients of performance function. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of sensitivities of statistical moments or reliability of structural system.