A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Static condensation based reduced order modelling of stochastically parametered large ordered systems

A surrogate stochastic reduced order model is developed for the analysis of randomly parametered structural systems with complex geometries. It is assumed that the mathematical model is available in terms of large ordered finite element (FE) matrices. The structure material properties are assumed to have spatial random inhomogeneities and are modelled as non-Gaussian random fields. A polynomial chaos expansion (PCE) based framework is developed for modelling the random fields directly from measurements and for uncertainty quantification of the response. Difficulties in implementing PCE due to geometrical complexities are circumvented by adopting PCE on a geometrically regular domain that bounds the physical domain and are shown to lead to mathematically equivalent representation. The static condensation technique is subsequently extended for stochastic cases based on PCE formalism to obtain reduced order stochastic FE models. The efficacy of the method is illustrated through two numerical examples.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

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.     

Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.     

Uncertainty analysis based on sensitivity applied to angle-ply composite structures

This article describes a finite element-based formulation for the statistical analysis of the response of stochastic structural composite systems whose material properties are described by random fields. A first-order technique is used to obtain the second-order statistics for the structural response considering means and variances of the displacement and stress fields of plate or shell composite structures. Propagation of uncertainties depends on sensitivities taken as measurement of variation effects. The adjoint variable method is used to obtain the sensitivity matrix. This method is appropriated for composite structures due to the large number of random input parameters. Dominant effects on the stochastic characteristics are studied analyzing the influence of different random parameters. In particular, a study of the anisotropy influence on uncertainties propagation of angle-ply composites is carried out based on the proposed approach.     

Lower bounds on the stress and strain fields inside random two-phase elastic media

Summary The heterogeneous media under consideration are isotropic composites consisting of two well-ordered elastic isotropic phases and subjected to uniform macroscopic loading. By extending a method due to Lipton [2], lower bounds on the stress and strain fields inside each phase are explicitly established in terms of the phase volume fractions and properties. These bounds on the second moments turn out to be optimal, since they are achieved by the relevant stress and strain fields inside the finite-rank laminates which, constructed by Francfort and Murat [6], attain the Hashin-Shtrikman lower and upper bounds on the elastic bulk and shear moduli.     

Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment

This paper presents the stochastic nonlinear free vibration response of elastically supported functionally graded materials (FGMs) plate resting on two parameter Pasternak foundation having Winkler cubic nonlinearity with random system properties subjected to uniform and nonuniform temperature changes with temperature independent (TID) and dependent (TD) material properties. System properties such as material properties of each constituent’s material, volume fraction index and foundation parameters are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strains using modified C⁰ continuity. A direct iterative based nonlinear finite element method in conjunction with first order perturbation technique (FOPT) developed by last two authors for the composite plate is extended for FGM plate to compute the second order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency. The present outlined approach has been validated with those results available in the literature and independent Monte Carlo simulation (MCS).     

分片响应面法在随机振动响应分析中的应用

在分析结构的随机振动响应时,响应面法(Response Surface Method)可有效地降低随机仿真的计算代价。然而,当随机变量存在大变异系数时,传统的响应面法无法满足所要求的精度。分片响应面基于对随机变量变异系数进行合理分块的原则,缩小响应面的近似范围,并对分割后的响应面进行独立分析,从而提高了响应面在该空间的近似精度。首先采用分块响应面法结合蒙特卡洛MC抽样技术,以单质点振子模型的随机振动响应为算例,对分块响应面法的正确性进行验证。计算结果表明,在随机变量的变异系数不大时,分片响应面法的计算精度不低于传统响应面法,而当随机变量具有大变异系数时,分片响应面法的近似精度远高于传统响应面法。此外,以随机地震动作用下的桥墩随机振动为背景,将该方法进行了进一步地推广及应用。     

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.     

Recent results on least squares-based adaptive control of linear stochastic systems in white noise

Recently, progress has been made on establishing the stability and performance of linear stochastic systems when they are adaptively controlled in a certainty equivalent fashion using least squares- or extended least squares-based parameter estimates. Here we provide an overview of these results. We consider first the case of white gaussian noise, where the convergence of the parameter estimates can be established for generically all systems. Then we provide an account of the stability and performance of certainty equivalent controllers for which parameter convergence has been established. Next we turn to the white non-gaussian case, and obtain upper bounds for the parameter error and the normalized prediction error. Finally we exploit these bounds for the self-tuning regulator when “b ₀” is known and the delay equals one. The research reported here has been supported by the US Army Research Office under Contract No. DAAL-03-88-K-0046, by the Joint Services Electronics Program under Contract No. N00014-90-J-1270, and by an International Paper Fellowship for the first author.     

Efficient structural reliability analysis via a weak-intrusive stochastic finite element method

This paper presents a novel methodology for structural reliability analysis by means of the stochastic finite element method (SFEM). The key issue of structural reliability analysis is to determine the limit state function and corresponding multidimensional integral that are usually related to the structural stochastic displacement and/or its derivative, e.g., the stress and strain. In this paper, a novel weak-intrusive SFEM is first used to calculate structural stochastic displacements of all spatial positions. In this method, the stochastic displacement is decoupled into a combination of a series of deterministic displacements with random variable coefficients. An iterative algorithm is then given to solve the deterministic displacements and the corresponding random variables. Based on the stochastic displacement obtained by the SFEM, the limit state function described by the stochastic displacement (and/or its derivative) and the corresponding multidimensional integral encountered in reliability analysis can be calculated in a straightforward way. Failure probabilities of all spatial positions can be obtained at once since the stochastic displacements of all spatial points have been known by using the proposed SFEM. Furthermore, the proposed method can be applied to high-dimensional stochastic problems without any modification. One of the most challenging problems encountered in high-dimensional reliability analysis, known as the curse of dimensionality, can be circumvented with great success. Three numerical examples, including low- and high-dimensional reliability analysis, are given to demonstrate the good accuracy and the high efficiency of the proposed method.     

Assessment of the local stress state through macroscopic variables

Macroscopic quantities beyond effective elastic tensors are presented that can be used to assess the local state of stress within a composite in the linear elastic regime. These are presented in a general homogenization context. It is shown that the gradient of the effective elastic property can be used to develop a lower bound on the maximum pointwise equivalent stress in the fine-scale limit. Upper bounds are more sensitive and are correlated with the distribution of states of the equivalent stress in the finescale limit. The upper bounds are given in terms of the macrostress modulation function. This function gauges the magnitude of the actual stress. For 1 相似文献   

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

Stochastic finite element method for elasto‐plastic body

This paper proposes a Stochastic Finite Element Method (SFEM) for non‐linear elasto‐plastic bodies, as a generalization of the SFEM for linear elastic bodies developed by Ghanem and Spanos who applied the Karhunen–Loeve expansion and the polynomial chaos expansion for stochastic material properties and field variables, respectively. The key feature of the proposed SFEM is the introduction of two fictitious bodies whose behaviours provide upper and lower bounds for the mean of field variables. The two bounding bodies are rigorously obtained from a given distribution of material properties. The deformation of an ideal elasto‐plastic body of the Huber–von Mises type is computed as an illustrative example. The results are compared with Monte‐Carlo simulation. It is shown that the proposed SFEM can satisfactorily estimate means, variances and other probabilistic characteristics of field variables even when the body has a larger variance of the material properties. Copyright © 1999 John Wiley & Sons, Ltd.     

Stochastic optimization using a sparse grid collocation scheme

In computational sciences, optimization problems are frequently encountered in solving inverse problems for computing system parameters based on data measurements at specific sensor locations, or to perform design of system parameters. This task becomes increasingly complicated in the presence of uncertainties in boundary conditions or material properties. The task of computing the optimal probability density function (PDF) of parameters based on measurements of physical fields of interest in the form of a PDF, is posed as a stochastic optimization problem. This stochastic optimization problem is solved by dividing it into two problems—an auxiliary optimization problem to construct stochastic space representations from the PDF of measurement data, and a stochastic optimization problem to compute the PDF of problem parameters. The auxiliary optimization problem is solved using a downhill simplex method, whilst a gradient based approach is employed for solving the stochastic optimization problem. The gradients required for stochastic optimization are defined, using appropriate stochastic sensitivity problems. A computationally efficient sparse grid collocation scheme is utilized to compute the solution of these stochastic sensitivity problems. The implementation discussed, requires minimum intrusion into existing deterministic solvers, and it is thus applicable to a variety of problems. Numerical examples involving stochastic inverse heat conduction problems, contamination source identification problems and large deformation robust design problems are discussed.     

An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics

A generalization for non-Gaussian random variables of the well-known Kazakov relationship is reported in this work. If applied to the stochastic linearization of non-linear systems under non-Gaussian excitations, this relationship allows us to define the significance of the linearized stiffness coefficient. It is the sum of that one known in the literature (the mean of the tangent stiffness) and of terms taking into account the non-Gaussianity of the response. Moreover, the relationship here given is used for finding alternative formulae between the moments and the quasi-moments. Lastly, it is used in the framework of the moment equation approach, coupled with a quasi-moment neglect closure, for solving non-linear systems under Gaussian or non-Gaussian forces. In this way an iterative procedure based on the solution of a linear differential equation system, in which the values of the response mean and variance are those of the precedent iteration, is originated. It reveals a good level of accuracy and a fast convergence.     

Quantization dimension estimate of probability measures on hyperbolic recurrent sets

In this paper, under the strong open set condition we have determined the bounds of the lower and the upper quantization dimensions of a probability measure supported by the limit set of a hyperbolic recurrent iterated function system. Moreover, we have shown that the lower and upper bounds are related with the temperature functions of the thermodynamic formalism corresponding to the lower and upper contractive ratios of the hyperbolic maps.