Variability response functions for effective material properties

The variability response function (VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.     

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

This paper is the second 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. 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 the first paper for the special case of material property variations modeled by random variables to more general problems involving random fields. Specifically, a hierarchy of spectral- and probability-distribution-free upper bounds on the mean, variance, and exceedance values of the response of stochastic systems is established when only the coefficient of variation and lower limit of the stochastic material properties are known. Furthermore, a hierarchy of probability-distribution-free upper bounds on these quantities is established when the spectral density function describing the stochastic material properties is known in addition to the coefficient of variation and the lower limit.     

Efficient uncertainty quantification of dynamical systems with local nonlinearities and uncertainties

Idealized modeling of most engineering structures yields linear mathematical models, i.e., linear ordinary or partial differential equations. However, features like nonlinear dampers and/or springs can render nonlinear an otherwise linear model. Often, the connectivity of these nonlinear elements is confined to only a few degrees-of-freedom (DOFs) of the structure. In such cases, treating the entire structure as nonlinear results in very computationally expensive solutions. Moreover, if system parameters are uncertain, their stochastic nature can render the analysis even more computationally costly. This paper presents an approach for computing the response of such systems in a very efficient manner. The proposed solution procedure first segregates the DOFs appearing in the nonlinear and/or stochastic terms from those DOFs that involve only linear deterministic operations. Second, the responses of nonlinear/stochastic terms are determined using a non-standard form of a nonlinear Volterra integral equation (NVIE). Finally, the responses of the remaining DOFs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. Three examples are presented to demonstrate the efficacy and accuracy of the proposed method. It is shown that, even for moderately sized systems (∼1000 DOFs), the proposed method is about three orders of magnitude faster than a conventional Monte Carlo sampling method (i.e., solving the system of ODEs repeatedly).     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.     

Random vibrations: A spectral method for linear and nonlinear structures

The time-dependent power spectral density of any linear response to an input modulated stochastic process is calculated by a single numerical integration. Thus, computer costs solely depend on the efficient computation of the frequency response function. The spectral method can be extended to elasto-plastic structures with certain approximations on the nonlinear yielding process which allow the determination of the mean yielding-deformation increment. Time- and frequency dependent envelope functions of the power density of modal driving forces of the associated linear system are computed in a time-stepping procedure. Results compare favourably well with simulations in the stationary limit.     

An adaptive spectral Galerkin stochastic finite element method using variability response functions

A methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

An explicit closed-form solution for linear systems subjected to nonstationary random excitation

Explicit, closed-form solutions are presented for the correlation matrix and evolutionary power spectral density matrix of the response of a linear, classically damped MDOF system subjected to a uniformly modulated random process with the gamma envelope function. The effects of the statistical cross-modal correlations on the evolutionary mean square responses are investigated. A simple MDOF system subjected to ground motion is used as an illustrative example. Through this study, additional insight is gained into the nonstationary behavior of linear dynamic systems.     

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。     

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。     

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.     

Multimodal nonlinear spectral response of a beam with impact under random input

A linearization procedure for estimating the spectral response of a randomly excited beam—stop system is proposed. The elastic stop is replaced by a spring with a stiffness depending on the amplitude of the deflection at the impact location. The probability density function of the amplitude is obtained using the stochastic averaging principle. Next, an estimate of the nonlinear response spectrum is derived providing the expectation of the spectral density function of the random spring linear system with respect to the probability density function of the amplitude response (assumed to be a random variable). The efficiency of the method is checked by comparing results with those of numerical simulations.     

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.     

Response variability for a structure with soil–structure interactions and uncertain soil properties

A non-classical modal analysis based formulation is used to quantify the effect of uncertain soil-foundation properties on structural response in seismically excited soil–structure interacting (SSI) systems. This formulation allows the response of the interacting system to be represented as a superposition of the responses of uncoupled modal equations which include an additional random vector to represent the system uncertainties. The method is implemented by an effective Gaussian quadrature integration method to find the frequency-domain stochastic response properties of the SSI systems. Numerical examples of MDOF SSI systems illustrate that the main effect of uncertain soil properties on such SSI systems is to alter the magnitudes of modal response near the system resonant frequencies, rather than to shift the resonant frequencies. When the uncertainty of soil-foundation properties is not negligible, there may be significant variations of the transfer functions for modal response and significant uncertainty about the spectral density of structural response may occur near the system’s resonant frequencies.     

Stochastic dynamic analysis of a functionally graded thick hollow cylinder with uncertain material properties subjected to shock loading

This paper presents an investigation of the stochastic dynamic response of a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading. The mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. To obtain the radial displacement in each point, the Navier equation in displacement form is derived using linear functionally graded elements. The Galerkin finite element and Newmark finite difference methods along with the Monte Carlo simulation are employed to deal with the statistical response of the FG cylinder. The mean and variance of radial displacements are calculated in various points across thickness for different values of volume fraction exponents. The results are used to quantify the effects of variations in the mechanical properties on the dynamic response and safety within the FG cylinder.     

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.     

Approximate analysis of nonlinear stochastic differential equations using certain generalized quasi-moment functions

Summary The application and the advantages of the method of certain generalized quasi-moment functions are demonstrated by way of a simple mechanical example. The stress of the considered viscoelastic beam, subjected to a stochastically variable temperature, is described by a nonlinear (casea) or a linear (caseb) equation of first order with stochastic coefficients resulting by passing Gaussian white noise through a linear shaping filter. As a result, the final differential equation system is nonlinear also in caseb. The mean value, the variance (for the casea andb), the covariance function and the spectral density (for caseb only) of the stress are estimated by means of linear quasi-moment equations with good convergence. In contrast to this, the results which were obtained by the normal distribution method, here being used as the basic approximation, are affected with great deviations.With 6 Figures     

Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input: Part II: equivalent linear system with random matrices and power spectral density matrix

We present a method for estimating the (power spectral density) PSD matrix of the stationary response of lightly damped randomly excited multi-degree of fredom mechanical systems with strong non-linear asymmetrical restoring forces. The PSD matrix is defined as the mean value of the PSD matrix response of an equivalent linear system (ELS) whose damping and stiffness matrices depend on non-linear vibration modes of the associated conservative system, the frequencies and modes shapes being amplitude dependent. Based on a generalized van der Pol transformation and using a stochastic averaging principle, as developed in a companion paper, a stationary probability density function for the amplitude process is derived to characterize the ELS fully. Some possible simplifications of the method, such as modal reduction and/or local linearization, are also discussed. The results obtained are in good agreement with those of direct numerical simulations taking two typical examples.     

一种平稳地震地面运动的改进金井清谱模型

该文提出了一种描述平稳地震地面运动过程随机特性的功率谱密度函数模型。假定基岩地震动为某种过滤白谱过程, 通过两个特定频率参数控制该过滤白谱的低频和高频含量, 从而实现对基岩地震动物理特性的模拟。利用金井清滤波器对基岩过滤白谱进行过滤, 得到地震地面加速度的谱密度模型。通过数学推导解释了该模型的物理机制, 选择汶川地震强震观测数据拟合模型中的相关参数, 给出了汶川地震场地相应的模型参数建议取值。该文模型在物理机制上可以认为是金井清谱模型的改进方案, 但修正了金井清谱模型在零频处奇异的缺陷。     

Uncertainty quantification in non-linear dynamic response of functionally graded materials plate

This study deals with the stochastic non-linear dynamic response of functionally graded materials (FGMs) plate with uncertain system properties subjected to time-dependent uniformly distributed transverse load in thermal environments. System properties, such as material properties of each constituent's material, volume fraction index, and transverse load, are taken as uncorrelated random input variables. Material properties are assumed as temperature dependent (TD). The formulation is based on higher-order shear deformation theory (HSDT) with von-Karman non-linear strain kinematics using modified C° continuity. A Newton–Raphson-based non-linear finite element method along with a first-order perturbation technique (FOPT) and Monte Carlo sampling (MCS) is outlined to examine the second-order statistics (mean, standard deviation (SD), and probability density function (PDF)) of the non-linear dynamic response of the FGM plate. The governing dynamic equation is solved by Newmark's time integration scheme. The effects of volume fraction index, load parameters, plate thickness ratios, and temperature changes with random system properties are examined through parametric studies. The present outlined approach is validated with the results available in the literature and by MCS.