On the local average of random field in stochastic finite element analysis

The distinguishing feature of stochastic finite element analysis is that it involves the discretization of the parameter space of random fields of material properties, the geometry of structure and / or the loads. It is shown in earlier investigations that a reasonable procedure of discretization is to take the local averages of the random fields on each element. In the present paper the formulae for the covariance of the local averages of a homogeneous random vector field on rectangular elements are generalized by relaxing the condition. For an inhomogeneous random field and /or non-rectangular elements, a procedure of using Gaussian quadrature to evaluate the means and covariances of the local averages is proposed. Thus, the stochastic finite element method (SFEM) based on the local averages of random fields is adapted to a structure with irregular shape and / or inhomogeneous random fields. The effects of the mesh geometry, the ratio of element size to the correlation scale as well as the number of Gaussian quadrature points on the convergence of SFEM are discussed. It is found that even better results could be obtained by utilizing appropriate Gaussian quadrature instead of exact local average.Project supported by National Natural Science Foundation of China.     

具有随机参数的含裂纹板弯曲应力强度因子的统计分析

本文首次应用随机有限元法研究了具有随机参数的含裂纹板裂纹尖端弯曲应力强度因子的统计性质。文中首先给出了杂交模式的裂纹尖端奇异单元的刚度矩阵,然后基于随机场的局部平均理论和一阶泰勒展开得到了应力强度因子均值和方差的计算公式。作为数例,详细讨论了杨氏模量、泊松比及板厚度的不确定性对应力强度因子的影响。     

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation‐based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1‐D as well as the 3‐D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1‐D problems. Copyright © 2007 John Wiley & Sons, Ltd.     

具有随机参数的含裂纹板弯曲应力强度因子的统计分析

本文首次应用随机有限元法研究了具有随机参数的含裂纹板裂纹尖端弯曲应力强度因子的统计性质。文中首先给出了杂交模式的裂纹尖端奇异单元的刚度矩阵,然后基于随机场的局部平均理论和一阶泰勒展开得到了应力强度因子均值和方差的计算公式。作为数例,详细讨论了杨氏模量、泊松比及板厚度的不确定性对应力强度因子的影响。     

精细积分时域平均法和随机扩阶系统法

讨论含随机参数结构的动力响应的计算问题,发展了精细积分时域平均法（ＴＡＰＩＭ）,它可以用来计算确定性系统受到随机激励时的动力响应;结合随机扩阶系统方法与随机有限元法,将ＴＡＰＩＭ方法应用于计算随机参数结构的动力响应,取得了较好的结果。结出了数值算例,结果表明随机扩阶系统法,随机有限元法与精细积分时域平均法的结合是计算 随机参数结构动力响应的有效方法。     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

考虑空间变异性的框架结构可靠度分析方法

空间变异性是结构参数的固有属性,对于工程结构的随机响应和可靠度分析具有重要影响。结合随机场离散的局部平均理论和随机响应分析的摄动随机有限元法,提出一种考虑参数空间变异性的平面框架结构可靠度分析方法,并定量分析了参数空间变异性对结构可靠度的影响规律。首先,考虑随机因素的空间变异性,采用二维线性随机场离散的局部平均理论将平面框架结构的连续随机场离散为一组随机变量,并通过理论推导建立了随机场局部平均间协方差矩阵的二重积分表达式;然后,采用摄动随机有限元法分析结构随机响应及其对基本随机变量的梯度向量,并利用可靠度分析的梯度优化法计算结构可靠指标,从而提出了一种考虑参数空间变异性的平面框架结构可靠度分析方法。分析表明,该方法具有较高的计算精度和计算效率;随机场离散的局部平均理论对相关结构类型不敏感;随着随机场相关偏度和变异性的增大,框架结构的可靠指标逐渐减小,说明结构参数的空间变异性对结构可靠度的影响不容忽视。     

Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials

Boundary-layer effects on the effective response of fibre-reinforced media are analysed. The distribution of the fibres is assumed random. A methodology is presented for obtaining non-local effective constitutive operators in the vicinity of a boundary. These relate ensemble averaged stress to ensemble averaged strain. Operators are also developed which re-construct the local fields from their ensemble averages. These require information on the local configuration of the medium. Complete information is likely not to be available, but averages of these operators conditional upon any given local information generate corresponding conditional averages of the fields. Explicit implementation is performed within the framework of an approximation of Hashin-Shtrikman type. Two types of geometry are considered in examples: a half-space and a crack in an infinite heterogeneous medium. These are representative, asymptotically, of the field in the vicinity of any smooth boundary, and in the vicinity of a crack tip, respectively. Results have been obtained for the case of anti-plane deformation, realized by the imposition of either Dirichlet or Neumann conditions on the boundary; those for the Neumann condition are presented and discussed explicitly. The stresses in both fibre and matrix adjacent to a crack tip are shown to differ substantially from the values that would be predicted by ordinary homogenization.     

Spectral strip-element method for beams treated with active constrained layer damping

For vibration analysis of beams fully treated with active constrained layer damping (ACLD), a new approach called spectral strip-element method (SSEM) based on the spectral finite element method (SFEM) is proposed. It can avoid difficulties in solving the characteristic equation with higher orders and unknown parameters for wave numbers when using the SFEM; simultaneously, advantages of a very few elements and high accuracy of the SFEM are kept. A numerical example shows that the proposed method is very effective and reliable, compared with the exact solutions resulted from the spectral transfer matrix method (STMM).     

线性随机结构在随机激励下动力响应分析

利用虚拟激励法对随机结构正交展开理论进行扩展,并在Ritz向量子空间中对扩阶系统方程进行动力聚缩,提出了一类可以快速高效地进行线性随机结构复合随机振动分析的计算方法．算例分析表明,该法可以方便地分析随机结构在平稳或非平稳随机激励下的复合随机振动问题,且分析结果与 Monte Carlo模拟分析结果符合良好;与均值参数确定性结构传统随机振动分析计算结果相比,随机结构在相同随机激励下响应自谱密度曲线具有峰值降低、谱宽增大的特点．     

Coherent structure identification from wavelet analysis of particle image velocimetry data

 A new technique based on wavelet transform is applied to bidimensional velocity fields obtained by particle image velocimetry (PIV) measurements, in order to extract and characterize swirling motion associated with coherent structures. The proposed technique is based on the selectivity property of the wavelet transform and permits the detection of regions of the flow field associated with coherent structures and their spatial localization. Furthermore, being the method based on the analysis of the local energy content at separated scales, it is possible to extract the typical wavenumber associated with structures and therefore the typical length-scale. The procedure is validated by the application to velocity vector fields obtained from PIV measurements in different flow conditions and turbulence levels. Results are compared with those obtained by other more standard procedures, and the advantages and limitations of the proposed method are then discussed. Received: 16 October 2000 / Accepted: 18 June 2001 Published online: 29 November 2001     

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.     

Modeling the effects of material non-linearity using moving window micromechanics

In the analysis of materials with random heterogeneous microstructure the assumption is often made that material behavior can be represented by homogenized or effective properties. While this assumption yields accurate results for the bulk behavior of composite materials, it ignores the effects of the random microstructure. The spatial variations in these microstructures can focus, initiate and propagate localized non-linear behavior, subsequent damage and failure. In previous work a computational method, moving window micromechanics (MW), was used to capture microstructural detail and characterize the variability of the local and global elastic response. Digital images of material microstructure described the microstructure and a local micromechanical analysis was used to generate spatially varying material property fields. The strengths of this approach are that the material property fields can be consistently developed from digital images of real microstructures, they are easy to import into finite element models (FE) using regular grids, and their statistical characterizations can provide the basis for simulations further characterizing stochastic response. In this work, the moving window micromechanics technique was used to generate material property fields characterizing the non-linear behavior of random materials under plastic yielding; specifically yield stress and hardening slope, post yield. The complete set of material property fields were input into FE models of uniaxial loading. Global stress strain curves from the FE–MW model were compared to a more traditional micromechanics model, the generalized method of cells. Local plastic strain and local stress fields were produced which correlate well to the microstructure. The FE–MW method qualitatively captures the inelastic behavior, based on a non-linear flow rule, of the sample continuous fiber composites in transverse uniaxial loading.     

Dispersion error reduction for acoustic problems using the smoothed finite element method (SFEM)

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell‐based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two‐dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell‐based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.     

土性参数各向异性随机场的表征与建模方法

土性参数在空间上的相关性具有各向异性,因此对各向异性随机场表征与建模方法的研究具有重要的意义.本文首先通过对相关函数的分析,将各向异性相关的问题归结到相关距离函数的探讨上,给出了一种描述参数各向异性相关的方法;其次分析了 目前常用的两种随机场反演方法在处理各向异性问题所面临的问题,(1)局部平均划分法只适合横观各向同性...     

论不确定性结构力学的发展

结构力学是直接为工程结构设计服务的,而工程设计中包含大量不确定性因素和信息,包括随机性、模糊性和未确知性信息.由于引用统计数学方法处理随机性问题,在结构力学中产生了可靠性理论和随机振动理论.随着模糊性和未确知性问题的提出和研究,近年来在结构力学中又逐渐形成了一些新的分支,包括结构的不确定性优化设计,广义(模糊随机)可靠性理论,模糊随机振动理论和模糊地震工程学等.本文简要地介绍这四个新的分支的发展情况,主要介绍作者及其学生们的工作,并着重介绍基本概念.      

Bounds for the non-local effective properties of random media

The paper deals with a random medium subjected to a static scalar field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the “wave vector” k of the mean field. The variational methods of P.H. Dederichs and R. Zeller (1973) are generalized to derive upper and lower bounds for scalar effective material parameters as functions of k. In the limit k → 0 (homogeneous mean fields), bounds of the Hashin-Shtrikman type are reproduced. For k → ∞, the bounds coincide with the exact result. In the general case, a two-point moment of the stochastic material parameter is involved. Especially, composites with cell structure and binary mixtures are considered. Detailed calculations are carried out for effective dielectricity, relating mean electric displacement to the mean electric field (which is mathematically equivalent to electrical and thermal conductivities and other scalar parameters), of a binary system composed of nearly spherical grains of equal size.     

Three-dimensional unsaturated flow in heterogeneous systems and implications on groundwater contamination: A stochastic approach

A stochastic approach for modeling transient unsaturated flow in large-scale spatially variable soils is developed in order to overcome the problem of limited information about the local details of spatial soil variability. It is assumed that local soil properties are realizations of three-dimensional stationary random fields, and a large-scale model representation is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields. The three-dimensionality of the local flow equations and the nonlinear dependence of the local flow output on the local soil properties are considered. The resulting mean representation (structure) is in the form of a partial differential equation in which averaged or effective model parameters occur. These effective model parameters are evalutated using a quasi-linearized fluctuation equation and a spectral representation of stationary processes. The large-scale model structure considers the large-scale effects of soil variability and have relatively few parameters which should be identifiable from a realistic data set. The general stochastic theory is then applied to the case of flow in stratified soil formations, which is of practical importance in applications such as waste disposal control. An important finding of this study is that spatial variability of the hydraulic soil properties produces significant large-scale effects, such as large-scale hysteresis and anisotropy of the effective parameters. These large-scale effects should be considered in field applications such as for predicting the movement of liquid wastes in the unsaturated zone.     

论不确定性结构力学的发展

结构力学是直接为工程结构设计服务的,而工程设计中包含大量不确定性因素和信息,包括随机性、模糊性和未确知性信息．由于引用统计数学方法处理随机性问题,在结构力学中产生了可靠性理论和随机振动理论．随着模糊性和未确知性问题的提出和研究,近年来在结构力学中又逐渐形成了一些新的分支,包括结构的不确定性优化设计,广义(模糊随机)可靠性理论,模糊随机振动理论和模糊地震工程学等．本文简要地介绍这四个新的分支的发展情况,主要介绍作者及其学生们的工作,并着重介绍基本概念．