Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients

This paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single‐phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen–Loève expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss–Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen–Loève expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Copyright © 2013 John Wiley & Sons, Ltd.     

随机杆系结构几何非线性分析的递推求解方法

建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程. 将和 位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式，运用传统的摄动方法获 得了关于非正交多项式展式的待定系数的确定性的递推方程. 在求解了待定系数后，利用非 正交多项式展开式和正交多项式展开式的关系矩阵，可以很方便地得到未知响应量的二阶统计矩. 两杆结构和平面桁架拱的算例结果表明，当随机量涨落较大时，递推随机有限元方法比基于 二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果，显示了该方法对几何非线性 随机问题求解的有效性.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

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.     

随机有限元方法在断裂分析中的应用

在幂律非线性随机有限元基础上,以单边裂纹板为例给出计算含量钢继裂参数,J（J积分）,δ（裂纹张开位移）,Δ（由裂纹引起的裂纹板上下底面相对位移）,θ（由裂纹引起的裂纹板上下底在相对转角）及其对基本随机变量变化率的方法和分析算例。     

基于侵入混沌多项式法的随机多孔介质内顺磁性流体热磁对流不确定度量化

目前流体流动与传热问题的研究大都基于确定性工况条件,而现实流体流动与传热问题中存在着大量不确定性因素,计算流体力学的不确定性量化提供了一种理解流体物性、边界条件与初始条件等不确定性因素对模拟结果影响的能力.为揭示随机多孔介质内顺磁性流体热磁对流的传播规律与演化特征,本文发展了一种基于侵入式多项式混沌展开法的热磁对流不确...     

随机有限元方法与结构可靠性

简要介绍和综合论述了随机结构的有限元分析及可靠性分析方法的研究及发展情况．包括随机结构的离散化、随机有限元控制方程的求解、随机有限元可靠性方法及随机结构的可靠性分析等．讨论了随机结构分析中存在的一些问题．概要论述了可能的发展方向．     

基于随机有限元的非线性结构稳健性优化设计

结合结构优化技术和摄动随机有限元方法研究了非线性结构稳健设计问题。将结构稳健性优化设计问题构造为双目标优化问题。优化目标包含结构性能函数的期望值和标准差。约束函数的变异也给予考虑，并采用基于函数梯度的算法进行求解。为对具有路径相关特征的非线性结构性能及结构响应的平均值及标准差进行分析。本文采用缩减的随机变量，提出了基于增量法的摄动随机有限元计算格式。在此框架下，进一步提出以一般泛函形式表达的结构性能的平均值和方差及其灵敏度的计算格式。为显示方法的有效性。文中给出几个数值算例。     

含随机参数的非线性黏弹性有限元计算

进行了粗粒土与结构接触面单调和循环加载试验，基于宏细观测量结果, 扩展了 损伤概念以 描述该类接触面在受载过程中的物态演化, 及由于物态演化导致的力学特性从初始状态到最终 稳定状态的连续变化过程. 揭示了接触面损伤的细观物理基础主要是接触面内土的颗粒破碎 和剪切压密这两种物态演化；指出接触面的剪胀体应变可以划分为可逆性和不可逆性剪胀体 应变两部分，其中不可逆性剪胀体应变可作为接触面损伤发展的宏观量度，因此其归一化 形式可作为一种损伤因子的定义；提出了建立粗粒土与结构接触面一种损伤本构关系的基本思路.     

基于观点法的谱随机有限元分析——随机响应面法

提出了一种基于配点法的谱随机有限元分析方法-随机响应面法(SRSM),这种方法与已有的谱随机有限元方法(SSFEM)类似,都用Karhunen-Loeve级数扩展式表示输入随机场而计算结果的输出用多项式混沌展式表达。然而这两种方法采用了不同的方法确定多项式混沌展式中的系数:SRSM利用概率最小二乘配点法而SSFEM利用概率Galerkin法。与解析的SSFEM相比,SRSM的优势在于有限元计算和随机分析计算不耦合,即可把通用有限元程序作为黑箱进行求解。与黑箱版的SSFEM相比,SRSM需要的样本计算更少。SRSM中的各配点来自高概率的区域并使均方差最小化,从而可用少量的样本计算获得较高的计算精度。算例突出了本文提出的方法的特点并显示此方法是有效的且有较高的计算精度。     

Stochastic finite element analysis of local average random fields

This paper discussed the concepts underlying a new approach to representation and analysis of homogeneous random Fields. An orthogonalized isoparametric local average model of random fields is proposed, and the optimum selection of random variables is determined based on the theory of nonlinear programming. The finite element method is used to develop a methodology for random field analysis. Numerical examples showing the accuracy and efficiency of the method are studied.     

Stochastic finite element method based on local averages of random fields

The stochastic finite element method (SFEM) based on the local, averages of random fields, which was proposed in [5], is now generalized to analyze the structures with several correlated random parameters. The covariance matrix of the local averages of a random vector field is derived. The SFEM based on the local averages of random vector fields is formulated. The numerical examples show that the generalized SFEM preserves the advantages of the original one, i. e., rapid convergence, good accuracy and insensitivity to the correlation structures of random parameters.Project supported by National Natural Science Foundation of China.     

混凝土结构温度徐变应力的首次超越可靠度

基于随机有限元法，充分考虑大体积混凝土结构徐变温度应力计算中的各种随机因素，视混凝土温度场，弹模，徐变度以及抗力等为非平稳随机过程，针对非线性功能函数情况，首次提出了大体积混凝土结构随机温度徐变应力首次超越可靠度的计算方法，该方法可以方便地退化为求解任一时刻混凝土结构的静态可靠度。     

等参局部平均随机场和Neumann随机有限元法

本文提出了二维连续平稳随机场的一种新的表述方法—等参局部平均随机场模型,并将它引入Neumann随机有限元法。算例表明,本文的方法是有效的且能提高精度和计算效率。     

随机结构有限元分析的递推求解方法的改进

将随机结构有限元分析的递推求解方法和伽辽金投影方法相结合,提出了求解随机静力响应的改进的递推求解方法。利用随机收敛的非正交多项式展开表示由于材料、外部荷载或构件几何尺寸的随机性导致的结构随机响应。采用递推求解方法得到响应多项式展开的初始系数,并运用定义的数学算子显式地表达出来。然后,通过定义修正系数,应用伽辽金方法对随机力平衡方程在非正交多项式基上进行投影,得到了和响应展开阶次个数相同的确定的有限元方程,并进行求解得到了修正系数。数值算例表明,通过对递推求解方法中响应表达式系数的修正,以很小的计算代价较大地提高了随机响应的计算精度;与基于正交多项式展开的随机有限元方法相比,在精度相当的前提下新方法耗费的计算时间大大降低。     

非确定性结构静动态特性稳健优化设计

本文研究了考虑参数随机性的结构静动态特性稳健性优化设计问题的数学模型和数值求解。在考虑结构设计变量和其研究了考虑参数随机性的结构静动态特性稳健性优化设计问题的数学模型和数值求 解. 在考虑结构设计变量和其他参数随机分布的前二阶矩的条件下，采用基于二阶摄动法的 随机有限元方法对结构响应的平均值和方差进行近似求解. 在摄动法有限元分析的框架下， 提出以一般函数形式表达的结构性能的平均值和标准差及其灵敏度的计算格式. 将结构 稳健性优化设计问题构造为双目标优化问题，优化目标包含结构性能函数的期望值和标准 差，约束函数的变异也给予考虑. 优化问题采用基于函数梯度的算法进行求解. 文中给出的数值算例显示了方法的有效性.     

HIERARCHICAL STOCHASTIC FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS

In this paper, the hierarchical approach is adopted for series representation of the stochastic nodal displacement vector using the hierarchical basis vectors, while the Karhunen-Lòeve series expansion technique is employed to discretize the random field into a set of random variables. A set of hierarchical basis vectors are defined to approximate the stochastic response quantities. The stochastic variational principle instead of the projection scheme is adopted to develop a hierarchical stochastic finite element method (HSFEM) for stochastic structures under stochastic loads. Simplified expressions of coefficients of governing equations and the first two statistical moments of the response quantities in the schemes of the HSFEM are developed, so that the time consumed for computation can be greatly reduced. Investigation in this paper suggests that the HSFEM yields a series of stiffness equations with similar dimensionality as the perturbation stochastic finite element method (PSFEM). Two examples are presented for numerical study on the performance of the HSFEM in elastic structural problems with stochastic Young’s Modulus and external loads. Results show that the proposed method can achieve higher accuracy than the PSFEM for cases with large coefficients of variation, and yield results agreeing well with those obtained by the Monte Carlo simulation (MCS).     

Sensitivity analysis for 3-D elastoplastic structural reliability

By taking the elastoplastic effect of structural material into account and based on 3-D elastoplastic stochastic finite element method, methods for sensitivity analysis with respect to both the distribution parameters of random variables and parameters in the limit state function are suggested. In the incremental iterative calculation, the sub-increment changingK, method and the corresponding formulas for accelerating convergence are used. The sensitivity of 3-D structural system reliability with respect to random variables is also studied.     

Bifurcation and Chaos Analysis of Stochastic Duffing System Under Harmonic Excitations

The Chebyshev polynomial approximation is applied to the dynamic response problem of a stochastic Duffing system with bounded random parameters, subject to harmonic excitations. The stochastic Duffing system is first reduced into an equivalent deterministic non-linear one for substitution. Then basic non-linear phenomena, such as stochastic saddle-node bifurcation, stochastic symmetry-breaking bifurcation, stochastic period-doubling bifurcation, coexistence of different kinds of steady-state stochastic responses, and stochastic chaos, are studied by numerical simulations. The main feature of stochastic chaos is explored. The suggested method provides a new approach to stochastic dynamic response problems of some dissipative stochastic systems with polynomial non-linearity.     

随机桁架结构几何非线性问题的混合摄动-伽辽金法求解

提出应用混合摄动$\!$-$\!$-$\!$伽辽金法求解随机桁架结构的几何非线性问题.将含位移项的随机割线弹性模量以及随机响应表示为幂多项式展开,利用高阶摄动方法确定随机结构几何非线性响应的幂多项式展开的各项系数.将随机响应的各阶摄动项假定为伽辽金试函数,运用伽辽金投影对试函数系数进行求解,从而得到随机桁架结构几何非线性响应的显式表达式.同已有的随机伽辽金法相比,本文所给的试函数由摄动解的线性组合而成,在求解非线性问题时,试函数的获取具有自适应性.数值算例结果表明,对于具有不同概率分布的多随机变量问题,本文方法无需对随机变量的概率分布形式进行转换,避免了转换误差,因而比同阶的广义正交多项式方法(generalizedpolynomial chaos, GPC)计算精度高.同时,在结果精度相当时,和GPC方法相比,本文方法得到的试函数系数的非线性方程维度不大,方程的求解工作量小且更易求解.当随机量涨落较大时,混合摄动$\!$-$\!$-$\!$伽辽金法计算所得的结构响应的各阶统计矩比高阶摄动法所得结果更逼近于蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性.