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.     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

随机参数结构的统计特征对

结合非正交多项式展式和传统的摄动技巧研究了随机参数结构的统计特征对问题,建立了和摄动法类似的一系列确定的递推方程,并用有限元方法进行了求解,得到了包括特征值和特征向量的特征对的统计值．最后,用算例验证了此方法的正确性．     

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。     

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。     

Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen-Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss-Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.     

基于有限材料参数试验数据的横观各向同性板结构随机有限元分析与验证

当材性试验数据有限时,为了研究各力学参数的离散性和不确定性对结构性能计算的影响,需要对材料参数采用随机变量建模并基于概率理论构建刚度矩阵的随机模型。为此,首先将随机弹性张量分解为一组基张量和由材料参数构成的随机系数的线性组合,以考虑刚度矩阵各分量间的统计相关性;并利用最大熵原理确定由上述随机系数组成的随机向量的概率密度函数。采用基于Metropolis-Hasting算法的马尔科夫链蒙特卡罗方法用于计算与之相关的概率模型的拉格朗日乘子,并通过Matlab生成材料参数的随机样本。最后采用蒙特卡罗随机有限元法对横观各向同性材料构成的板式结构在不同荷载下的力学行为进行了数值分析。以刨花板材料为典型案例,与试验结果对比,验证了本文方法的效果和实用性。     

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.      

边界约束刚度不确定的结构振动特征值

利用摄动法 ,将随机的微分方程和边界条件化为一系列的确定性微分方程和边界条件。运用有限元离散方法 ,推导了统计特征值的二阶摄动近似表达 ,用算例对本文方法进行了说明并和 Monte-Carlo模拟法结果进行了比较     

随机梁式结构静力损伤识别的一种改进方法

针对已有的损伤识别方法会出现损伤误识别的问题,本文在已有方法的基础上发展了一种随机梁式结构静力损伤识别的改进方法。假定静力荷载下梁式结构初始模型参数（如弹性模量和几何尺寸等）及测量误差为随机量,给出已有的基于随机有限元模型的梁式结构静力损伤识别方法,并进一步提出了一种改进方法。该方法通过设定损伤概率指标的阈值和反复迭代对结构损伤识别进行改进。数值算例和简支梁静力试验表明,考虑初始模型的不确定性以及静力响应测量误差,本文方法相较已有方法可以更有效地识别梁式结构的损伤。     

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...     

OPTIMUM DESIGN BASED ON RELIABILITY IN STOCHASTIC STRUCTURE SYSTEMS

IntroductionIn traditional optimum structural math model,the target function and the constraintfunction are all considered as the certain value.But it does not accord to the actual forcedenvironment condition and structural status.Thus the optimum results…     

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.     

随机参数天线结构的概率优化设计

研究了具有随机参数天线结构基于概率的优化设计问题。同时考虑了结构物理参数、几何尺寸和作用荷载等的随机性;应用有限元方法与结构可靠性理论对抛物面天线反射面精度的可靠性和结构单元强度、稳定性和可靠性进行了分析;在此基础上构建了基于概率的天线背架结构优化设计模型。对8m口径天线结构的优化结果表明:文中提出的天线结构分析和优化设计模型与方法是合理可行的。     

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．     

A computational strategy for the random response of assemblies of structures

This paper presents a dedicated approach to the calculation of the random response of assemblies with uncertain interface characteristics. The random response is constructed using a polynomial chaos expansion (PCE). A decomposition of the assemblies into substructures and interfaces is defined and associated with a dedicated computational strategy which leads to a local/global algorithm enabling the treatments of the substructure and of the interface problems to be uncoupled. Since the only uncertain parameters are those which appear in the interface equations, this approach results in a drastic reduction of the computational costs. This paper first presents the classical stochastic finite element strategy for this kind of problem, then details the proposed dedicated approach. The applications concern structures assembled with uncertain elastic bonded joints. The proposed approach is compared to the Monte Carlo method and to the stochastic finite element method.     

Eléments finis stochastiques en élasticité linéaireA stochastic finite element method in linear mechanics

The stochastic finite element method presented in this Note consists in representing in a probabilistic form the response of a linear mechanical system whose material properties and loading are random. Each input random variable is expanded into a Hermite polynomial series in standard normal random variables. The response (e.g., the nodal displacement vector) is expanded onto the so-called polynomial chaos. The coefficients of the expansion are obtained by a Galerkin-type method in the space of probability. To cite this article: B. Sudret et al., C. R. Mecanique 332 (2004).     

旋转弹性体的有限元线法分析

本文将有限元线法应用于一般荷载作用下的旋转弹性体的分析。文中将任意荷载沿环向展开为Ｆｏｕｒｉｅｒ级数，利用正交性，将问题转变为一系列旋转子午面上二维问题的叠加。文中对任一环向谐波建立了旋转面上的曲线曲边单元，导出了相应的常微分方程体系，并对由柱坐标引起的ｒ＝０处的奇异性问题做了完备合理的处理。文中给出了具有代表性的数值算例，用以展示本法的出色的表现     

渗流-应力耦合随机性分析方法与应用

基于摄动随机有限元法计算参数随机性对渗流-应力耦合问题的影响, 以渗透系数和弹性模量为基本随机量, 推导了渗流-应力耦合摄动随机有限元迭代格式, 在此基础上对现有计算渗流场和应力场的单场随机计算程序进行改造. 使用改造程序对某大坝渗流-应力耦合问题进行分析, 并与商用软件计算结果进行对比.结果表明, 两种方法计算结果基本一致, 揭示了工程实际中的流固耦合规律, 改造的程序还可以输出商用软件未涉及的随机特征.     

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。