The stochastic Galerkin scaled boundary finite element method on random domain

The research work extends the scaled boundary finite element method to non‐deterministic framework defined on random domain wherein random behaviour is exhibited in the presence of the random‐field uncertainties. The aim is to blend the scaled boundary finite element method into the Galerkin spectral stochastic methods, which leads to a proficient procedure for handling the stress singularity problems and crack analysis. The Young's modulus of structures is considered to have random‐field uncertainty resulting in the stochastic behaviour of responses. Mathematical expressions and the solution procedure are derived to evaluate the statistical characteristics of responses (displacement, strain, and stress) and stress intensity factors of cracked structures. The feasibility and effectiveness of the presented method are demonstrated by particularly chosen numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Non-Deterministic Structural Response and Reliability Analysis Using a Hybrid Perturbation-Based Stochastic Finite Element and Quasi-Monte Carlo Method

The random interval response and probabilistic interval reliability of structures with a mixture of random and interval properties are studied in this paper. Structural stiffness matrix is a random interval matrix if some structural parameters and loads are modeled as random variables and the others are considered as interval variables. The perturbation-based stochastic finite element method and random interval moment method are employed to develop the expressions for the mean value and standard deviation of random interval structural displacement and stress responses. The lower bound and upper bound of the mean value and standard deviation of random interval structural responses are then determined by the quasi-Monte Carlo method. The structural reliability is not a deterministic value but an interval as the structural stress responses are random interval variables. Using a combination of the first order reliability method and interval approach, the lower and upper bounds of reliability for structural elements, series, parallel, parallel-series and series-parallel systems are investigated. Three numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method.     

区间桁架结构有限元分析的区间因子法

研究了具有区间参数的桁架结构在区间力作用下的有限元分析方法。利用区间因子法,桁架结构材料物理参数、几何尺寸和外荷载均可表达为其区间因子和其确定性量的乘积,进而结构的位移和应力响应也可表达成区间因子们的函数。利用区间算法,推导出了结构位移和应力响应的上、下限和均值的计算表达式。通过算例,分析了结构参数和外荷载的不确定性对结构响应的影响,并验证了模型和方法的合理性与可行性。该方法的优点是能够反映结构某一参数的不确定性对结构响应的影响。     

An interval random perturbation method for structural‐acoustic system with hybrid uncertain parameters

An interval random model is introduced for the response analysis of structural‐acoustic systems that lack sufficient information to construct the precise probability distributions of uncertain parameters. In the interval random model, the uncertain parameters are treated as random variables, whereas some distribution parameters of random variables with limited information are expressed as interval variables instead of precise values. On the basis of the interval random model, the interval random structural‐acoustic finite element equation is constructed, and an interval random perturbation method for solving this interval random equation is proposed. In the proposed method, the interval random matrix and vector are expanded by the first‐order Taylor series, and the response vector of the structural‐acoustic system is calculated by the matrix perturbation method. According to the linear monotonicity of the response vector, the lower and upper bounds of the response vector are calculated by the vertex method. On the basis of the lower and upper bounds, the intervals of expectation and standard variance of the response vector are obtained by the random interval moment method. The numerical results on a shell structural‐acoustic model and an automobile passenger compartment with flexible front panel demonstrate the effectiveness and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Interval Finite Element Analysis using Interval Factor Method

A new method called the interval factor method for the finite element analysis of truss structures with interval parameters is presented in this paper. The structural parameters and applied forces can be considered as interval variables by using the interval factor method, the structural stiffness matrix can then be divided into the product of two parts corresponding to the interval factors and the deterministic value. From the static governing equations of interval finite element method of structures, the structural displacement and stress responses are expressed as the functions of the interval factors. The computational expressions for lower and upper bounds, mean value and interval change ratio of structural static responses are derived by means of the interval operations. The effect of the uncertainty of the structural parameters and applied forces on the structural displacement and stress responses is demonstrated by truss structures.     

随机-区间型天线结构有限元及可靠性分析

构建了对随机-区间混合型天线结构的有限元及可靠性分析模型,提出了一种新的处理不确定性因素的结构有限元分析方法,给出了结构保精度和保强度两工况的概率描述。同时考虑了结构的物理参数、几何参数的随机性和作用风载荷的区间性。首先将随机变量固定,利用区间因子法求得结构位移和应力响应的区间范围,然后在区间内任意点处利用随机因子法求结构响应的随机分布范围。构造了天线反射面位移响应和结构单元应力响应不确定变量的数字特征计算公式,进而得到结构各响应量的可靠性指标。对一8m口径天线结构进行了分析,分析结果表明文中所提方法具有合理性和可行性。     

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces

An eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial differential equations defined on random domains. This method is based on a marriage between the eXtended Finite Element Method and spectral stochastic methods. In this article, we propose an extension of this method for the numerical simulation of random multi‐phased materials. The random geometry of material interfaces is described implicitly by using random level set functions. A fixed deterministic finite element mesh, which is not conforming to the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

不均匀随机参数桁架结构的随机反应分析

以随机荷载作用下的不均匀随机参数桁架结构为研究对象,提出了求解结构反应数字特征的矩法。从结构有限元方程出发,推导出结构刚度矩阵对各单元的弹性模量、横截面积以及各节点坐标的导数,进一步导得结构位移反应对各随机参数的敏度。利用随机变量函数的矩法,导出了结构位移反应的均值和方差,依据单元位移应力的转换表达式,分析了应力反应的均值和方差。算例表明,结构反应的方差取决于各随机参数的分散性和参数间的相关性。     

复合材料层合板非线性系统随机振动的样条有限元分析

工程结构中的复合材料的几何参数往往具有随机性质,如何研究随机参数非线性系统的随机响应及统计特性,对结构的可靠性设计和优化设计有着非常重要的意义。应用摄动法、随机中心差分法和线化和校正法,建立了复合材料非线性系统的振动方程和计算模型,采用样条有限元法研究了复合材料层合板具有随机参数的非线性系统在确定性荷载下的随机响应,数值算例说明了本算法的正确性。     

基于非概率集合理论的屋面板风致疲劳寿命估计

现有疲劳分析中,通常将结构材料参数、几何尺寸等定义为确定性参数;实际结构中,相关参数均为有界但不确定变量,如按确定性参数估计结构的疲劳寿命是偏于不安全的。该文将结构体系中不确定参数定义为区间变量,在线性疲劳损伤累积理论基础上,提出一种仅需一次动力响应分析即可计算不确定结构在动力荷载作用下疲劳损伤的新方法。该方法将金属屋面板弹性模量和屋面板板厚等由于施工误差等因素引起的不确定参数定义为区间变量,通过摄动法和区间动力响应分析,计算屋面板在脉动风荷载作用下的应力响应区间;结合屋面板材料的S-N曲线,采用修正Miner疲劳线性累积准则对屋面板的疲劳损伤和寿命区间进行估计。结果表明:该文方法可有效计算考虑结构参数不确定条件下金属屋面板的疲劳损伤和寿命区间;与顶点法比较,该文方法仅需一次动力响应分析就可计算金属屋面板风致疲劳损伤和寿命区间。     

Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen–Loève expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by k ?version/p ?version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We derive a priori error estimates, which are uniform with respect to the Lamè constant λ ∈(0,+∞ ). Finally, we provide some numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

钢筋砼结构非线性随机地震响应分析

考虑地震作用和结构参数的随机性,建立了钢筋砼结构药非线性随机动力学模型。文中导出了随机结构动力分析的非线性随机有限元法的增量列式,并据此对多层钢筋砼结构进行了弹塑性随机地震响应分析。计算结果与该建筑物的实际震害作了对比,效果良好。还讨论了动力模型中随机变量对响应量的影响。     

消能伸臂体系减震性能的参数变异性影响分析

为研究结构参数变异性对消能伸臂体系减震性能的影响,推导了该体系的有限元振动方程,同时提出地震响应简化分析的“Maxwell型阻尼器计算法”,并结合Gauss-Hermite降维算法分析了确定性激励、随机地震激励作用的参数变异影响情况。结果表明:Gauss-Hermite降维算法既能保证精度又能提高计算效率;确定性激励时,传统伸臂体系与消能伸臂体系的楼层响应标准差随变异系数的增大而增大,但后者的标准差更小;随机激励时,消能伸臂体系仍具有良好的减震性能,且对参数变异敏感性弱,鲁棒性更好。     

基于区间分析的不确定性结构动力学模型修正方法

提出了一种基于区间分析的不确定性有限元模型修正方法。在区间参数结构特征值分析理论和确定性有限元模型修正方法基础上,假设不确定性与初始有限元模型误差均较小,采用灵敏度方法推导了待修正参数区间中点值和不确定区间的迭代格式。以三自由度弹簧-质量系统和复合材料板为例,采用拉丁超立方抽样构造仿真试验模态参数样本,开展仿真研究。结果表明,当仿真试验样本能准确反映结构模态参数的区间特性时,方法的收敛精度和效率均较高;修正后计算模态参数能准确反映试验数据的区间特性。所提出方法适用于解决试验样本较少,仅能得到试验模态参数区间的有限元模型修正问题。     

随机结构重特征值分析的递推随机有限元法

利用递推随机有限元方法研究了具有随机参数结构的重特征值问题。采用随机收敛的非正交多项式展式表示未知的随机重特征值和随机特征向量，建立了和摄动法类似的一系列确定的递推方程，通过求解这些速推方程，得到了重特征值的统计值。算例表明，同基于二阶泰勒展开的摄动随机有限元法相比，递推随机有限元法的结果能在较宽的随机涨落范围内更好地逼近蒙特卡洛模拟结果。     

A perturbation method for stochastic meshless analysis in elastostatics

A stochastic meshless method is presented for solving boundary‐value problems in linear elasticity that involves random material properties. The material property was modelled as a homogeneous random field. A meshless formulation was developed to predict stochastic structural response. Unlike the finite element method, the meshless method requires no structured mesh, since only a scattered set of nodal points is required in the domain of interest. There is no need for fixed connectivities between nodes. In conjunction with the meshless equations, classical perturbation expansions were derived to predict second‐moment characteristics of response. Numerical examples based on one‐ and two‐dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and Monte Carlo simulation. Since mesh generation of complex structures can be a far more time‐consuming and costly effort than the solution of a discrete set of equations, the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Comparison of static response of structures using convex models and interval analysis method

In this paper, by combining the finite element analysis and non‐probabilistic convex models, we present the numerical algorithm of non‐probabilistic convex models and interval analysis method for the static displacement of structures with uncertain‐but‐bounded parameters. Under the condition of the box or interval vector determined from the ellipsoid of the uncertain‐but‐bounded structural parameter vector, by comparing the numerical algorithm of non‐probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example, we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non‐probabilistic convex models is tighter than those produced by the interval analysis method. Copyright © 2003 John Wiley & Sons, Ltd.     

Supersensitivity due to uncertain boundary conditions

We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi‐polynomial‐chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth‐order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright © 2004 John Wiley & Sons, Ltd.     

随机有限元及其进展──Ⅰ．随机场的离散和反应矩的计算

本文讨论了随机有限元方法近二十年的进展。全文分为独立的两篇，本文为第一篇，其中讨论了随机场的离散，改进的模拟法，随机有限元方程的建立及随机反应各阶矩的计算．