有界不确定参数结构位移范围的区间摄动法

当结构参数具有误差或有界不确定性时,区间数学可以在不同知识不确定变量的概率分布的情况下定量地考察不确定参数对结构响应的影响。为计算出不确定结构参数对结构位移影响范围的上下界,文中提出了的两种区骚动国方法。     

Interval parameter perturbation method for evaluating the bounds on natural frequencies of structures with interval parameters

For structural parameters with uncertainties, interval mathematics can, in the case where the probabilistic distribution density of uncertain variables is unavailable, deal with the influence of uncertainties in structural parameters on the response of structures. In order to evaluate the region containing natural frequencies of structures with interval parameters, the interval parameter perturbation method is presented in this paper. The advantage of the present method is its computational efficiency in evaluating the region containing natural frequencies. A numerical example is used to illustrate the efficiency of the method proposed. The project is supported by National Youth Natural Science Foundation of China and National Outstanding Youth Science Foundation of China.     

随机和区间非齐次线性哈密顿系统的比较研究及其应用

随着计算机技术的飞速发展,更高效、更稳定和长时间模拟能力更强的数值算法需求迫切.哈密顿系统辛算法与传统算法相比在稳定性和长期模拟方面具有显著优越性.但动力系统中不可避免地存在大量不同程度的不确定性,动力学分析中需要考虑这些不确定性的影响以确保合理有效性. 然而,目前考虑参数不确定性的哈密顿系统响应分析的研究基础还比较薄弱. 为此,本文考虑随机和区间参数不确定性,对两种不确定性非齐次线性哈密顿系统分析计算结果进行了比较研究,从而突破了传统哈密顿系统的局限性, 并应用于结构动力响应评估中. 首先,针对确定性非齐次线性哈密顿系统, 提出了考虑确定性扰动的参数摄动法;在此基础上, 分别提出了随机、区间非齐次线性哈密顿系统的参数摄动法,得到了它们响应界限的数学表达; 随后,用数学理论推导得到了区间响应范围包含随机响应范围的相容性结论; 最后,两个数值算例在较小时间步长下验证了所提方法在结构动力响应中的可行性和有效性,体现了随机、区间哈密顿系统响应结果之间的包络关系,并在较大时间步长下与传统方法相比较凸显了哈密顿系统辛算法的数值计算优势、与蒙特卡洛模拟方法相比较验证了所提方法的精度.      

Exact bounds for the static response set of structures with uncertain-but-bounded parameters

The numerical estimation of the static displacement bounds of structures with uncertain-but-bounded parameters is considered in this paper. By representing each uncertain-but-bounded parameter as an interval number or vector, a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations, in which the coefficient matrix and the right-hand side term are, respectively, the interval matrix and the interval vector. In this study, we present two new simple mathematical proofs of the vertex solution theorem using Cramer’s rule for solving linear interval equations, different from the other proof methods, to find the upper and lower bounds on the set of solutions. The first takes advantage of optimization theory, while the second is based on interval extension. By means of a typical example considered first by Hansen, it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the Oettli–Prager criterion. Examples of a three-stepped beam and a 10-bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method.     

不确定非线性结构动力响应的区间分析方法

研究多自由度非线性不确定参数系统的动力响应问题. 以区间数学为基础,将不确定 性参数用区间进行定量化,借助一阶Taylor级数,给出了近似估计非线性振动系统动力响 应范围的区间分析方法. 从数学证明和数值算例两方面,将其与概率摄动有限元法进行了比 较,结果显示区间分析方法对不确定参数先验信息具有要求较少、精度较高的优点.     

区间参数结构的动力响应优化

讨论区间参数结构的动态响应问题的区间优化方法．利用摄动理论和函数区间扩张,将区间优化问题转化为近似的确定性优化问题．由于区间设汁变量的中值和不确定性半径均可取作优化参数,昕以可得到比确定性优化更多的优化信息．将该方法应j用于桁架结构,算例表明该方法是有效的．     

基于Taylor展式的不确定结构复特征值问题两种非概率方法比较研究

代替传统的处理不确定问题的概率统计方法,将利用区问数学和凸模型理论研究具有有界不确定参数的非比例阻尼结构复特征值所在区域问题．区间数学将有界不确定结构参数用超长方体即区问向量进行定量化,而凸模型理论则用椭球对有界不确定参数进行定量化．在不用知道不确定变量的概率统计特性的条件下,区间分析方法和凸模型理论都可以确定出有界不确定结构参数的非比例阻尼结构复特征值所在区域．通过数学证明和数值算例来说明,在凸模型理论中的椭球在由区间分析中的超长方体—区间向量来确定的条件下,由区间数学所确定出不确定结构复特征值实部和虚部的宽度要比凸模型所确定出的范围的宽度要小,而这正是工程技术人员所要求的结果。     

DYNAMIC OPTIMIZATION FOR UNCERTAIN STRUCTURES USING INTERVAL METHOD

An interval optimization method for the dynamic response of structures with interval parameters is presented. The matrices of structures with interval parameters are given. Combining the interval extension with the perturbation, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure. The numerical results show that the method is effective.     

计算不确定结构系统静态响应的一种可靠方法

不确定性广泛存在于工程结构分析和设计过程之中，不能简单地予以忽略。目前，概率方法、模糊方法和区间方法是不确定性建模的三种主要方法。本文把具有不确定性的结构材料参数、几何参数和所受外力用区间数描述，通过求解线性区间方程组准确地计算了结构静态响应。计算结果易于扩张是区间计算的一个主要缺陷，本文提出了一种有效避免这一问题的方法。该方法把区间函数的计算和区间线性方程组的求解转化为相应的全局优化问题，来确定解中的每个区间元素的边界值，并采用一种智能性算法(实数编码遗传算法)来求解这些全局优化问题。本文首先采用数学和结构分析算例对该方法的正确性和有效性进行了验证，然后把该方法与有限元方法相结合计算不确定结构系统的响应范围，并和求解同类问题的方法进行了比较。     

基于二阶摄动法求解区间参数结构动力响应

在处理区间参数结构动力响应问题时,现有的分析方法大多局限于一阶区间分析方法. 如果参数的不确定量稍大,采用一阶区间分析方法对结构动力响应范围进行估计可能会失效,所以需要考虑二阶区间分析方法.但是采用基于区间运算的二阶区间分析方法得到的结果将会对动力响应范围过分高估. 为了克服以上缺点,首先基于二阶摄动法得到结构动力响应广义函数. 然后通过求解此动力响应函数的最大和最小值,将结构动力响应区间的问题转化为序列低维箱型约束下的二次规划问题. 最后采用DC 算法(di erence of convex functionsalgorithm) 对这些箱型约束下的二次规划问题进行求解. 这样可以在不引入过多计算量的情况下,避免了对动力响应范围的过分估计. 通过数值算例,将该方法和其他区间分析方法进行比较,验证了该方法的有效性与精确性.      

含随机参数的多体系统动力学分析

基于Lagrange方程建立了含随机参数的多体系统的动力学 模型,利用广义坐标分离法将随机微分代数方程转化为随机纯微分方程,利用Newmark法进行数值解算. 应用随机因子法求解系 统随机响应的数字特征,获得统计意义下的解. 以旋转杆滑块系统为例,考虑系统中载荷、物理和几何参数的随机性,通过与Monte Carlo法结果的对比验证了文中方法的正确性和有效性. 计算结果表明,部分随机参数的分散性对多体系统动力响应的影响不可忽略,利用随机参数的动力学模型将能客观地反映出系统的动力学行为.     

改进的区间截断法及基于区间分析的非概率可靠性分析方法

从工程应用的角度出发 ,提出了用区间表示参数的不确定性时 ,线性系统的非概率可靠度指标 ,此可靠度指标在试验数据较少时比概率可靠度指标更合理 ;另外本文还提出了计算响应参量变化范围的区间截断法 ,研究了截断参数 t的选取范围 ,算例分析表明 ,当 t取为所有输入参数的最大相对变化量时 ,由区间截断法算得的结果近似等于精确解 ,而随着 t的增大 ,区间截断法的解逐渐平稳地趋近于由区间算术运算所求得的直接解。     

Interval analysis method and convex models for impulsive response of structures with uncertain-but-bounded external loads

Two non-probabilistic, set-theoretical methods for determining the maximum and minimum impulsive responses of structures to uncertain-but-bounded impulses are presented. They are, respectively, based on the theories of interval mathematics and convex models. The uncertain-but-bounded impulses are assumed to be a convex set, hyper-rectangle or ellipsoid. For the two non-probabilistic methods, less prior information is required about the uncertain nature of impulses than the probabilistic model. Comparisons between the interval analysis method and the convex model, which are developed as an anti-optimization problem of finding the least favorable impulsive response and the most favorable impulsive response, are made through mathematical analyses and numerical calculations. The results of this study indicate that under the condition of the interval vector being determined from an ellipsoid containing the uncertain impulses, the width of the impulsive responses predicted by the interval analysis method is larger than that by the convex model; under the condition of the ellipsoid being determined from an interval vector containing the uncertain impulses, the width of the interval impulsive responses obtained by the interval analysis method is smaller than that by the convex model.The project supported by the National Outstanding Youth Science Foundation of China (10425208), the National Natural Science Foundation of China and Institute of Engineering Physics of China (10376002) The English text was polished by Keren Wang.     

具有随机路径的振动传递路径系统的随机响应分析

基于一般概率摄动有限元法,解决了具有随机路径的振动传递路径系统的响应分析问题.应用Kronecker代数,矩阵微分理论,向量值和矩阵值函数的二阶矩技术,矩阵摄动理论和概率统计方法,提出了振动传递路径系统的随机响应分析方法,在考虑工程中的不确定因素以后,在时域内清晰地描述了振动传递路径的随机响应.     

复合材料层合板的摄动随机Cell-Based光滑有限元

随机性是实际工程结构的固有特性,如何更真实地描述含随机参数结构的随机响应及统计特性,对工程结构的可靠性设计具有非常重要的意义。本文基于Cell-Based光滑有限元,采用四边形单元,推导了基于一阶剪切变形理论的复合材料层合板的光滑有限元公式,降低了网格划分要求,适应不规则网格,并采用离散剪切间隙有效地消除了剪切自锁;结合摄动法和随机场理论,导出了复合材料层合板的摄动随机光滑有限元平衡方程,并给出了结构随机响应数字特征的计算公式,求解了材料属性含随机性的复合材料层合板的随机响应问题,数值算例结果表明了本方法的有效性和准确性。     

An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters

This paper proposes two interval analysis methods, called the first-order interval parameter perturbation method (FIPPM) and the modified interval parameter perturbation method (MIPPM), for use in exterior acoustic field prediction when there are uncertainties in both the material properties and the external load. Interval variables are used to quantitatively describe the uncertain parameters in the face of limited information. The conventional first-order Taylor expansion and perturbation terms are employed in the FIPPM, while the MIPPM introduces modified Taylor series to approximate the non-linear interval matrix and vector. The high-order terms of the Neumann expansion are retained to calculate the interval matrix inverse. A numerical example is given by comparing the results with a Monte Carlo simulation to demonstrate the feasibility and effectiveness of the proposed methods at evaluating the sound pressure ranges in an exterior acoustic field.     

Change-of-variable interval stochastic perturbation method for hybrid uncertain structural-acoustic systems with random and interval variables

The response analysis from hybrid uncertain structural-acoustic systems with random and interval variables (HUSAS) plays an important role in the optimal design of structural-acoustic systems. In this work, a hybrid uncertain numerical method known as the change-of-variable interval stochastic perturbation method (CVISPM) is proposed to predict the interval of the response probability density function and the response confidence interval of a HUSAS. This method is based on perturbation analysis and the change-of-variable technique. In the proposed method, the response of a HUSAS is approximated as a linear function of random variables using the stochastic perturbation analysis. According to the approximated linear relationships between the response and the random variables, the change-of-variable technique is introduced to calculate the response probability density function. Based on the response probability density function, the interval perturbation approach is used to predict the interval of the response probability density function and the response confidence interval. A numerical example of a shell structural-acoustic system with random and interval variables was employed to verify the effectiveness and precision of the proposed method.     

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.     

区间参数振动系统的动力优化

对具有区间参数的多自由度振动系统的不确定性优化问题，提出一种新的区间优化方法．利用泰勒展开和函数的区间扩张，将区间优化问题转化为近似的确定性优化问题．该方法应用于多自由度线性扭振系统，并把区间设计变量的中值和不确定性半径取作优化参数．算例表明该方法是有效的．