基于改进统计矩点估计法和最大熵原理的结构整体可靠度分析

结构整体可靠度评估一直以来是结构可靠度领域研究的热点与难点。该文将结构整体可靠度分类,并给出其对应功能函数的统一描述;结合提出的有效维度两步分析法和共轭无迹变换法,发展了改进统计矩点估计法;结合最大熵原理和改进统计矩点估计法,提出了适用于两类结构整体可靠度的统一分析方法;通过2个数值算例对该文方法进行了验证。算例分析结果表明:同一精度水平下,该文方法的计算效率较传统的三变量降维近似统计矩点估计法高2.3倍~2.6倍;该文方法具有高的精度水平,其最大相对误差低于2%,适用于结构整体可靠度评估。     

简化三阶矩拟正态变换及其在结构可靠度分析中的应用

针对三阶矩拟正态变换理论公式系数形式复杂及现有相关系数的转换公式适用范围未明确的问题,通过对公式系数进行简化和对相关系数的讨论,提出了独立随机变量和相关随机变量的简化三阶矩拟正态变换模型,并给出了相关系数转换公式的简明适用范围。通过将提出的简化三阶矩拟正态变换模型与一阶可靠度分析方法 (FORM)结合,发展了随机变量分布未知条件下的可靠度分析方法,并采用数值算例验证了该方法的准确性和适用性。研究结果表明,所提出的简化三阶矩拟正态变换模型具有较高的准确性和适用性,能够与FORM分析方法结合,实现随机变量分布未知条件下的结构可靠度分析。     

复合随机振动系统的动力可靠性矩独立重要性测度及态相关参数解法

针对激励与结构参数同时存在不确定性的复合随机振动系统,由随机结构无条件动力可靠度表达式出发,利用条件概率密度函数解析变换给出衡量基本随机变量对动力可靠性影响的矩独立重要性测度指标。该指标可表征不确定性随机变量对动力可靠度响应量分布的平均影响程度,可全面反映随机变量对响应分布影响。基于状态依存参数模型,提出求解矩独立重要性测度指标的态相关参数(SDP)法。利用算例分析结构动力可靠性参数的矩独立重要性测度,并与直接Monte-Carlo法对比。所提方法可在保证计算精度同时大幅度提高计算效率,适用于分析复合随机振动系统非线性可靠性响应。     

概率信息不完全系统的统计矩估计方法

根据已知变量概率信息的不同,概率信息不完全系统可分为子类I、子类II和子类III。现有的统计矩点估计法可以方便地用于概率信息完全系统和概率信息不完全系统子类I,但是对可能出现的概率信息不完全系统子类II和子类III无能为力。为此,该文在重点研究子类III的等效相关系数求解方法的同时给出了子类II等效相关系数的简化方法,并发展了适用于一般概率信息不完全系统的广义Nataf变换;在此基础上,结合多变量函数的单变量降维近似模型,提出了概率信息不完全系统的统计矩估计方法,并讨论了参考点选择、变量排序等对计算效率的影响;最后,通过算例对建议方法进行了系统的验证。算例结果表明:该文建议的等效相关系数求解方法准确有效、变量排序策略切实可行,统计矩估计法具有广泛适用性且对于低阶矩具有较理想的精度。     

基于分解法的最大熵随机有限元法

摘要：提出了一种新的基于分解法的最大熵随机有限元方法,利用单变量分解将多维随机响应函数表述为单维随机响应函数的组合形式,从而将求解随机结构响应统计矩的多维积分表达式转化为单维积分式,对单维积分采用高斯-埃尔米特积分格式求解。在获得结构响应的统计矩之后,利用最大熵原理求得结构响应的概率密度函数解析表达式。该法不涉及求导运算,对于非线性随机问题非常适用。算例结果表明,本文方法具有较好的精度与计算效率。
     

非平稳地震作用下强非线性结构动力可靠度分析

针对结构-地震动双重不确定影响下强非线性结构地震可靠度难以精确求解的问题,基于极值分布理论,提出了一种非平稳地震作用下非线性结构地震可靠度分析的高效数值方法。首先采用随机函数的降维思想对非平稳地震动进行降维模拟,将谱表达方法模拟非平稳地震动时需要的上千个高维随机变量减少到2个基本随机变量,大大减少了概率空间的维数;然后提出了一种改进的伪相关性折减拉丁超立方抽样方法确定积分点,从而对结构非线性地震响应极值分布分数矩进行有效估计,最后采用分数阶矩最大熵原理获得结构非线性地震响应极值分布,实现了小失效概率水平下非线性结构地震可靠度的精确估计。数值算例表明,该方法通过300次的动力分析就能够对结构-地震动双重不确定影响下强非线性结构地震响应的极值分布进行估计,其能够在兼顾效率和计算精度时对小失效概率水平下结构的动力可靠度进行精确计算,与既有方法相比,其计算量仅为现在方法的1/5,该方法能够为结构的地震风险评估提供一种有效途径。     

经验Cressie-Read统计量与广义矩方法估计

本文利用经验Cressie-Read统计量和广义矩方法,研究了参数、分布函数和Lagrange乘子的有效估计问题,并得到了它们的渐近正态性,证明了经验Cressie-Read统计量具有渐近χ2分布。最后举例说明文中结果。     

单变量函数统计矩的点估计法性能比较

点估计法是随机系统响应量统计矩计算的方法之一,由于简单、高效而颇受关注,其中单变量函数的统计矩估计则是点估计法的基础.虽然各研究者对各自提出的点估计方法均进行了算例验证,但这些算例验证的普适性值得商榷.该文通过详细、系统的研究,对已有的单变量函数统计矩的点估计方法进行全面的影响因素分析和计算性能评价.大量的算例分析结果表明:1) 函数的非线性程度、随机变量的类型及其变异系数是点估计算法精度的主要影响因素,变量均值影响较小,且本质上是通过改变函数的非线性程度间接影响精度;2) Zhou & Nowak方法(5 个计算点)精度最优;3) 当函数非线性程度较强、变量变异系数较大时,各方法精度均不够理想,此时应慎用点估计法.     

Ⅰ型和Ⅱ型Dugdale模型解析元列式及其半解析有限元法

利用弹性平面扇形域哈密顿体系的方程,通过分离变量法及共轭辛本征函数向量展开法,推导了两个圆形奇异超级解析单元列式,这两个超级单元能够分别准确地描述Ⅰ型和Ⅱ型Dugdale模型平面裂纹尖端场。将该解析元与有限元相结合,构成半解析的有限元法,可求解任意几何形状和载荷的Ⅰ型或Ⅱ型裂纹基于Dugdale模型的裂纹尖端塑性区尺寸和裂纹尖端张开位移(CTOD)或裂纹尖端滑开位移(CTSD)的计算问题。对典型算例的计算结果表明方法简单有效,具有令人满意的精度。     

基于线性化Nataf变换的一次可靠度方法

首先引入等概率边缘变换的基本原理,证明了常用的Rackwitz-Fiessler变换是等概率边缘变换的一次近似形式,将当量正态化原理和线性变换相结合,提出了扩展的Rackwitz-Fiessler变换,并指出其存在的缺点。然后针对Nataf变换的非线性特征,提出了线性化Nataf变换,并将该变换与改进的HLRF算法相结合,给出了基于线性化Nataf变换和iHLRF算法的一次可靠度方法。将Nataf变换、线性化Nataf变换和扩展的Rackwitz-Fiessler变换通过算例进行了对比分析,结果表明:采用线性化Nataf变换的结构可靠度分析结果收敛于采用Nataf变换的计算结果,而采用扩展的Rackwitz-Fiessler变换的计算结果则有较大的误差。     

基于双变量降维模型和Kriging近似的统计矩点估计法

对复杂随机系统进行统计矩分析时,双变量降维近似模型一定程度上可以缓解"维数灾难"。但当系统维数较高时,双变量分量函数较多,计算量仍然较大。为此,该文将降维近似和Kriging代理模型有机结合起来,提出了一类高效、合理的改进点估计法。充分考虑函数逼近和数值积分中积分点的特点,提出了"米"字形的选点策略,并基于此发展了双变量分量函数的Kriging近似模型;将此近似模型用于原函数和矩函数的双变量降维近似模型中双变量分量函数的近似,分别建立了基于原函数近似和矩函数近似的统计矩改进点估计法;通过多个算例对该文提出方法进行了效率和精度的分析。算例分析结果表明:基于"米"字形选点策略的双变量分量函数的Kriging近似具有较高的精度;相比于已有的基于双变量降维近似模型的统计矩点估计法,建议方法仅需较少的结构分析即可达到与已有方法相当的精度,能更好地体现精度和效率的平衡。     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

涉及离散变量的函数统计矩点估计法

点估计法对于仅包含连续随机变量的函数和系统的随机分析具有原理简洁清晰、操作简单易行的优点,并可以直接给出除均值和标准差之外的其他低阶统计矩。然而,对于客观存在的或者是需处理为的涉及离散随机变量的系统,现有的点估计法无能为力。为解决这一问题,该文基于一般随机系统的形式解析解,导出了涉及离散变量函数和系统的统计矩估计的理论表达式;然后,将其与现有的点估计法相结合,给出了涉及离散变量的函数和系统的低阶矩估计的点估计法;最后,通过理论推导和算例分析两种方式验证了建议方法的合理性和有效性,且指出该方法对包含离散变量的一般工程随机系统分析的适用性。     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient approach for statistical moments estimation of structural response based on a novel adaptive hybrid dimension-reduction method

Statistical moments estimation is one of the main topics for the analysis of a stochastic system, but the balance among the accuracy, efficiency, and versatility for different methods of statistical moments estimation still remains a challenge. In this paper, a novel point estimate method (PEM) based on a new adaptive hybrid dimension-reduction method (AH-DRM) is proposed. Firstly, the adaptive cut-high-dimensional model representation (cut-HDMR) is briefly reviewed, and a novel AH-DRM is developed, where the high-order component functions of the adaptive cut-HDMR are further approximated by multiplicative forms of the low-order component functions. Secondly, a new point estimation method (PEM) based on the AH-DRM is proposed for statistical moments estimation. Finally, several examples are investigated to demonstrate the performance of the proposed PEM. The results show the proposed PEM has fairly high accuracy and good versatility for statistical moments estimation.     

A system reliability estimation method by the fourth moment saddle point approximation and copula functions

Despite many advances in the field of computational system reliability analysis, estimating the joint probability distribution of correlated non-normal state variables on the basis of incomplete statistical data brings great challenges for engineers. To avoid multidimensional integration, system reliability estimation usually requires the calculation of marginal failure probability and joint failure probability. The current article proposed an integrated approach for estimating system reliability on the basis of the high moment method, saddle point approximation, and copulas. First, the statistic moment estimation based on the stochastic perturbation theory is presented. Thereafter, by constructing CGF (concise cumulant generating function) for the state variable with its first four statistical moments, a fourth moment saddle point approximation method is established for the component reliability estimation. Second, the copula theory is briefly introduced and extensively utilized two-dimensional copulas are presented. The best fit copula for estimating the probability of system failure is selected according to the AIC (Akaike Information Criterion). Finally, the derived method is applied to three numerical examples for the sake of a comprehensive validation.     

A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

A new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S?N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.