摘要
混沌多项式展开是一种广泛使用的方法,用于建立功能函数的代理模型,以方便对随机结构进行不确定性量化及可靠度分析。然而,在某些可靠度分析问题中经常需要混沌多项式展开模型的概率密度函数作为随机变量的完整表达,但在一般情况下难以准确计算混沌多项式展开模型的概率密度函数。为研究二阶混沌多项式展开模型的概率密度函数计算方法,通过正交变换消除模型中的交叉项,推导出二阶混沌多项式展开其特征函数的显式表达式,然后利用快速傅里叶变换求得二阶混沌多项式展开的概率密度函数,并通过数值算例验证了所提方法在结构可靠度应用中的准确性和适用性。研究结果表明:所提方法能获得二阶混沌多项式模型的概率密度函数与累积分布函数,计算结果与理论精确解吻合,获得的非中心卡方分布的累积分布函数尾部可与精确值在10^(-8)水平上保持一致,且适用于高维情形。同时,所提方法能高效准确地给出不同响应阈值下的结构失效概率,即使是在10^(-8)水平上的小失效概率情形。相较于前四阶矩方法,所提方法计算精度更高,对于输出变量具有强非高斯性的情况依然适用。此外,由于二阶混沌多项式展开模型代理强非线性功能函数存在一定误差,因此所提方法对于强非线性问题存在一定局限性。
Polynomial chaos expansion(PCE)is a widely used approach for establishing the surrogate model of a performance function for the convenience of uncertainty quantification and reliability analy-sis of a stochastic structure.However,it remains difficult to calculate the probability density function of the PCE accurately for general cases,though the probability density function,as a complete repre-sentation of a random variable,is often required in some reliability analysis problems.In order to study the calculation method of the probability density function for the second-order polynomial chaos expansion model,the cross terms in the model are eliminated through an orthogonal transformation.The explicit formulation of the characteristic function of the second-order PCE is derived and then the corresponding probability density function of the PCE is computed via inverse fast Fourier transforma-tion.The accuracy and applicability of the proposed method for structural reliability problems are veri-fied through numerical examples.The results show that:the proposed method can provide the proba-bility density function and cumulative distribution function of the second-order PCE model,and the calculation results are accurately consistent with the theoretical solution.The tails of the cumulative distribution function of the obtained non-centered chi-square distribution can be consistent with the ex-act value at the 10^(-8) level and are suitable for high-dimensional cases.At the same time,the proposed method can efficiently and accurately give the structural failure probability at different response thresh-olds,even in the small failure probability cases.Compared with the first four-moment-based method,the proposed method is of higher accuracy and is applicable to strongly non-Gaussian output variables.In addition,the proposed method has some limitations for strongly nonlinear problems due to the error in the second-order chaos polynomial expansion model to represent strongly nonlinear performance functions.The research results provide a theoretical basis for the uncertainty quantification of PCE models and their application in structural reliability analysis and optimization design.
作者
刘腾
翁叶耀
张玄一
赵衍刚
LIU Teng;WENG Yeyao;ZHANG Xuanyi;ZHAO Yangang(Faculty of Architecture,Civil and Transportation Engineering,Beijing University of Technology,Beijing 100124,China)
出处
《防灾减灾工程学报》
CSCD
北大核心
2024年第1期28-38,共11页
Journal of Disaster Prevention and Mitigation Engineering
基金
国家自然科学基金项目(51738001,51820105014,U1934217)资助。
关键词
结构可靠度
混沌多项式展开
特征函数
概率密度函数
傅里叶变换
structural reliability
polynomial chaos expansion
characteristic function
probability den-sity function
Fourier transformation
