Accurate Calculation of Probability Density for Second‑order Chaos Polynomials and Its Application in Structural Reliability

LIU Teng; WENG Yeyao; ZHANG Xuanyi; ZHAO Yangang

doi:10.13409/j.cnki.jdpme.20230203001

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Accurate Calculation of Probability Density for Second‑order Chaos Polynomials and Its Application in Structural Reliability

更新时间：2025-09-25

- Accurate Calculation of Probability Density for Second‑order Chaos Polynomials and Its Application in Structural Reliability
- Journal of Disaster Prevention and Mitigation Engineering Vol. 44, Issue 1, Pages: 28-38(2024)
- 作者机构：
  
  北京工业大学城市建设学部，北京 100124
- 作者简介：
- 基金信息：
- DOI：10.13409/j.cnki.jdpme.20230203001
  CLC： U213.2
- Received：03 February 2023，
  
  Revised：2023-05-13，
  
  Published：15 February 2024
- 稿件说明：
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刘腾,翁叶耀,张玄一等.二阶混沌多项式的概率密度精确计算及其结构可靠度应用[J].防灾减灾工程学报,2024,44(01):28-38.

LIU Teng,WENG Yeyao,ZHANG Xuanyi,et al.Accurate Calculation of Probability Density for Second‑order Chaos Polynomials and Its Application in Structural Reliability[J].Journal of Disaster Prevention and Mitigation Engineering,2024,44(01):28-38.
刘腾,翁叶耀,张玄一等.二阶混沌多项式的概率密度精确计算及其结构可靠度应用[J].防灾减灾工程学报,2024,44(01):28-38. DOI： 10.13409/j.cnki.jdpme.20230203001.

LIU Teng,WENG Yeyao,ZHANG Xuanyi,et al.Accurate Calculation of Probability Density for Second‑order Chaos Polynomials and Its Application in Structural Reliability[J].Journal of Disaster Prevention and Mitigation Engineering,2024,44(01):28-38. DOI： 10.13409/j.cnki.jdpme.20230203001.

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。

Abstract

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 analysis 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 representation 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 transformation. The accuracy and applicability of the proposed method for structural reliability problems are verified through numerical examples. The results show that： the proposed method can provide the probability 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 exact 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 thresholds， 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.

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