An adaptive polynomial chaos expansion for high-dimensional reliability analysis
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RESEARCH PAPER
An adaptive polynomial chaos expansion for high-dimensional reliability analysis Wanxin He 1 & Yan Zeng 1 & Gang Li 1 Received: 11 December 2019 / Revised: 16 March 2020 / Accepted: 30 March 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Efficiency is greatly concerned in reliability analysis community, especially for the problems with high-dimensional input random variables, because the computation cost of common reliability analysis methods may increase sharply with respect to the dimension of the problem. This paper proposes a novel meta-model based on the concepts of polynomial chaos expansion (PCE), dimension-reduction method (DRM), and information-theoretic entropy. Firstly, a PCE method based on DRM is developed to approximate the original function by a series of PCEs of univariate components. Compared with the PCE of the original function, the DRM-based PCE can reduce the computational cost. Before constructing the meta-model, a prior of the degree of the PCE is required, which determines the accuracy and efficiency of the PCE. However, the prior is usually determined by experience. According to the maximum entropy principle, this paper proposes an adaptive method for the selection of the polynomial chaos basis efficiently. With the adaptive PCE method based on DRM, a novel meta-model method is proposed, with which the reliability analysis can be achieved by Monte Carlo simulation efficiently. In order to verify the performance of the proposed method, three numerical examples and one structural dynamics engineering example are tested, with good accuracy and efficiency. Keywords Polynomial chaos expansion . Dimension-reduction method . Entropy . Structural reliability analysis
1 Introduction Due to the uncertainties frequently involved in industrial application, such as uncertainties of material, loads, and geometry, it has been well recognized that the assessment of structural safety based on probabilistic theory plays an important role in practical engineering (Du and Chen 2004; Youn et al. 2005). One of the common ways to assess structural safety is reliability analysis, which is usually modeled by the following mathematical formulation: Responsible Editor: Pingfeng Wang * Gang Li [email protected] Wanxin He [email protected] Yan Zeng [email protected] 1
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
g
t P f ¼ PrðgðxÞ < gt Þ ¼ ∫−∞ pðgÞdg ¼ ∫gðxÞ< gt f ðxÞdx
ð1Þ
where Pf is the failure probability, Pr(•) is the probability of an event, gt is the threshold for the definition of failure, x is the vector of input variables with the joint probability density function (PDF) of f(x), and g(x) is the interested response of input variables with the PDF of p(g). Based on Eq. (1), extensive methods have been developed to achieve the structural reliability analysis with good efficiency and/or accuracy, including the sampling-based methods
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