New collocation method for stochastic response surface reliability analyses
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ORIGINAL ARTICLE
New collocation method for stochastic response surface reliability analyses Peng Zeng1 · Tianbin Li1 · Yu Chen1 · Rafael Jimenez2 · Xianda Feng3 · Salvador Senent2 Received: 5 September 2018 / Accepted: 4 June 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract The stochastic response surface method (SRSM) is widely used in engineering reliability analyses due to its efficiency and accuracy. The selection of collocation points in the SRSM has great significance, as it may strongly affect the computed results. This paper investigates the performance of different selection strategies in SRSM, and proposes a new collocation method. First, two commonly used collocation methods—the regression-based collocation method and the linearly independent collocation method—are briefly reviewed; and their limitations in application to reliability analysis are discussed. Then, an improved collocation method that achieves a better tradeoff between efficiency and accuracy is proposed. Four examples are employed to test the performance of the proposed collocation method; and a comparative study is conducted to demonstrate its advantages with respect to some other existing collocation methods. Keywords Reliability analysis · Stochastic response surface method · Polynomial chaos expansion · Collocation point · Probability of failure List of symbols p Order of PCE X A vector of random variables in physical space U A vector of uncorrelated standard normal random variables y Random output of the model Γp(·) Multidimensional Hermite polynomials of order p n Number of random variables in PCE a A vector of unknown coefficients T Hermite polynomial information matrix T Transpose matrix operator Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00366-019-00793-2) contains supplementary material, which is available to authorized users. * Tianbin Li [email protected] 1
State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology), 1#, Dongsanlu, Erxianqiao, Chengdu 610059, Sichuan, China
2
Department of Ground Engineering, Technical University of Madrid, Madrid, Spain
3
School of Civil Engineering and Architecture, University of Jinan, Jinan, China
Na Number of unknown coefficients Pi Selected collocation point P′i Symmetric point of Pi with respect to the origin ζ Asymmetrical ratio of the selected collocation points Δ Relative error with respect to MCS or LHS Np Number of selected collocation points or limit state function evaluations COV Coefficient of variation Pf Probability of failure μ Mean value SD Standard deviation σt Applied support pressure at the tunnel face σc,partial Collapse pressure provided by partial collapse mechanism σc,global Collapse pressure provided by global collapse mechanism σc,max Maximum collapse pressure provided by partial collapse or global collapse c Cohesion φ Friction angle ρ Correlation coefficient
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