Efficient computational method based on AK-MCS and Bayes formula for time-dependent failure probability function

  • Kaixuan Feng
  • Zhenzhou LuEmail author
  • Chunyan Ling
  • Wanying Yun
Research Paper


The time-dependent failure probability function (TDFPF) is defined as a function of the time-dependent failure probability (TDFP) varying with the design parameters and the service time, and it is useful in the reliability-based design optimization for the time-dependent problem. For the lack of method estimating TDFPF, the direct Monte Carlo simulation (DMCS) and an adaptive Kriging-MCS based on Bayes formula (shorten as AK-MCS-Bay) are developed to estimate TDFPF. The DMCS is time-consuming, but its convergent solution can be used as reference to validate other methods. In the AK-MCS-Bay, the TDFPF is primarily transformed into the estimation of the augmented TDFP and the conditional probability density function (PDF) of design parameters on the time-dependent failure event. Then, a single AK model is constructed to efficiently identify the failure samples in the MCS sample pool at different service times. By using these identified failure samples, the TDFPs under different service times can be estimated by the double-loop MCS without any extra model evaluations, and the conditional PDF of design parameters can be also acquired by the kernel density estimation method. The numerical and engineering examples indicate the efficiency and accuracy of the proposed method.


Time-dependent failure probability function Bayes formula Adaptive Kriging Kernel density estimation 


Funding information

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 51775439) and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant No. CX201934).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Kaixuan Feng
    • 1
  • Zhenzhou Lu
    • 1
    Email author
  • Chunyan Ling
    • 1
  • Wanying Yun
    • 1
  1. 1.School of AeronauticsNorthwestern Polytechnical UniversityXi’anChina

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