# An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability

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## Abstract

A vital challenge problem of structural reliability analysis is how to estimate the small failure probability with a minimum number of model evaluations. The Adaptive Kriging combined with Importance Sampling method (AK-IS) which is developed from the adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is a viable method to deal with this challenge. The aim of this paper is to reduce the number of model evaluations of the existing AK-IS algorithm. Firstly, we use a contributive weight function to divide the candidate samples of model input variables generated in AK-IS. The candidate samples are used to select the best next sample to update the Kriging model in AK-IS. Secondly, select the best next sample only in the important area obtained according to the contributive weight value to failure probability to update the Kriging model until the stopping condition is satisfied. Thirdly, use the Kriging model constructed in the important area to predict the other area and update the important area by adding the point with the maximum contributive weight value in the area except the important area ceaselessly until the probability of the accurate identification on the limit state function’s signs (positive limit state value or negative limit state value) of all the importance sampling points satisfies a criterion. Finally, the updated Kriging model is used to estimate the failure probability especially for the small failure probability. The proposed method uses a thought from local to global in order to reduce the computational cost of AK-IS and simultaneously guarantees the accuracy of estimation. A non-linear oscillator system, a roof truss structure and a planar ten-bar structure are analyzed by the proposed method. The results demonstrate the efficiency and accuracy of the proposed method in structural reliability analysis especially for small failure probability.

## Keywords

Reliability analysis Contributive weight function Kriging Importance sampling Small failure probability## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 51,475,370, 51,775,439), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201708), and the Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1007) behind (Grant CX201708).

## References

- Au SK, Beck JL (1999) A new adaptive importance sampling scheme. Struct Saf 21:135–158CrossRefGoogle Scholar
- Au SK, Beck JL (2002) Importance sampling in high dimensions. Struct Saf 25:139–163CrossRefGoogle Scholar
- Balesdent M, Morio J, Marzat J (2013) Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct Saf 44:1–10CrossRefGoogle Scholar
- Bourinet JM, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33(6):343–353CrossRefGoogle Scholar
- Der Kiureghian A, Stefano M (1991) Efficient algorithm for second-order reliability analysis. J Eng Mech ASCE 117(2):2904–2923CrossRefGoogle Scholar
- Echard B, Gayton N, Lemaire M (2011) AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation. Struct Saf 33:145–154CrossRefGoogle Scholar
- Echard B, Gayton N, Lemaire M, Relun N (2013) A combined importance sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab Eng Syst Saf 111:232–240CrossRefGoogle Scholar
- Fauriat W, Gayton N (2014) AK-SYS: An adaptation of the AK-MCS method for system reliability. Reliab Eng Syst Saf 123:137–144CrossRefGoogle Scholar
- Grooteman F (2008) Adaptive radial-based importance sampling method for structural reliability. Struct Saf 30:533–542CrossRefGoogle Scholar
- Harbitz A (1986) An efficient sampling method for probability of failure calculation. Struct Saf 3:109–115CrossRefGoogle Scholar
- Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. J Eng Mech ASCE 100(1):111–121Google Scholar
- Hu Z, Mahadevan S (2016) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53:501–521MathSciNetCrossRefGoogle Scholar
- Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19(1):3–19CrossRefGoogle Scholar
- Melchers RE (1989) Importance sampling in structural system. Struct Saf 6:3–10CrossRefGoogle Scholar
- Papadrakakis M, Lagaros N (2001) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507CrossRefzbMATHGoogle Scholar
- Rajashekhar MR, Ellingwood BR (1993) A new look at the response surface approach for reliability analysis. Struct Saf 12(3):205–220CrossRefGoogle Scholar
- Rubinstein RY, Kroese DP (2016) Simulation and the Monte Carlo method. Wiley, HokobenCrossRefzbMATHGoogle Scholar
- Sacks J, Schiller SB, Welch WJ (1989) Design for computer experiment. Technometrics 31(1):41–47MathSciNetCrossRefGoogle Scholar
- Schueremans L, Van GD (2005) Benefit of splines and neural networks in simulation based structural reliability analysis. Struct Saf 37(3):246–261CrossRefGoogle Scholar
- Song H, Choi KK, Lee CI, Zhao L, Lamb D (2013) Adaptive virtual support vector machine for reliability analysis of high-dimensional problems. Struct Multidiscip Optim 47:479–491MathSciNetCrossRefzbMATHGoogle Scholar
- Xu L, Cheng GD (2003) Discussion on: moment methods for structural reliability. Struct Saf 25:193–199CrossRefGoogle Scholar
- Yun WY, Lu ZZ, Jiang X (2017) A modified importance sampling method for structural reliability and its global reliability sensitivity analysis. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-017-1832-z
- Zhao YG, Ono T (1999a) A general procedure for first/s-order reliability method (FORM/SORM). Struct Saf 21(2):95–112CrossRefGoogle Scholar
- Zhao YG, Ono T (1999b) New approximations for SORM: part 1. J Eng Mech ASCE 125(1):79–85CrossRefGoogle Scholar
- Zhao YG, Ono T (2001) Moment method for structural reliability. Struct Saf 23(1):47–75CrossRefGoogle Scholar
- Zhao YG, Ono T (2004) On the problems of the fourth moment method. Struct Saf 26(3):343–347CrossRefGoogle Scholar
- Zhao YG, Lu ZH, Ono T (2006) A simple-moment method for structural reliability. J Asian Archit Build Eng 5(1):129–136CrossRefGoogle Scholar
- Zhao HL, Yue ZF, Liu YS, Gao ZZ, Zhang YS (2015) An efficient reliability method combining adaptive importance sampling and Kriging metamodel. Appl Math Model 39:1853–1866MathSciNetCrossRefGoogle Scholar