# AK-SYSi: an improved adaptive Kriging model for system reliability analysis with multiple failure modes by a refined *U* learning function

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## Abstract

Due to multiple implicit limit state functions needed to be surrogated, adaptive Kriging model for system reliability analysis with multiple failure modes meets a big challenge in accuracy and efficiency. In order to improve the accuracy of adaptive Kriging meta-model in system reliability analysis, this paper mainly proposes an improved AK-SYS by using a refined *U* learning function. The improved AK-SYS updates the Kriging meta-model from the most easily identifiable failure mode among the multiple failure modes, and this strategy can avoid identifying the minimum mode or the maximum mode by the initial and the in-process Kriging meta-models and eliminate the corresponding inaccuracy propagating to the final result. By analyzing three case studies, the effectiveness and the accuracy of the proposed refined *U* learning function are verified.

## Keywords

System reliability analysis Refined*U*learning function Easily identifiable failure mode Independency of the initial Kriging meta-model

## Notes

### Funding information

This work was supported by the National Natural Science Foundation of China (Grant 51775439, 11602197) and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201708).

## References

- Ambartzumian R, Kiureghian AD, Ohanian B, Sukiasian H (1997) Multinormal probability by sequential conditioned importance sampling. In: Advances in safety and reliability, proceedings ESREL 92, vol 2, Lisbon, p. 1261–1268Google Scholar
- Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16(4):263–277CrossRefGoogle Scholar
- Au SK, Beck JL (2002) Important sampling in high dimensions. Struct Saf 25(2):139–163CrossRefGoogle Scholar
- Bichon BJ, Eldred MS, Swiler LP, Mahadevan S, McFarland JM (2008) Efficient global reliability analysis for nonlinear implicit performance function. AIAA J 46:2459–2568CrossRefGoogle Scholar
- Bichon B, McFarlan J, Mahadevan S (2011) Efficient surrogate models for reliability analysis of systems with multiple failure modes. Reliab Eng Syst Saf 96:1386–1395CrossRefGoogle Scholar
- Cadini F, Santos F, Zio E (2014) An improved adaptive Kriging-based importance technique for sampling multiple failure regions of low probability. Reliab Eng Syst Saf 131:109–117CrossRefGoogle Scholar
- Dai HZ, Wang W (2009) Application of low-discrepancy sampling method in structural reliability analysis. Struct Saf 32:55–64CrossRefGoogle Scholar
- Depina I, Le TMH, Fenton G, Eiksund G (2016) Reliability analysis with metamodel line sampling. Struct Saf 60:1–15CrossRefGoogle Scholar
- Ditlevsen O (1979) Narrow reliability bounds for structural systems. J Struct Mech 7:453–472CrossRefGoogle Scholar
- Du XP (2010) System reliability analysis with saddlepoint approximation. Struct Multidiscip Optim 42(2):193–208MathSciNetCrossRefGoogle 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 probability ties 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
- Gong C, Zhou W (2018) Importance sampling-based system reliability analysis of corroding pipelines considering multiple failure modes. Reliab Eng Syst Saf 169:199–208CrossRefGoogle Scholar
- Grooteman F (2008) Adaptive radial-based importance sampling method for structural reliability. Struct Saf 30:533–542CrossRefGoogle Scholar
- Hohenbichler M, Rackwitz R (1983) First-order concepts in system reliability. Struct Saf 1:177–188CrossRefGoogle Scholar
- Hu Z, Du XP (2015) Mixed efficient global optimization for time-dependent reliability analysis. J Mech Des ASME 137:051401 1–051401-9CrossRefGoogle Scholar
- Hu Z, Mahadevan S (2016) A single-loop kriging surrogate modeling for time-dependent reliability analysis. J Mech Des ASME 138(6):061406 1-061406-10CrossRefGoogle Scholar
- Huang XX, Chen JQ, Zhu HP (2016) Assessing small failure probabilities by AK-SS: an active learning method combining Kriging and subset simulation. Struct Saf 59:86–95CrossRefGoogle Scholar
- Li DQ, Yang ZY, Cao ZJ, Au SK, Phoon KK (2017) System reliability analysis of slope stability using generalized subset simulation. Appl Math Model 46:650–664MathSciNetCrossRefGoogle Scholar
- Lu ZZ, Song SF, Yue ZF, Wang J (2008) Reliability sensitivity method by line sampling. Struct Saf 30(6):517–532CrossRefGoogle Scholar
- Lv ZY, Lu ZZ, Wang P (2015) A new learning function for Kriging and its applications to solve reliability problems in engineering. Comput Math Appl 70:1182–1197MathSciNetCrossRefGoogle Scholar
- Miao F, Ghosn M (2011) Modified subset simulation method for reliability analysis of structural systems. Struct Saf 33:251–260CrossRefGoogle Scholar
- Pandey M (1998) An effective approximation to evaluate multinormal integrals. Struct Saf 20:51–67CrossRefGoogle Scholar
- Rabhi N, Guedri M, Hassis H, Bouhaddi N (2011) Structure dynamic reliability: a hybrid approach and robust meta-models. Mech Syst Signal Process 25(7):2313–2323CrossRefGoogle Scholar
- Sacks J, Schiller SB, Welch WJ (1989) Design for computer experiment. Technometrics 31(1):41–47MathSciNetCrossRefGoogle Scholar
- Sobol IM (1976) Uniformly distributed sequences with additional uniformity properties. USSR Comput Math Math Phys 16:236–242CrossRefGoogle Scholar
- Sobol IM (1998) On quasi-Monte Carlo integrations. Math Comput Simul 47:103–112MathSciNetCrossRefGoogle Scholar
- Sun ZL, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliab Eng Syst Saf 157:152–165CrossRefGoogle Scholar
- Tong C, Sun Z, Zhao Q, Wang Q, Wang S (2015) A hybrid algorithm for reliability analysis combining kriging and subset simulation importance sampling. J Mech Sci Technol 29(8):3183–3193CrossRefGoogle Scholar
- Yun WY, Lu ZZ, Jiang X (2018a) An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-018-1975-6
- Yun WY, Lu ZZ, Jiang X (2018b) A modified importance sampling method for structural reliability and its global reliability sensitivity analysis. Struct Multidiscip Optim 57:1625–1641MathSciNetCrossRefGoogle Scholar
- Zhang LG, Lu ZZ, Wang P (2015) Efficient structural reliability analysis method based on advanced Kriging model. Appl Math Model 39(2):781–793MathSciNetCrossRefGoogle Scholar