An efficient method combining active learning kriging and Monte Carlo simulation for profust failure probability.

*(English)*Zbl 1452.62732Summary: For more and more complicated engineering structures, it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption. By combining active learning Kriging with Monte Carlo simulation (AK-MCS), an efficient method is proposed to estimate the profust failure probability. Firstly, the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption. This integral is further reorganized as a weighted sum of a series of classical failure probabilities by Gaussian quadrature, and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds. Secondly, MCS is used according to the probability input distribution to generate the sample pool, in which the active learning Kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds. An improved learning function is proposed by minimizing the U-learning function minima corresponding to all limit state functions, so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the Kriging model. Once the updating process of the Kriging model converges, all limit state functions can be identified by the Kriging model, and the profust failure probability can be estimated by using the Kriging model without any extra model evaluation. Several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability.

##### MSC:

62N05 | Reliability and life testing |

62N86 | Fuzziness, and survival analysis and censored data |

65C05 | Monte Carlo methods |

68T05 | Learning and adaptive systems in artificial intelligence |

##### Keywords:

profust failure probability; fuzzy state; active learning kriging; Monte Carlo simulation; Gaussian quadrature; U-learning function
Full Text:
DOI

##### References:

[1] | Cai, K. Y.; Wen, C. Y.; Zhang, M. L., Fuzzy reliability modelling of gracefully degradable computing systems, Reliab. Eng. Syst. Saf., 33, 1, 141-157 (1991) |

[2] | Cai, K. Y.; Wen, C. Y.; Zhang, M. L., Fuzzy states as a basis for a theory of fuzzy reliability, Microelectron. Reliab., 33, 15, 2253-2263 (1993) |

[3] | Cutello, V.; Montero, J.; Yanez, J., Structure functions with fuzzy states, Fuzzy Sets Syst., 83, 2, 189-202 (1996) |

[4] | Zhao, Z.; Quan, Q.; Cai, K. Y., A profust reliability based approach to prognostics and health management, IEEE Trans. Reliab., 63, 1, 26-41 (2014) |

[5] | Cai, K. Y.; Wen, C. Y.; Zhang, M. L., Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets Syst., 42, 2, 145-172 (1991) · Zbl 0743.90059 |

[6] | Cai, K. Y.; Wen, C. Y.; Zhang, M. L., Posbist reliability behavior of fault-tolerant systems, Microelectron. Reliab., 35, 1, 49-56 (1995) |

[7] | Fan, C. Q.; Lu, Z. Z.; Shi, Y., Time-dependent failure possibility analysis under consideration of fuzzy uncertainty, Fuzzy Sets Syst. (2019), in press · Zbl 1426.62383 |

[8] | Cai, K. Y., System failure engineering and fuzzy methodology: an introductory overview, Fuzzy Sets Syst., 83, 2, 113-133 (1996) |

[9] | Wang, J. Q.; Lu, Z. Z.; Shi, Y., Aircraft icing safety analysis method in presence of fuzzy inputs and fuzzy state, Aerosp. Sci. Technol., 82, 83, 172-184 (2018) |

[10] | Lu, H.; Shangguan, W. B.; Yu, D. J., A unified method and its application to brake instability analysis involving different types of epistemic uncertainties, Appl. Math. Model., 56, 158-171 (2018) · Zbl 07166680 |

[11] | Lu, H.; Shangguan, W. B.; Yu, D. J., An imprecise probability approach for squeal instability analysis based on evidence theory, J. Sound Vib., 387, 20, 96-113 (2017) |

[12] | Lu, H.; Shangguan, W. B.; Yu, D. J., A unified approach for squeal instability analysis of disc brakes with two types of random-fuzzy uncertainties, Mech. Syst. Signal Process., 93, 281-298 (2017) |

[13] | Lu, H.; Shangguan, W. B.; Yu, D. J., Uncertainties quantification of squeal instability under two fuzzy-interval cases, Fuzzy Sets Syst., 328, 70-82 (2017) |

[14] | Pandey, D.; Tyagi, S. K., Profust reliability of a gracefully degradable system, Fuzzy Sets Syst., 158, 7, 794-803 (2007) · Zbl 1116.90337 |

[15] | Cai, K. Y., Profust Reliability Theory, vol. 363, 87-134 (1996), Springer US |

[16] | Jensen, J. J., Efficient estimation of extreme non-linear roll motions using the first-order reliability method (FORM), J. Mar. Sci. Technol., 12, 4, 191-202 (2007) |

[17] | Koyluoglu, H. U.; Nielsen, S. R.K., New approximation for SORM integrals, Struct. Saf., 13, 235-246 (1994) |

[18] | Cadini, F.; Santos, F.; Zio, E., An improved adaptive Kriging-based importance technique for sampling multiple failure regions of low probability, Reliab. Eng. Syst. Saf., 131, 3, 109-117 (2014) |

[19] | Liu, J. S., Monte Carlo Strategies in Scientific Computing (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0991.65001 |

[20] | Melchers, R., Importance sampling in structural systems, Struct. Saf., 6, 1, 3-10 (1989) |

[21] | Grooteman, F., Adaptive radial-based importance sampling method for structural reliability, Struct. Saf., 30, 6, 533-542 (2008) |

[22] | Au, S. K.; Beck, J. L., Estimation of small failure probabilities in high dimensions by subset simulation, Probab. Eng. Mech., 16, 263-277 (2001) |

[23] | Echard, B.; Gayton, N.; Lemaire AK-MCS, M., An active learning reliability method combining Kriging and Monte Carlo simulation, Struct. Saf., 33, 2, 145-154 (2011) |

[24] | Kaymaz, R.; McMahon, C. A., A response surface method based on weighted regression for structural reliability analysis, Probab. Eng. Mech., 20, 11-17 (2005) |

[25] | Sudret, B., Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Syst. Saf., 93, 7, 964-979 (2008) |

[26] | Yun, W. Y.; Lu, Z. Z.; Jiang, X.; Zhang, L. G., Borgonovo moment independent global sensitivity analysis by Gaussian radial basis function meta-model, Appl. Math. Model., 54, 378-392 (2018) · Zbl 07166598 |

[27] | Amouzgar, K.; Stromberg, N., Radial basis functions as surrogate models with a priori bias in comparison with a posteriori bias, Struct. Multidiscip. Optim., 55, 4, 1453-1469 (2017) |

[28] | Cheng, K.; Lu, Z. Z.; Zhou, Y. C.; Shi, Y.; Wei, Y. H., Global sensitivity analysis using support vector regression, Appl. Math. Model., 49, 587-598 (2017) · Zbl 07163501 |

[29] | Cheng, K.; Lu, Z. Z., Sparse polynomial chaos expansion based on D-MORPH regression, Appl. Math. Comput., 323, 17-30 (2018) |

[30] | Cheng, K.; Lu, Z. Z., Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression, Comput. Struct., 194, 86-96 (2018) |

[31] | Shao, Q.; Younes, A.; Fahs, M.; Mara, T. A., Bayesian sparse polynomial chaos expansion for global sensitivity analysis, Comput. Methods Appl. Mech. Eng., 318, 474-496 (2017) |

[32] | Yun, W. Y.; Lu, Z. Z.; Jiang, X., AK-SYSi: an improved adaptive Kriging model for system reliability analysis with multiple failure modes by refined U learning function, Struct. Multidiscip. Optim., 59, 1, 263-278 (2019) |

[33] | Sacks, J.; Welch, W. J.; Mitchell, T. J.; Wynn, H. P., Design and analysis of computer experiments, Stat. Sci., 4, 409-423 (1989) · Zbl 0955.62619 |

[34] | Zhou, Y. C.; Lu, Z. Z.; Cheng, K.; Yun, W. Y., A Bayesian Monte Carlo-based method for efficient computation of global sensitivity indices, Mech. Syst. Signal Process., 117, 5, 498-516 (2019) |

[35] | Viana, F. A.; Simpson, T. W.; Balabanov, V.; Toropov, V., Special section on multidisciplinary design optimization: metamodeling in multidisciplinary design optimization: how far have we really come?, AIAA J., 52, 4, 670-690 (2014) |

[36] | Zhang, L. G.; Lu, Z. Z.; Wang, P., Efficient structural reliability analysis method based on advanced Kriging model, Appl. Math. Model., 39, 781-793 (2015) · Zbl 1432.90050 |

[37] | Bichon, B. J.; Eldred, M. S.; Swiler, L. P.; Mahadevan, S.; McFarland, J. M., Efficient global reliability analysis for nonlinear implicit performance functions, AIAA J., 46, 2459-2468 (2008) |

[38] | Suzuki, H., Fuzzy sets and membership functions, Fuzzy Sets Syst., 58, 2, 123-132 (1993) · Zbl 0789.94011 |

[39] | Inuiguchi, M.; Ichihashi, H.; Kume, Y., A solution algorithm for fuzzy linear programming with piecewise linear membership functions, Fuzzy Sets Syst., 34, 1, 15-31 (1990) · Zbl 0693.90064 |

[40] | Feng, K. X.; Lu, Z. Z.; Yun, W. Y., Aircraft icing severity analysis with hybrid parameters under considering epistemic uncertainty, AIAA J. (2019), in press |

[41] | Zhang, F.; Huang, Z.; Yao, H. J.; Zhai, W. H.; Gao, T. F., Icing severity forecast algorithm under both subjective and objective parameters uncertainties, Atmos. Environ., 128, 263-267 (2016) |

[42] | Ling, C. Y.; Lu, Z. Z.; Feng, K. X.; Sun, B., Efficient numerical simulation methods for estimating fuzzy failure probability based importance measure indices, Struct. Multidiscip. Optim. (2018) |

[43] | Feng, K. X.; Lu, Z. Z.; Pang, C.; Yun, W. Y., Efficient numerical algorithm of profust reliability analysis: an application to wing box structure, Aerosp. Sci. Technol., 80, 203-211 (2018) |

[44] | Liu, B. D., Uncertainty Theory (2002), Springer: Springer New York |

[45] | Liu, H. T.; Cai, J. F.; Ong, Y. S., An adaptive sampling approach for Kriging metamodeling by maximizing expected prediction error, Comput. Chem. Eng. (2017) |

[46] | Jones, D. R.; Schonlau, M.; Welch, W. J., Efficient Global Optimization of Expensive Black-Box Functions, vol. 13(4), 455-492 (1998), Kluwer Academic Publishers · Zbl 0917.90270 |

[47] | Schueremans, L.; Gemert, D. V., Benefit of splines and neural networks in simulation based structural reliability analysis, Struct. Saf., 27, 246-261 (2005) |

[48] | Xiao, S. N.; Lu, Z. Z., Structural reliability sensitivity analysis based on classification of model output, Aerosp. Sci. Technol., 71, 52-61 (2017) |

[49] | Shi, Y.; Lu, Z. Z.; Cheng, K.; Zhou, Y. C., Temporal and spatial multi-parameter dynamic reliability and global reliability sensitivity analysis based on the extreme value moments, Struct. Multidiscip. Optim., 56, 1, 117-129 (2017) |

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