Abstract
This paper proposes an efficient Kriging-based subset simulation (KSS) method for hybrid reliability analysis under random and interval variables (HRA-RI) with small failure probability. In this method, Kriging metamodel is employed to replace the true performance function, and it is smartly updated based on the samples in the first and last levels of subset simulation (SS). To achieve the smart update, a new update strategy is developed to search out samples located around the projection outlines on the limit-state surface. Meanwhile, the number of samples in each level of SS is adaptively adjusted according to the coefficients of variation of estimated failure probabilities. Besides, to quantify the Kriging metamodel uncertainty in the estimation of the upper and lower bounds of the small failure probability, two uncertainty functions are defined and the corresponding termination conditions are developed to control Kriging update. The performance of KSS is tested by four examples. Results indicate that KSS is accurate and efficient for HRA-RI with small failure probability.
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References
Alvarez DA, Uribe F, Hurtado JE (2018) Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory. Mech Syst Signal Process 100:782–801
An D, Choi JH (2012) Efficient reliability analysis based on Bayesian framework under input variable and metamodel uncertainties. Struct Multidiscip Optim 46(4):533–547
Apley DW, Liu J, Chen W (2006) Understanding the effects of model uncertainty in robust design with computer experiments. AMSE J Mech Des 128(4):945–958
Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16(4):263–277
Au SK, Wang Y (2014) Engineering risk assessment with subset simulation. Wiley, Hoboken
Balesdent M, Morio J, Marzat J (2013) Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct Saf 44:1–10
Basudhar A, Missoum S (2010) An improved adaptive sampling scheme for the construction of explicit boundaries. Struct Multidiscip Optim 42(4):517–529
Brevault L, Lacaze S, Balesdent M, Missoum S (2016) Reliability analysis in the presence of aleatory and epistemic uncertainties, application to the prediction of a launch vehicle fallout zone. AMSE J Mech Des 138(11):11401
Dai H, Xue G, Wang W (2014) An adaptive wavelet frame neural network method for efficient reliability analysis. Comput Aided Civ Inf Eng 29(10):801–814
Du X (2008) Unified uncertainty analysis by the first order reliability method. AMSE J Mech Des 130(9):091401
Du X, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. ASME J Mech Des 127(6):1068–1076
Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo Simulation. Struct Saf 33(2):145–154
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(2):232–240
Guo J, Du X (2009) Reliability sensitivity analysis with random and interval variables. Int J Numer Methods Eng 78:1585–1617
Jiang C, Han X, Liu W, Liu J, Zhang Z (2012) A hybrid reliability approach based on probability and interval for uncertain structures. AMSE J Mech Des 134(3):031001
Jiang C, Zhang Z, Han X, Liu J (2013) A novel evidence-theory-based reliability analysis method for structures with epistemic uncertainty. Comput Struct 129:1–12
Jiang C, Zheng J, Han X (2018) Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: a review. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-017-1864-4
Lelièvre N, Beaurepaire P, Mattrand C, Gayton N (2018) AK-MCSi: a Kriging-based method to deal with small failure probabilities and time-consuming models. Struct Saf 73:1–11
Li F, Luo Z, Rong J, Zhang N (2013) Interval multi-objective optimization of structures using adaptive kriging approximations. Comput Struct 119(4):68–84
Liu X, Kuang Z, Yin L, Hu L (2017) Structural reliability analysis based on probability and probability box hybrid model. Struct Saf 6:73–84
Lophaven SN, Nielsen HB, Sondergaard J (2002) DACE, a matlab Kriging toolbox, version 2.0. Tech. Rep. IMM-TR-2002-12; Technical University of Denmark; http://www2.imm.dtu.dk/hbn/dace/
Mourelatos ZP, Zhou J (2005) Reliability estimation with insufficient data based on possibility theory. AIAA J 43(8):1696–1705
Picheny V, Ginsbourger D, Roustant O, Haftka RT, Kim NH (2010) Adaptive designs of experiments for accurate approximation of a target region. AMSE J Mech Des 132(7):461–471
Sadoughi M, Li M, Hu C, Mackenzie C, Lee S, Eshghi AT (2018) A high-dimensional reliability analysis method for simulation-based design under uncertainty. AMSE J Mech Des 140(7):071401
Seong S, Hu C, Lee S (2017) Design under uncertainty for reliable power generation of piezoelectric energy harvester. J Intell Mater Syst Struct 28(17):2437–2449
Shan S, Wang GG (2006) Failure surface frontier for reliability assessment on expensive performance function. AMSE J Mech Des 128(6):1227–1235
Sun Z, 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 157:152–165
Wang ZQ, Wang PF (2014) A maximum confidence enhancement based sequential sampling scheme for simulation-based design. AMSE J Mech Des 136(2):021006
Wang L, Wang X, Wang R, Chen X (2016) Reliability-based design optimization under mixture of random, interval and convex uncertainties. Arch Appl Mech 86(7):1341–1367
Wang L, Wang X, Su H, Lin G (2017) Reliability estimation of fatigue crack growth prediction via limited measured data. Int J Mech Sci 121:44–57
Wang L, Xiong C, Wang X, Li Y, Xu M (2018a) Hybrid time-variant reliability estimation for active control structures under aleatory and epistemic uncertainties. J Sound Vib 419:469–492
Wang L, Xiong C, Yang Y (2018b) A novel methodology of reliability-based multidisciplinary design optimization under hybrid interval and fuzzy uncertainties. Comput Methods Appl Mech Eng 337:439–457
Wu J, Luo Z, Zhang Y, Zhang N, Chen L (2013) Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. Int J Numer Methods Eng 95(7):608–630
Xiao M, Gao L, Xiong H, Luo Z (2015) An efficient method for reliability analysis under epistemic uncertainty based on evidence theory and support vector regression. J Eng Des 26(10–12):1–25
Yang XF, Liu YS, Gao Y, Zhang YS, Gao ZZ (2015) An active learning kriging model for hybrid reliability analysis with both random and interval variables. Struct Multidiscip Optim 51(5):1003–1016
Zhang X, Huang H (2010) Sequential optimization and reliability assessment for multidisciplinary design optimization under aleatory and epistemic uncertainties. Struct Multidiscip Optim 40(1):165–175
Zhang SL, Zhu P, Chen W, Arendt P (2013) Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design. Struct Multidiscip Optim 47(1):63–76
Zhang Z, Jiang C, Wang GG, Han X (2015) First and second order approximate reliability analysis methods using evidence theory. Reliab Eng Syst Saf 137:40–49
Zhang JH, Xiao M, Gao L, Fu JJ (2018a) A novel projection outline based active learning method and its combination with Kriging metamodel for hybrid reliability analysis with random and interval variables. Comput Methods Appl Mech Eng 341:32–52
Zhang JH, Xiao M, Gao L, Qiu HB, Yang Z (2018b) An improved two-stage framework of evidence-based design optimization. Struct Multidiscip Optim 58(4):1673–1693
Zhu Z, Du X (2016) Reliability analysis with Monte Carlo simulation and dependent kriging predictions. AMSE J Mech Des 138(12):121403
Zuev KM, Beck JL, Au SK, Katafygiotis LS (2012) Bayesian post-processor and other enhancements of subset simulation for estimating failure probabilities in high dimensions. Comput Struct 92:283–296
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This research was financially supported by the National Natural Science Foundation of China [grant numbers 51675196 and 51721092] and the Program for HUST Academic Frontier Youth Team.
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Xiao, M., Zhang, J., Gao, L. et al. An efficient Kriging-based subset simulation method for hybrid reliability analysis under random and interval variables with small failure probability. Struct Multidisc Optim 59, 2077–2092 (2019). https://doi.org/10.1007/s00158-018-2176-z
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DOI: https://doi.org/10.1007/s00158-018-2176-z