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Subset simulation method including fitness-based seed selection for reliability analysis

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Abstract

Probability estimation of rare events is a challenging task in the reliability theory. Subset simulation (SS) is a robust simulation technique that transforms a rare event into a sequence of multiple intermediate failure events with large probabilities and efficiently approximates the mentioned probability. Proper handling of a reliability problem by this method requires employing a suitable sampling approach to transmit samples toward the failure set. Markov Chain Monte Carlo (MCMC) is a suitable sampling approach that solves the SS transition phase using the failed sample of each simulation level as the seed of next samples. This paper is aimed to study the seed selection effect on the SS accuracy through several seed selection approaches inspired by the genetic algorithm and particle filter and using the main PDF of the variables to assign a mass function probability to each subset sample in the failure domain. Roulette wheel (I, II), tournament and proportional probability techniques are then employed to choose the weighed samples as seeds to be placed in the MCMC to transmit the samples. To examine the capability of each approach, reliabilities of some engineering problems were investigated and results showed that the proposed approaches could find proper failure sets better than the original SS method, especially in problems with several failure domains.

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References

  1. 1.

    Zeng P, Li T, Chen Y et al (2019) New collocation method for stochastic response surface reliability analyses. Eng Comput. https://doi.org/10.1007/s00366-019-00793-2

  2. 2.

    Ghohani Arab H, Rashki M, Rostamian M et al (2019) Refined first-order reliability method using cross-entropy optimization method. Eng Comput 35:1507–1519. https://doi.org/10.1007/s00366-018-0680-9

  3. 3.

    Melchers RE (1999) Structural reliability analysis and prediction. John, Chichester

  4. 4.

    Koutsourelakis P-S, Pradlwarter HJ, Schueller GI (2004) Reliability of structures in high dimensions, part I: algorithms and applications. Probab Eng Mech 19:409–417. https://doi.org/10.1016/j.probengmech.2004.05.001

  5. 5.

    Au S-K, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16:263–277. https://doi.org/10.1016/S0266-8920(01)00019-4

  6. 6.

    Katafygiotis LS, Cheung JSH (2002) A new efficient MCMC based simulation methodology for reliability calculations. In: Proceedings of the fifth world congress on computational mechanics

  7. 7.

    Guan XL, Melchers RE (2001) Effect of response surface parameter variation on structural reliability estimates. Struct Saf 23:429–444. https://doi.org/10.1016/S0167-4730(02)00013-9

  8. 8.

    Katafygiotis LS, Cheung SH (2007) Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study. Struct Saf 29:194–207. https://doi.org/10.1016/j.strusafe.2006.07.003

  9. 9.

    Tee KF, Khan LR, Li H (2014) Application of subset simulation in reliability estimation of underground pipelines. Reliab Eng Syst Saf 130:125–131. https://doi.org/10.1016/j.ress.2014.05.006

  10. 10.

    Phoon K-K (2008) Reliability-based design in geotechnical engineering: computations and applications. CRC Press, Boca Raton

  11. 11.

    Santoso AM, Phoon K-K, Quek S-T (2011) Effects of soil spatial variability on rainfall-induced landslides. Comput Struct 89:893–900. https://doi.org/10.1016/j.compstruc.2011.02.016

  12. 12.

    Li D-Q, Xiao T, Cao Z-J et al (2016) Enhancement of random finite element method in reliability analysis and risk assessment of soil slopes using subset simulation. Landslides 13:293–303. https://doi.org/10.1007/s10346-015-0569-2

  13. 13.

    Pellissetti MF, Schuëller GI, Pradlwarter HJ et al (2006) Reliability analysis of spacecraft structures under static and dynamic loading. Comput Struct 84:1313–1325. https://doi.org/10.1016/j.compstruc.2006.03.009

  14. 14.

    Thunnissen DP, Au SK, Swenka ER (2007) Uncertainty quantification in conceptual design via an advanced Monte Carlo method. J Aerosp Comput Inf Commun 4:902–917. https://doi.org/10.2514/1.28307

  15. 15.

    Thunnissen DP, Au SK, Tsuyuki GT (2007) Uncertainty quantification in estimating critical spacecraft component temperatures. J Thermophys Heat Transf 21:422–430. https://doi.org/10.2514/1.23979

  16. 16.

    Au SK, Beck JL (2000) Subset simulation: a new approach to calculating small failure probabilities. In: Proceedings of the international conference on Monte Carlo simulation, pp 287–293

  17. 17.

    Au SK, Beck JL (2000) Calculation of first excursion probabilities by subset simulation. In: Proceedings of the 8th ASCE conference on probabilistic mechanics and structural reliability, p 101

  18. 18.

    Au SK, Beck JL (2003) Subset simulation and its application to seismic risk based on dynamic analysis. J Eng Mech 129:901–917. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:8(901)

  19. 19.

    Guédé Z, Tantar A, Tantar E, Del Moral P (2012) Application of a particle filter-based subset simulation method to the reliability assessment of a marine structure. In: Structures, safety and reliability, vol 2. American Society of Mechanical Engineers, pp 203–210

  20. 20.

    Cadini F, Avram D, Pedroni N, Zio E (2012) Subset Simulation of a reliability model for radioactive waste repository performance assessment. Reliab Eng Syst Saf 100:75–83

  21. 21.

    Jensen HA, Jerez DJ (2018) A stochastic framework for reliability and sensitivity analysis of large scale water distribution networks. Reliab Eng Syst Saf 176:80–92. https://doi.org/10.1016/j.ress.2018.04.001

  22. 22.

    Li H-S, Cao Z-J (2016) Matlab codes of subset simulation for reliability analysis and structural optimization. Struct Multidiscip Optim 54:391–410. https://doi.org/10.1007/s00158-016-1414-5

  23. 23.

    Li H-S, Ma Y-Z (2015) Discrete optimum design for truss structures by subset simulation algorithm. J Aerosp Eng 28:04014091. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000411

  24. 24.

    Song S, Lu Z, Qiao H (2009) Subset simulation for structural reliability sensitivity analysis. Reliab Eng Syst Saf 94:658–665. https://doi.org/10.1016/j.ress.2008.07.006

  25. 25.

    Ahmed A (2012) Simplified and advanced approaches for the probabilistic analysis of shallow foundations. Dissertation, University of Nantes

  26. 26.

    Papadopoulos V, Giovanis DG, Lagaros ND, Papadrakakis M (2012) Accelerated subset simulation with neural networks for reliability analysis. Comput Methods Appl Mech Eng 223–224:70–80. https://doi.org/10.1016/j.cma.2012.02.013

  27. 27.

    Zuev KM, Beck JL, Au S-K, Katafygiotis LS (2012) Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions. Comput Struct 92–93:283–296. https://doi.org/10.1016/j.compstruc.2011.10.017

  28. 28.

    Papaioannou I, Betz W, Zwirglmaier K, Straub D (2015) MCMC algorithms for subset simulation. Probab Eng Mech 41:89–103. https://doi.org/10.1016/j.probengmech.2015.06.006

  29. 29.

    Ullmann E, Papaioannou I (2015) Multilevel estimation of rare events. SIAM/ASA J Uncertain Quantif 3:922–953. https://doi.org/10.1137/140992953

  30. 30.

    Au SK, Ching J, Beck JL (2007) Application of subset simulation methods to reliability benchmark problems. Struct Saf 29:183–193. https://doi.org/10.1016/j.strusafe.2006.07.008

  31. 31.

    Bourinet J-M, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33:343–353. https://doi.org/10.1016/j.strusafe.2011.06.001

  32. 32.

    Dubourg V, Sudret B, Bourinet J-M (2011) Reliability-based design optimization using kriging surrogates and subset simulation. Struct Multidiscip Optim 44:673–690. https://doi.org/10.1007/s00158-011-0653-8

  33. 33.

    Li L, Bect J, Vazquez E (2012) Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure. arXiv Prepr arXiv12071963

  34. 34.

    Chiachio M, Beck JL, Chiachio J, Rus G (2014) Approximate Bayesian computation by subset simulation. SIAM J Sci Comput 36:A1339–A1358. https://doi.org/10.1137/130932831

  35. 35.

    Hsu W-C, Ching J (2010) Evaluating small failure probabilities of multiple limit states by parallel subset simulation. Probab Eng Mech 25:291–304. https://doi.org/10.1016/j.probengmech.2010.01.003

  36. 36.

    Rashki M (2018) Hybrid control variates-based simulation method for structural reliability analysis of some problems with low failure probability. Appl Math Model 60:220–234. https://doi.org/10.1016/j.apm.2018.03.009

  37. 37.

    Li H-S, Ma Y-Z, Cao Z (2015) A generalized Subset Simulation approach for estimating small failure probabilities of multiple stochastic responses. Comput Struct 153:239–251. https://doi.org/10.1016/j.compstruc.2014.10.014

  38. 38.

    Zio E, Pedroni N (2009) Estimation of the functional failure probability of a thermal–hydraulic passive system by subset simulation. Nucl Eng Des 239:580–599. https://doi.org/10.1016/j.nucengdes.2008.11.005

  39. 39.

    Metropolis N, Rosenbluth AW, Rosenbluth MN et al (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092. https://doi.org/10.1063/1.1699114

  40. 40.

    Robert CP, Casella G (2004) Monte Carlo optimization. In: Monte Carlo statistical methods. Springer texts in statistics, 3rd edn. Springer, New York, pp 157–204

  41. 41.

    Liu JS (2001) Monte Carlo strategies in statistical computing. Springer, Berlin

  42. 42.

    Hastions WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109

  43. 43.

    Breitung K (2019) The geometry of limit state function graphs and subset simulation: counterexamples. Reliab Eng Syst Saf 182:98–106. https://doi.org/10.1016/j.ress.2018.10.008

  44. 44.

    Rashki M, Miri M, Azhdary Moghaddam M (2012) A new efficient simulation method to approximate the probability of failure and most probable point. Struct Saf 39:22–29. https://doi.org/10.1016/j.strusafe.2012.06.003

  45. 45.

    Jahani E, Shayanfar MA, Barkhordari MA (2013) Structural reliability based on genetic algorithm-Monte Carlo (GAMC). Adv Struct Eng 16:419–426

  46. 46.

    Cheng H, Shuku T, Thoeni K, Yamamoto H (2017) Calibration of micromechanical parameters for DEM simulations by using the particle filter. EPJ Web Conf 140:12011. https://doi.org/10.1051/epjconf/201714012011

  47. 47.

    Goldberg DE, Deb K (1991) A comparative analysis of selection schemes used in genetic algorithms. In: Spatz B M (ed) Foundations of genetic algorithms. Elsevier, Amsterdam, pp 69–93

  48. 48.

    Bect J, Li L, Vazquez E (2017) Bayesian subset simulation. SIAM/ASA J Uncertain Quantif 5:762–786. https://doi.org/10.1137/16M1078276

  49. 49.

    Gayton N, Bourinet JM, Lemaire M (2003) CQ2RS: a new statistical approach to the response surface method for reliability analysis. Struct Saf 25:99–121. https://doi.org/10.1016/S0167-4730(02)00045-0

  50. 50.

    Bijlaard PP (1957) Buckling under external pressure of cylindrical shells evenly stiffened by rings only. J Aeronaut Sci 24:437–447. https://doi.org/10.2514/8.3874

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Correspondence to Mehdi Azhdary Moghaddam.

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Abdollahi, A., Azhdary Moghaddam, M., Hashemi Monfared, S.A. et al. Subset simulation method including fitness-based seed selection for reliability analysis. Engineering with Computers (2020). https://doi.org/10.1007/s00366-020-00961-9

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Keywords

  • Reliability analysis
  • Subset simulation
  • Probability mass function
  • Seed selection