Structural and Multidisciplinary Optimization

, Volume 57, Issue 4, pp 1625–1641 | Cite as

A modified importance sampling method for structural reliability and its global reliability sensitivity analysis

RESEARCH PAPER
First Online:
Received:
Revised:
Accepted:

Abstract

The importance sampling method is an extensively used numerical simulation method in reliability analysis. In this paper, a modification to the importance sampling method (ISM) is proposed, and the modified ISM divides the sample set of input variables into different subsets based on the contributive weight of the importance sample defined in this paper and the maximum super-sphere denoted by β-sphere in the safe domain defined by the truncated ISM. By this proposed modification, only samples with large contributive weight and locating outside of the β-sphere need to call the limit state function. This amelioration remarkably reduces the number of limit state function evaluations required in the simulation procedure, and it doesn’t sacrifice the precision of the results by controlling the level of relative error. Based on this modified ISM and the space-partition idea in variance-based sensitivity analysis, the global reliability sensitivity indices can be estimated as byproducts, which is especially useful for reliability-based design optimization. This process of estimating the global reliability sensitivity indices only needs the sample points used in reliability analysis and is independent of the dimensionality of input variables. A roof truss structure and a composite cantilever beam structure are analyzed by the modified ISM. The results demonstrate the efficiency, accuracy, and robustness of the proposed method.

Keywords

Reliability analysis Sensitivity analysis Contributive weight Modification Importance sampling Sample subset Sample reduction 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant 51475370, 51775439) and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201708).

References

  1. Au SK, Beck JL (1999) A new adaptive importance sampling scheme. Struct Saf 21:135–158CrossRefGoogle Scholar
  2. Au SK, Beck JL (2002) Importance sampling in high dimensions. Struct Saf 25:139–163CrossRefGoogle Scholar
  3. 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
  4. Cui LJ, Lu ZZ, Zhao XP (2010) Moment-independent importance measure of basic random variable and its probability density evolution solution. SCIENCE CHINA Technol Sci 53(4):1138–1145CrossRefMATHGoogle Scholar
  5. De Der Kiureghian A (1991) stefano M. Efficient algorithm for second-order reliability analysis. J Eng Mech ASCE 117(2):2904–2923CrossRefGoogle Scholar
  6. 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
  7. Grooteman F (2008) Adaptive radial-based importance sampling method for structural reliability. Struct Saf 30:533–542CrossRefGoogle Scholar
  8. Harbitz A (1986) An efficient sampling method for probability of failure calculation. Struct Saf 3:109–115CrossRefGoogle Scholar
  9. Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. J Eng Mech ASCE 100(1):111–121Google Scholar
  10. Hu Z, Mahadevan S (2016) Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis. Struct Multidiscip Optim 53:501–521MathSciNetCrossRefGoogle Scholar
  11. Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19(1):3–19CrossRefGoogle Scholar
  12. Li CZ, Mahadevan S (2016) An efficient modularized sample-based method to estimate the first-order Sobol' index. Reliab Eng Syst Saf 153:110–121CrossRefGoogle Scholar
  13. Li LY, Lu ZZ, Feng J, Wang BT (2012) Moment-independent importance measure of basic variable and its state dependent parameter solution. Struct Saf 38:40–47CrossRefGoogle Scholar
  14. Lu ZZ, Feng YS (1995) An importance sampling function for computing failure probability for a limit state equation with multiple design point. Acta Aeronautica et Astronautica Sinica 16(4):484–487 (in Chinese)Google Scholar
  15. Melchers RE (1989) Importance sampling in structural system. Struct Saf 6:3–10CrossRefGoogle Scholar
  16. Papadrakakis M, Lagaros N (2001) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507CrossRefMATHGoogle Scholar
  17. Rashki M, Miri M, Moghaddam MA (2012) A new efficient simulation method to approximate the probability of failure and most probable point. Struct Saf 39:22–29CrossRefGoogle Scholar
  18. Sobol IM (1976) Uniformly distributed sequences with additional uniformity properties. USSR Comput Math Math Phys 16:236–242CrossRefMATHGoogle Scholar
  19. Sobol IM (1998) On quasi-Monte Carlo integrations. Math Comput Simul 47:103–112MathSciNetCrossRefGoogle Scholar
  20. 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 47:479–491MathSciNetCrossRefMATHGoogle Scholar
  21. Wei P, Lu ZZ, Hao WR, Feng J, Wang BT (2012) Efficient sampling methods for global reliability sensitivity analysis. Comput Phys Commun 183:1728–1743MathSciNetCrossRefMATHGoogle Scholar
  22. Wei PF, Lu ZZ, Song JW (2014) Extended Monte Carlo simulation for parametric global sensitivity analysis and optimization. AIAA J 52(4):867–878CrossRefGoogle Scholar
  23. Xu L, Cheng GD (2003) Discussion on: moment methods for structural reliability. Struct Saf 25:193–199CrossRefGoogle Scholar
  24. Zhai QQ, Yang J, Zhao Y (2014) Space-partition method for the variance-based sensitivity analysis: Optimal partition scheme and comparative study. Reliab Eng Syst Saf 131:66–82CrossRefGoogle Scholar
  25. Zhang XF, Pandey MD (2013) Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct Saf 43:28–40CrossRefGoogle Scholar
  26. Zhang XF, Pandey MD, Zhang YM (2014) Computationally efficient reliability analysis of mechanisms based on a multiplicative dimensional reduction method. J Mech Des ASME 136(6):061006CrossRefGoogle Scholar
  27. Zhao YG, Ono T (1999a) A general procedure for first/second-order reliability method (FORM/SORM). Struct Saf 21(2):95–112CrossRefGoogle Scholar
  28. Zhao YG, Ono T (1999b) New approximations for SORM: part 1. J Eng Mech ASCE 125(1):79–85CrossRefGoogle Scholar
  29. Zhao YG, Ono T (2001) Moment method for structural reliability. Struct Saf 23(1):47–75CrossRefGoogle Scholar
  30. Zhao YG, Ono T (2004) On the problems of the fourth moment method. Struct Saf 26(3):343–347CrossRefGoogle Scholar
  31. Zhao YG, Lu ZH, Ono T (2006) A simple-moment method for structural reliability. J Asian Archit Build Eng 5(1):129–136CrossRefGoogle Scholar
  32. Zhou CC, Lu ZZ, Zhang F, Yue ZF (2015) An adaptive reliability method combining relevance vector machine and importance sampling. Struct Multidiscip Optim 52:945–957MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of AeronauticsNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Aircraft Flight Test Technology InstituteChinese Flight Test EstablishmentXi’anChina

Personalised recommendations