Skip to main content
SpringerLink
Log in

Hybrid reliability analysis with both random and probability-box variables

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Hybrid reliability analysis (HRA) with both aleatory and epistemic uncertainties is investigated in this paper. The aleatory uncertainties are described by random variables, and the epistemic uncertainties are described by a probability-box (p-box) model. Although tremendous efforts have been devoted to propagating random or p-box uncertainties, much less attention has been paid to analyzing the hybrid reliability with both of them. For HRA, optimization-based Interval Monte Carlo Simulation (OIMCS) is available to estimate the bounds of failure probability, but it requires enormous computational performance. A new method combining the Kriging model with OIMCS is proposed in this paper. When constructing the Kriging model, we only locally approximate the performance function in the region where the sign is prone to be wrongly predicted. It is based on the idea that a surrogate model only exactly predicting the sign of performance function could satisfy the demand of accuracy for HRA. Then OIMCS can be effectively performed based on the Kriging model. Three numerical examples and an engineering application are investigated to demonstrate the performance of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Helton J.C.: Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty. J. Stat. Comput. Simul. 57, 3–76 (1997)

    Article  MATH  Google Scholar 

  2. Bae H.R., Grandhi R.V., Canfield R.A.: Epistemic uncertainty quantification techniques including evidence theory for large-scale structures. Comput. Struct. 82, 1101–1112 (2004)

    Article  Google Scholar 

  3. Der Kiureghian A.: Analysis of structural reliability under parameter uncertainties. Probab. Eng. Mech. 23, 351–358 (2008)

    Article  Google Scholar 

  4. Choi J., An D., Won J.: Bayesian approach for structural reliability analysis and optimization using the Kriging Dimension Reduction Method. J. Mech. Des. 132, 051003 (2010)

    Article  Google Scholar 

  5. Davis J.P., Hall J.W.: A software-supported process for assembling evidence and handling uncertainty in decision-making. Decis. Support Syst. 35, 415–433 (2003)

    Article  Google Scholar 

  6. Augustin T.: Optimal decisions under complex uncertainty-basic notions and a general algorithm for data-based decision making with partial prior knowledge described by interval probability. ZAMM-J. Appl. Math. Mech. 84, 678–687 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Troffaes M.C.M., Miranda E., Destercke S.: On the connection between probability boxes and possibility measures. Inf. Sci. 224, 88–108 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Crespo L.G., Kenny S., Giesy D.: Reliability analysis of polynomial systems subject to p-box uncertainties. Mech. Syst. Signal Process. 37, 121–136 (2013)

    Article  Google Scholar 

  9. Jiang C., Zhang Z., Han X., Liu J.: A novel evidence-theory-based reliability analysis method for structures with epistemic uncertainty. Comput. Struct. 129, 1–12 (2013)

    Article  Google Scholar 

  10. Beer, M., Ferson, S.: Fuzzy probability in engineering analyses. In: Proceedings of the First International Conference on Vulnerability and Risk Analysis and Management (ICVRAM 2011) and the Fifth International Symposium on Uncertainty Modeling and Analysis (ISUMA 2011) (2011)

  11. Hurtado J.E., Alvarez D.A., Ramírez J.: Fuzzy structural analysis based on fundamental reliability concepts. Comput. Struct. 112, 183–192 (2012)

    Article  Google Scholar 

  12. Jiang C., Bi R.G., Lu G.Y., Han X.: Structural reliability analysis using non-probabilistic convex model. Comput. Methods Appl. Mech. Eng. 254, 83–98 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Luo Y.J., Alex L., Kang Z.: Reliability-based design optimization of adhesive bonded steel–concrete composite beams with probabilistic and non-probabilistic uncertainties. Eng. Struct. 33, 2110–2119 (2011)

    Article  Google Scholar 

  14. Elishakoff I.: Essay on uncertainties in elastic and viscoelastic structures: from AM Freudenthal’s criticisms to modern convex modeling. Comput. Struct. 56, 871–895 (1995)

    Article  MATH  Google Scholar 

  15. Beer M., Ferson S., Kreinovich V.: Imprecise probabilities in engineering analyses. Mech. Syst. Signal Process. 37, 4–29 (2013)

    Article  Google Scholar 

  16. Ferson S., Kreinovich V., Ginzburg L., Myers D.S., Sentz K.: Constructing probability boxes and Dempster–Shafer structures, vol. 835. Sandia National Laboratories, Albuquerque (2002)

    Google Scholar 

  17. Alvarez D.A.: On the calculation of the bounds of probability of events using infinite random sets. Int. J. Approx. Reason. 43, 241–267 (2006)

    Article  MATH  Google Scholar 

  18. Alvarez D.A.: A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type. Fuzzy Sets Syst. 160, 384–401 (2009)

    Article  MATH  Google Scholar 

  19. Batarseh, O.G., Wang, Y.: Reliable simulation with input uncertainties using an interval-based approach. In: Simulation Conference, 2008. WSC 2008. Winter 2008, pp. 344–352

  20. Zhang H., Mullen R.L., Muhanna R.L.: Interval Monte Carlo methods for structural reliability. Struct. Saf. 32, 183–190 (2010)

    Article  Google Scholar 

  21. Zhang H.: Interval importance sampling method for finite element-based structural reliability assessment under parameter uncertainties. Struct. Saf. 38, 1–10 (2012)

    Article  MATH  Google Scholar 

  22. Zhang H., Dai H., Beer M., Wang W.: Structural reliability analysis on the basis of small samples: an interval quasi-Monte Carlo method. Mech. Syst. Signal Process. 37, 137–151 (2013)

    Article  Google Scholar 

  23. Jiang C., Li W.X., Han X., Liu L.X., Le P.H.: Structural reliability analysis based on random distributions with interval parameters. Comput. Struct. 89, 2292–2302 (2011)

    Article  Google Scholar 

  24. Hurtado J.E.: Assessment of reliability intervals under input distributions with uncertain parameters. Probab. Eng. Mech. 32, 80–92 (2013)

    Article  Google Scholar 

  25. Xiao N.C., Huang H.Z., Wang Z., Pang Y., He L.: Reliability sensitivity analysis for structural systems in interval probability form. Struct. Multidisc. Optim. 44, 691–705 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Qiu Z., Yang D., Elishakoff I.: Probabilistic interval reliability of structural systems. Int. J. Solids Struct. 45, 2850–2860 (2008)

    Article  MATH  Google Scholar 

  27. Eldred M.S., Swiler L.P., Tang G.: Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation. Reliab. Eng. Syst. Saf. 96, 1092–1113 (2011)

    Article  Google Scholar 

  28. Du X.: Reliability-based design optimization with dependent interval variables. Int. J. Numer. Methods Eng. 91, 218–228 (2012)

    Article  MATH  Google Scholar 

  29. Luo Y.J., Kang Z., Alex L.: Structural reliability assessment based on probability and convex set mixed model. Comput. Struct. 87, 1408–1415 (2009)

    Article  Google Scholar 

  30. Du X.: Unified uncertainty analysis by the first order reliability method. J. Mech. Des. 130, 091401–091410 (2008)

    Article  Google Scholar 

  31. Yao W., Chen X., Huang Y., Tooren M.: An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization. Reliab. Eng. Syst. Saf. 116, 28–37 (2013)

    Article  Google Scholar 

  32. Balu A.S., Rao B.N.: Multicut-high dimensional model representation for structural reliability bounds estimation under mixed uncertainties. Comput. Aid. Civ. Infrastruct. Eng. 27, 419–438 (2012)

    Article  Google Scholar 

  33. Guo J., Du X.: Sensitivity analysis with mixture of epistemic and aleatory uncertainties. AIAA J. 45, 2337–2349 (2007)

    Article  Google Scholar 

  34. Moens D., Hanss M.: Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances. Finite Elem. Anal. Des. 47, 4–16 (2011)

    Article  Google Scholar 

  35. Ang A., Tang W.: Probability Concepts in Engineering Planning and Design, vol. 1. Wiley, New York (1975)

    Google Scholar 

  36. Dai H., Wang W.: Application of low-discrepancy sampling method in structural reliability analysis. Struct. Saf. 31, 55–64 (2009)

    Article  Google Scholar 

  37. Vanhatalo, J.: GPstuff: Gaussian process models for Bayesian analysis V2.0. (2010). http://pmtksupport.googlecode.com/svn/trunk/GPstuff-2.0/dist/hammersley.m

  38. Gablonsky, J.: Implementation of the DIRECT Algorithm. Center for Research in Scientific Computation, Technical Rept. CRSC-TR98-29, North Carolina State Univ., Raleigh, NC (1998)

  39. 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)

    Article  Google Scholar 

  40. Echard B., Gayton N., Lemaire M.: AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct. Saf. 33, 145–154 (2011)

    Article  Google Scholar 

  41. Bect J., Ginsbourger D., Li L., Picheny V., Vazquez E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22, 773–793 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  42. Balesdent M., Morio J., Marzat J.: Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct. Saf. 44, 1–10 (2013)

    Article  Google Scholar 

  43. Picheny V., Ginsbourger D., Roustant O., Haftka R.T., Kim N.H.: Adaptive designs of experiments for accurate approximation of a target region. J. Mech. Des. 132, 071008 (2010)

    Article  Google Scholar 

  44. Dubourg V., Sudret B., Bourinet J.M.: Reliability-based design optimization using Kriging surrogates and subset simulation. Struct. Multidiscip. Optim. 44, 673–690 (2011)

    Article  Google Scholar 

  45. Ranjan P., Bingham D., Michailidis G.: Sequential experiment design for contour estimation from complex computer codes. Technometrics 50, 527–541 (2008)

    Article  MathSciNet  Google Scholar 

  46. Jones D., Schonlau M., Welch W.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  47. Yang, X., Liu, Y., Gao, Y., Zhang, Y., Gao, Z.: An active learning Kriging model for hybrid reliability analysis with both random and interval variables. Struct. Multidisc. Optim. (accepted). doi:10.1007/s00158-014-1189-5

  48. Matheron G.: The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  49. Dumas A., Echard B., Gayton N., Rochat O., Dantan J.Y., Veen S.V.D.: AK-ILS: an active learning method based on kriging for the inspection of large surfaces. Precis. Eng. 37, 1–9 (2013)

    Article  Google Scholar 

  50. Lophaven, S.N., Nielsen, H.B., Sondergaard, J.: DACE, a MATLAB Kriging toolbox, version 2.0. Tech. Rep. IMM-TR-2002-12; Technical University of Denmark (2002)

  51. Luo X., Li X., Zhou J., Cheng T.: A Kriging-based hybrid optimization algorithm for slope reliability analysis. Struct. Saf. 34, 401–406 (2012)

    Article  Google Scholar 

  52. Kaymaz I.: Application of kriging method to structural reliability problems. Struct. Saf. 27, 133–151 (2005)

    Article  Google Scholar 

  53. Jiang C., Wang B., Li Z.R., Han X., Yu D.J.: An evidence-theory model considering dependence among parameters and its application in structural reliability analysis. Eng. Struct. 57, 12–22 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Institute of Aircraft Reliability Engineering Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Xufeng Yang, Yongshou Liu, Yishang Zhang & Zhufeng Yue

Authors
  1. Xufeng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongshou Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yishang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhufeng Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongshou Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, X., Liu, Y., Zhang, Y. et al. Hybrid reliability analysis with both random and probability-box variables. Acta Mech 226, 1341–1357 (2015). https://doi.org/10.1007/s00707-014-1252-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-014-1252-8

Keywords

Search

Navigation