Advertisement

Structural and Multidisciplinary Optimization

, Volume 44, Issue 5, pp 673–690 | Cite as

Reliability-based design optimization using kriging surrogates and subset simulation

  • Vincent Dubourg
  • Bruno Sudret
  • Jean-Marc Bourinet
Research Paper

Abstract

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.

Keywords

Reliability-based design optimization (RBDO) Kriging Surrogate modeling Subset simulation Adaptive refinement 

Notes

Acknowledgements

The first author was funded by a CIFRE grant from Phimeca Engineering S.A. subsidized by the ANRT (convention number 706/2008). The financial support from DCNS is also gratefully acknowledged.

References

  1. Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidisc Optim 41(2):277–294MathSciNetCrossRefGoogle Scholar
  2. Au S, Beck J (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16(4): 263–277CrossRefGoogle Scholar
  3. Au SK (2005) Reliability-based design sensitivity by efficient simulation. Comput Struct 83(14):1048–1061CrossRefGoogle Scholar
  4. Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite elements: a non intrusive approach by regression. Eur J Comput Mech 15(1–3): 81–92zbMATHGoogle Scholar
  5. Bichon B, Eldred M, Swiler L, Mahadevan S, McFarland J (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459–2468CrossRefGoogle Scholar
  6. Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C R Méc 336(6):518–523zbMATHCrossRefGoogle Scholar
  7. Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab Eng Mech 25(2):183–197CrossRefGoogle Scholar
  8. Bourinet JM, Mattrand C, Dubourg V (2009) A review of recent features and improvements added to FERUM software. In: Proc. ICOSSAR’09, int conf. on structural safety and reliability. Osaka, JapanGoogle Scholar
  9. Bourinet JM, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf, submittedGoogle Scholar
  10. Bucher C, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7(1):57–66CrossRefGoogle Scholar
  11. Chateauneuf A, Aoues Y (2008) Structural design optimization considering uncertainties. Taylor & Francis, chap 9, pp 217–246Google Scholar
  12. Cressie N (1993) Statistics for spatial data, revised edition. WileyGoogle Scholar
  13. Das PK, Zheng Y (2000) Cumulative formation of response surface and its use in reliability analysis. Probab Eng Mech 15(4): 309–315CrossRefGoogle Scholar
  14. Deheeger F (2008) Couplage mécano-fiabiliste, 2SMART méthodologie d’apprentissage stochastique en fiabilité. PhD thesis, Université Blaise Pascal - Clermont IIGoogle Scholar
  15. Deheeger F, Lemaire M (2007) Support vector machine for efficient subset simulations: 2SMART method. In: Proc. 10th int. conf. on applications of stat. and prob. in civil engineering (ICASP10). Tokyo, JapanGoogle Scholar
  16. Der Kiureghian A, Ditlevsen O (2009) Aleatory or epistemic? Does it matter? Struct Saf 31(2):105–112CrossRefGoogle Scholar
  17. Ditlevsen O, Madsen H (1996) Structural reliability methods. Internet (v2.3.7, June–Sept 2007) edn. Wiley, ChichesterGoogle Scholar
  18. Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126:225–233CrossRefGoogle Scholar
  19. Enevoldsen I, Sorensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15(3):169–196CrossRefGoogle Scholar
  20. Faravelli L (1989) Response surface approach for reliability analysis. J Eng Mech 115(12):2763–2781CrossRefGoogle Scholar
  21. Hurtado J (2004) An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Struct Saf 26:271–293CrossRefGoogle Scholar
  22. Jensen H, Valdebenito M, Schuëller G, Kusanovic D (2009) Reliability-based optimization of stochastic systems using line search. Comput Methods Appl Mech Eng 198(49–52):3915–3924CrossRefGoogle Scholar
  23. Jones D, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetzbMATHCrossRefGoogle Scholar
  24. Kharmanda G, Mohamed A, Lemaire M (2002) Efficient reliability-based design optimization using a hybrid space with application to finite element analysis. Struct Multidisc Optim 24(3): 233–245CrossRefGoogle Scholar
  25. Kim SH, Na SW (1997) Response surface method using vector projected sampling points. Struct Saf 19(1):3–19CrossRefGoogle Scholar
  26. Kirjner-Neto C, Polak E, Der Kiureghian A (1998) An outer approximation approach to reliability-based optimal design of structures. J Optim Theory Appl 98:1–16MathSciNetzbMATHCrossRefGoogle Scholar
  27. Kuschel N, Rackwitz R (1997) Two basic problems in reliability-based structural optimization. Math Methods Oper Res 46(3):309–333MathSciNetzbMATHCrossRefGoogle Scholar
  28. Lebrun R, Dutfoy A (2009) An innovating analysis of the Nataf transformation from the copula viewpoint. Probab Eng Mech 24(3):312–320CrossRefGoogle Scholar
  29. Lee T, Jung J (2008) A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling. Comput Struct 86(13–14):1463–1476CrossRefGoogle Scholar
  30. Lophaven S, Nielsen H, Søndergaard J (2002) DACE, a Matlab kriging toolbox. Technical University of DenmarkGoogle Scholar
  31. MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Le Cam J, Neyman LM (eds) Proc. 5th Berkeley symp. on math. stat. & prob, vol 1. University of California Press, Berkeley, pp 281–297Google Scholar
  32. Neal R (2003) Slice sampling. Ann Stat 31:705–767MathSciNetzbMATHCrossRefGoogle Scholar
  33. Oakley J (2004) Estimating percentiles of uncertain computer code outputs. J R Stat Soc Ser C 53(1):83–93MathSciNetzbMATHCrossRefGoogle Scholar
  34. Papadrakakis M, Lagaros N (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191(32):3491–3507zbMATHCrossRefGoogle Scholar
  35. Picheny V, Ginsbourger D, Roustant, Haftka R (2010) Adaptive designs of experiments for accurate approximation of a target region. J Mech Des 132(7):071008 (9 pages)CrossRefGoogle Scholar
  36. Polak E (1997) Optimization algorithms and consistent approximations. SpringerGoogle Scholar
  37. Ranjan P, Bingham D, Michailidis G (2008) Sequential experiment design for contour estimation from complex computer codes. Technometrics 50(4):527–541MathSciNetCrossRefGoogle Scholar
  38. Rasmussen C, Williams C (2006) Gaussian processes for machine learning, Internet edn. Adaptive computation and machine learning. MIT Press, CambridgeGoogle Scholar
  39. Robert C, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer series in statistics. SpringerGoogle Scholar
  40. Royset J, Polak E (2004a) Reliability-based optimal design using sample average approximations. Probab Eng Mech 19:331–343CrossRefGoogle Scholar
  41. Royset JO, Polak E (2004b) Implementable algorithm for stochastic optimization using sample average approximations. J Optim Theory Appl 122(1):157–184MathSciNetzbMATHCrossRefGoogle Scholar
  42. Royset JO, Der Kiureghian A, Polak E (2001) Reliability-based optimal structural design by the decoupling approach. Reliab Eng Syst Saf 73(3):213–221CrossRefGoogle Scholar
  43. Santner T, Williams B, Notz W (2003) The design and analysis of computer experiments. Springer series in statistics. SpringerGoogle Scholar
  44. Severini T (2005) Elements of distribution theory. Cambridge series in statistical and probabilistic mathematics. Cambridge University PressGoogle Scholar
  45. Shan S, Wang G (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93(8):1218–1230CrossRefGoogle Scholar
  46. Song S, Lu Z, Qiao H (2009) Subset simulation for structural reliability sensitivity analysis. Reliab Eng Syst Saf 94(2):658–665CrossRefGoogle Scholar
  47. Sudret B, Der Kiureghian A (2002) Comparison of finite element reliability methods. Probab Eng Mech 17:337–348CrossRefGoogle Scholar
  48. Taflanidis A, Beck J (2009a) Life-cycle cost optimal design of passive dissipative devices. Struct Saf 31(6):508–522CrossRefGoogle Scholar
  49. Taflanidis A, Beck J (2009b) Stochastic subset optimization for reliability optimization and sensitivity analysis in system design. Comput Struct 87(5–6):318–331CrossRefGoogle Scholar
  50. Tu J, Choi K, Park Y (1999) A new study on reliability-based design optimization. J Mech Des 121:557–564CrossRefGoogle Scholar
  51. Vazquez E, Bect J (2009) A sequential Bayesian algorithm to estimate a probability of failure. In: 15th IFAC symposium on system identification. IFAC, Saint-MaloGoogle Scholar
  52. Waarts PH (2000) Structural reliability using finite element methods: an appraisal of DARS: directional adaptive response surface sampling. PhD thesis, Technical University of Delft, The NetherlandsGoogle Scholar
  53. Welch W, Buck R, Sacks J, Wynn H, Mitchell T, Morris M (1992) Screening, predicting, and computer experiments. Technometrics 34(1):15–25CrossRefGoogle Scholar
  54. Youn B, Choi K (2004) Selecting probabilistic approaches for reliability-based design optimization. AIAA J 42:124–131CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Vincent Dubourg
    • 1
    • 2
  • Bruno Sudret
    • 1
    • 2
  • Jean-Marc Bourinet
    • 2
  1. 1.Phimeca Engineering, Centre d’Affaires du ZénithCournon d’AuvergneFrance
  2. 2.IFMA, EA 3867, Laboratoire de Mécanique et IngénieriesClermont UniversitéClermont-FerrandFrance

Personalised recommendations