Structural and Multidisciplinary Optimization

, Volume 48, Issue 2, pp 235–248 | Cite as

Equivalent target probability of failure to convert high-reliability model to low-reliability model for efficiency of sampling-based RBDO

Research Paper


This study presents a methodology to convert an RBDO problem requiring very high reliability to an RBDO problem requiring relatively low reliability by appropriately increasing the input standard deviations for efficient computation in sampling-based RBDO. First, for linear performance functions with independent normal random inputs, an exact probability of failure is derived in terms of the ratio of the input standard deviation, which is denoted by \(\boldsymbol {\delta } \). Then, the probability of failure estimation is generalized for other types of random inputs and performance functions. For the generalization of the probability of failure estimation, two types of coefficients need to be determined by equating the probability of failure and its sensitivities with respect to the input standard deviation at the given design point. The sensitivities of the probability of failure with respect to the standard deviation are obtained using the first-order score function for the standard deviation. To apply the proposed method to an RBDO problem, a concept of an equivalent target probability of failure, which is an increased target probability of failure corresponding to the increased input standard deviations, is also introduced. Numerical results indicate that the proposed method can estimate the probability of failure accurately as a function of the input standard deviation compared to the Monte Carlo simulation results. As anticipated, the sampling-based RBDO using equivalent target probability of failure helps find the optimum design very efficiently while yielding reasonably accurate optimum design, which is close to the one obtained using the original target probability of failure.


Very small probability of failure Sampling-based RBDO Monte Carlo simulation Score function Copula Surrogate model 


  1. Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Struct Saf 21(2):135–158CrossRefGoogle Scholar
  2. Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16:263–277CrossRefGoogle Scholar
  3. Bjerager P (1988) Probability integration by directional simulation. J Eng Mech 114:1285–1302CrossRefGoogle Scholar
  4. Buranathiti T, Cao J, Chen W, Baghdasaryan L, Xia ZC (2004) Approaches for model validation: methodology and illustration on a sheet metal flanging process. SME J Manuf Sci Eng 126:2009–2013Google Scholar
  5. Denny M (2001) Introduction to importance sampling in rare-event simulations. Eur J Phys 22:403–411CrossRefGoogle Scholar
  6. Ditlevsen O, Madsen HO (1996) Structural reliability methods. John Wiley & Sons Ltd, ChichesterGoogle Scholar
  7. Gu L, Yang RJ, Tho CH, Makowskit M, Faruquet O, Li Y (2001) Optimization and robustness for crashworthiness of side impact. Int J Veh Des 26(4):348–360CrossRefGoogle Scholar
  8. Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. John Wiley & Sons, New YorkGoogle Scholar
  9. Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. ASCE J Eng Mech Div 100(1):111–121Google Scholar
  10. Helton JC, Davis FJ (2003) Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab Eng Syst Saf 81(1):23–69CrossRefGoogle Scholar
  11. Helton JC, Johnson JD, Sallaberry CJ, Storlie CB (2006) Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab Eng Syst Saf 91:1175–1209CrossRefGoogle Scholar
  12. Hu C, Youn BD (2011a) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidisc Optim 43(3):419–442MathSciNetCrossRefGoogle Scholar
  13. Hu C, Youn BD (2011b) An asymmetric dimension-adaptive tensor-product method for reliability analysis. Struct Saf 33:218–231CrossRefGoogle Scholar
  14. Huntington DE, Lyrintzis CS (1998) Improvements to and limitations of latin hypercube sampling. Probab Eng Mech 13(4):245–253CrossRefGoogle Scholar
  15. Kim C, Choi KK (2008) Reliability-based design optimization using response surface method with prediction interval estimation. ASME J Mech Des 130(12):1–12CrossRefGoogle Scholar
  16. Lee I, Choi KK, Du L, Gorsich D (2008) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Mech Eng 198(1):14–27MATHCrossRefGoogle Scholar
  17. Lee I, Choi KK, Gorsich D (2011a) Equivalent standard deviation to convert high-reliability model to low-reliability model for efficiency of sampling-based RBDO. In: 37th ASME Design automation conference, Washington, D.C., August, 28–31Google Scholar
  18. Lee I, Choi KK, Noh Y, Zhao L (2011b) Sampling-based stochastic sensitivity analysis using score functions for RBDO problems with correlated random variables. J Mech Des 133(2):21003CrossRefGoogle Scholar
  19. Lee I, Choi KK, Zhao L (2011c) Sampling-based RBDO using the dynamic kriging (D-kriging) method and stochastic sensitivity analysis. Struct Multidisc Optim 44(3):299–317MathSciNetCrossRefGoogle Scholar
  20. Li J, Xiu D (2010) Evaluation of failure probability via surrogate models. J Comput Phys 229(23):8966–8980MathSciNetMATHCrossRefGoogle Scholar
  21. McDonald M, Mahadevan S (2008) Design optimization with system-level reliability constraints. J Mech Des 130(2):21403CrossRefGoogle Scholar
  22. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245MathSciNetMATHGoogle Scholar
  23. Nelsen RB (1999) An introduction to copulas. Springer, New YorkMATHCrossRefGoogle Scholar
  24. Nie J, Ellingwood BR (2000) Directional methods for structural reliability. Struct Saf 22:233–249CrossRefGoogle Scholar
  25. Noh Y, Choi KK, Lee I (2009) Reduction of transformation ordering effect in RBDO using MPP-based dimension reduction method. AIAA J 47(4):994–1004CrossRefGoogle Scholar
  26. Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using the Bayesian method for RBDO. Struct Multidisc Optim 40(1):35–51MathSciNetCrossRefGoogle Scholar
  27. Olsson A, Sandberg G, Dahlblom O (2003) On latin hypercube sampling for structural reliability analysis. Struct Saf 25:47–68CrossRefGoogle Scholar
  28. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28CrossRefGoogle Scholar
  29. Rahman S (2009) Stochastic sensitivity analysis by dimensional decomposition and score functions. Probab Eng Mech 24:278–287CrossRefGoogle Scholar
  30. Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472MathSciNetMATHCrossRefGoogle Scholar
  31. Rubinstein RY (1981) Simulation and Monte Carlo method. John Wiley & Sons, New YorkMATHCrossRefGoogle Scholar
  32. Rubinstein RY, Shapiro A (1993) Discrete event systems—sensitivity analysis and stochastic optimization by the score function method. John Wiley & Sons, New YorkMATHGoogle Scholar
  33. Viana ACF, Haftka RT, Steffen V (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscip Optim 39(4):439–457CrossRefGoogle Scholar
  34. Wei DL, Cui ZS, Chen J (2008) Uncertainty quantification using polynomial chaos expansion with points of monomial cubature rules. Comput Struct 86(23–24):2102–2108CrossRefGoogle Scholar
  35. Xiong F, Greene S, Chen W, Xiong Y, Yang S (2010) A new sparse grid based method for uncertainty propagation. Struct Multidisc Optim 41(3):335–349MathSciNetCrossRefGoogle Scholar
  36. Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82(2–3):241–256CrossRefGoogle Scholar
  37. Zhang T, Choi KK, Rahman S, Cho K, Perry B, Shakil M, Heitka D (2006) A response surface and pattern search based hybrid optimization method and application to microelectronics. Struct Multidisc Optim 32(4):327–345CrossRefGoogle Scholar
  38. Zhao L, Choi KK, Lee I (2011) Metamodeling method using dynamic kriging for design optimization. AIAA J 49(9):2034–2046CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mechanical Engineering Department, School of EngineeringUniversity of ConnecticutStorrsUSA
  2. 2.Department of Mechanical & Industrial Engineering, College of EngineeringThe University of IowaIowa CityUSA

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