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Structural and Multidisciplinary Optimization

, Volume 44, Issue 3, pp 299–317 | Cite as

Sampling-based RBDO using the stochastic sensitivity analysis and Dynamic Kriging method

  • Ikjin Lee
  • K. K. ChoiEmail author
  • Liang Zhao
Research Paper

Abstract

This paper presents a sampling-based RBDO method using surrogate models. The Dynamic Kriging (D-Kriging) method is used for surrogate models, and a stochastic sensitivity analysis is introduced to compute the sensitivities of probabilistic constraints with respect to independent or correlated random variables. For the sampling-based RBDO, which requires Monte Carlo simulation (MCS) to evaluate the probabilistic constraints and stochastic sensitivities, this paper proposes new efficiency and accuracy strategies such as a hyper-spherical local window for surrogate model generation, sample reuse, local window enlargement, filtering of constraints, and an adaptive initial point for the pattern search. To further improve computational efficiency of the sampling-based RBDO method for large-scale engineering problems, parallel computing is proposed as well. Once the D-Kriging accurately approximates the responses, there is no further approximation in the estimation of the probabilistic constraints and stochastic sensitivities, and thus the sampling-based RBDO can yield very accurate optimum design. In addition, newly proposed efficiency strategies as well as parallel computing help find the optimum design very efficiently. Numerical examples verify that the proposed sampling-based RBDO can find the optimum design more accurately than some existing methods. Also, the proposed method can find the optimum design more efficiently than some existing methods for low dimensional problems, and as efficient as some existing methods for high dimensional problems when the parallel computing is utilized.

Keywords

Sampling-based RBDO Surrogate model Stochastic sensitivity analyses Score functions  Monte Carlo simulation Copula 

Notes

Acknowledgments

Research is jointly supported by the ARO Project W911NF-09-1-0250 and the Automotive Research Center, which is sponsored by the U.S. Army TARDEC. These supports are greatly appreciated.

References

  1. Acar E, Haftka RT (2007) Reliability-based aircraft structural design pays, even with limited statistical data. J Aircr 44(3):812–823CrossRefGoogle Scholar
  2. Annis C (2004) Probabilistic life prediction isn’t as easy as it looks. J ASTM Int 1(2):3–14MathSciNetCrossRefGoogle Scholar
  3. Breitung K (1984) Asymptotic approximations for multinormal integrals. ASCE J Eng Mech 110(3):357–366CrossRefGoogle Scholar
  4. Browder A (1996) Mathematical analysis: an introduction. Springer, New YorkGoogle Scholar
  5. 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
  6. Burkardt J, Gunzburger M, Peterson J, Brannon R (2002) User manual and supporting information for library of codes for Centroidal Voronoi placement and associated Zeroth, first, and second moment determination. Sandia National Laboratories Technical Report, vSAND2002–0099Google Scholar
  7. Center for Computer-Aided Design, College of Engineering (1999a) DRAW concept manual. The University of Iowa, Iowa CityGoogle Scholar
  8. Center for Computer-Aided Design, College of Engineering (1999b) DRAW user reference. The University of Iowa, Iowa CityGoogle Scholar
  9. Chan KY, Skerlos SJ, Papalambros P (2007) An adaptive sequential linear programming algorithm for optimal design problems with probabilistic constraints. J Mech Des 129(2):140–149CrossRefGoogle Scholar
  10. Chiles JP, Delfiner P (1999) Geostatistics:-modeling spatial uncertainty. Wiley, New YorkzbMATHGoogle Scholar
  11. Ditlevsen O, Madsen HO (1996) Structural reliability methods. Wiley, ChichesterGoogle Scholar
  12. Dong J, Choi KK, Wang A, Zhang W, Vlahopoulos N (2005) Parametric design sensitivity analysis of high frequency structural-acoustic problems using energy finite element method. Int J Numer Methods Eng 62:83–121zbMATHCrossRefGoogle Scholar
  13. Dong J, Choi KK, Vlahopoulos N, Wang A, Zhang W (2007) Design sensitivity analysis and optimization of high frequency radiation problems using energy finite element and energy boundary element methods. AIAA J 45(6):1187–1198CrossRefGoogle Scholar
  14. 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
  15. Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. Wiley, New YorkGoogle Scholar
  16. Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. ASCE J Eng Mech Div 100(1):111–121Google Scholar
  17. Hijab O (1997) Introduction to calculus and classical analysis. Springer, New YorkzbMATHGoogle Scholar
  18. Hohenbichler M, Rackwitz R (1988) Improvement of second-order reliability estimates by importance sampling. ASCE J Eng Mech 114(12):2195–2199CrossRefGoogle Scholar
  19. Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidiscpl Optim 23(1):1–13CrossRefGoogle Scholar
  20. Kim C, Choi KK (2008) Reliability-based design optimization using response surface method with prediction interval estimation. J Mech Des 130(12):1–12CrossRefGoogle Scholar
  21. Kim NH, Wang H, Queipo NV (2006) Adaptive reduction of design variables using global sensitivity in reliability-based optimization. Int J Reliab Appl 1(1–2):102–119Google Scholar
  22. 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):14–27zbMATHCrossRefGoogle Scholar
  23. Lee I, Choi KK, Gorsich D (2010) System reliability-based design optimization using the MPP-based dimension reduction method. J Struct Multidiscipl Optim 41(6):823–839CrossRefGoogle Scholar
  24. Lee I, Choi KK, Noh Y, Zhao L, Gorsich D (2011) Sampling-based stochastic sensitivity analysis using score functions for RBDO problems with correlated random variables. J Mech Des 133(2):21003CrossRefGoogle Scholar
  25. Lee SH, Chen W, Kwak BM (2009) Robust design with arbitrary distribution using gauss-type Quadrature formula. Struct Multidiscipl Optim 39(3):227–243MathSciNetCrossRefGoogle Scholar
  26. Lewis RM, Torczon V (1999) Pattern search algorithms for bound constrained minimization. SIAM J Optim 9(4):1082–1099MathSciNetzbMATHCrossRefGoogle Scholar
  27. Madsen HO, Krenk S, Lind NC (1986) Methods of structural safety. Prentice-Hall, Englewood CliffsGoogle Scholar
  28. Martin JD, Simpson TW (2005) Use of kriging models to approximate deterministic computer models. AIAA J 43(4):853–863CrossRefGoogle Scholar
  29. McDonald M, Mahadevan S (2008) Design optimization with system-level reliability constraints. J Mech Des 130(2):21403CrossRefGoogle Scholar
  30. Meggiolaro MA, Castro JTP (2004) Statistical evaluation of strain-life fatigue crack initiation predictions. Int J Fatigue 26(5):463–476CrossRefGoogle Scholar
  31. Noh Y, Choi KK, Du L (2009a) Reliability based design optimization of problems with correlated input variables using a Gaussian Copula. Struct Multidiscipl Optim 38(1):1–16CrossRefGoogle Scholar
  32. Noh Y, Choi KK, Lee I (2009b) Reduction of ordering effect in RBDO using dimension reduction method. AIAA J 47(4):994–1004CrossRefGoogle Scholar
  33. Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidiscipl Optim 40(1):35–51MathSciNetCrossRefGoogle Scholar
  34. 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
  35. Rahman S (2009) Stochastic sensitivity analysis by dimensional decomposition and score functions. Probab Eng Mech 24:278–287CrossRefGoogle Scholar
  36. Rahman S, Wei D (2006) A univariate approximation at most probable point for higer-order reliability analysis. Int J Solids Struct 43:2820–2839zbMATHCrossRefGoogle Scholar
  37. Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472MathSciNetzbMATHCrossRefGoogle Scholar
  38. Rubinstein RY (1981) Simulation and Monte Carlo method. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  39. Simpson T, Lin D, Chen W (2001a) Sampling strategies for computer experiments: design and analysis. Int J Reliab Appl 2(3):209–240Google Scholar
  40. Simpson TW, Mauery TM, Korte JJ, Mistree F (2001b) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241CrossRefGoogle Scholar
  41. Swanson Analysis System (1989) AnSYS engineering analysis system user’s manual, vol. I, II. Swanson Analysis System, HoustonGoogle Scholar
  42. Viana ACF, Haftka RT, Steffen V (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscipl Optim 39(4):439–457CrossRefGoogle Scholar
  43. Wei D (2006) A univariate decomposition method for higher-order reliability analysis and design optimization. Ph. D. Thesis, The University of IowaGoogle Scholar
  44. Yi K, Choi KK, Kim NH, Botkin ME (2007) Design sensitivity analysis and optimization for minimizing springback of sheet-formed part. Int J Numer Methods Eng 71:1483–1511zbMATHCrossRefGoogle Scholar
  45. Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82(2–3):241–256CrossRefGoogle Scholar
  46. Youn BD, Choi KK, Yang R-J, Gu L (2004) Reliability-based design optimization for crashworthiness of vehicle side impact. Struct Multidiscipl Optim 26(3–4):272–283CrossRefGoogle Scholar
  47. Youn BD, Choi KK, Yi K (2005a) Performance moment integration (PMI) method for quality assessment in reliability-based robust optimization. Mech Based Des Struct Mach 33(2):185–213CrossRefGoogle Scholar
  48. Youn BD, Choi KK, Tang J (2005b) Structural durability design optimization and its reliability assessment. Int J Prod Dev 1(3/4):383–401CrossRefGoogle Scholar
  49. Youn BD, Choi KK, Du L (2005c) Enriched performance measure approach (PMA+) for reliability-based design optimization. AIAA J 43(4):874–884CrossRefGoogle Scholar
  50. Yu X, Du X (2006) Reliability-based multidisciplinary optimization for aircraft wing design. Struct Infrastruct E 2(3/4):277–289CrossRefGoogle Scholar
  51. 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 Multidiscipl Optim 32(4):327–345CrossRefGoogle Scholar
  52. Zhao L, Choi KK, Lee I, Gorsich D (2010) A metamodel method using dynamic Kriging and sequential sampling. 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Fort Worth, Texas, September 13–15Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial Engineering, College of EngineeringThe University of IowaIowa CityUSA

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