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

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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.

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References

  • Acar E, Haftka RT (2007) Reliability-based aircraft structural design pays, even with limited statistical data. J Aircr 44(3):812–823

    Article  Google Scholar 

  • Annis C (2004) Probabilistic life prediction isn’t as easy as it looks. J ASTM Int 1(2):3–14

    Article  MathSciNet  Google Scholar 

  • Breitung K (1984) Asymptotic approximations for multinormal integrals. ASCE J Eng Mech 110(3):357–366

    Article  Google Scholar 

  • Browder A (1996) Mathematical analysis: an introduction. Springer, New York

    Google Scholar 

  • 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 126:2009–2013

    Google Scholar 

  • 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, vSAND2002–0099

  • Center for Computer-Aided Design, College of Engineering (1999a) DRAW concept manual. The University of Iowa, Iowa City

  • Center for Computer-Aided Design, College of Engineering (1999b) DRAW user reference. The University of Iowa, Iowa City

  • 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):140–149

    Article  Google Scholar 

  • Chiles JP, Delfiner P (1999) Geostatistics:-modeling spatial uncertainty. Wiley, New York

    MATH  Google Scholar 

  • Ditlevsen O, Madsen HO (1996) Structural reliability methods. Wiley, Chichester

    Google Scholar 

  • 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 62:83–121

    Article  MATH  Google Scholar 

  • 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):1187–1198

    Article  Google Scholar 

  • 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):348–360

    Article  Google Scholar 

  • Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. Wiley, New York

    Google Scholar 

  • Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. ASCE J Eng Mech Div 100(1):111–121

    Google Scholar 

  • Hijab O (1997) Introduction to calculus and classical analysis. Springer, New York

    MATH  Google Scholar 

  • Hohenbichler M, Rackwitz R (1988) Improvement of second-order reliability estimates by importance sampling. ASCE J Eng Mech 114(12):2195–2199

    Article  Google Scholar 

  • Jin R, Chen W, Simpson TW (2001) Comparative studies of metamodeling techniques under multiple modeling criteria. Struct Multidiscpl Optim 23(1):1–13

    Article  Google Scholar 

  • Kim C, Choi KK (2008) Reliability-based design optimization using response surface method with prediction interval estimation. J Mech Des 130(12):1–12

    Article  Google Scholar 

  • 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):102–119

    Google Scholar 

  • 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):14–27

    Article  MATH  Google Scholar 

  • 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):823–839

    Article  Google Scholar 

  • 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):21003

    Article  Google Scholar 

  • Lee SH, Chen W, Kwak BM (2009) Robust design with arbitrary distribution using gauss-type Quadrature formula. Struct Multidiscipl Optim 39(3):227–243

    Article  MathSciNet  Google Scholar 

  • Lewis RM, Torczon V (1999) Pattern search algorithms for bound constrained minimization. SIAM J Optim 9(4):1082–1099

    Article  MathSciNet  MATH  Google Scholar 

  • Madsen HO, Krenk S, Lind NC (1986) Methods of structural safety. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Martin JD, Simpson TW (2005) Use of kriging models to approximate deterministic computer models. AIAA J 43(4):853–863

    Article  Google Scholar 

  • McDonald M, Mahadevan S (2008) Design optimization with system-level reliability constraints. J Mech Des 130(2):21403

    Article  Google Scholar 

  • Meggiolaro MA, Castro JTP (2004) Statistical evaluation of strain-life fatigue crack initiation predictions. Int J Fatigue 26(5):463–476

    Article  Google Scholar 

  • 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):1–16

    Article  Google Scholar 

  • Noh Y, Choi KK, Lee I (2009b) Reduction of ordering effect in RBDO using dimension reduction method. AIAA J 47(4):994–1004

    Article  Google Scholar 

  • Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidiscipl Optim 40(1):35–51

    Article  MathSciNet  Google Scholar 

  • Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28

    Article  Google Scholar 

  • Rahman S (2009) Stochastic sensitivity analysis by dimensional decomposition and score functions. Probab Eng Mech 24:278–287

    Article  Google Scholar 

  • Rahman S, Wei D (2006) A univariate approximation at most probable point for higer-order reliability analysis. Int J Solids Struct 43:2820–2839

    Article  MATH  Google Scholar 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472

    Article  MathSciNet  MATH  Google Scholar 

  • Rubinstein RY (1981) Simulation and Monte Carlo method. Wiley, New York

    Book  MATH  Google Scholar 

  • Simpson T, Lin D, Chen W (2001a) Sampling strategies for computer experiments: design and analysis. Int J Reliab Appl 2(3):209–240

    Google Scholar 

  • Simpson TW, Mauery TM, Korte JJ, Mistree F (2001b) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241

    Article  Google Scholar 

  • Swanson Analysis System (1989) AnSYS engineering analysis system user’s manual, vol. I, II. Swanson Analysis System, Houston

    Google Scholar 

  • 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):439–457

    Article  Google Scholar 

  • Wei D (2006) A univariate decomposition method for higher-order reliability analysis and design optimization. Ph. D. Thesis, The University of Iowa

  • 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 71:1483–1511

    Article  MATH  Google Scholar 

  • Youn BD, Choi KK (2004) A new response surface methodology for reliability-based design optimization. Comput Struct 82(2–3):241–256

    Article  Google Scholar 

  • 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):272–283

    Article  Google Scholar 

  • 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):185–213

    Article  Google Scholar 

  • Youn BD, Choi KK, Tang J (2005b) Structural durability design optimization and its reliability assessment. Int J Prod Dev 1(3/4):383–401

    Article  Google Scholar 

  • Youn BD, Choi KK, Du L (2005c) Enriched performance measure approach (PMA+) for reliability-based design optimization. AIAA J 43(4):874–884

    Article  Google Scholar 

  • Yu X, Du X (2006) Reliability-based multidisciplinary optimization for aircraft wing design. Struct Infrastruct E 2(3/4):277–289

    Article  Google Scholar 

  • 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):327–345

    Article  Google Scholar 

  • 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 13–15

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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.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, College of Engineering, The University of Iowa, Iowa City, IA, 52242, USA

    Ikjin Lee, K. K. Choi & Liang Zhao

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  1. Ikjin Lee
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  2. K. K. Choi
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  3. Liang Zhao
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Correspondence to K. K. Choi.

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Lee, I., Choi, K.K. & Zhao, L. Sampling-based RBDO using the stochastic sensitivity analysis and Dynamic Kriging method. Struct Multidisc Optim 44, 299–317 (2011). https://doi.org/10.1007/s00158-011-0659-2

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  • DOI: https://doi.org/10.1007/s00158-011-0659-2

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