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Reliability-based design optimization for problems with interval distribution parameters

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Abstract

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.

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References

  • Alibrandi U, Koh CG (2015) First-order reliability method for structural reliability analysis in the presence of random and interval variables. ASCE-ASME J Risk Uncertain Eng Syst Part B Mech Eng 1(4):041006

    Article  Google Scholar 

  • Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14(4):227–245

    Article  Google Scholar 

  • Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainties in applied mechanics. Elsevier Science, Amsterdam

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Burden RL, Faires JD (1985) Numerical analysis. Prindle, Weber & Schmidt, Boston

    MATH  Google Scholar 

  • Chen ZZ, Qiu HB, Gao L, Li P (2013) An optimal shifting vector approach for efficient probabilistic design. Struct Multidiscip Optim 47(6):905–920

    Article  Google Scholar 

  • Cheng YS, Zhong YX, Zeng GW (2005) Structural robust design based on hybrid probabilistic and non-probabilistic models. Chin J Comput Mech 22(4):501–505 (in Chinese)

    Google Scholar 

  • Cheng GD, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84(21):1353–1367

    Article  Google Scholar 

  • Du XP (2007) Interval reliability analysis. ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference American Society of Mechanical Engineers, pp.1103–1109

  • Du XP (2012) Reliability-based design optimization with dependent interval variables. Int J Numer Methods Eng 91(2):218–228

    Article  MathSciNet  MATH  Google Scholar 

  • Du XP, Chen W (2004) Sequential optimization and reliability assessment method for efficient arobabilistic design. ASME J Mech Des 126(2):225–233

    Article  Google Scholar 

  • Du XP, Sudjianto A (2003) Reliability-based design with the mixture of random and interval variables. Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Chicago, Illinois, USA

  • Elishakoff I (1995) Discussion on a non-probabilistic concept of reliability”. Struct Saf 17(3):195–199

    Article  Google Scholar 

  • Elishakoff I, Colombi P (1993) combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Comput Methods Appl Mech Eng 104(2):187–209

    Article  MathSciNet  MATH  Google Scholar 

  • Elishakoff I, Colombi P (1994) Ideas of probability and convexity combined for analyzing parameter uncertainty. In: Proceedings of the 6th international conference on structural safety and reliability, Rotterdam: Balkema Publishers

  • Enevoldsen I, Sørensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15(3):169–196

    Article  Google Scholar 

  • Fang Y, Xiong J, Tee KF (2014) An iterative hybrid random-interval structural reliability analysis. Earthq Struct 7(6):1061–1070

    Article  Google Scholar 

  • Guo J, Du XP (2009) Reliability sensitivity analysis with random and interval variables. Int J Numer Meth Eng 78(13):1585–1617

    Article  MathSciNet  MATH  Google Scholar 

  • Guo SX, Lu ZZ (2002) Hybrid probabilistic and non-probabilistic model of structural reliability. Chin J Mech Strength 24(4):524–526 (in Chinese)

    Google Scholar 

  • Hadim H, Suwa T (2008) Multidisciplinary design and optimization methodologies in electronics packaging: state-of-the-art review. J Electron Packag 130(3):1504–1508

    Article  Google Scholar 

  • Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121

    Google Scholar 

  • Hirohata K, Hisano K, Takahashi H (2006) Reliability design method for solder joints based on coupled thermal-stress analysis of electronics packaging structure (Thermal Design/ Mechanical Stress Design and Simulation Technologies to Support System JISSO). J Jpn Inst Electron Packag 2006(9):405–412

    Article  Google Scholar 

  • Huang ZL, Jiang C, Zhou YS, Luo Z, Zhang Z (2016) An incremental shifting vector approach for reliability-based design optimization. Struct Multidisc Optim 53(3):523–543

  • Jiang C, Deng SL (2014) Multi-objective optimization and design considering automotive high-speed and low-speed crashworthiness. Chin J Comput Mech 31(04):474–479 (in Chinese)

    Google Scholar 

  • Jiang C, Li W, Han X, Liu LX (2011) Structural reliability analysis based on random distributions with interval parameters. Comput Struct 89(23–24):2292–2302

    Article  Google Scholar 

  • Jiang C, Han X, Li WX, Zhang Z (2012) A hybrid reliability approach based on probability and interval for uncertain structures. ASME J Mech Des 134(3):1–11

    Article  Google Scholar 

  • Kang Z, Luo YJ (2010) Reliability-based structural optimization with probability and convex set hybrid models. Struct Multidiscip Optim 42(1):89–102

    Article  Google Scholar 

  • Kim C (2008) Reliability-based design optimization using response surface method with prediction interval estimation. ASME J Mech Des 130(12):1786–1787

    Article  Google Scholar 

  • Kuschel N, Rackwitz R (1997) Two basic problems in reliability-based structural optimization. Math Meth Oper Res 46(3):309–333

    Article  MathSciNet  MATH  Google Scholar 

  • Li G, Cheng GD (2001) Optimal decision for the target value of performance based structural system reliability. Struct Multidiscip Optim 22(4):261–267

  • Li F, Luo Z, Rong J, Hu L (2013) A non-probabilistic reliability-based optimization of structures using convex models. Comput Model Eng Sci 95(6):453–482

    MathSciNet  Google Scholar 

  • Liang J, Mourelatos ZP, Tu J (2004) A single-loop method for reliability-based design optimization. Proceedings of ASME Design Engineering Technical Conferences, Salt Lake City, UT

  • Luo YJ, Kang Z, Luo Z, Li A (2008) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscip Optim 39(3):297–310

    Article  MathSciNet  MATH  Google Scholar 

  • Madsen HO, Krenk S, Lind NC (2006) Methods of structural safety. Courier Corporation

  • Penmetsa RC, Grandhi RV (2002) Efficient estimation of structural reliability for problems with uncertain intervals. Comput Struct 80(12):1103–1112

    Article  Google Scholar 

  • Qiu ZP, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. Int J Solids Struct 45(10):2850–2860

    Article  MATH  Google Scholar 

  • Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494

    Article  MATH  Google Scholar 

  • Roger F (2000) Practical methods of optimization. John Wiley & Sons Ltd., New York

    Google Scholar 

  • Shan S, Wang G (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Sys Saf 93(8):1218–1230

    Article  Google Scholar 

  • Shan SQ, Wang GG (2009) Reliable space pursuing for reliability-based design optimization with black-box performance functions. Chin J Mech Eng 01(1):27–35

    Article  Google Scholar 

  • Wang J, Qiu Z (2010) The reliability analysis of probabilistic and interval hybrid structural system. Appl Math Model 34(11):3648–3658

    Article  MathSciNet  MATH  Google Scholar 

  • Wu YT, Wang W (1998) Efficient probabilistic design by converting reliability constraints to一个例子approximately equivalent deterministic constraints. J Integr Des Process Sci 2(4):13–21

    Google Scholar 

  • Wu YT, Millwater HR, Cruse TA (1990) Advanced probabilistic structural analysis method for implicit performance functions. AIAA J 28(9):1663–1669

    Article  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 

  • Zhang H, Dai H, Beer M, Wang W (2013) Structural reliability analysis on the basis of small samples: an interval Quasi-Monte Carlo method. Mech Syst Signal Process 37(1):137–151

    Article  Google Scholar 

  • Zhu LP, Elishakoff I (1996) Hybrid probabilistic and convex modeling of excitation and response of periodic structures. Math Prob Eng 2(2):143–163

    Article  MATH  Google Scholar 

  • Zhuang XT, Pan R (2012) A sequential sampling strategy to improve reliability-based design optimization with implicit constraint functions. ASME J Mech Des 134(2):55–58

    Article  Google Scholar 

Download references

Acknowledgments

This study is supported by the National Science Foundation for Excellent Young Scholars (51222502), the Key Project of Chinese National Programs for Fundamental Research and Development (2012AA111710), and the National Science Foundation of China (11172096).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha City, People’s Republic of China, 410082

    Z. L. Huang, C. Jiang, Y. S. Zhou, J. Zheng & X. Y. Long

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  1. Z. L. Huang
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  2. C. Jiang
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  3. Y. S. Zhou
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  4. J. Zheng
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  5. X. Y. Long
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Correspondence to C. Jiang.

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Huang, Z.L., Jiang, C., Zhou, Y.S. et al. Reliability-based design optimization for problems with interval distribution parameters. Struct Multidisc Optim 55, 513–528 (2017). https://doi.org/10.1007/s00158-016-1505-3

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  • DOI: https://doi.org/10.1007/s00158-016-1505-3

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