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An improved decomposition method in probabilistic analysis using Chebyshev approximations

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Abstract

This paper presents an effective method for solving stochastic problems commonly encountered in structural engineering and applied mechanics. Design of experiments is conducted to obtain the response values of interest, where Chebyshev nodes are applied to collect discrete samples. Then Chebyshev polynomials are applied to approximate the univariate functional relationship between each variable and the structural response. The univariate dimension reduction method and saddlepoint approximation with truncated cumulant generating functions are employed to estimate the statistic cumulants, probability density function and cumulative distribution function of the response. The results of numerical examples indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the probabilistic characteristics of random mathematical functions or the responses of structural systems.

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References

  • Butler RW (2007) Saddlepoint approximations with applications. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Choi J, An D, Won J (2010) Bayesian approach for structural reliability analysis and optimization using Kriging dimension reduction method. ASME J Mech Des 132(5):051003

    Article  Google Scholar 

  • Daniels HE (1954) Saddlepoint approximations in statistics. Ann Math Stat 25(4):631–650

    Article  MathSciNet  MATH  Google Scholar 

  • Du X (2008) Saddlepoint approximation for sequential optimization and reliability analysis. ASME J Mech Des 130(1):011011

    Article  Google Scholar 

  • Du X (2010) System reliability analysis with saddlepoint approximation. Struct Multidiscipl Optim 42(2):193–208

    Article  Google Scholar 

  • Du XP, Chen W (2001) A most probable point based method for uncertainty analysis. J Design Manuf Autom 4(1):47–66

    Article  Google Scholar 

  • Du X, Sudjianto A (2004) The first order saddlepoint approximation for reliability analysis. AIAA J 42(6):1199–1207

    Article  Google Scholar 

  • Easton GS, Ronchetti E (1986) General saddlepoint approximations with applications to L statistics. J Am Stat Assoc 81:321–337

    Article  MathSciNet  Google Scholar 

  • Engelund S, Rackwitz R (1993) A benchmark study on importance sampling techniques in structural reliability. Struct Saf 4(12):255–276

    Article  Google Scholar 

  • Gillespie CS, Renshaw E (2007) An improved saddlepoint approximation. Math Biosci 208(2):359–374

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Hohenbichler M, Gollwitzer S, Kruse W et al (1987) New light on first- and second-order reliability methods. Struct Saf 4(4):267–284

    Article  Google Scholar 

  • Huang BQ, Du XP (2006) Uncertainty analysis by dimension reduction integration and saddlepoint approximations. ASME J Mech Des 128(1):26–33

    Article  Google Scholar 

  • Huang B, Du X (2008) Probabilistic uncertainty analysis by mean-value first order saddlepoint approximation. Reliab Eng Syst Saf 93(2):325–336

    Article  MathSciNet  Google Scholar 

  • Kiureghian AD (2008) Analysis of structural reliability under parameter uncertainties. Probab Eng Mech 23(4):351–358

    Article  Google Scholar 

  • Laumakis PJ, Harlow G (2002) Structural reliability and Monte Carlo simulation. Int J Math Educ Sci Technol 33(3):377–387

    Article  Google Scholar 

  • Lee SH, Chen W (2009) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidiscipl Optim 37(3):239–253

    Article  MathSciNet  Google Scholar 

  • Lee I, Choi KK, Du L et al (2008) Dimension reduction method for reliability-based robust design optimization. Comput Struct 86(13–14):1550–1562

    Article  Google Scholar 

  • Lee I, Choi KK, Gorsich D (2010) Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO). Int J Numer Methods Eng 82(1):26–46

    MATH  Google Scholar 

  • Li J, Chen JB (2009) Stochastic dynamics of structures. Wiley, Singapore

    Book  MATH  Google Scholar 

  • Lugannani R, Rice SO (1980) Saddlepoint approximation for the distribution of the sum of independent random variables. Adv Appl Probab 12:475–490

    Article  MathSciNet  MATH  Google Scholar 

  • Mason JC, Handscomb DC (2003) Chebyshev polynomials. Champman & Hall, New York

    MATH  Google Scholar 

  • Mathews JH, Fink KD (2004) Numerical methods using MATLAB, 4th edn. Prentice-Hall, Upper Saddle River

    Google Scholar 

  • Melchers RE, Ahammed M (2004) A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability. Comput Struct 82(1):55–61

    Article  Google Scholar 

  • Pellissetti MF, Schuëller GI, Pradlwarter HJ, Calvi A, Fransen S, Klein M (2006) Reliability analysis of spacecraft structures under static and dynamic loading. Comput Struct 84(21):1313–1325

    Article  Google Scholar 

  • Picheny V, Kim NH, Haftka RT (2010) Application of bootstrap method in conservative estimation of reliability with limited samples. Struct Multidiscipl Optim 41(2):205–217

    Article  MathSciNet  Google Scholar 

  • Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408

    Article  Google Scholar 

  • Resenblueth E (1981) Two-point estimates in probabilities. Appl Math Model 5(5):329–335

    Article  Google Scholar 

  • Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23(3):470–472

    Article  MathSciNet  MATH  Google Scholar 

  • Smyth GK, Podlich HM (2002) An improved saddlepoint approximation based on the negative binomial distribution for the general birth process. Comput Stat 17(1):17–27

    Article  MathSciNet  MATH  Google Scholar 

  • Süli E, Mayers DF (2003) An introduction to numerical analysis. Cambridge University Press, New York

    MATH  Google Scholar 

  • Wang S (1992) General saddlepoint approximations in the bootstrap. Probab Stat Lett 13(1):61–66

    Article  Google Scholar 

  • Wang L, Beeson D, Wiggs G (2004) Efficient and accurate point estimate method for moments and probability distributions. Proc. 10th AIAA/ISSMO Multidisciplinary analysis and optimization conference, Albany, NY, August 30–September 1, AIAA 2004–4359

  • Wei D, Rahman S (2007) Structural reliability analysis by univariate decomposition and numerical integration. Probab Eng Mech 22(1):27–38

    Article  Google Scholar 

  • Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Methods Eng 61(12):1992–2019

    Article  MATH  Google Scholar 

  • Xu H, Rahman S (2005) Decomposition methods for structural reliability analysis. Probab Eng Mech 20(3):239–250

    Article  MathSciNet  Google Scholar 

  • Youn BD, Xi ZM (2009) Reliability-based robust design optimization using the eigenvector dimension reduction (EDR) method. Struct Multidiscipl Optim 37(5):475–492

    Article  MathSciNet  Google Scholar 

  • Youn BD, Choi KK, Du L (2005) Adaptive probability analysis using an enhanced hybrid mean value method. Struct Multidiscipl Optim 29(2):134–148

    Article  Google Scholar 

  • Youn BD, Xi ZM, Wang PF (2008) Eigenvector dimension reduction method for sensitivity-free probability analysis. Struct Multidiscipl Optim 37(1):13–28

    Article  MathSciNet  Google Scholar 

  • Zhang YM, Chen SH, Liu QL, Liu TQ (1996) Stochastic perturbation finite elements. Comput Struct 59(3):425–429

    Article  MATH  Google Scholar 

  • Zhao YG, Ono T (2000) New point estimates for probability moments. J Eng Mech 126(4):433–436

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of Fundamental Research Funds for the Central Universities No. N090603002, Key National Science & Technology Special Project on “High-Grade CNC Machine Tools and Basic Manufacturing Equipment” No. 2010ZX04014-014 and and Program for Changjiang Scholars and Innovative Research Team in University No. IRT0816. Moreover, all comments and suggestions from editors and reviewers are gratefully acknowledged.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, People’s Republic of China

    Xianzhen Huang, Yimin Zhang, Yajuan Jin & Hao Lu

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  1. Xianzhen Huang
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  2. Yimin Zhang
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  3. Yajuan Jin
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  4. Hao Lu
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Correspondence to Xianzhen Huang.

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Huang, X., Zhang, Y., Jin, Y. et al. An improved decomposition method in probabilistic analysis using Chebyshev approximations. Struct Multidisc Optim 43, 785–797 (2011). https://doi.org/10.1007/s00158-010-0606-7

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  • DOI: https://doi.org/10.1007/s00158-010-0606-7

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