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A multi-fidelity integration rule for statistical moments and failure probability evaluations

Abstract

This paper presents a multi-fidelity integration rule for statistical moments and failure probability evaluations. The contribution-degree analysis is first conducted for dividing the random inputs as relatively important and unimportant ones, where the multi-dimensional Gaussian-weighted integral respect to moments estimation can be separated into two lower-dimensional integrals in an additive form. A flexible spherical-radial cubature rule is derived to evaluate the integral consisting of important random variables, where the free parameter is optimally determined via a moment-matching strategy. A low-degree spherical-radial cubature rule, whose algebraic degree of accuracy is between 3 and 5, is then applied to estimate the integral related to unimportant variables. In this regard, a multi-fidelity integration rule, where different numerical schemes are employed, is established accordingly for estimating the statistical moments of the limit state function, which can ensure the balance of precision and efficiency. The maximum entropy method is then applied to obtain the entire probability distribution of the limit state function based on the statistical moments, where the failure probability can be straightforwardly assessed. The efficiency and accuracy of the proposed method are demonstrated through five numerical examples for both the statistical moments and failure probability evaluations.

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References

  • Ahmadabadi M, Poisel R (2015) Assessment of the application of point estimate methods in the probabilistic stability analysis of slopes. Comput Geotech 69:540–550

    Article  Google Scholar 

  • Arasaratnam I, Haykin S (2009) Cubature kalman filters. IEEE Trans Autom Control 54 (6):1254–1269

    MathSciNet  Article  Google Scholar 

  • Bordbari MJ, Seifi AR, Rastegar M (2018) Probabilistic energy consumption analysis in buildings using point estimate method. Energy 142:716–722

    Article  Google Scholar 

  • Borgonovo E (2010) Sensitivity analysis with finite changes: An application to modified eoq models. Eur J Oper Res 200(1):127–138

    Article  Google Scholar 

  • Bressolette P, Fogli M, Chauvière C (2010) A stochastic collocation method for large classes of mechanical problems with uncertain parameters. Probab Eng Mechan 25(2):255–270

    Article  Google Scholar 

  • Chang CH, Yang JC, Tung YK (1997) Uncertainty analysis by point estimate methods incorporating marginal distributions. J Hydraul Eng 123(3):244–250

    Article  Google Scholar 

  • Che-Hao C, Yeou-Koung T, Jinn-Chuang Y (1995) Evaluation of probability point estimate methods. Appl Math Model 19(2):95– 105

    Article  Google Scholar 

  • Chen X, Lin Z (2014). Theory, Structural nonlinear analysis program opensees. Tutorial China Architecture & Building press: Beijing, China 87–89

  • Fan W, Liu R, Ang AH-S, Li Z (2018) A new point estimation method for statistical moments based on dimension-reduction method and direct numerical integration. Appl Math Model 62:664–679

    MathSciNet  Article  Google Scholar 

  • Fan W, Wei J, Ang AH-S, Li Z (2016) Adaptive estimation of statistical moments of the responses of random systems. Probab Eng Mechan 43:50–67

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Harr ME (1989) Probabilistic estimates for multivariate analyses. Appl Math Model 13(5):313–318

    Article  Google Scholar 

  • He J, Gao S, Gong J (2014) A sparse grid stochastic collocation method for structural reliability analysis. Struct Saf 51:29–34

    Article  Google Scholar 

  • He W, Li G, Hao P, Zeng Y (2019) Maximum entropy method-based reliability analysis with correlated input variables via hybrid dimension-reduction method. J Mechan Design 141(10)

  • Jia XY, Jiang C, Fu CM, Ni BY, Wang CS, Ping MH (2019) Uncertainty propagation analysis by an extended sparse grid technique. Front Mechan Eng 14(1):33–46

    Article  Google Scholar 

  • Jia B, Xin M, Cheng Y (2013) High-degree cubature kalman filter. Automatica 49(2):510–518

    MathSciNet  Article  Google Scholar 

  • Keshtegar B, Zhu S-P (2019) Three-term conjugate approach for structural reliability analysis. Appl Math Model 76:428–442

    MathSciNet  Article  Google Scholar 

  • Kiureghian AD, Stefano MD (1991) Efficient algorithm for second-order reliability analysis. J Eng Mech 117(12):2904–2923

    Article  Google Scholar 

  • Liu R, Fan W, Wang Y, Ang AH-S, Li Z (2019) Adaptive estimation for statistical moments of response based on the exact dimension reduction method in terms of vector. Mech Syst Signal Process 126:609–625

    Article  Google Scholar 

  • Liu J, Hu Y, Xu C, Jiang C, Han X (2016) Probability assessments of identified parameters for stochastic structures using point estimation method. Reliab Eng Syst Safe 156:51–58

    Article  Google Scholar 

  • Meng D, Miao L, Shao H, Shen J (2018) A seventh-degree cubature kalman filter. Asian J Control 20(1):250–262

    MathSciNet  Article  Google Scholar 

  • Minzliana Z, Lixin W, Weiwei Q (2018) Seventh-degree spherical simplex-radial cubature kalman filter. In: 2018 37th Chinese Control Conference (CCC), pp 965–970

  • Mysovskikh I (1980) The approximation of multiple integrals by using interpolatory cubature formulae. In: Quantitative Approximation, pp 217–243

  • Rahman S, Wei D (2006) A univariate approximation at most probable point for higher-order reliability analysis. Int J Solids Struct 43(9):2820–2839

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Rosenblueth E (1975) Point estimates for probability moments. Proc Natl Acad Sci U S A 72 (10):3812–3814

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Wang S, Feng Y, Duan S, Wang L (2017) Mixed-degree spherical simplex-radial cubature kalman filter. Math Prob Eng

  • Wang S, Feng J, Tse CK (2014) Spherical simplex-radial cubature kalman filter. IEEE Signal Process Lett 21(1):43–46

    Article  Google Scholar 

  • Wang Y, Tung Y-K (2009) Improved probabilistic point estimation schemes for uncertainty analysis. Appl Math Model 33(2):1042–1057

    Article  Google Scholar 

  • Zhang WX, Lu ZZ (2018) An inequality unscented transformation for estimating the statistical moments. Appl Math Model 62:21–37

    MathSciNet  Article  Google Scholar 

  • Xiao Q, He Y, Chen K, Yang Y, Lu Y (2017) Point estimate method based on univariate dimension reduction model for probabilistic power flow computation, IET Generation. Transmis Distribut 11 (14):3522–3531

    Article  Google Scholar 

  • Xiao S, Lu Z (2018) Reliability analysis by combining higher-order unscented transformation and fourth-moment method, vol 4 , p 04017034

  • Xiong F, Greene S, Chen W, Xiong Y, Yang S (2009) A new sparse grid based method for uncertainty propagation. Struct Multidiscipl Optim 41(3):335–349

    MathSciNet  Article  Google Scholar 

  • Xu J (2016) A new method for reliability assessment of structural dynamic systems with random parameters. Struct Saf 60:130– 143

    Article  Google Scholar 

  • Xu J, Dang C (2019) A new bivariate dimension reduction method for efficient structural reliability analysis. Mech Syst Signal Process 115:281–300

    Article  Google Scholar 

  • Xu J, Kong F (2019) An efficient method for statistical moments and reliability assessment of structures. Struct Multidiscipl Optim 58(5):2019–2035

    MathSciNet  Article  Google Scholar 

  • Xu J, Lu Z-H (2017) Evaluation of moments of performance functions based on efficient cubature formulation. J Eng Mechan 143(8):06017007

    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  Google Scholar 

  • Xu J, Zhang Y, Dang C (2020) A novel hybrid cubature formula with pearson system for efficient moment-based uncertainty propagation analysis, vol 140, p 106661

  • Zhang Y, Huang Y, Wu Z, Li N (2014) Seventh-degree spherical simplex-radial cubature kalman filter. In: Proceedings of the 33rd Chinese Control Conference, pp 2513–2517

  • Zhao Y-G, Ono T (2000) Third-moment standardization for structural reliability analysis. J Struct Eng 126(6):724–732

    Article  Google Scholar 

  • Zhou Q, Li Z, Fan W, Ang AH-S, Liu R (2017) System reliability assessment of deteriorating structures subjected to time-invariant loads based on improved moment method. Struct Saf 68:54– 64

    Article  Google Scholar 

  • Zhou J, Nowak AS (1988) Integration formulas to evaluate functions of random variables. Struct Saf 5(4):267–284

    Article  Google Scholar 

Download references

Funding

The National Natural Science Foundation of China (No. 51978253) and the Fundamental Research Funds for the Central Universities (No. 531107040110) are gratefully appreciated for the financial support of this research.

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Correspondence to Jun Xu.

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Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

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Responsible Editor: Yoojeong Noh

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Xu, J., Du, Y. & Zhou, L. A multi-fidelity integration rule for statistical moments and failure probability evaluations. Struct Multidisc Optim 64, 1305–1326 (2021). https://doi.org/10.1007/s00158-021-02919-x

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  • DOI: https://doi.org/10.1007/s00158-021-02919-x

Keywords

  • Flexible spherical-radial cubature rule
  • Contribution-degree analysis
  • Statistical moments
  • Probability distribution
  • Failure probability