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A two-step methodology to apply low-discrepancy sequences in reliability assessment of structural dynamic systems

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Abstract

This study introduces various low-discrepancy sequences and then develops a new methodology for reliability assessment for structural dynamic systems. In this methodology, a two-step algorithm is first proposed, in which the most uniformly scattered point set among the low-discrepancy sequences is selected according to the centered L2-discrepancy (CL2 discrepancy) and then rearranged to minimize the generalized F-discrepancy (GF discrepancy). After that, the developed point set is incorporated into the maximum entropy method to capture the fractional moments for deriving the extreme value distribution for reliability assessment of structural dynamic systems. Numerical examples are investigated, where the results are compared with those obtained from Monte Carlo simulations, demonstrating the accuracy and efficiency of the proposed methodology.

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References

  • Au S, Beck J (2003) Subset simulation and its application to seismic risk based on dynamic analysis. J Eng Mech 129(8):901–917

    Article  Google Scholar 

  • Brandimarte P (2014) Low-Discrepancy Sequences. Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics, pp 379–401

  • Bratley P, Fox BL (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator. ACM Trans Math Softw (TOMS) 14(1):88–100

    Article  MATH  Google Scholar 

  • Burkardt J (2015) MATLAB Source Codes. http://people.sc.fsu.edu/~jburkardt/m_src/m_src.html

  • Chen JB, Li J (2007) The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Struct Saf 29(2):77–93

    Article  Google Scholar 

  • Chen Jb, Zhang Sh (2013) Improving point selection in cubature by a new discrepancy. SIAM J Sci Comput 35(5):A2121–A2149

    Article  MathSciNet  MATH  Google Scholar 

  • Chen JB, Ghanem R, Li J (2009) Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures. Probabilist Eng Mech 24(1):27–42

    Article  Google Scholar 

  • Chen J, Yang J, Li J (2016) A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters. Struct Saf 59:20–31

    Article  Google Scholar 

  • Conway JH, Sloane NJA (2013) Sphere packings, lattices and groups, vol 290. Springer Science & Business Media, Berlin

    MATH  Google Scholar 

  • Dai H, Wang W (2009) Application of low-discrepancy sampling method in structural reliability analysis. Struct Saf 31(1):55–64

    Article  Google Scholar 

  • Dick J, Pillichshammer F (2010) Digital nets and sequences: discrepancy theory and quasi–Monte Carlo integration. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Faure H (1992) Good permutations for extreme discrepancy. J Number Theory 42(1):47–56

    Article  MathSciNet  MATH  Google Scholar 

  • Goller B, Pradlwarter HJ, Schuller GI (2013) Reliability assessment in structural dynamics. J Sound Vib 332(10):2488–2499

    Article  Google Scholar 

  • Halton JH (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2(1):84–90

    Article  MathSciNet  MATH  Google Scholar 

  • Harald N (1992) Random number generation and quasi-Monte Carlo methods. Society for lndustrial and Applied Mathematics, Philadelphia

    MATH  Google Scholar 

  • Hess S, Polak J (2003) An alternative method to the scrambled Halton sequence for removing correlation between standard Halton sequences in high dimensions. Plant Cell, 15(3):760-770.

  • Hickernell F, Wang X (2002) The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension. Math Comput 71(240):1641–1661

    Article  MathSciNet  MATH  Google Scholar 

  • Hu Z, Du X (2013) A sampling approach to extreme value distribution for time-dependent reliability analysis. J Mech Des 135(7):071003

    Article  Google Scholar 

  • Hu Z, Du X (2015a) First order reliability method for time-variant problems using series expansions. Struct Multidiscip Optim 51(1):1–21

    Article  MathSciNet  Google Scholar 

  • Hu Z, Du X (2015b) Mixed efficient global optimization for time-dependent reliability analysis. J Mech Des 137(5):051401

    Article  Google Scholar 

  • Hua L-K, Wang Y (2012) Applications of number theory to numerical analysis. Springer Science & Business Media, Berlin

    Google Scholar 

  • Hua LK, Yuan W (1981) Applications of number theory to numerical analysis. Springer, Berlin

    MATH  Google Scholar 

  • Iourtchenko DV, Mo E, Naess A (2006) Response probability density functions of strongly non-linear systems by the path integration method. Int J Non Linear Mech 41(5):693–705

    Article  MathSciNet  MATH  Google Scholar 

  • Joe S, Kuo FY (2003) Remark on algorithm 659: implementing Sobol's quasirandom sequence generator. ACM Trans Math Softw (TOMS) 29(1):49–57

    Article  MathSciNet  MATH  Google Scholar 

  • Joe S, Kuo FY (2008) Notes on generating Sobol sequences. Technical report, University of New South Wales.

  • Kapur JN, Kesavan HK (1992) Entropy optimization principles with applications. Academic Pr, Cambridge

    Book  MATH  Google Scholar 

  • Kocis L, Whiten WJ (1997) Computational investigations of low-discrepancy sequences. ACM Trans Math Softw (TOMS) 23(2):266–294

    Article  MATH  Google Scholar 

  • Kougioumtzoglou IA, Spanos PD (2012) Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. J Eng Mech 139(9):1207–1217

    Article  Google Scholar 

  • Li J, Chen JB (2009) Stochastic dynamics of structures. John Wiley & Sons, Hoboken

    Book  MATH  Google Scholar 

  • Li J, Chen JB, Fan WL (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 29(2):112–131

    Article  Google Scholar 

  • Madsen PH, Krenk S (1984) An integral equation method for the first-passage problem in random vibration. J Appl Mech 51(3):674–679

    Article  MathSciNet  MATH  Google Scholar 

  • Mourelatos ZP, Majcher M, Pandey V, Baseski I (2015) Time-dependent reliability analysis using the Total probability theorem. J Mech Des 137(3):031405

    Article  Google Scholar 

  • Naess A, Iourtchenko D, Batsevych O (2011) Reliability of systems with randomly varying parameters by the path integration method. Probabilist Eng Mech 26(1):5–9

    Article  MATH  Google Scholar 

  • Nie J, Ellingwood BR (2004) A new directional simulation method for system reliability. Part I: application of deterministic point sets. Probabilist Eng Mech 19(4):425–436

    Article  Google Scholar 

  • Preumont A (1985) On the peak factor of stationary Gaussian processes. J Sound Vib 100(1):15–34

    Article  MathSciNet  Google Scholar 

  • Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 24(1):46–156

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson D, Atcitty C (1999) Comparison of quasi- and pseudo-Monte Carlo sampling for reliability and uncertainty analysis. In: Proceedings of the AIAA probabilistic methods conference, St. Louis, MO. AIAA99-1589.

  • Singh A, Mourelatos Z, Nikolaidis E (2011) Time-dependent reliability of random dynamic systems using time-series modeling and importance sampling. SAE Technical Paper

  • Song PY, Chen JB (2015) Point selection strategy based on minimizing GL2-discrepancy and its application to multi-dimensional integration. Chin Sci 45:547–558 (in Chinese)

    Google Scholar 

  • Spanos PD, Kougioumtzoglou IA (2014) Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation. J Appl Mech 81(5):051016

    Article  Google Scholar 

  • Tont G, Vladareanu L, Munteanu MS, Tont DG (2010) Markov approach of adaptive task assignment for robotic system in non-stationary environments. WSEAS Trans Circuits Syst 9(3):273–282

    MATH  Google Scholar 

  • Tuffin B (1996) On the use of low discrepancy sequences in Monte Carlo methods. Monte Carlo Methods Appl 2:295–320

    Article  MathSciNet  MATH  Google Scholar 

  • van Noortwijk JM, van der Weide JA, Kallen M-J, Pandey MD (2007) Gamma processes and peaks-over-threshold distributions for time-dependent reliability. Reliab Eng Syst Safe 92(12):1651–1658

    Article  Google Scholar 

  • Vanmarcke EH (1975) On the distribution of the first-passage time for normal stationary random processes. J Appl Mech 42(1):215–220

    Article  MATH  Google Scholar 

  • Wang X, Hickernell FJ (2000) Randomized halton sequences. Math Comput Model 32(7):887–899

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Z, Wang P (2012) Reliability-based product design with time-dependent performance deterioration. Prognostics and Health Management (PHM), 2012 I.E. Conference on, IEEE

  • Wen Y-K (1976) Method for random vibration of hysteretic systems. J Eng Mech Div 102(2):249–263

    Google Scholar 

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

  • Xu J, Chen JB, Li J (2012) Probability density evolution analysis of engineering structures via cubature points. Comput Mech 50(1):135–156

    Article  MathSciNet  MATH  Google Scholar 

  • Xu J, Zhang W, Sun R (2016) Efficient reliability assessment of structural dynamic systems with unequal weighted quasi-Monte Carlo simulation. Comput Struct 175:37–51

    Article  Google Scholar 

  • Xu J, Dang C, Kong F (2017) Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods. Mech Syst Signal Process 95:58–76

    Article  Google Scholar 

  • Zhang X, Pandey MD (2013) Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct Saf 43:28–40

    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 

  • Zhang X, Pandey MD, Zhang Y (2014) Computationally efficient reliability analysis of mechanisms based on a multiplicative dimensional reduction method. J Mech Des 136(6):061006

    Article  Google Scholar 

Download references

Acknowledgements

The support of the National Natural Science Foundation of China (Grant No.: 51608186) and the Fundamental Research Funds for the Central Universities (No.531107040890) is highly appreciated. The anonymous reviewers are greatly acknowledged for their constructive criticisms to the original version of the paper.

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Xu, J., Wang, D. A two-step methodology to apply low-discrepancy sequences in reliability assessment of structural dynamic systems. Struct Multidisc Optim 57, 1643–1662 (2018). https://doi.org/10.1007/s00158-017-1834-x

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  • DOI: https://doi.org/10.1007/s00158-017-1834-x

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