Structural and Multidisciplinary Optimization

, Volume 57, Issue 4, pp 1643–1662 | Cite as

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

  • Jun Xu
  • Ding Wang


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.


Low-discrepancy sequence Reliability Structural dynamic systems CL2 discrepancy GF discrepancy Extreme value distribution Fractional moments 



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.


  1. Au S, Beck J (2003) Subset simulation and its application to seismic risk based on dynamic analysis. J Eng Mech 129(8):901–917CrossRefGoogle Scholar
  2. Brandimarte P (2014) Low-Discrepancy Sequences. Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics, pp 379–401Google Scholar
  3. Bratley P, Fox BL (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator. ACM Trans Math Softw (TOMS) 14(1):88–100CrossRefzbMATHGoogle Scholar
  4. Burkardt J (2015) MATLAB Source Codes.
  5. Chen JB, Li J (2007) The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Struct Saf 29(2):77–93CrossRefGoogle Scholar
  6. Chen Jb, Zhang Sh (2013) Improving point selection in cubature by a new discrepancy. SIAM J Sci Comput 35(5):A2121–A2149MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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):27–42CrossRefGoogle Scholar
  8. 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 59:20–31CrossRefGoogle Scholar
  9. Conway JH, Sloane NJA (2013) Sphere packings, lattices and groups, vol 290. Springer Science & Business Media, BerlinzbMATHGoogle Scholar
  10. Dai H, Wang W (2009) Application of low-discrepancy sampling method in structural reliability analysis. Struct Saf 31(1):55–64CrossRefGoogle Scholar
  11. Dick J, Pillichshammer F (2010) Digital nets and sequences: discrepancy theory and quasi–Monte Carlo integration. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  12. Faure H (1992) Good permutations for extreme discrepancy. J Number Theory 42(1):47–56MathSciNetCrossRefzbMATHGoogle Scholar
  13. Goller B, Pradlwarter HJ, Schuller GI (2013) Reliability assessment in structural dynamics. J Sound Vib 332(10):2488–2499CrossRefGoogle Scholar
  14. Halton JH (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2(1):84–90MathSciNetCrossRefzbMATHGoogle Scholar
  15. Harald N (1992) Random number generation and quasi-Monte Carlo methods. Society for lndustrial and Applied Mathematics, PhiladelphiazbMATHGoogle Scholar
  16. 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.Google Scholar
  17. Hickernell F, Wang X (2002) The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension. Math Comput 71(240):1641–1661MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hu Z, Du X (2013) A sampling approach to extreme value distribution for time-dependent reliability analysis. J Mech Des 135(7):071003CrossRefGoogle Scholar
  19. Hu Z, Du X (2015a) First order reliability method for time-variant problems using series expansions. Struct Multidiscip Optim 51(1):1–21MathSciNetCrossRefGoogle Scholar
  20. Hu Z, Du X (2015b) Mixed efficient global optimization for time-dependent reliability analysis. J Mech Des 137(5):051401CrossRefGoogle Scholar
  21. Hua L-K, Wang Y (2012) Applications of number theory to numerical analysis. Springer Science & Business Media, BerlinGoogle Scholar
  22. Hua LK, Yuan W (1981) Applications of number theory to numerical analysis. Springer, BerlinzbMATHGoogle Scholar
  23. 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):693–705MathSciNetCrossRefzbMATHGoogle Scholar
  24. Joe S, Kuo FY (2003) Remark on algorithm 659: implementing Sobol's quasirandom sequence generator. ACM Trans Math Softw (TOMS) 29(1):49–57MathSciNetCrossRefzbMATHGoogle Scholar
  25. Joe S, Kuo FY (2008) Notes on generating Sobol sequences. Technical report, University of New South Wales.Google Scholar
  26. Kapur JN, Kesavan HK (1992) Entropy optimization principles with applications. Academic Pr, CambridgeCrossRefzbMATHGoogle Scholar
  27. Kocis L, Whiten WJ (1997) Computational investigations of low-discrepancy sequences. ACM Trans Math Softw (TOMS) 23(2):266–294CrossRefzbMATHGoogle Scholar
  28. Kougioumtzoglou IA, Spanos PD (2012) Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. J Eng Mech 139(9):1207–1217CrossRefGoogle Scholar
  29. Li J, Chen JB (2009) Stochastic dynamics of structures. John Wiley & Sons, HobokenCrossRefzbMATHGoogle Scholar
  30. Li J, Chen JB, Fan WL (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 29(2):112–131CrossRefGoogle Scholar
  31. Madsen PH, Krenk S (1984) An integral equation method for the first-passage problem in random vibration. J Appl Mech 51(3):674–679MathSciNetCrossRefzbMATHGoogle Scholar
  32. Mourelatos ZP, Majcher M, Pandey V, Baseski I (2015) Time-dependent reliability analysis using the Total probability theorem. J Mech Des 137(3):031405CrossRefGoogle Scholar
  33. Naess A, Iourtchenko D, Batsevych O (2011) Reliability of systems with randomly varying parameters by the path integration method. Probabilist Eng Mech 26(1):5–9CrossRefzbMATHGoogle Scholar
  34. 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):425–436CrossRefGoogle Scholar
  35. Preumont A (1985) On the peak factor of stationary Gaussian processes. J Sound Vib 100(1):15–34MathSciNetCrossRefGoogle Scholar
  36. Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 24(1):46–156MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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.Google Scholar
  38. Singh A, Mourelatos Z, Nikolaidis E (2011) Time-dependent reliability of random dynamic systems using time-series modeling and importance sampling. SAE Technical PaperGoogle Scholar
  39. 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
  40. Spanos PD, Kougioumtzoglou IA (2014) Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation. J Appl Mech 81(5):051016CrossRefGoogle Scholar
  41. 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):273–282zbMATHGoogle Scholar
  42. Tuffin B (1996) On the use of low discrepancy sequences in Monte Carlo methods. Monte Carlo Methods Appl 2:295–320MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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):1651–1658CrossRefGoogle Scholar
  44. Vanmarcke EH (1975) On the distribution of the first-passage time for normal stationary random processes. J Appl Mech 42(1):215–220CrossRefzbMATHGoogle Scholar
  45. Wang X, Hickernell FJ (2000) Randomized halton sequences. Math Comput Model 32(7):887–899MathSciNetCrossRefzbMATHGoogle Scholar
  46. Wang Z, Wang P (2012) Reliability-based product design with time-dependent performance deterioration. Prognostics and Health Management (PHM), 2012 I.E. Conference on, IEEEGoogle Scholar
  47. Wen Y-K (1976) Method for random vibration of hysteretic systems. J Eng Mech Div 102(2):249–263Google Scholar
  48. Xu J (2016) A new method for reliability assessment of structural dynamic systems with random parameters. Struct Saf 60:130–143Google Scholar
  49. Xu J, Chen JB, Li J (2012) Probability density evolution analysis of engineering structures via cubature points. Comput Mech 50(1):135–156MathSciNetCrossRefzbMATHGoogle Scholar
  50. Xu J, Zhang W, Sun R (2016) Efficient reliability assessment of structural dynamic systems with unequal weighted quasi-Monte Carlo simulation. Comput Struct 175:37–51CrossRefGoogle Scholar
  51. 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 95:58–76CrossRefGoogle Scholar
  52. Zhang X, Pandey MD (2013) Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct Saf 43:28–40CrossRefGoogle Scholar
  53. 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):137–151CrossRefGoogle Scholar
  54. 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):061006CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Structural Engineering, College of Civil Engineering & Hunan Provincial Key Lab on Damage Diagnosis for Engineering StructuresHunan UniversityChangshaPeople’s Republic of China
  2. 2.School of Civil Engineering & MechanicsYanshan UniversityQinhuangdaoPeople’s Republic of China

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