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.
Similar content being viewed by others
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
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
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
Chen Jb, Zhang Sh (2013) Improving point selection in cubature by a new discrepancy. SIAM J Sci Comput 35(5):A2121–A2149
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
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
Conway JH, Sloane NJA (2013) Sphere packings, lattices and groups, vol 290. Springer Science & Business Media, Berlin
Dai H, Wang W (2009) Application of low-discrepancy sampling method in structural reliability analysis. Struct Saf 31(1):55–64
Dick J, Pillichshammer F (2010) Digital nets and sequences: discrepancy theory and quasi–Monte Carlo integration. Cambridge University Press, Cambridge
Faure H (1992) Good permutations for extreme discrepancy. J Number Theory 42(1):47–56
Goller B, Pradlwarter HJ, Schuller GI (2013) Reliability assessment in structural dynamics. J Sound Vib 332(10):2488–2499
Halton JH (1960) On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer Math 2(1):84–90
Harald N (1992) Random number generation and quasi-Monte Carlo methods. Society for lndustrial and Applied Mathematics, Philadelphia
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
Hu Z, Du X (2013) A sampling approach to extreme value distribution for time-dependent reliability analysis. J Mech Des 135(7):071003
Hu Z, Du X (2015a) First order reliability method for time-variant problems using series expansions. Struct Multidiscip Optim 51(1):1–21
Hu Z, Du X (2015b) Mixed efficient global optimization for time-dependent reliability analysis. J Mech Des 137(5):051401
Hua L-K, Wang Y (2012) Applications of number theory to numerical analysis. Springer Science & Business Media, Berlin
Hua LK, Yuan W (1981) Applications of number theory to numerical analysis. Springer, Berlin
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
Joe S, Kuo FY (2003) Remark on algorithm 659: implementing Sobol's quasirandom sequence generator. ACM Trans Math Softw (TOMS) 29(1):49–57
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
Kocis L, Whiten WJ (1997) Computational investigations of low-discrepancy sequences. ACM Trans Math Softw (TOMS) 23(2):266–294
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
Li J, Chen JB (2009) Stochastic dynamics of structures. John Wiley & Sons, Hoboken
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
Madsen PH, Krenk S (1984) An integral equation method for the first-passage problem in random vibration. J Appl Mech 51(3):674–679
Mourelatos ZP, Majcher M, Pandey V, Baseski I (2015) Time-dependent reliability analysis using the Total probability theorem. J Mech Des 137(3):031405
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
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
Preumont A (1985) On the peak factor of stationary Gaussian processes. J Sound Vib 100(1):15–34
Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 24(1):46–156
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)
Spanos PD, Kougioumtzoglou IA (2014) Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation. J Appl Mech 81(5):051016
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
Tuffin B (1996) On the use of low discrepancy sequences in Monte Carlo methods. Monte Carlo Methods Appl 2:295–320
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
Vanmarcke EH (1975) On the distribution of the first-passage time for normal stationary random processes. J Appl Mech 42(1):215–220
Wang X, Hickernell FJ (2000) Randomized halton sequences. Math Comput Model 32(7):887–899
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
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
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
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
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
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
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
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1834-x