Series System Reliability of Uncertain Linear Structures under Gaussian Excitation by Cross Entropy–Based Importance Sampling
Publication: Journal of Engineering Mechanics
Volume 148, Issue 1
Abstract
We present an adaptive importance sampling (IS) method to estimate the reliability of linear structures with parameter uncertainties that are subjected to Gaussian process excitation. Structural failure is defined as a union of multiple first-passage failure events. The main contribution is the introduction of an efficient IS density for the uncertain structural parameters. This density is determined by minimizing the cross-entropy (CE) between the theoretically optimal IS density of the structural parameters and a chosen parametric family of probability distributions. The CE minimization procedure requires evaluation of the system failure probability conditional on fixed values of the uncertain parameters. A closed-form analytical approximation of this conditional failure probability was derived based on an upper bound on the out-crossing rate. Finally, a joint IS density of the random excitation and the uncertain structural parameters was introduced to estimate the series system failure probability involving parameter uncertainties. The accuracy and efficiency of the proposed method was demonstrated by means of two examples: a Gaussian white noise–excited two-story linear shear frame; and a six-story, three-bay moment-resisting steel frame subjected to a filtered nonstationary Gaussian excitation.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work is supported by the Alexander von Humboldt Foundation.
References
Au, S. K., and J. L. Beck. 2001b. “First excursion probabilities for linear systems by very efficient importance sampling.” Probab. Eng. Mech. 16 (3): 193–207. https://doi.org/10.1016/S0266-8920(01)00002-9.
Au, S. K., and J. L. Beck. 2003. “Importance sampling in high dimensions.” Struct. Saf. 25 (2): 139–163. https://doi.org/10.1016/S0167-4730(02)00047-4.
Au, S.-K., and J. L. Beck. 2001a. “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech. 16 (4): 263–277. https://doi.org/10.1016/S0266-8920(01)00019-4.
Barbato, M., and J. P. Conte. 2011. “Structural reliability applications of nonstationary spectral characteristics.” J. Eng. Mech. 137 (5): 371–382. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000238.
Belyaev, Y. K. 1968. “On the number of exits across the boundary of a region by a vector stochastic process.” Theory Probab. Appl. 13 (2): 320–324. https://doi.org/10.1137/1113036.
Byun, J.-E., and J. Song. 2020. “Bounds on reliability of larger systems by linear programming with delayed column generation.” J. Eng. Mech. 146 (4): 04020008. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001717.
Der Kiureghian, A., and P.-L. Liu. 1986. “Structural reliability under incomplete probability information.” J. Eng. Mech. 112 (1): 85–104. https://doi.org/10.1061/(ASCE)0733-9399(1986)112:1(85).
Di Paola, M. 1985. “Transient spectral moments of linear systems.” SM Arch. 10 (3): 225–243.
dos Santos, K. R. M., I. A. Kougioumtzoglou, and P. D. Spanos. 2019. “Hilbert transform–based stochastic averaging technique for determining the survival probability of nonlinear oscillators.” J. Eng. Mech. 145 (10): 04019079. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001651.
Fujimura, K., and A. Der Kiureghian. 2007. “Tail-equivalent linearization method for nonlinear random vibration.” Probab. Eng. Mech. 22 (1): 63–76. https://doi.org/10.1016/j.probengmech.2006.08.001.
Geyer, S., I. Papaioannou, and D. Straub. 2019. “Cross entropy-based importance sampling using Gaussian densities revisited.” Struct. Saf. 76 (Jan): 15–27. https://doi.org/10.1016/j.strusafe.2018.07.001.
Hohenbichler, M., and R. Rackwitz. 1981. “Non-normal dependent vectors in structural safety.” J. Eng. Mech. Div. 107 (6): 1227–1238. https://doi.org/10.1061/JMCEA3.0002777.
Jensen, H. A., and M. A. Valdebenito. 2007. “Reliability analysis of linear dynamical systems using approximate representations of performance functions.” Struct. Saf. 29 (3): 222–237. https://doi.org/10.1016/j.strusafe.2006.07.004.
Kanjilal, O., and C. S. Manohar. 2019. “Estimation of time-variant system reliability of nonlinear randomly excited systems based on the Girsanov transformation with state-dependent controls.” Nonlinear Dyn. 95 (2): 1693–1711. https://doi.org/10.1007/s11071-018-4655-6.
Kanjilal, O., I. Papaioannou, and D. Straub. 2020. “Series system reliability estimation of randomly excited uncertain linear structures by cross entropy-based importance sampling.” In Proc., 7th Asian-Pacific Symp. on Sructural Reliability and Its Applications, 136–141. Tokyo: Asian-Pacific Symposium on Structural Reliability and Its Applications.
Kanjilal, O., I. Papaioannou, and D. Straub. 2021. “Cross entropy-based importance sampling for first-passage probability estimation of randomly excited linear structures with parameter uncertainty.” Struct. Saf. 91 (Jul): 102090. https://doi.org/10.1016/j.strusafe.2021.102090.
Katafygiotis, L., and S. H. Cheung. 2004. “Wedge simulation method for calculating the reliability of linear dynamical systems.” Probab. Eng. Mech. 19 (3): 229–238. https://doi.org/10.1016/j.probengmech.2004.02.006.
Katafygiotis, L., and S. H. Cheung. 2006. “Domain decomposition method for calculating the failure probability of linear dynamic systems subjected to Gaussian stochastic loads.” J. Eng. Mech. 132 (5): 475–486. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:5(475).
Katafygiotis, L. S., and K. M. Zuev. 2008. “Geometric insight into the challenges of solving high-dimensional reliability problems.” Probab. Eng. Mech. 23 (2–3): 208–218. https://doi.org/10.1016/j.probengmech.2007.12.026.
Koutsourelakis, P. S., H. J. Pradlwarter, and G. I. Schuëller. 2004. “Reliability of structures in high dimensions, part I: Algorithms and applications.” Probab. Eng. Mech. 19 (4): 409–417. https://doi.org/10.1016/j.probengmech.2004.05.001.
Latz, J., I. Papaioannou, and E. Ullmann. 2018. “Multilevel sequential Monte Carlo for Bayesian inverse problems.” J. Comput. Phys. 368 (Sep): 154–178. https://doi.org/10.1016/j.jcp.2018.04.014.
Li, C. Q., and R. E. Melchers. 1993. “Outcrossings from convex polyhedrons for nonstationary Gaussian processes.” J. Eng. Mech. 119 (11): 2354–2361. https://doi.org/10.1061/(ASCE)0733-9399(1993)119:11(2354).
Melchers, R. E., and A. T. Beck. 2018. Structural reliability analysis and prediction. Hoboken, NJ: Wiley.
Michaelov, G., S. Sarkani, and L. Lutes. 1999. “Spectral characteristics of nonstationary random processes—A critical review.” Struct. Saf. 21 (3): 223–244. https://doi.org/10.1016/S0167-4730(99)00022-3.
Misraji, M. A., M. A. Valdebenito, H. A. Jensen, and C. F. Mayorga. 2020. “Application of directional importance sampling for estimation of first excursion probabilities of linear structural systems subject to stochastic Gaussian loading.” Mech. Syst. Sig. Process. 139 (May): 106621. https://doi.org/10.1016/j.ymssp.2020.106621.
Owen, D. B. 1980. “A table of normal integrals.” Commun. Stat.- Simul. Comput. 9 (4): 389–419. https://doi.org/10.1080/03610918008812164.
Papaioannou, I., K. Breitung, and D. Straub. 2018. “Reliability sensitivity estimation with sequential importance sampling.” Struct. Saf. 75 (Nov): 24–34. https://doi.org/10.1016/j.strusafe.2018.05.003.
Papaioannou, I., S. Geyer, and D. Straub. 2019. “Improved cross entropy-based importance sampling with a flexible mixture model.” Reliab. Eng. Syst. Saf. 191 (Nov): 106564. https://doi.org/10.1016/j.ress.2019.106564.
Papaioannou, I., C. Papadimitriou, and D. Straub. 2016. “Sequential importance sampling for structural reliability analysis.” Struct. Saf. 62 (Sep): 66–75. https://doi.org/10.1016/j.strusafe.2016.06.002.
Pradlwarter, H. J., and G. I. Schuëller. 2010. “Uncertain linear structural systems in dynamics: Efficient stochastic reliability assessment.” Comput. Struct. 88 (1–2): 74–86. https://doi.org/10.1016/j.compstruc.2009.06.010.
Rice, S. O. 1944. “Mathematical analysis of random noise.” Bell Syst. Tech. J. 23 (3): 282–332. https://doi.org/10.1002/j.1538-7305.1944.tb00874.x.
Rubinstein, R. Y., and D. P. Kroese. 2016. Vol. 10 of Simulation and the Monte Carlo method. Hoboken, NJ: Wiley.
Schuëller, G. I., H. J. Pradlwarter, and P.-S. Koutsourelakis. 2004. “A critical appraisal of reliability estimation procedures for high dimensions.” Probab. Eng. Mech. 19 (4): 463–474. https://doi.org/10.1016/j.probengmech.2004.05.004.
Song, J., and A. Der Kiureghian. 2003. “Bounds on system reliability by linear programming.” J. Eng. Mech. 129 (6): 627–636. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:6(627).
Song, J., and A. Der Kiureghian. 2006. “Joint first-passage probability and reliability of systems under stochastic excitation.” J. Eng. Mech. 132 (1): 65–77. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:1(65).
Valdebenito, M. A., H. A. Jensen, and A. A. Labarca. 2014. “Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load.” Comput. Struct. 138 (Jul): 36–48. https://doi.org/10.1016/j.compstruc.2014.02.010.
Vanmarcke, E. H. 1975. “On the distribution of the first-passage time for normal stationary random processes.” J. Appl. Mech. 42 (1): 215–220. https://doi.org/10.1115/1.3423521.
Wang, Z., and J. Song. 2016. “Cross entropy-based adaptive importance sampling using von Mises-Fisher mixture for high dimensional reliability analysis.” Struct. Saf. 59 (Mar): 42–52. https://doi.org/10.1016/j.strusafe.2015.11.002.
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© 2021 American Society of Civil Engineers.
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Received: Apr 19, 2021
Accepted: Jul 28, 2021
Published online: Nov 10, 2021
Published in print: Jan 1, 2022
Discussion open until: Apr 10, 2022
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