Operator Norm-Based Statistical Linearization to Bound the First Excursion Probability of Nonlinear Structures Subjected to Imprecise Stochastic Loading
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 8, Issue 1
Abstract
This paper presents a highly efficient approach for bounding the responses and probability of failure of nonlinear models subjected to imprecisely defined stochastic Gaussian loads. Typically, such computations involve solving a nested double-loop problem, where the propagation of the aleatory uncertainty has to be performed for each realization of the epistemic parameters. Apart from near-trivial cases, such computation is generally intractable without resorting to surrogate modeling schemes, especially in the context of performing nonlinear dynamical simulations. The recently introduced operator norm framework allows for breaking this double loop by determining those values of the epistemic uncertain parameters that produce bounds on the probability of failure a priori. However, the method in its current form is only applicable to linear models due to the adopted assumptions in the derivation of the involved operator norms. In this paper, the operator norm framework is extended and generalized by resorting to the statistical linearization methodology to account for nonlinear systems. Two case studies are included to demonstrate the validity and efficiency of the proposed approach.
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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
Peihua Ni and Michael Beer acknowledge the support from the German Research Foundation under Grant Nos. BE 2570/7-1 and MI 2459/1-1. Danko Jerez acknowledges the support from ANID (National Agency for Research and Development, Chile) and DAAD (German Academic Exchange Service) under CONICYT-PFCHA/Doctorado Acuerdo Bilateral DAAD Becas Chile/2018-6218007. Vasileios Fragkoulis gratefully acknowledges the support from the German Research Foundation under Grant No. FR 4442/2-1. Matthias Faes acknowledges the financial support of the Research Foundation Flanders (FWO) under postdoctoral Grant No. 12P3519N, as well as the Alexander von Humboldt foundation. Marcos Valdebenito acknowledges the support by ANID under its program FONDECYT, Grant No. 1180271.
References
Au, S. 2009. “Stochastic control approach to reliability of elasto-plastic structures.” J. Struct. Eng. Mech. 32 (1): 21–36. https://doi.org/10.12989/sem.2009.32.1.021.
Beer, M. 2004. “Uncertain structural design based on nonlinear fuzzy analysis.” J. Appl. Math. Mech. 84 (10–11): 740–753. https://doi.org/10.1002/zamm.200410154.
Beer, M., S. Ferson, and V. Kreinovich. 2013. “Imprecise probabilities in engineering analyses.” Mech. Syst. Sig. Process. 37 (1–2): 4–29. https://doi.org/10.1016/j.ymssp.2013.01.024.
Betz, W., I. Papaioannou, and D. Straub. 2014. “Numerical methods for the discretization of random fields by means of the Karhunen-Loève expansion.” Comput. Methods Appl. Mech. Eng. 271: 109–129. https://doi.org/10.1016/j.cma.2013.12.010.
Chen, J., and J. Li. 2013. “Optimal determination of frequencies in the spectral representation of stochastic processes.” Comput. Mech. 51 (5): 791–806. https://doi.org/10.1007/s00466-012-0764-0.
Chopra, A. 1995. Dynamics of structures: Theory and applications to earthquake engineering Englewood Cliff, NJ: Prentice Hall.
de Angelis, M., E. Patelli, and M. Beer. 2015. “Advanced line sampling for efficient robust reliability analysis.” Struct. Saf. 52 (Part B): 170–182. https://doi.org/10.1016/j.strusafe.2014.10.002.
Elishakoff, I., and L. Andriamasy. 2012. “The tale of stochastic linearization technique: Over half a century of progress.” In Nondeterministic mechanics, 115–189. New York: Springer.
Faes, M., and D. Moens. 2019. “Imprecise random field analysis with parametrized kernel functions.” Mech. Syst. Sig. Process. 134: 106334. https://doi.org/10.1016/j.ymssp.2019.106334.
Faes, M., and D. Moens. 2020. “Recent trends in the modeling and quantification of non-probabilistic uncertainty.” In Archives of computational methods in engineering. New York: Springer.
Faes, M., and M. Valdebenito. 2021. “Fully decoupled reliability-based optimization of linear structures subject to Gaussian dynamic loading considering discrete design variables.” Mech. Syst. Sig. Process. 156 (Jul): 107616. https://doi.org/10.1016/j.ymssp.2021.107616.
Faes, M. G., and M. A. Valdebenito. 2020. “Fully decoupled reliability-based design optimization of structural systems subject to uncertain loads.” Comput. Methods Appl. Mech. Eng. 371 (Nov): 113313. https://doi.org/10.1016/j.cma.2020.113313.
Faes, M. G., M. A. Valdebenito, D. Moens, and M. Beer. 2020. “Bounding the first excursion probability of linear structures subjected to imprecise stochastic loading.” Comput. Struct. 239 (Oct): 106320. https://doi.org/10.1016/j.compstruc.2020.106320.
Faes, M. G. R., M. Daub, S. Marelli, E. Patelli, and M. Beer. 2021a. “Engineering analysis with probability boxes: A review on computational methods.” Struct. Saf. 93 (Nov): 102092. https://doi.org/10.1016/j.strusafe.2021.102092.
Faes, M. G. R., M. A. Valdebenito, D. Moens, and M. Beer. 2021b. “Operator norm theory as an efficient tool to propagate hybrid uncertainties and calculate imprecise probabilities.” Mech. Syst. Sig. Process. 152 (May): 107482. https://doi.org/10.1016/j.ymssp.2020.107482.
Fragkoulis, V. C., I. A. Kougioumtzoglou, and A. A. Pantelous. 2016a. “Linear random vibration of structural systems with singular matrices.” J. Eng. Mech. 142 (2): 04015081. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001000.
Fragkoulis, V. C., I. A. Kougioumtzoglou, and A. A. Pantelous. 2016b. “Statistical linearization of nonlinear structural systems with singular matrices.” J. Eng. Mech. 142 (9): 04016063. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001119.
Fragkoulis, V. C., I. A. Kougioumtzoglou, A. A. Pantelous, and M. Beer. 2019. “Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation.” Nonlinear Dyn. 97 (4): 2291–2303. https://doi.org/10.1007/s11071-019-05124-0.
Gautschi, W. 2012. Numerical analysis. 2nd ed. Boston: Birkhäuser.
Imholz, M., M. Faes, D. Vandepitte, and D. Moens. 2020. “Robust uncertainty quantification in structural dynamics under scarse experimental modal data: A Bayesian-interval approach.” J. Sound Vib. 467 (Feb): 114983. https://doi.org/10.1016/j.jsv.2019.114983.
Jensen, H., and M. 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.
Kolda, T., R. Lewis, and V. Torczon. 2003. “Optimization by direct search: New perspectives on some classical and modern methods.” SIAM Rev. 45 (3): 385–482. https://doi.org/10.1137/S003614450242889.
Kougioumtzoglou, I., V. Fragkoulis, A. Pantelous, and A. Pirrotta. 2017. “Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach.” J. Sound Vib. 404 (Sep): 84–101. https://doi.org/10.1016/j.jsv.2017.05.038.
Lee, J., and M. Verleysen. 2007. Nonlinear dimensionality reduction. New York: Springer.
Li, J., and J. Chen. 2009. Stochastic dynamics of structures. Singapore: Wiley.
Moens, D., and D. Vandepitte. 2004. “An interval finite element approach for the calculation of envelope frequency response functions.” Int. J. Numer. Methods Eng. 61 (14): 2480–2507. https://doi.org/10.1002/nme.1159.
Ni, P., V. C. Fragkoulis, F. Kong, I. P. Mitseas, and M. Beer. 2021. “Response determination of nonlinear systems with singular matrices subject to combined stochastic and deterministic excitations.” ASCE-ASME J. Risk Uncertainty Eng. Syst. Part A: Civ. Eng. 7 (4): 04021049. https://doi.org/10.1061/AJRUA6.0001167.
Pasparakis, G., V. Fragkoulis, and M. Beer. 2021. “Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear structural systems with singular matrices.” Mech. Syst. Sig. Process. 149 (Feb): 107203. https://doi.org/10.1016/j.ymssp.2020.107203.
Pradlwarter, H., G. Schuëller, P. Koutsourelakis, and D. Charmpis. 2007. “Application of line sampling simulation method to reliability benchmark problems.” Struct. Saf. 29 (3): 208–221. https://doi.org/10.1016/j.strusafe.2006.07.009.
Roberts, J. B., and P. D. Spanos. 2003. Random vibration and statistical linearization. New York: Dover Publications.
Schenk, C., and G. Schuëller. 2005. Uncertainty assessment of large finite element systems. New York: Springer.
Schöbi, R., and B. Sudret. 2017. “Structural reliability analysis for p-boxes using multi-level meta-models.” Probab. Eng. Mech. 48 (Apr): 27–38. https://doi.org/10.1016/j.probengmech.2017.04.001.
Schuëller, G., and H. Pradlwarter. 2007. “Benchmark study on reliability estimation in higher dimensions of structural systems—An overview.” Struct. Saf. 29 (3): 167–182. https://doi.org/10.1016/j.strusafe.2006.07.010.
Shinozuka, M., and Y. Sato. 1967. “Simulation of nonstationary random process.” J. Eng. Mech. Div. 93 (1): 11–40. https://doi.org/10.1061/JMCEA3.0000822.
Socha, L. 2007. Vol. 730 of Linearization methods for stochastic dynamic systems. New York: Springer.
Spanos, P. D., A. Di Matteo, Y. Cheng, A. Pirrotta, and J. Li. 2016. “Galerkin scheme-based determination of survival probability of oscillators with fractional derivative elements.” J. Appl. Mech. 83 (12): 121003. https://doi.org/10.1115/1.4034460.
Spanos, P. D., and I. A. Kougioumtzoglou. 2014. “Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation.” J. Appl. Mech. 81 (5): 051016. https://doi.org/10.1115/1.4026182.
Spanos, P. D., and G. Malara. 2020. “Nonlinear vibrations of beams and plates with fractional derivative elements subject to combined harmonic and random excitations.” Probab. Eng. Mech. 59 (Jan): 103043. https://doi.org/10.1016/j.probengmech.2020.103043.
Stefanou, G. 2009. “The stochastic finite element method: Past, present and future.” Comput. Methods Appl. Mech. Eng. 198 (9–12): 1031–1051. https://doi.org/10.1016/j.cma.2008.11.007.
Tropp, J. A. 2004. “Topics in sparse approximation.” Ph.D. thesis, Dept. of Computer Science and Mathematics, Univ. of Texas at Austin.
Vanmarcke, E., and M. Grigoriu. 1983. “Stochastic finite element analysis of simple beams.” J. Eng. Mech. 109 (5): 1203–1214. https://doi.org/10.1061/(ASCE)0733-9399(1983)109:5(1203).
Wei, P., F. Liu, M. Valdebenito, and M. Beer. 2021. “Bayesian probabilistic propagation of imprecise probabilities with large epistemic uncertainty.” Mech. Syst. Sig. Process. 149 (Feb): 107219. https://doi.org/10.1016/j.ymssp.2020.107219.
Wei, P., J. Song, S. Bi, M. Broggi, M. Beer, Z. Lu, and Z. Yue. 2019. “Non-intrusive stochastic analysis with parameterized imprecise probability models: I. Performance estimation.” Mech. Syst. Sig. Process. 124 (Jun): 349–368. https://doi.org/10.1016/j.ymssp.2019.01.058.
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Published In
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 8 • Issue 1 • March 2022
Copyright
© 2021 American Society of Civil Engineers.
History
Received: May 20, 2021
Accepted: Oct 27, 2021
Published online: Dec 15, 2021
Published in print: Mar 1, 2022
Discussion open until: May 15, 2022
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Cited by
- Marc Fina, Celine Lauff, Matthias G.R. Faes, Marcos A. Valdebenito, Werner Wagner, Steffen Freitag, Bounding imprecise failure probabilities in structural mechanics based on maximum standard deviation, Structural Safety, 10.1016/j.strusafe.2022.102293, 101, (102293), (2023).