Structural Engineering and Mechanics

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Markov Chain Monte Carlo simulation based Bayesian updating of model parameters and their uncertainties

  • Sengupta, Partha (Department of Civil Engineering, Indian Institute of Engineering Science and Technology) ;
  • Chakraborty, Subrata (Department of Civil Engineering, Indian Institute of Engineering Science and Technology)
  • Received : 2020.04.22
  • Accepted : 2021.11.20
  • Published : 2022.01.10
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Abstract

The prediction error variances for frequencies are usually considered as unknown in the Bayesian system identification process. However, the error variances for mode shapes are taken as known to reduce the dimension of an identification problem. The present study attempts to explore the effectiveness of Bayesian approach of model parameters updating using Markov Chain Monte Carlo (MCMC) technique considering the prediction error variances for both the frequencies and mode shapes. To remove the ergodicity of Markov Chain, the posterior distribution is obtained by Gaussian Random walk over the proposal distribution. The prior distributions of prediction error variances of modal evidences are implemented through inverse gamma distribution to assess the effectiveness of estimation of posterior values of model parameters. The issue of incomplete data that makes the problem ill-conditioned and the associated singularity problem is prudently dealt in by adopting a regularization technique. The proposed approach is demonstrated numerically by considering an eight-storey frame model with both complete and incomplete modal data sets. Further, to study the effectiveness of the proposed approach, a comparative study with regard to accuracy and computational efficacy of the proposed approach is made with the Sequential Monte Carlo approach of model parameter updating.

Keywords

Acknowledgement

The first author acknowledge the funding in the form of Institute research fellowship.

References

  1. Abhinav, S. and Manohar, C.S. (2015), "Bayesian parameter identification in dynamic state space models using modified measurement equations", Int. J. Nonlin. Mech., 71, 89-103. https://doi.org/10.1016/j.ijnonlinmec.2015.02.003.
  2. Au, S. (2011), "Fast Bayesian FFT method for ambient modal identification with separated modes", J. Eng. Mech., 137(3), 214-226. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000213.
  3. Au, S. (2012a), "Fast Bayesian ambient modal identification in the frequency domain, Part I: Posterior most probable value", Mech. Syst. Signal Pr., 26, 60-75. https://doi.org/10.1016/j.ymssp.2011.06.017.
  4. Au, S. (2012b), "Fast Bayesian ambient modal identification in the frequency domain, Part II: Posterior uncertainty", Mech. Syst. Signal Pr., 26, 76-90. https://doi.org/10.1016/j.ymssp.2011.06.019.
  5. Au, S. and Wang, Y. (2014), Engineering Risk Assessment with Subset Simulation, John Wiley & Sons Pte.Ltd., Singapore.
  6. Au, S., Zhang, F. and Ni, Y. (2013), "Bayesian operational modal analysis: Theory, computation, practice", Compos. Struct., 126, 3-14. https://doi.org/10.1016/j.compstruc.2012.12.015.
  7. Beck, J.L. (2010), "Bayesian system identification based on probability logic", Struct. Control Hlth. Monit., 17, 825-847. https://doi.org/10.1002/stc.424.
  8. Beck, J.L. and Au, S. (2002), "Bayesian updating of structural models and reliability using Markov Chain Monte Carlo Simulation", J. Eng. Mech., 128(4), 380. https://doi.org/10.1061/(ASCE)0733-9399(2002)128,4(380).
  9. Beck, J.L. and Katafygiotis, L.S. (1998), "Updating models and their uncertainties. I: Bayesian statistical framework", J. Eng. Mech., 124(4), 455-461. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455).
  10. Catanach, T.A. and Beck, J.L. (2017), "Bayesian system identification using auxiliary stochastic dynamical systems", Int. J. Nonlin. Mech., 94, 72-83. https://doi.org/10.1016/j.ijnonlinmec.2017.03.012.
  11. Chen, Z.P. and Ling, Y. (2017), "A novel PSO-based algorithm for structural damage detection using Bayesian multi-sample objective function", Struct. Eng. Mech., 63(6), 825-835. https://doi.org/10.12989/sem.2017.63.6.825.
  12. Cheung, S.H. and Bansal, S. (2017), "A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data", Mech. Syst. Signal Pr., 92, 156-172. https://doi.org/10.1016/j.ymssp.2017.01.015.
  13. Cheung, S.H. and Beck, J.L. (2009), "Bayesian model updating using Hybrid Monte Carlo Simulation with application to structural dynamics models with many uncertain parameters", J. Eng. Mech., 135, 243-255. https://doi.org/10.1061/(ASCE)0733-9399(2009)135:4(243).
  14. Ching, J. and Chen, Y. (2007), "Transitional Markov Chain Monte Carlo Method for Bayesian model updating, model class selection, and model averaging", J. Eng. Mech., 133(7), 816. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(816).
  15. Clarence, W.S. (2007), Vibration Monitoring, Testing, and Instrumentation, CRC Press Taylor & Francis Group, U.K.
  16. Clarke, B.S. and Clarke, J.L. (2018), Predictive Statistics: Analysis and Inference beyond Models, Cambridge University Press, New York, United States.
  17. Doebling, S.W., Farrar, C.R., Prime, M.B. and Shevitz, D.W. (1996), "Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review", Tech. Report No. LA-13070-MS, Los Alamos National laboratory.
  18. Gomes, G.F., Mendez, Y.A.D., Alexandrino, P.D.S.L., da Cunha Jr, S.S. and Ancelotti Jr, A.C. (2018), "The use of intelligent computational tools for damage detection and identification with an emphasis on composites-A review", Compos. Struct., 196, 44-54. https://doi.org/10.1016/j.compstruct.2018.05.002.
  19. Gustafson, P. (1998), "A guided walk Metropolis algorithm", Stat. Comput., 8, 357-364. https://doi.org/10.1023/A:1008880707168.
  20. Hasan, K., Martinez, R. and Haldar, A. (2005), "Health assessment at local level with unknown input excitation", J. Struct. Eng., 131(6), 956. https://10.1061/(ASCE)0733-9445(2005)131:6(956).
  21. Hu, Q. Heung, F.L., Hong, P.Z. and Stephen, A.A. (2018), "Bayesian ballast damage detection utilizing a modified evolutionary algorithm", Smart Struct. Syst., 21(4), 435-448. https://doi.org/10.12989/sss.2018.21.4.435.
  22. Huang, Y., Beck, J.L. and Li, H. (2017), "Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment", Comput. Meth. Appl. Mech. Eng., 318, 382-411. https://doi.org/10.1016/j.cma.2017.01.030.
  23. Huang, Y., Shao, C., Wu, B., Beck, J.L. and Li, H. (2018), "State-of-the-art review on Bayesian inference in structural system identification and damage assessment", Adv. Struct. Eng., 22(6), 1-23. https://doi.org/10.1177/1369433218811540.
  24. Lam, H., Hu, J. and Yang, J. (2017), "Bayesian operational modal analysis and Markov chain Monte Carlo-based model updating of a factory building", Eng. Struct., 132, 314-336. https://doi.org/10.1016/j.engstruct.2016.11.048.
  25. Li, H., Ren, L., Jia, Z., Yi, T. and Li, D. (2016), "State-of-the-art in structural health monitoring of large and complex civil infrastructures", J. Civil Struct. Hlth. Monit., 6, 3-16. https://doi.org/10.1007/s13349-015-0108-9.
  26. Lophaven, S.N., Nielsen, H.B. and Sondergaard, J. (2002), "Aspects of the MATLAB toolbox DACE. Informatics and Mathematical modelling", Technical Report IMM-REP-2002-13, University of Denmark, Lyngby, Denmark.
  27. Lyngdoh, G.A., Rahman, M.A. and Mishra, S.K. (2019), "Bayesian updating of structural model with a conditionally heteroscedastic error distribution", J. Eng. Mech., 145(12), 04019091. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001668.
  28. Mares, C., Dratz, B., Mottershead, J.E. and Friswell, M.I. (2006), "Model updating using Bayesian estimation", International Conference on Noise and Vibration Engineering, ISMA2006, Katholieke Universiteit Leuven, Leuven, September.
  29. Marwala, T., Boulkaibet, I. and Adhikari, S. (2017), Probabilistic Finite Element Model Updating using Bayesian Statistics Applications to Aeronautical and Mechanical Engineering, John Wiley & Sons, UK.
  30. Mehdi, V., Shahram, V., Rahimian, M., Jamshidi, N. and Kanee, A.T. (2019), "Evolutionary-base finite element model updating and damage detection using modal testing results", Struct. Eng. Mech., 70(3), 339-350. https://doi.org/10.12989/sem.2019.70.3.339.
  31. Moral, P.D., Doucet, A. and Jasra, A. (2006), "Sequential Monte Carlo samplers". J. Roy. Stat. Soc. Stat. Methodol. Ser. B, 68(3), 411-436. https://doi.org/10.1111/j.1467-9868.2006.00553.x
  32. Prabhu, S.R., Lee, Y.J. and Park, Y.C. (2019), "A new Bayesian approach to derive Paris' law parameters from S-N curve data", Struct. Eng. Mech., 69(4), 361-369. https://doi.org/10.12989/sem.2019.69.4.361.
  33. Prajapat, K. and Ray-Chaudhuri, S. (2018), "Detection of multiple damages employing best achievable eigenvectors under Bayesian inference", J. Sound Vib., 422, 237-263. https://doi.org/10.1016/j.jsv.2018.02.012.
  34. Proppe, C. (2017), "Markov Chain Monte Carlo simulation methods for structural reliability analysis", Procedia Eng., 199, 1122-1127. https://doi.org/10.1016/j.proeng.2017.09.226.
  35. Robert, C.P. and Casella, G. (2004), Monte Carlo Statistical Methods, Springer, Heidelberg Germany.
  36. Sohn, H., Farrar, C.R., Hemez, F.M., Shunk, D.D., Stinemates, D.W., Nadler, B.R. and Czarnecki, J.J. (2004), "A review of structural health monitoring literature: 1996-2001", Tech. Report No. LA-13976-MS, Los Alamos National Laboratory.
  37. Sun, H. and Buyukozturk, O. (2016), "Probabilistic updating of building models using incomplete modal data", Mech. Syst. Signal Pr., 75, 27-40. https://doi.org/10.1016/j.ymssp.2015.12.024.
  38. Yang, J. and Lam, H. (2018), "An efficient adaptive sequential Monte Carlo method for Bayesian model updating and damage detection", Struct. Control Hlth. Monit., 25(12), e2260. https://doi.org/10.1002/stc.2260.