Skip to main content
SpringerLink
Log in

Bayesian Finite Element Model Updating

  • Chapter
  • First Online:
Sub-structure Coupling for Dynamic Analysis

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 89))

Abstract

In this chapter, the implementation of the reduced-order models within Bayesian finite element model updating is explored. The Bayesian framework for model parameter estimation, model selection, and robust predictions of output quantities of interest is first presented. Bayesian asymptotic approximations and sampling algorithms are then outlined. The framework is implemented for updating linear and nonlinear finite element models in structural dynamics using vibration measurements consisting of either identified modal frequencies or measured response time histories. For asymptotic approximations based on modal properties, the formulation for the posterior distribution is presented with respect to the modal properties of the reduced-order model. In addition, analytical expressions for the required gradients with respect to the model parameters are provided using adjoint methods. Two applications demonstrate that drastic reductions in computational demands can be achieved without compromising the accuracy of the model updating results. In the first application, a high-fidelity linear finite element model of a full-scale bridge with hundreds of thousands of degrees-of-freedom (DOFs) is updated using experimentally identified modal properties. In the second application, a nonlinear model of a base-isolated building is updated using acceleration response time histories.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Angelikopoulos, C. Papadimitriou, P. Koumoutsakos, Bayesian uncertainty quantification and propagation in molecular dynamics simulations: a high performance computing framework. J. Chem. Phys. 137(14) (2012)

    Google Scholar 

  2. Y. Aoyama, G. Yagawa, Component mode synthesis for large scale structural eigenanalysis. Comput. Struct. 79(6), 605–615 (2001)

    Google Scholar 

  3. C. Argyris, Bayesian Uncertainty Quantification and Optimal Experimental Design in Data-Driven Simulations of Engineering Systems, Ph.D. thesis, Department of Mechanical Engineering, University of Thessaly, 2017

    Google Scholar 

  4. S.K. Au, J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16, 263–277 (2011)

    Google Scholar 

  5. S.K. Au, Assembling mode shapes by least squares. Mech. Syst. Signal Process. 25, 163–179 (2010)

    Google Scholar 

  6. S.K. Au, Fast Bayesian ambient modal identification in the frequency domain, part II: posterior uncertainty. Mech. Syst. Signal Process. 26, 76–90 (2012)

    Google Scholar 

  7. S.K. Au, Operational Modal Analysis: Modeling, Bayesian Inference, Uncertainty Laws (Springer, Berlin, 2017)

    MATH  Google Scholar 

  8. R.R. Craig Jr, M.C.C. Bampton, Coupling of substructures for dynamic analysis. AIAA J 6(5), 678–685 (1965)

    Google Scholar 

  9. R.M. Bampton, A modal combination program for dynamic analysis of structures. Technical memorandum 33–920 (Jet Propulsion Laboratory, Pasadena, CA, 1 July 1967)

    Google Scholar 

  10. K.J. Bathe, Finite Element Procedures (Prentice Hall, New Jersey, 2006)

    MATH  Google Scholar 

  11. M. Barbato, A. Zona, J.P. Conte, Finite element response sensitivity analysis using three-field mixed formulation: general theory and application to frame structures. Int. J. Numer. Methods Eng. 69(1), 114–161 (2007)

    MATH  Google Scholar 

  12. M. Barbato, J.P. Conte, Finite element response sensitivity analysis: a comparison between force-based and displacement-based frame element models. Comput. Methods Appl. Mech. Eng. 194(12–16), 1479–1512 (2005)

    MATH  Google Scholar 

  13. J.L. Beck, L.S. Katafygiotis, Updating models and their uncertainties. I: Bayesian statistical framework. ASCE J. Eng. Mech. 124(4), 455–461 (1998)

    Google Scholar 

  14. J.L. Beck, A. Taflanidis, Prior and posterior robust stochastic predictions for dynamical systems using probability logic. Int. J. Uncertain. Quantif. 3(4), 271–288 (2013)

    MathSciNet  Google Scholar 

  15. J.L. Beck, Bayesian system identification based on probability logic. Struct. Control. Health Monit. 17(7), 825–847 (2010)

    Google Scholar 

  16. J.L. Beck, K.V. Yuen, Model selection using response measurements: Bayesian probabilistic approach. ASCE J. Eng. Mech. 130(2), 192–203 (2004)

    Google Scholar 

  17. J.L. Beck, S.K. Au, Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. ASCE J. Eng. Mech. 128(4), 380–391 (2002)

    Google Scholar 

  18. E.N. Chatzi, C. Papadimitriou (eds.), Identification Methods for Structural Health Monitoring, Series: CISM-International Centre for Mechanical Sciences (Springer, Berlin, 2016)

    Google Scholar 

  19. K. Christodoulou, C. Papadimitriou, Structural identification based on optimally weighted modal residuals. Mech. Syst. Signal Process. 21(1), 4–23 (2007)

    Google Scholar 

  20. S.H. Cheung, J.L. Beck, Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters. ASCE J. Eng. Mech. 135(4), 243–255 (2009)

    Google Scholar 

  21. J.Y. Ching, Y.C. Chen, Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. ASCE J. Eng. Mech. 133(7), 816–832 (2007)

    Google Scholar 

  22. O. Ditlevsen, H.O. Madsen, Structural Reliability Methods (Wiley, Chichester, 1996)

    Google Scholar 

  23. R.L. Fox, M.P. Kapoor, Rate of change of eigenvalues and eigenvectors. AIAA J. 6(12), 2426–2429 (1968)

    MATH  Google Scholar 

  24. D. Giagopoulos, C. Salpistis, S. Natsiavas, Effect of nonlinearities in the identification and fault detection of Gear-pair systems. Int. J. Non-Linear Mech. 41, 213–230 (2006)

    Google Scholar 

  25. D. Giagopoulos, D.-C. Papadioti, C. Papadimitriou, S. Natsiavas, Bayesian uncertainty quantification and propagation in nonlinear structural dynamics. in International Modal Analysis Conference (IMAC), Topics in Model Validation and Uncertainty Quantification (2013), pp. 33–41

    Google Scholar 

  26. B. Goller, J.L. Beck, G.I. Schueller, Evidence-based identification of weighting factors in Bayesian model updating using modal data. ASCE J. Eng. Mech. 138(5), 430–440 (2012)

    Google Scholar 

  27. B. Goller, H.J. Pradlwarter, G.I. Schueller, An interpolation scheme for the approximation of dynamical systems. Comput. Methods Appl. Mech. Eng. 200(1–4), 414–423 (2011)

    MathSciNet  MATH  Google Scholar 

  28. P.L. Green, E.J. Cross, K. Worden, Bayesian system identification of dynamical systems using highly informative training data. Mech. Syst. Signal Process. 56, 109–122 (2015)

    Google Scholar 

  29. P.L. Green, Bayesian system identification of a nonlinear dynamical system using a novel variant of simulated annealing. Mech. Syst. Signal Process. 52, 133–146 (2015)

    Google Scholar 

  30. P.E. Hadjidoukas, P. Angelikopoulos, C. Papadimitriou, P. Koumoutsakos, \(\Pi \)4U: A high performance computing framework for Bayesian uncertainty quantification of complex models. J. Comput. Phys. 284(1), 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  31. N. Hansen, S.D. Muller, P. Koumoutsakos, Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11(1), 1–18 (2003)

    Google Scholar 

  32. W.K. Hastings, Monte-Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)

    MathSciNet  MATH  Google Scholar 

  33. H.A. Jensen, E. Millas, D. Kusanovic, C. Papadimitriou, Model reduction techniques for Bayesian finite element model updating using dynamic response data. Comput. Methods Appl. Mech. Eng. 279, 301–324 (2014)

    MathSciNet  MATH  Google Scholar 

  34. H.A. Jensen, C. Vergara, C. Papadimitriou, E. Millas, The use of updated robust reliability measures in stochastic dynamical systems. Comput. Methods Appl. Mech. Eng. 267, 293–317 (2013)

    MathSciNet  MATH  Google Scholar 

  35. H.A. Jensen, F. Mayorga, C. Papadimitriou, Reliability sensitivity analysis of stochastic finite element models. Comput. Methods Appl. Mech. Eng. 296, 327–351 (2015)

    MathSciNet  MATH  Google Scholar 

  36. H.A. Jensen, A. Munoz, C. Papadimitriou, C. Vergara, An enhanced substructure coupling technique for dynamic re-analyses: application to simulation-based problems. Comput. Methods Appl. Mech. Eng. 307, 215–234 (2016)

    MathSciNet  MATH  Google Scholar 

  37. H.A. Jensen, D.S. Kusanovic, M. Papadrakakis, Reliability-based characterization of base-isolated structural systems, in European Congress on Computational Methods in Applied Sciences and Engineering (Vienna, Austria, 2012)

    Google Scholar 

  38. L.S. Katafygiotis, H.F. Lam, Tangential-projection algorithm for manifold representation in unidentifiable model updating problems. Earthq. Eng. Struct. Dyn. 31(4), 791–812 (2002)

    Google Scholar 

  39. L.S. Katafygiotis, H.F. Lam, C. Papadimitriou, Treatment of unidentifiability in structural model updating. Adv. Struct. Eng.-Int. J. 3(1), 19–39 (2000)

    Google Scholar 

  40. L.S. Katafygiotis, J.L. Beck, Updating models and their uncertainties. II: model identifiability. ASCE J. Eng. Mech. 124(4), 463–467 (1998)

    Google Scholar 

  41. J.N. Lyness, C.B. Moler, Generalized Romberg methods for integrals of derivatives. Numer. Math. 14(1), 1–13 (1969)

    MathSciNet  MATH  Google Scholar 

  42. P. Metallidis, G. Verros, S. Natsiavas, C. Papadimitriou, Fault detection and optimal sensor location in vehicle suspensions. J. Vib. Control. 9(3–4), 337–359 (2003)

    MATH  Google Scholar 

  43. P. Metallidis, S. Natsiavas, Parametric identification and health monitoring of complex ground vehicle models. J. Vib. Control. 14(7), 1021–1036 (2008)

    MATH  Google Scholar 

  44. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1092 (1953)

    MATH  Google Scholar 

  45. M. Muto, J.L. Beck, Bayesian updating and model class selection for hysteretic structural models using stochastic simulation. J. Vib. Control. 14(1–2), 7–34 (2008)

    MATH  Google Scholar 

  46. R.B. Nelson, Simplified calculation of eigenvector derivatives. AIAA J. 14(9), 1201–1205 (1976)

    MathSciNet  MATH  Google Scholar 

  47. E. Ntotsios, C. Papadimitriou, Multi-objective optimization algorithms for finite element model updating, in Proceedings of International Conference on Noise and Vibration Engineering (ISMA) (2008), 1895–1909

    Google Scholar 

  48. J.T. Oden, T. Belytschko, J. Fish, T.J.R. Hughes, C. Johnson, D. Keyes, A. Laud, L. Petzold, D. Srolovitz, S. Yip, Simulation-Based Engineering Science (SBES) Revolutionizing Engineering Science through Simulation, Report of the NSF, Blue Ribbon Panel on SBES (2006)

    Google Scholar 

  49. C. Papadimitriou, D.C. Papadioti, Component mode synthesis techniques for finite element model updating. Comput. Struct. 126, 15–28 (2013)

    Google Scholar 

  50. C. Papadimitriou, L.S. Katafygiotis, Bayesian modeling and updating, in Engineering Mechanics Reliability Handbook, ed. by E. Nikolaidis, D.M. Ghiocel, S. Singhal (CRC Press, Boca Raton, 2004)

    Google Scholar 

  51. C. Papadimitriou, J.L. Beck, L.S. Katafygiotis, Asymptotic expansions for reliability and moments of uncertain dynamic systems. ASCE J. Eng. Mech. 123(12), 1219–1229 (1997)

    Google Scholar 

  52. C. Papadimitriou, L.S. Katafygiotis, A Bayesian methodology for structural integrity and reliability assessment. Int. J. Adv. Manuf. Syst. 4(1), 93–100 (2001)

    Google Scholar 

  53. C. Papadimitriou, J.L. Beck, L.S. Katafygiotis, Updating robust reliability using structural test data. Probab. Eng. Mech. 16(2), 103–113 (2001)

    Google Scholar 

  54. C. Papadimitriou, E. Ntotsios, D. Giagopoulos, S. Natsiavas, Variability of updated finite element models and their predictions consistent with vibration measurements. Struct. Control. Health Monit. 19(5), 630–654 (2011)

    Google Scholar 

  55. I. Papaioannou, W. Betz, K. Zwirglmaier, D. Straub, MCMC algorithms for subset simulation. Probab. Eng. Mech. 41, 89–103 (2015)

    Google Scholar 

  56. E. Simoen, B. Moaveni, J.L. Conte, G. Lombaert, Uncertainty quantification in the assessment of progressive damage in a 7-story full-scale building slice. ASCE J. Eng. Mech. 139(12), 1818–1830 (2013)

    Google Scholar 

  57. E. Simoen, C. Papadimitriou, G. Lombaert, On prediction error correlation in Bayesian model updating. J. Sound Vib. 332(18), 4136–4152 (2013)

    Google Scholar 

  58. Y.C Tan, M.P. Castanier, C. Pierre. Characteristic mode based component mode synthesis for power flow analysis in complex structures, in Proceedings of the 41st AIAA/ASME/ASCE/AHS/ASC structures, Structural Dynamics and Materials Conference and Exhibit, Reston VA, vol. 1908–917 (2000)

    Google Scholar 

  59. L. Tierney, J.B. Kadane, Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81(393), 82–86 (1986)

    MathSciNet  MATH  Google Scholar 

  60. S. Wu, P. Angelikopoulos, C. Papadimitriou, P. Koumoutsakos, Bayesian Annealed Sequential Importance Sampling (BASIS): an unbiased version of Transitional Markov Chain Monte Carlo. ASCE-ASME J. Risk Uncert. Eng. Sys. Part B Mech. Eng. 4, 011008–1 (2018)

    Google Scholar 

  61. M.W. Vanik, J.L. Beck, S.K. Au, Bayesian probabilistic approach to structural health monitoring. ASCE J. Eng. Mech. 126(7), 738–745 (2000)

    Google Scholar 

  62. W.-J. Yan, L.S. Katafygiotis, A novel Bayesian approach for structural model updating utilizing statistical modal information from multiple setups. Struct. Saf. 52(Part B), 260–271 (2015)

    Google Scholar 

  63. K.V. Yuen, J.L. Beck, L.S. Katafygiotis, Efficient model updating and health monitoring methodology using incomplete modal data without mode matching. Struct. Control. Health Monit. 13, 91–107 (2006)

    Google Scholar 

  64. K.-V. Yuen, Bayesian Methods for Structural Dynamics and Civil Engineering (Wiley, New York, 2010)

    Google Scholar 

  65. K.V. Yuen, J.L. Beck, L.S. Katafygiotis, Efficient model updating and health monitoring methodology using incomplete modal data without mode matching. Struct. Control. Health Monit. 13(1), 91–107 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Federico Santa María Technical University, Valparaiso, Chile

    Hector Jensen

  2. University of Thessaly, Volos, Greece

    Costas Papadimitriou

Authors
  1. Hector Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Costas Papadimitriou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hector Jensen .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Jensen, H., Papadimitriou, C. (2019). Bayesian Finite Element Model Updating. In: Sub-structure Coupling for Dynamic Analysis. Lecture Notes in Applied and Computational Mechanics, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-030-12819-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-12819-7_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-12818-0

  • Online ISBN: 978-3-030-12819-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation