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Bayesian inference to identify parameters in viscoelasticity

Abstract

This contribution discusses Bayesian inference (BI) as an approach to identify parameters in viscoelasticity. The aims are: (i) to show that the prior has a substantial influence for viscoelasticity, (ii) to show that this influence decreases for an increasing number of measurements and (iii) to show how different types of experiments influence the identified parameters and their uncertainties. The standard linear solid model is the material description of interest and a relaxation test, a constant strain-rate test and a creep test are the tensile experiments focused on. The experimental data are artificially created, allowing us to make a one-to-one comparison between the input parameters and the identified parameter values. Besides dealing with the aforementioned issues, we believe that this contribution forms a comprehensible start for those interested in applying BI in viscoelasticity.

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References

  1. Abhinav, S., Manohar, C.: Bayesian parameter identification in dynamic state space models using modified measurement equations. Int. J. Non-Linear Mech. 71, 89–103 (2015)

    Article  Google Scholar 

  2. Alvin, K.: Finite element model update via Bayesian estimation and minimization of dynamic residuals. AIAA J. 35(5), 879–886 (1997)

    Article  MATH  Google Scholar 

  3. Andrieu, C., De Freitas, N., Doucet, A., Jordan, M.I.: An introduction to MCMC for machine learning. Mach. Learn. 50(1–2), 5–43 (2003)

    Article  MATH  Google Scholar 

  4. Babuška, I., Sawlan, Z., Scavino, M., Szabó, B., Tempone, R.: Bayesian inference and model comparison for metallic fatigue data. Comput. Methods Appl. Math. 304, 171–196 (2016)

    MathSciNet  Google Scholar 

  5. Banks, H.T., Hu, S., Kenz, Z.R.: A brief review of elasticity and viscoelasticity for solids. Adv. Appl. Math. Mech. 3(1), 1–51 (2011)

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Beex, L.A.A., Verberne, C.W., Peerlings, R.H.J.: Experimental identification of a lattice model for woven fabrics: application to electronic textile. Composites, Part A, Appl. Sci. Manuf. 48, 82–92 (2013)

    Article  Google Scholar 

  9. Brooks, S., Gelman, A., Jones, G., Meng, X.L.: Handbook of Markov Chain Monte Carlo. CRC Press, Boca Raton (2011)

    Book  MATH  Google Scholar 

  10. Calvetti, D., Somersalo, E.: An Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing, vol. 2. Springer, New York (2007)

    MATH  Google Scholar 

  11. Chaparro, B.M., Thuillier, S., Menezes, L.F., Manach, P.Y., Fernandes, J.V.: Material parameters identification: gradient-based, genetic and hybrid optimization algorithms. Compos. Mater. Sci. 44(2), 339–346 (2008)

    Article  Google Scholar 

  12. Chiachío, J., Chiachío, M., Saxena, A., Sankararaman, S., Rus, G., Goebel, K.: Bayesian model selection and parameter estimation for fatigue damage progression models in composites. Int. J. Fatigue 70, 361–373 (2015)

    Article  Google Scholar 

  13. Daghia, F., de Miranda, S., Ubertini, F., Viola, E.: Estimation of elastic constants of thick laminated plates within a Bayesian framework. Compos. Struct. 80(3), 461–473 (2007)

    Article  Google Scholar 

  14. Elster, C., Wübbeler, G.: Bayesian regression versus application of least squares an example. Metrologia 53(1), S10–S16 (2016)

    Article  Google Scholar 

  15. Fitzenz, D.D., Jalobeanu, A., Hickman, S.H.: Integrating laboratory creep compaction data with numerical fault models: a Bayesian framework. J. Geophys. Res., Solid Earth 112, B08410 (2007)

    Article  Google Scholar 

  16. Gelman, A., Roberts, G.O., Gilks, W.R.: Efficient Metropolis jumping rules. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 5, pp. 599–607. Oxford Univ. Press, New York (1996)

    Google Scholar 

  17. Genovese, K., Lamberti, L., Pappalettere, C.: Improved global-local simulated annealing formulation for solving non-smooth engineering optimization problems. Int. J. Solids Struct. 42(1), 203–237 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. Gogu, C., Haftka, R., Molimard, J., Vautrin, A.: Introduction to the Bayesian approach applied to elastic constants identification. AIAA J. 48(5), 893–903 (2010)

    Article  Google Scholar 

  19. Gogu, C., Yin, W., Haftka, R., Ifju, P., Molimard, J., Le Riche, R., Vautrin, A.: Bayesian identification of elastic constants in multi-directional laminate from Moiré interferometry displacement fields. Exp. Mech. 53(4), 635–648 (2013)

    Article  Google Scholar 

  20. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison–Wesley, Reading (1989)

    MATH  Google Scholar 

  21. Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14(3), 375–396 (1999)

    Article  MATH  Google Scholar 

  22. Hernandez, W.P., Borges, F.C.L., Castello, D.A., Roitman, N., Magluta, C.: Bayesian inference applied on model calibration of fractional derivative viscoelastic model. In: Steffen, V. Jr, Rade, D.A., Bessa, W.M. (eds.) Proceedings of the XVII International Symposium on Dynamic Problems of Mechanics, DINAME 2015, Natal (2015)

    Google Scholar 

  23. Higdon, D., Lee, H., Bi, Z.: A Bayesian approach to characterizing uncertainty in inverse problems using. coarse and fine scale information. IEEE Trans. Signal Process. 50, 388–399 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  24. Isenberg, J.: Progressing from least squares to Bayesian estimation. In: Proceedings of the 1979 ASME Design Engineering Technical Conference, pp. 1–11 (1979)

    Google Scholar 

  25. Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems, vol. 160. Springer, New York (2007)

    MATH  Google Scholar 

  26. Kenz, Z.R., Banks, H.T., Smith, R.C.: Comparison of frequentist and Bayesian confidence analysis methods on a viscoelastic stenosis model. SIAM/ASA J. Uncertain. Quantificat. 1(1), 348–369 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. Koutsourelakis, P.S.: A novel Bayesian strategy for the identification of spatially varying material properties and model validation: an application to static elastography. Int. J. Numer. Methods Eng. 91(3), 249–268 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  28. Kristensen, J., Zabaras, N.: Bayesian uncertainty quantification in the evaluation of alloy properties with the cluster expansion method. Comput. Phys. Commun. 185(11), 2885–2892 (2014)

    MathSciNet  Article  Google Scholar 

  29. Lai, T.C., Ip, K.H.: Parameter estimation of orthotropic plates by Bayesian sensitivity analysis. Compos. Struct. 34(1), 29–42 (1996)

    Article  Google Scholar 

  30. Liu, P., Au, S.K.: Bayesian parameter identification of hysteretic behavior of composite walls. Probab. Eng. Mech. 34, 101–109 (2013)

    Article  Google Scholar 

  31. Liu, X.Y., Chen, X.F., Ren, Y.H., Zhan, Q.Y., Wang, C., Yang, C.: Alveolar type II cells escape stress failure caused by tonic stretch through transient focal adhesion disassembly. Int. J. Biol. Sci. 7(5), 588–599 (2011)

    Article  Google Scholar 

  32. Madireddy, S., Sista, B., Vemaganti, K.: A Bayesian approach to selecting hyperelastic constitutive models of soft tissue. Comput. Methods Appl. Math. 291, 102–122 (2015)

    MathSciNet  Google Scholar 

  33. Magorou, L.L., Bos, F., Rouger, F.: Identification of constitutive laws for wood-based panels by means of an inverse method. Compos. Sci. Technol. 62(4), 591–596 (2002)

    Article  Google Scholar 

  34. Marwala, T., Sibusiso, S.: Finite element model updating using Bayesian framework and modal properties. J. Aircr. 42(1), 275–278 (2005)

    Article  Google Scholar 

  35. Marzouk, Y.M., Najm, H.N., Rahn, L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224(2), 560–586 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  36. Mehrez, L., Kassem, E., Masad, E., Little, D.: Stochastic identification of linear-viscoelastic models of aged and unaged asphalt mixtures. J. Mater. Civ. Eng. 27(4), 04014149 (2015)

    Article  Google Scholar 

  37. Miles, P., Hays, M., Smith, R., Oates, W.: Bayesian uncertainty analysis of finite deformation viscoelasticity. Mech. Mater. 91, 35–49 (2015)

    Article  Google Scholar 

  38. Most, T.: Identification of the parameters of complex constitutive models: least squares minimization vs. Bayesian updating. In: Straub, D. (ed.) Reliability and Optimization of Structural Systems, pp. 119–130. CRC Press, Boca Raton (2010)

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  40. Nichols, J.M., Link, W.A., Murphy, K.D., Olson, C.C.: A Bayesian approach to identifying structural nonlinearity using free-decay response: application to damage detection in composites. J. Sound Vib. 329(15), 2995–3007 (2010)

    Article  Google Scholar 

  41. Oden, J.T., Prudencio, E.E., Hawkins-Daarud, A.: Selection and assessment of phenomenological models of tumor growth. Math. Models Methods Appl. Sci. 23(7), 1309–1338 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  42. Oh, C.K., Beck, J.L., Yamada, M.: Bayesian learning using automatic relevance determination prior with an application to earthquake early warning. J. Eng. Mech. 134(12), 1013–1020 (2008)

    Article  Google Scholar 

  43. Orosz, M., Molnarka, G., Monos, E.: Curve fitting methods and mechanical models for identification of viscoelastic parameters of vascular wall—a comparative study. Med. Sci. Monit. 3(4), MT599–MT604 (1997)

    Google Scholar 

  44. Rappel, H., Beex, L.A.A., Hale, J.S., Bordas, S.P.A.: Bayesian inference for the stochastic identification of elastoplastic material parameters: introduction, misconceptions and additional insight. arXiv:1606.02422 (2016)

  45. Rosić, B.V., Kčerová, A., Sýkora, J., Pajonk, O., Litvinenko, A., Matthies, H.G.: Parameter identification in a probabilistic setting. Eng. Struct. 50, 179–196 (2013)

    Article  Google Scholar 

  46. Sarkar, S., Kosson, D.S., Mahadevan, S., Meeussen, J.C.L., van der Sloot, H., Arnold, J.R., Brown, K.G.: Bayesian calibration of thermodynamic parameters for geochemical speciation modeling of cementitious materials. Cem. Concr. Res. 42(7), 889–902 (2012)

    Article  Google Scholar 

  47. Ulrych, T.J., Sacchi, M.D., Woodbury, A.: A Bayes tour of inversion: a tutorial. Geophysics 66(1), 55–69 (2001)

    Article  Google Scholar 

  48. Wang, J., Zabaras, N.: A Bayesian inference approach to the inverse heat conduction problem. Int. J. Heat Mass Transf. 47(17–18), 3927–3941 (2004)

    Article  MATH  Google Scholar 

  49. Zhang, E., Chazot, J.D., Antoni, J., Hamdi, M.: Bayesian characterization of Young’s modulus of viscoelastic materials in laminated structures. J. Sound Vib. 332(16), 3654–3666 (2013)

    Article  Google Scholar 

  50. Zhao, X., Pelegri, A.A.: A Bayesian approach for characterization of soft tissue viscoelasticity in acoustic radiation force imaging. Int. J. Numer. Methods Biomed. Eng. 32(4), e02741 (2016)

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support from the University of Luxembourg, the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled “Towards real time multiscale simulation of cutting in nonlinear materials with applications to surgical simulation and computer guided surgery” and the Luxembourg National Research Fund (project No. INTER/FNRS/15/11019432).

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Correspondence to Stéphane P. A. Bordas.

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Rappel, H., Beex, L.A.A. & Bordas, S.P.A. Bayesian inference to identify parameters in viscoelasticity. Mech Time-Depend Mater 22, 221–258 (2018). https://doi.org/10.1007/s11043-017-9361-0

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Keywords

  • Bayesian inference
  • Bayes’ theorem
  • Statistical identification
  • Parameter identification
  • Viscoelasticity