29 Non-unique Estimates in Material Parameter Identification of Nonlinear. . . 265 Fig. 29.7 Predicted versus true response using θk|k at (a) tk =0 s, (b) tk =3 s, (c) tk =10 s, and (d) tk =20 s predicted (using the material parameter estimates) and true response time histories. This implies that the set of all eleven material parameters defining the cap plasticity model is not identifiable. In other words, the mathematical inverse problem involves a many-to-one function, i.e., many sets of parameter values result in the same FE predicted response. Therefore, for all practical purposes, any such set of parameter estimates can be used for dam (structural) response prediction, damage diagnosis and prognosis. In the future, structural and practical identifiability analysis of such multiaxial plasticity models should be performed to investigate non-identifiability, if any, of parameters before solving a parameter estimation problem. Acknowledgements Funding for this work was provided by the U.S. Army Corps of Engineers through the U.S. Army Engineer Research and Development Center Research Cooperative Agreement. References 1. Friswell, M., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics. Springer Science & Business Media, Berlin (2013) 2. Ebrahimian, H., Astroza, R., Conte, J.P., de Callafon, R.A.: Nonlinear finite element model updating for damage identification of civil structures using batch Bayesian estimation. Mech. Syst. Signal Process. 84, 194–222 (2017) 3. Astroza, R., Ebrahimian, H., Conte, J.P.: Material parameter identification in distributed plasticity FE models of frame-type structures using nonlinear stochastic filtering. J. Eng. Mech. ASCE. 141(5), 04014149–04011/17 (2014) 4. Yuen, K.-V.: Bayesian Methods for Structural Dynamics and Civil Engineering. John Wiley & Sons, New York, NY (2010) 5. Hofstetter, G., Simo, J.C., Taylor, R.L.: A modified cap model: closest point solution algorithms. Comput. Struct. 46(2), 203–214 (1993) 6. Sandler, I.S., Dimaggio, F.L., Baladi, G.Y.: Generalized cap model for geological materials. J. Geotech. Eng. Div. ASCE. 102(7), 683–699 (1976) 7. Simo, J.C., Ju, J.-W., Pister, K.S., Taylor, R.L.: Assessment of cap model: consistent return algorithms and rate-dependent extension. J. Eng. Mech. ASCE. 114(2), 191–218 (1988) 8. Hemez, F.M., Farrar, C.R.: A brief history of 30 years of model updating in structural dynamics. Special Topics Struct. Dynam. 6, 53–71., Springer (2014) 9. Van Der Merwe, R.: Sigma-point Kalman filters for probabilistic inference in dynamic state-space models. Ph.D. Thesis, Department of Electrical and Computer Engineering, OGI School of Science & Engineering, Oregon Health & Science University, Beaverton, Oregon (2004) 10. Hall, J.F.: Study of the earthquake response of Pine Flat dam. Earthq. Eng. Struct. Dynam. 14(2), 281–295 (1986) 11. Rea, D., Liaw, C.Y., Chopra, A.K.: Mathematical models for the dynamic analysis of concrete gravity dams. Earthq. Eng. Struct. Dynam. 3(3), 249–258 (1974) 12. Chen, W.-F., Saleeb, A.F.: Constitutive equations for engineering materials. In: Elasticity and Modeling, vol. 1, Revised edn. Elsevier, Amsterdam (2014)
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