Bayesian nonlinear finite element (FE) model updating can estimate the material parameters of multiaxial plasticity laws from seismic input-output measurement data, but for multiaxial constitutive models such as the eleven-parameter cap plasticity model for plain concrete the resulting parameter estimates are non-unique: different parameter sets yield equally accurate response predictions because parameter errors compensate one another. The open problem is to perform structural and practical identifiability analysis of multiaxial plasticity models before solving the parameter estimation problem: determine a priori which parameters or parameter combinations are uniquely recoverable from the noise-free model map and from realistic noisy data, and explain the observed non-unique yet predictively accurate estimates. The source study disregarded several uncertainty sources (model class error, boundary conditions, fluid-structure interaction, foundation flexibility) and did not examine the transformation of input measurement noise into colored output noise. Subsequent work by the authors of the source study delivered local practical identifiability analysis (local sensitivity with the Fisher information matrix) and a Sobol global-sensitivity proxy for the dam testbed, and information-theoretic estimators of global practical identifiability now exist, but no structural identifiability analysis of a multiaxial plasticity model in FE updating and no global practical identifiability application to this model class has been published.

Background

Finite element (FE) model updating is the calibration of a numerical FE model of a structure so that its simulated response matches measured response data; in the Bayesian formulation, unknown model parameters are treated as random variables and inferred probabilistically from noisy measurements. For structures responding inelastically under strong dynamic loads, such as a concrete gravity dam under earthquake excitation, the FE model is governed by a multiaxial plasticity constitutive law, which describes material stress-strain behavior under combined multi-directional stress states. A representative example is the cap plasticity model for plain concrete, a non-smooth multi-surface plasticity model whose response is controlled by eleven time-invariant material parameters: shear modulus GG, bulk modulus KK, tensile strength TT, and parameters X0X_0, DD, WW, RR, λ\lambda, θ\theta, β\beta, α\alpha governing hardening, softening, and the geometry of the cap yield surface. The unscented Kalman filter (UKF) is a recursive nonlinear Bayesian filter that propagates a small deterministic set of sample points (sigma points) through the nonlinear model to estimate states and parameters online; transitional Markov chain Monte Carlo (TMCMC) is a batch, offline Bayesian sampling counterpart.

In the source study, the authors applied a UKF to update all eleven cap-plasticity parameters of a plane-strain FE model of the Pine Flat dam (a concrete gravity dam on King's River near Fresno, California) using numerically simulated, noisy seismic input and output (acceleration and displacement) data. The central finding was that the parameter estimates are non-unique: the filter converges to different parameter sets, yet every set yields very similar and accurate predictions of both measured and unmeasured seismic response quantities, meaning the errors in different parameters compensate one another. This is an identifiability failure. A parameter vector is called structurally identifiable if it is uniquely determined by the idealized noise-free input-output map of the model, and practically identifiable if it can be estimated with acceptable uncertainty from realistically sparse and noisy data. Classical diagnostic tools include local sensitivity analysis combined with the Fisher information matrix (the expected information the data carry about the parameters; its rank and eigenvalues reveal locally weakly identifiable directions) and variance-based global sensitivity analysis using Sobol indices (which decompose output variance into contributions from individual parameters and their interactions).

Prior identifiability work in this area assessed nonlinear structural FE models through information-theoretic entropy-gain measures usable for parameter selection, sensor placement, and experiment design (Ebrahimian, Astroza, Conte and Bitmead, 2019), used global sensitivity analysis to design nonlinear identification experiments before data collection (Lund, Dyke, Song and Bilionis, 2019), and performed Bayesian identification of uniaxial viscoplastic-damage constitutive parameters with a discussion of identifiability (Adeli, Rosić, Matthies, Reinstädler and Dinkler, 2020). However, no identifiability analysis existed for multiaxial plasticity models in nonlinear FE updating. The source study also left several effects unquantified: the modeling uncertainty it disregarded (incorrect model class, unknown boundary conditions, fluid-structure interaction, foundation flexibility), the transformation of input measurement noise into colored, i.e. temporally correlated, output noise, and generalization to other ground motions, dam geometries, and material models. These remain limitations of the benchmark setting.

Problem Statement

For a nonlinear finite element model of a civil structure governed by a multiaxial plasticity constitutive law - such as the eleven-parameter cap plasticity model for plain concrete used in a plane-strain FE model of the Pine Flat concrete gravity dam - updated with Bayesian estimation methods (e.g., the unscented Kalman filter, or transitional Markov chain Monte Carlo (TMCMC)) from seismic input and output measurement data, develop and perform an identifiability analysis of the material parameters that can be carried out before solving the parameter estimation problem: (a) structural identifiability: determine which parameters, or which functions or combinations of parameters, are uniquely determined by the idealized noise-free input-output map of the nonlinear FE model; (b) practical identifiability: determine which parameters or parameter subsets are identifiable from realistically sparse and noisy measurement data, globally over the full feasible parameter range rather than merely locally (local practical identifiability by local sensitivity and the Fisher information matrix is established background); and (c) use this analysis to explain the empirically observed non-unique parameter estimates that nonetheless yield accurate predictions of both measured and unmeasured response quantities, thereby establishing which parameters or parameter combinations can be meaningfully estimated and which must be fixed, reparameterized, or constrained before Bayesian parameter estimation is attempted. The three parts are logically linked and all are needed to characterise the identifiability of the model class; no single part alone resolves the problem. The analysis is required at the level of multiaxial plasticity models generally, with the eleven-parameter cap-plasticity model of the Pine Flat concrete gravity dam serving as the representative example of the model class; it is carried out for the class rather than for a single dataset.

References

  1. Ramancha, M. K., Madarshahian, R., Astroza, R., Conte, J. P. (2020). Non-unique Estimates in Material Parameter Identification of Nonlinear FE Models Governed by Multiaxial Material Models Using Unscented Kalman Filtering. In: Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, Springer, pp. 257-265. DOI: 10.1007/978-3-030-12075-7_29. https://doi.org/10.1007/978-3-030-12075-7_29
  2. Ramancha, M. K., Astroza, R., Madarshahian, R., Conte, J. P. (2022). Bayesian updating and identifiability assessment of nonlinear finite element models. Mechanical Systems and Signal Processing 167:108517. DOI: 10.1016/j.ymssp.2021.108517. https://doi.org/10.1016/j.ymssp.2021.108517
  3. Ebrahimian, H., Astroza, R., Conte, J. P., Bitmead, R. R. (2019). Information-Theoretic Approach for Identifiability Assessment of Nonlinear Structural Finite-Element Models. Journal of Engineering Mechanics 145(7):04019039. DOI: 10.1061/(ASCE)EM.1943-7889.0001590. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001590
  4. Bhola, S., Duraisamy, K. (2023). Estimating global identifiability using conditional mutual information in a Bayesian framework. Scientific Reports 13:18336. DOI: 10.1038/s41598-023-44589-3. https://doi.org/10.1038/s41598-023-44589-3
  5. Rojíček, J., Cienciala, J., Fusek, M. (2025). Structural identifiability of parameters of Anand material model. Scientific Reports 15:5663. DOI: 10.1038/s41598-025-89360-y. https://doi.org/10.1038/s41598-025-89360-y
  6. Ju, Y. S. (2026). Structural identifiability of constitutive relations from indirect field observations. PNAS Nexus, article pgag281. DOI: 10.1093/pnasnexus/pgag281. https://doi.org/10.1093/pnasnexus/pgag281
  7. Simbort Zeballos, E. G., Ramancha, M. K., Astroza, R., Conte, J. P. (2026). Bayesian nonlinear finite element model updating of a full-scale reinforced-concrete bridge column under low-to-moderate seismic excitation. International Journal of Non-Linear Mechanics 190:105404. DOI: 10.1016/j.ijnonlinmec.2026.105404. https://doi.org/10.1016/j.ijnonlinmec.2026.105404
  8. Lund, A., Dyke, S. J., Song, W., Bilionis, I. (2019). Global sensitivity analysis for the design of nonlinear identification experiments. Nonlinear Dynamics 98(1):375-394. DOI: 10.1007/s11071-019-05199-9. https://doi.org/10.1007/s11071-019-05199-9
  9. Adeli, E., Rosić, B., Matthies, H. G., Reinstädler, S., Dinkler, D. (2020). Comparison of Bayesian methods on parameter identification for a viscoplastic model with damage. Probabilistic Engineering Mechanics 62:103083. DOI: 10.1016/j.probengmech.2020.103083. https://doi.org/10.1016/j.probengmech.2020.103083
  10. Zhou, T., Wei, Y., Jie, Y., Zhang, Y. (2025). A novel Bayesian updating finite element model using enhanced unscented Kalman filter for time-dependent estimation of rockfill dam structure deformation. Engineering Structures 323:119231. DOI: 10.1016/j.engstruct.2024.119231. https://doi.org/10.1016/j.engstruct.2024.119231
  11. Kiran, R. P., Das, A., Bansal, S. (2025). A state-of-the-art review of Bayesian finite element model updating techniques for structural systems. Probabilistic Engineering Mechanics 80:103761. DOI: 10.1016/j.probengmech.2025.103761. https://doi.org/10.1016/j.probengmech.2025.103761