Bayesian Uncertainty Quantification for Nonlinear Constitutive Modeling in Engineering
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Constitutive models play a central role in engineering analysis by mathematically describing material behavior. Combined with conservation laws, they provide a mathematical model for the behavior of engineering systems. Unlike conservation laws, constitutive relations can be affected by significant uncertainties arising from imperfect knowledge of material behavior, experimental variability, and model simplifications. A goal of this doctoral research is to develop Bayesian methodologies for the calibration, selection, and uncertainty quantification of nonlinear constitutive models, thereby improving the reliability and interpretability of engineering predictions. This research is founded on Bayesian inference as a framework for solving inverse problems in constitutive modeling. Within this framework, unknown model parameters are represented by probability distributions that are updated using observations. This approach enables the systematic combination of prior knowledge and measurement data while providing rigorous quantification of uncertainty in model parameters and predictions. Markov Chain Monte Carlo (MCMC) sampling techniques are employed to characterize posterior distributions and assess predictive uncertainty in a statistically consistent manner. A first research line investigates the practical application of Bayesian inference to constitutive modeling problems in engineering. Specifically, it investigates the performance and computational efficiency of MCMC algorithms used for posterior exploration. Through experimental case studies in heat conduction and rheology, different sampling approaches are evaluated in terms of convergence, accuracy, computational effort, and reliability. This work provides practical guidelines for the application of Bayesian calibration methodologies to constitutive models with varying levels of complexity and computational cost. A second research line focuses on nonlinear heat conduction, where material properties depend on temperature. A Bayesian calibration framework has been developed to infer temperature-dependent thermal conductivity from transient or regime measurements. The methodology combines uncertainty quantification with adaptive refinement of both numerical discretization and constitutive model complexity. By linking model refinement strategies to measurement uncertainty and statistical model-selection criteria, the framework achieves accurate parameter estimation while preventing overfitting and unnecessary computational expense. The proposed methodology has been validated using both synthetic and experimental datasets and enables identification of nonlinear thermal constitutive behavior together with credible intervals as uncertainty bounds. A third research line investigates the integration of goal-oriented finite element methods with surrogate modeling techniques for efficient Bayesian inference. The proposed framework exploits information contained in the likelihood function to guide adaptive numerical discretization. To further reduce computational cost, multi-output Gaussian process surrogate models are employed to approximate the forward problem while preserving predictive uncertainty. The combination of goal-oriented discretization, surrogate modeling, and Bayesian inference aims to enable efficient estimation and uncertainty quantification of constitutive models in computationally demanding engineering applications. More broadly, this thesis investigates Bayesian uncertainty quantification for constitutive models across multiple engineering domains, including heat transfer, rheology, and electromagnetism. The developed domain-independent methodologies enable probabilistic calibration, model validation, and uncertainty propagation for nonlinear and history-dependent material behavior. The results demonstrate that Bayesian methods provide a rigorous framework for constitutive modeling, yielding both parameter estimates and quantified confidence in model predictions. By integrating experimental data, physical modeling, and uncertainty quantification within a unified probabilistic framework, this research contributes methodologies that support more reliable and trustworthy engineering simulations of complex material systems.
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