Discretization-optimized Bayesian model calibration for nonlinear constitutive modeling in heat conduction

TL;DR

This paper introduces an adaptive Bayesian calibration framework for nonlinear heat conduction models that optimizes discretization and model complexity to improve accuracy and efficiency.

Contribution

It proposes a novel algorithm that sequentially refines discretization and model complexity using an uncertainty-aware criterion, enhancing robustness and computational efficiency.

Findings

Accurately calibrates nonlinear conductivity with limited data.

Balances model complexity and discretization to prevent overfitting.

Demonstrates effectiveness on synthetic and experimental data.

Abstract

We present a Bayesian model calibration framework for inferring nonlinear constitutive relationships in heat conduction problems, with a focus on temperature-dependent thermal conductivity. The proposed framework integrates gradient-based optimization and uncertainty quantification (UQ) to address the inverse problem of estimating the conductivity function from transient temperature measurements. A key contribution is an adaptive algorithm that sequentially refines both the numerical discretization for model simulation, and the model complexity used to represent the conductivity curve. The discretization is optimized through the minimization of a loss function, and Morozov's discrepancy principle is used as an uncertainty-motivated stopping criterion. The model complexity is selected using an approach that balances maximizing the likelihood of the data with penalizing excessive model complexity. As a result, the numerical and modeling biases remain of the same order as the uncertainty imposed by the measurement noise, leading to robust and computationally efficient inference. The methodology is demonstrated on both synthetic and experimental data, showing that it enables accurate calibration of nonlinear constitutive models with minimal overfitting and limited computational cost.