Optimized Bayesian Framework for Inverse Heat Transfer Problems Using Reduced Order Methods

TL;DR

This work tackles the inverse heat transfer problem of estimating a transient boundary heat flux $g(t,\mathbf{X})$ in Continuous Casting molds by embedding Ensemble-based Simultaneous Input and State Filtering (EnSISF) within a Radial Basis Function (RBF) parameterization. The input is represented as a reduced set of RBF weights, enabling joint Bayesian estimation of temperatures and the unknown boundary flux in a nonlinear system with direct feed-through, using iterative ensemble updates and a Kalman-like gain. Numerical experiments on a 3D unsteady heat conduction CC mold show that Multiquadric (MQ) RBFs achieve higher accuracy with fewer ensembles than Gaussian kernels, achieving a spatiotemporal error as low as $6.31\%$ compared to $7.59\%$ for Gaussian, with optimal settings identified for hyperparameters such as $\eta$, $\Delta t$, and observation span. The approach provides probabilistic HF estimates suitable for real-time monitoring and control, and the authors discuss future directions toward reduced-order or surrogate models to further reduce online computational costs and broaden applicability to varied HF scenarios.

Abstract

A stochastic inverse heat transfer problem is formulated to infer the transient heat flux, treated as an unknown Neumann boundary condition. Therefore, an Ensemble-based Simultaneous Input and State Filtering as a Data Assimilation technique is utilized for simultaneous temperature distribution prediction and heat flux estimation. This approach is incorporated with Radial Basis Functions not only to lessen the size of unknown inputs but also to mitigate the computational burden of this technique. The procedure applies to the specific case of a mold used in Continuous Casting machinery, and it is based on the sequential availability of temperature provided by thermocouples inside the mold. Our research represents a significant contribution to achieving probabilistic boundary condition estimation in real-time handling with noisy measurements and errors in the model. We additionally demonstrate the procedure's dependence on some hyperparameters that are not documented in the existing literature. Accurate real-time prediction of the heat flux is imperative for the smooth operation of Continuous Casting machinery at the boundary region where the Continuous Casting mold and the molten steel meet which is not also physically measurable. Thus, this paves the way for efficient real-time monitoring and control, which is critical for preventing caster shutdowns.

Figures (14)

• Figure 1: a) Schematic of CC along with b) cross-sectional view of the mold, with the casting direction oriented perpendicular to the image.
• Figure 2: Transparent mold with labeled boundaries, showing centers of RBFs (red) and thermocouple locations (black).
• Figure 3: Position of 100 thermocouples at the plane y=0.02 m.
• Figure 4: RBF profiles center situated in the middle of $\Gamma_{S_\text{in}}$ for different $\eta_j$; (a) Gaussian (b) Multiquadric (c) computational nodes.
• Figure 5: Impact of the ensemble size on the spatiotemporal error; (a) Gaussian (b) Multiquadric.