Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem
<|reference_start|>Error Estimators for the Small-Biot Lumped Approximation for the Conduction Dunking Problem: We consider the dunking problem: a solid body at uniform temperature $T_{\text i}$ is placed in a environment characterized by farfield temperature $T_\infty$ and spatially uniform time-independent heat transfer coefficient. We permit heterogeneous material composition: spatially dependent density, specific heat, and thermal conductivity. Mathematically, the problem is described by a heat equation with Robin boundary conditions. The crucial parameter is the Biot number -- a nondimensional heat transfer (Robin) coefficient; we consider the limit of small Biot number.   We introduce first-order and second-order asymptotic approximations (in Biot number) for several quantities of interest, notably the spatial domain average temperature as a function of time; the first-order approximation is simply the standard engineering `lumped' model. We then provide asymptotic error estimates for the first-order and second-order approximations for small Biot number, and also, for the first-order approximation, alternative strict bounds valid for all Biot number. Companion numerical solutions of the heat equation confirm the effectiveness of the error estimates for small Biot number.   The second-order approximation and the first-order and second-order error estimates depend on several functional outputs associated to an elliptic partial differential equation; the latter is derived from Biot-sensitivity analysis of the heat equation eigenproblem in the limit of small Biot number. Most important is $\phi$, the only functional output required for the first-order error estimates; $\phi$ admits a simple physical interpretation in terms of conduction length scale. We investigate the domain and property dependence of $\phi$: most notably, we characterize spatial domains for which the standard lumped-model error criterion -- Biot number (based on volume-to-area length scale) small -- is deficient.<|reference_end|>
@article{kaneko2024error,
    title={Error Estimators for the Small-Biot Lumped Approximation for the
  Conduction Dunking Problem},
    author={Kento Kaneko (1) and Claude Le Bris (2) and Anthony T. Patera (1) ((1)
  Department of Mechanical Engineering, Massachusetts Institute of Technology,
  (2) Matherials project-team, 'Ecole des Ponts and Inria)},
    journal={arXiv preprint arXiv:2406.12047},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.12047},
    primaryClass={math.NA cs.NA}
}