Analytical bounds and finite element modeling of the temperature distribution on a thin plate heated by a localized radiant source

Abstract

This work investigates the steady-state temperature field induced on a thin plate by a small high-temperature radiative source placed at a finite distance. A nonlinear source-to-surface model is derived by coupling in-plane heat conduction, a view-factor-based incident radiative flux, and two- sided radiative losses governed by the Stefan-Boltzmann law. The resulting problem is written in weak form and solved with a Newton-linearized finite element formulation. Analytical upper and lower bounds are established for the temperature field, giving admissible limits that are used to assess the numerical results. The implementation is verified with the method of manufactured solutions, for which the relative L' and H' errors decrease monotonically under mesh refinement. An independent uniform-irradiation benchmark is reproduced to roundoff accuracy, and a separate refinement study for the most localized physical source shows changes below 0.01% between the production and reference meshes. Simulations show a strong dependence on source height: the peak temperature decreases from about 140 to about 22 K as the dimensionless height increases from 0.1 to 10, while the computed extrema remain within the analytical bounds. The rigorous derivation of the radiative source term is restricted to the flat reference plate. Corrugated and perforated geometries are included as illustrative tests of the adaptability of the surface finite element implementation, with corrugated cases interpreted only for very low-amplitude surfaces where self-radiative exchange is neglected.