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

Jose Lages da Silva Neto; Eduardo D. Correa; Rogerio M. S. Gama; Gustavo R. Anjos; Jose Lages da Silva Neto; Eduardo D. Correa; Rogerio M. S. Gama; Gustavo R. Anjos

doi:10.3934/math.20261100

AIMS Mathematics

2026, Volume 11, Issue 8: 27481-27507. doi: 10.3934/math.20261100

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

1.
Department of Mechanical Engineering, Universidade do Estado do Rio de Janeiro–UERJ, Rio de Janeiro, RJ 20550-900, Brazil
2.
Department of Mechanical Engineering/Coppe, Universidade Federal do Rio de Janeiro–UFRJ, Rio de Janeiro, RJ 21941-914, Brazil

Received: 25 June 2026 Revised: 15 August 2026 Accepted: 25 August 2026 Published: 31 August 2026
MSC : 35J61, 65N30, 80A20

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^2 $ and $ H^1 $ 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.
- radiative heat transfer,
- finite element method,
- Newton linearization,
- Stefan–Boltzmann law,
- method of manufactured solutions,
- a priori bounds
Citation: Jose Lages da Silva Neto, Eduardo D. Correa, Rogerio M. S. Gama, Gustavo R. Anjos. Analytical bounds and finite element modeling of the temperature distribution on a thin plate heated by a localized radiant source[J]. AIMS Mathematics, 2026, 11(8): 27481-27507. doi: 10.3934/math.20261100

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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^2 $ and $ H^1 $ 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.

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