A Robust Newton-Based Variational Framework for Nonlinear Conduction-Radiation PDEs with Localized Sources

Abstract

A steady radiative-conductive problem is considered in a thin rectangular plate irradiated by a small hot source positioned at height H. A standard thickness reduction leads from the 3D balance equation to a 2D nonlinear equation with fourth-power radiative emission and homogeneous Neumann boundary conditions. A weak variational formulation discretized by finite elements and solved via Newton’s method is employed. Analytical upper and lower bounds constrain the numerical solution, ensuring physical consistency. As H increases from 0.1 to 10.0 m, maximum temperatures decrease from approximately 10 to approximately 0.1 and the field approaches spatial uniformity.