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Finite element technique applied to heat conduction in solids with temperature dependent thermal conductivity
Author(s) -
AguirreRamirez G.,
Oden J. T.
Publication year - 1973
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620070311
Subject(s) - thermal conduction , heat flux , relativistic heat conduction , heat transfer , finite element method , heat equation , thermal diffusivity , thermal conductivity , mathematical analysis , dimensionless quantity , boundary value problem , thermodynamics , mathematics , mechanics , materials science , physics
Consider a solid heat conductor with a non‐linear constitutive equation for the heat flux. If the material is anisotropic and inhomogeneous, the heat conduction equation to be satisfied by the temperature field θ(x, t) is,\documentclass{article}\pagestyle{empty}\begin{document}$$ \rho c\frac{{\partial \theta }}{{\partial t}} = {\rm div}({\rm L}(\theta,{\rm x})[{\rm grad}\theta]) + q $$\end{document}Here L (θ, x ) [grad θ] is a vector‐valued function of θ, x , grad θ which is linear in grad θ, In the present paper, the application of the finite element method to the solution of this class of problems is demonstrated. General discrete models are developed which enable approximate solutions to be obtained for arbitrary three‐dimensional regions and the following boundary and initial conditions: (a) prescribed surface temperature, (b) prescribed heat flux at the surface and (c) linear heat transfer at the surface. Numerical examples involve a homogeneous solid with a dimensionless temperature‐diffusivity curve of the form κ = κ 0 (l + σT ). The resulting system of non‐linear differential equations is integrated numerically.