Abstract.We study the well-posechiess of a class of models describing heat transfer by conduction and radiation.For that purpose we propose an abstract mathematical framework that allows us to prove existence, uniqueness and the comparison principle for the weak solution with minimal or almost minimal a priori assumptions for the data. The theory covers different types of grey materials, that is, both semitransparent and opaque bodies as well as isotropic or nonisotropic scattering/reflection provided that the material properties do not depend on the wavelength of the radiation. To demonstrate the use of the abstract theory we consider in detail two examples, heat transfer between opaque bodies with diffuse-grey surfaces and a model with semitransparent material and specularly reflecting surfaces.
In this paper we consider numerical methods for stationary free boundary problems. We start by analysing systematically di erent shape optimization formulations of a model problem and show how the optimality conditions relate to construction of trial type methods. Shape sensitivity analysis of the free boundary leads also to the so-called total linearization method which combines the good properties of Newton method and trial methods, i.e. fast convergence and relative simplicity of implementation. Detailed implementation for a model problem together with numerical tests is presented.
We consider both stationary and time-dependent heat
equations for a non-convex body or
a collection of disjoint conducting bodies with Stefan-Boltzmann radiation
conditions on
the surface. The main novelty of the resulting problem is the non-locality
of the boundary
condition due to self-illuminating radiation on the surface. Moreover,
the
problem is nonlinear
and in the general case also non-coercive. We show that the non-local
boundary value problem
admits a maximum principle. Hence, we can prove the existence of a
weak solution assuming
the existence of upper and lower solutions. This result is then applied
to
prove existence under some hypotheses that guarantee the existence of sub-
and
supersolutions. Some special
cases where the problem is coercive are also discussed. Finally, the
analysis is extended to cases with nonlinear material properties.
This paper focuses on finite element error analysis for problems involving both conductive and radiative heat transfers. The radiative heat exchange is modeled with a nonlinear and nonlocal term that also makes the problem non-monotone. The continuous problem has a maximum principle which suggests the use of inverse monotone discretizations. We also estimate the error due to the approximation of the boundary by showing continuous dependence on the geometric data for the continuous problem. The final result of this paper is a rigorous justification and error analysis for methods that use the so-called view factors for numerical modeling of the heat radiation.
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