Journal of Applied Mathematics
Volume 2004 (2004), Issue 4, Pages 311-330

The heat radiation problem: three-dimensional analysis for arbitrary enclosure geometries

Naji Qatanani1 and Monika Schulz2

1College of Science and Technology, Al-Quds University, Abu Dis, P.O. Box 20002, Jerusalem, Palestinian Authority
2Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, Stuttgart 70569, Germany

Received 25 June 2003; Revised 22 February 2004

Copyright © 2004 Naji Qatanani and Monika Schulz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper gives very significant and up-to-date analytical and numerical results to the three-dimensional heat radiation problem governed by a boundary integral equation. There are two types of enclosure geometries to be considered: convex and nonconvex geometries. The properties of the integral operator of the radiosity equation have been thoroughly investigated and presented. The application of the Banach fixed point theorem proves the existence and the uniqueness of the solution of the radiosity equation. For a nonconvex enclosure geometries, the visibility function must be taken into account. For the numerical treatment of the radiosity equation, we use the boundary element method based on the Galerkin discretization scheme. As a numerical example, we implement the conjugate gradient algorithm with preconditioning to compute the outgoing flux for a three-dimensional nonconvex geometry. This has turned out to be the most efficient method to solve this type of problems.