Journal of Applied Mathematics

Use of the multigrid methods for heat radiation problem

Naji A. Qatanani

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We consider the integral equation arising as a result of heat radiation exchange in both convex and nonconvex enclosures of diffuse grey surfaces. For nonconvex geometries, the visibility function must be taken into consideration. Therefore, a geometrical algorithm has been developed to provide an efficient detection of the shadow zones. For the numerical realization of the Fredholm integral equation, a boundary element method based on Galerkin-Bubnov discretization scheme is implemented. Consequently, multigrid iteration methods, which are closely related to two-grid methods, are used to solve the system of linear equations. To demonstrate the high efficiency of these iterations, we construct some numerical experiments for different enclosure geometries.

Article information

J. Appl. Math., Volume 2003, Number 6 (2003), 305-317.

First available in Project Euclid: 26 May 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65R20: Integral equations 65N38
Secondary: 65F10 65N55


Qatanani, Naji A. Use of the multigrid methods for heat radiation problem. J. Appl. Math. 2003 (2003), no. 6, 305--317. doi:10.1155/S1110757X03301160.

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  • R. A. Białecki, Boundary element calculation of the radiative heat sources, Advanced Computational Methods in Heat Transfer II (I. C. Wrobel, C. A. Brebbia, and A. J. Nowak, eds.), vol. 1, Elsevier, London, 1992.
  • --------, Solving Heat Radiation Problems Using the Boundary Element Method, Topics in Engineering, vol. 15, Computational Mechanics Publications, Southampton, 1993. \CMP1+266+1651 266 165
  • J. Blobner, R. A. Białecki, and G. Kuhn, Transient non-linear heat the second author and last page in [1 according to the MathSciNet database.?] element formulation}, Internat. J. Numer. Methods Engrg. 46 (1999), no. 11, 1865--1882.
  • M. F. Cohen and J. R. Wallace, Radiosity and Realistic Image Synthesis the MathSciNet database.}, Academic Press, Massachusetts, 1993.
  • W. Hackbusch, Multigrid Methods and Applications, Springer Series in Computational Mathematics, vol. 4, Springer-Verlag, Berlin, 1985.
  • --------, Integralgleichungen. Theorie und Numerik [Integral Equations. Theory and Numerics], Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, 1989 (German).
  • F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1985.
  • M. Laitinen and T. Tiihonen, Conductive-radiative heat transfer in [8.?]}, Reports of the Department of Mathematical Information Technology Series B, Scientific Computing, no. B6/2000, University of Jyväskylä, Finland, 2000.
  • M. F. Modest, Radiative Heat Transfer, Mc-Graw Hill, New York, 1993.
  • N. Qatanani, Lösungsverfahren und Analysis der Integralgleichung für das Hohlraum-Strahlungs-Problem, Ph.D. thesis, Universität Stuttgart, Germany, 1996.
  • N. Qatanani, M. Schulz, and W. Wendland, Solution methods and analysis of heat radiation integral equation, in preparation.