# American Institute of Mathematical Sciences

January  2013, 18(1): 1-36. doi: 10.3934/dcdsb.2013.18.1

## Second order corrector in the homogenization of a conductive-radiative heat transfer problem

 1 CMAP, Ecole Polytechnique, 91128 Palaiseau, & DM2S, CEA Saclay, 91191 Gif sur Yvette, France 2 DM2S/SFME/LTMF, CEA Saclay, 91191 Gif sur Yvette, & CMAP, Ecole Polytechnique, 91128 Palaiseau, France

Received  March 2012 Revised  July 2012 Published  September 2012

This paper focuses on the contribution of the so-called second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. More precisely, heat is diffusing in a periodically perforated domain with a non-local boundary condition modelling the radiative transfer in each hole. If the source term is a periodically oscillating function (which is the case in our application to nuclear reactor physics), a strong gradient of the temperature takes place in each periodicity cell, corresponding to a large heat flux between the sources and the perforations. This effect cannot be taken into account by the homogenized model, neither by the first order corrector. We show that this local gradient effect can be reproduced if the second order corrector is added to the reconstructed solution.
Citation: Grégoire Allaire, Zakaria Habibi. Second order corrector in the homogenization of a conductive-radiative heat transfer problem. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 1-36. doi: 10.3934/dcdsb.2013.18.1
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  Google Scholar [2] G. Allaire and M. Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var., 4 (1999), 209-243.  Google Scholar [3] G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model. Simul., 7 (2008), 1148-1170.  Google Scholar [4] G. Allaire and Z. Habibi, "Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem," submitted. Internal report, 746, CMAP, Ecole Polytechnique (March 2012). Available from: http://www.cmap.polytechnique.fr/preprint/repository/746.pdf Google Scholar [5] A. A. Amosov, Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on radiation frequency, J. Math. Sci., 164 (2010), 309-344.  Google Scholar [6] N. Bakhvalov and G. Panasenko, "Homogenisation: Averaging Processes in Periodic Media," vol. 36 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. Leĭtes.  Google Scholar [7] A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic analysis for periodic structures," vol. 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978.  Google Scholar [8] J. F. Bourgat, "Numerical Experiments of the Homogenization Method for Operators with Periodic Coefficients," Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977), I, pp. 330-356, Lecture Notes in Math., 704, Springer, Berlin, 1979.  Google Scholar [9] J. F. Bourgat and A. Dervieux, "Méthode D'homogénéisation des Opérateurs à Coefficients Périodiques: Étude des Correcteurs Provenant du Développement Asymptotique," IRIA-LABORIA, Rapport 278, 1978. Google Scholar [10] S. Boyaval, Reduced-bases approach for homogenization beyond the periodic setting, Multiscale Model. Simul., 7 (2008), 466-494.  Google Scholar [11] J. Casado-Diaz, The asymptotic behaviour near the boundary of periodic homogenization problems via two-scale convergence, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 33-66.  Google Scholar [12] , Cast3M., Available from: , ().   Google Scholar [13] S. Chandrasekhar, "Radiative Transfer," Dover Publications Inc., New York, 1960.  Google Scholar [14] K. Cherednichenko and V. Smyshlyaev, On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems, Arch. Ration. Mech. Anal., 174 (2004), 385-442.  Google Scholar [15] D. Cioranescu and P. Donato, "An Introduction to Homogenization," vol. 17 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1999.  Google Scholar [16] CEA e-den, "Les réacteurs Nucléaires à Caloporteur Gaz," CEA Saclay et Le Moniteur Editions. Monographie Den, 2006. Available from: http://nucleaire.cea.fr/fr/publications/pdf/M0-fr.pdf. Google Scholar [17] C. Conca, R. Orive and M. Vanninathan, First and second corrector in homogenization by Bloch waves, Bol. Soc. Esp. Mat. Apl. S$\vec{\ e}$MA, (2008), 61-69.  Google Scholar [18] D. Gérard-Varet and N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc., 13 (2011), 1477-1503.  Google Scholar [19] D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layer,, to appear in Acta Mathematica., ().   Google Scholar [20] G. Griso, Interior error estimate for periodic homogenization, Anal. Appl. (Singap.), 4 (2006), 61-79.  Google Scholar [21] Z. Habibi, Homogenization of a conductive-radiative heat transfer problem, the contribution of a second order corrector,, ESAIM: Proc. Volume 35, (): 228.   Google Scholar [22] Z. Habibi, Homogénéisation et convergence à deux échelles lors d'échanges thermiques stationnaires et transitoires dans un cœur de réacteur à caloporteur gaz, PhD thesis, Ecole Polytechnique, 2011. Available from: http://tel.archives-ouvertes.fr/tel-00695638 Google Scholar [23] P. S. Heckbert., "Simulating Global Illumination Using Adaptive Meshing," PhD thesis, UC Berkeley, 1991. Google Scholar [24] V. H. Hoang and Ch. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul. 3 (2004/05), 3 (): 168.   Google Scholar [25] U. Hornung, "Homogenization and Porous Media," vol. 6 of "Interdisciplinary Applied Mathematics," Springer-Verlag, New York, 1997.  Google Scholar [26] J. R. Howell, "A Catalogue of Radiation Heat Transfer Factors," The university of Texas at Austin, Austin, Texas, 3 edition, (2010). Available from: http://www.engr.uky.edu/rtl/Catalog/. Google Scholar [27] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer Verlag, 1994.  Google Scholar [28] C. Kenig, F. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal. 203 (2012), 1009-1036, arXiv:1103.0023. Google Scholar [29] M. Laitinen and T. Tiihonen, Conductive-radiative heat transfer in grey materials, Quart. Appl. Math., 59 (2001), 737-768.  Google Scholar [30] J. L. Lions, "Some Methods in the Mathematical Analysis of Systems and Their Control," Kexue Chubanshe (Science Press), Beijing, 1981.  Google Scholar [31] Y. Maday, A. T. Patera and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, C. R. Math. Acad. Sci. Paris, 335 (2002), 289-294.  Google Scholar [32] F. M. Modest, "Radiative Heat Transfer," Academic Press, 2 edition, 2003. Google Scholar [33] S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of aperiodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299.  Google Scholar [34] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  Google Scholar [35] D. Onofrei and B. Vernescu, Error estimates for periodic homogenization with non-smooth coefficients, Asymptot. Anal., 54 (2007), 103-123.  Google Scholar [36] E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory," vol. 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.  Google Scholar [37] L. Tartar, "The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana," Springer Verlag, Berlin; UMI, Bologna, 7 edition, 2009.  Google Scholar [38] T. Tiihonen, Stefan-boltzman radiation on non-convex surfaces, Math. Methods Appl. Sci., 20 (1997), 47-57.  Google Scholar [39] T. Tiihonen, Finite element approximation of nonlocal heat radiation problems, Math. Models Methods Appl. Sci., 8 (1998), 1071-1089.  Google Scholar

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  Google Scholar [2] G. Allaire and M. Amar, Boundary layer tails in periodic homogenization, ESAIM Control Optim. Calc. Var., 4 (1999), 209-243.  Google Scholar [3] G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model. Simul., 7 (2008), 1148-1170.  Google Scholar [4] G. Allaire and Z. Habibi, "Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem," submitted. Internal report, 746, CMAP, Ecole Polytechnique (March 2012). Available from: http://www.cmap.polytechnique.fr/preprint/repository/746.pdf Google Scholar [5] A. A. Amosov, Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on radiation frequency, J. Math. Sci., 164 (2010), 309-344.  Google Scholar [6] N. Bakhvalov and G. Panasenko, "Homogenisation: Averaging Processes in Periodic Media," vol. 36 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. Leĭtes.  Google Scholar [7] A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic analysis for periodic structures," vol. 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978.  Google Scholar [8] J. F. Bourgat, "Numerical Experiments of the Homogenization Method for Operators with Periodic Coefficients," Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977), I, pp. 330-356, Lecture Notes in Math., 704, Springer, Berlin, 1979.  Google Scholar [9] J. F. Bourgat and A. Dervieux, "Méthode D'homogénéisation des Opérateurs à Coefficients Périodiques: Étude des Correcteurs Provenant du Développement Asymptotique," IRIA-LABORIA, Rapport 278, 1978. Google Scholar [10] S. Boyaval, Reduced-bases approach for homogenization beyond the periodic setting, Multiscale Model. Simul., 7 (2008), 466-494.  Google Scholar [11] J. Casado-Diaz, The asymptotic behaviour near the boundary of periodic homogenization problems via two-scale convergence, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 33-66.  Google Scholar [12] , Cast3M., Available from: , ().   Google Scholar [13] S. Chandrasekhar, "Radiative Transfer," Dover Publications Inc., New York, 1960.  Google Scholar [14] K. Cherednichenko and V. Smyshlyaev, On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems, Arch. Ration. Mech. Anal., 174 (2004), 385-442.  Google Scholar [15] D. Cioranescu and P. Donato, "An Introduction to Homogenization," vol. 17 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1999.  Google Scholar [16] CEA e-den, "Les réacteurs Nucléaires à Caloporteur Gaz," CEA Saclay et Le Moniteur Editions. Monographie Den, 2006. Available from: http://nucleaire.cea.fr/fr/publications/pdf/M0-fr.pdf. Google Scholar [17] C. Conca, R. Orive and M. Vanninathan, First and second corrector in homogenization by Bloch waves, Bol. Soc. Esp. Mat. Apl. S$\vec{\ e}$MA, (2008), 61-69.  Google Scholar [18] D. Gérard-Varet and N. Masmoudi, Homogenization in polygonal domains, J. Eur. Math. Soc., 13 (2011), 1477-1503.  Google Scholar [19] D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layer,, to appear in Acta Mathematica., ().   Google Scholar [20] G. Griso, Interior error estimate for periodic homogenization, Anal. Appl. (Singap.), 4 (2006), 61-79.  Google Scholar [21] Z. Habibi, Homogenization of a conductive-radiative heat transfer problem, the contribution of a second order corrector,, ESAIM: Proc. Volume 35, (): 228.   Google Scholar [22] Z. Habibi, Homogénéisation et convergence à deux échelles lors d'échanges thermiques stationnaires et transitoires dans un cœur de réacteur à caloporteur gaz, PhD thesis, Ecole Polytechnique, 2011. Available from: http://tel.archives-ouvertes.fr/tel-00695638 Google Scholar [23] P. S. Heckbert., "Simulating Global Illumination Using Adaptive Meshing," PhD thesis, UC Berkeley, 1991. Google Scholar [24] V. H. Hoang and Ch. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul. 3 (2004/05), 3 (): 168.   Google Scholar [25] U. Hornung, "Homogenization and Porous Media," vol. 6 of "Interdisciplinary Applied Mathematics," Springer-Verlag, New York, 1997.  Google Scholar [26] J. R. Howell, "A Catalogue of Radiation Heat Transfer Factors," The university of Texas at Austin, Austin, Texas, 3 edition, (2010). Available from: http://www.engr.uky.edu/rtl/Catalog/. Google Scholar [27] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer Verlag, 1994.  Google Scholar [28] C. Kenig, F. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal. 203 (2012), 1009-1036, arXiv:1103.0023. Google Scholar [29] M. Laitinen and T. Tiihonen, Conductive-radiative heat transfer in grey materials, Quart. Appl. Math., 59 (2001), 737-768.  Google Scholar [30] J. L. Lions, "Some Methods in the Mathematical Analysis of Systems and Their Control," Kexue Chubanshe (Science Press), Beijing, 1981.  Google Scholar [31] Y. Maday, A. T. Patera and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, C. R. Math. Acad. Sci. Paris, 335 (2002), 289-294.  Google Scholar [32] F. M. Modest, "Radiative Heat Transfer," Academic Press, 2 edition, 2003. Google Scholar [33] S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of aperiodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299.  Google Scholar [34] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  Google Scholar [35] D. Onofrei and B. Vernescu, Error estimates for periodic homogenization with non-smooth coefficients, Asymptot. Anal., 54 (2007), 103-123.  Google Scholar [36] E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory," vol. 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.  Google Scholar [37] L. Tartar, "The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana," Springer Verlag, Berlin; UMI, Bologna, 7 edition, 2009.  Google Scholar [38] T. Tiihonen, Stefan-boltzman radiation on non-convex surfaces, Math. Methods Appl. Sci., 20 (1997), 47-57.  Google Scholar [39] T. Tiihonen, Finite element approximation of nonlocal heat radiation problems, Math. Models Methods Appl. Sci., 8 (1998), 1071-1089.  Google Scholar
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