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The secondorder twoscale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials
1.  Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China 
2.  LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China, China 
References:
[1] 
S. T. Liu and Y. C. Zhang, Multiscale analysis method for thermal conductivity of composite material with radiation, Multidiscipline Modeling in Mat. and Str., 2 (2006), 327344. 
[2] 
G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model.Sim., 7 (2008), 11481170. doi: 10.1137/080714737. 
[3] 
N. S. Bakhvalov, Averaging of the heat transfer process in periodic media with radiative, Differ. Uraun., 17 (1981), 17651773. 
[4] 
T. Tiihonen, StefanBoltzmann radiation on nonconvex surfaces, Math. Method. Appl. Sci., 20 (1997), 4757. doi: 10.1002/(SICI)10991476(19970110)20:1<47::AIDMMA847>3.0.CO;2B. 
[5] 
N. Qatanani, Analysis of the heat equation with nonlocal radiation terms in a nonconvex diffuse and grey surfaces, Eur. J. Sci. Res., 15 (2006), 245254. 
[6] 
K. Daryabeigi, Analysis and testing of high temperature fibrous insulation for reusable launch vehicles, 37th AIAA Aerospace Sciences Meeting and Exhibit, January 1114, (1999), Reno, NV. doi: 10.2514/6.19991044. 
[7] 
L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties, second edition, Cambridge University Press, 1997. 
[8] 
K. El Ganaoui, Homogénéisation de Modéles de Transferts Thermiques et Radiatifs: Application au Coeur des Réacteurs A Caloporteur Gaz, Ph.D thesis, Ecole Polytechnique, 2006. 
[9] 
K. Terada, M. Kurumatani, T. Ushida and N. Kikuchi, A method of twoscale thermomechanical analysis for porous solids with microscale heat transfer, Comp. Mech., 46 (2010), 269285. doi: 10.1007/s0046600904009. 
[10] 
F. Su, J. Z. Cui and Z. Xu, A twoorder and twoscale computation method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients, Appl. Math. MechEngl., 30 (2009), 15791588. doi: 10.1007/s104830091209z. 
[11] 
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, NorthHolland, Amsterdam, 1978. 
[12] 
O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, NorthHolland, Amsterdam, 1992. 
[13] 
L. Q. Cao, J. Z. Cui and D. C. Zhu, Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equation with oscillating coefficients over general convex domains, SIAM J.Numer.Anal., 40 (2002), 543577. doi: 10.1137/S0036142900376110. 
[14] 
Z. Q. Yang, J. Z. Cui, Y. F. Nie and Q. Ma, The secondorder twoscale method for heat transfer performances of periodic porous materials with interior surface radiation, CMES: Comp. Model. Eng., 88 (2012), 419442. 
[15] 
J. Z. Cui, T. M. Shin and Y. L. Wang, Twoscale analysis method for bodies with small periodic configurations, Struct. Eng. Mech., 7 (1999), 601614. doi: 10.12989/sem.1999.7.6.601. 
[16] 
A. A. Amosov, Semidiscrete and asymptotic approximations for the nonstationary radiativeconductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 176 (2011), 361408. doi: 10.1007/s1095801103992. 
[17] 
A. A. Amosov, Nonstationary radiativeconductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 169 (2010), 145. doi: 10.1007/s1095801000374. 
[18] 
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications II, SpringerVerlag, Berlin, 1972. 
[19] 
G. Allaire and Z. Habibi, Homogenization of a conductive, convective and radiative heat transfer problem in a heterogeneous domain, SIAM J. Math. Anal., 45 (2013), 11361178. doi: 10.1137/110849821. 
[20] 
L. Q. Cao and J. Z. Cui, The twoscale asymptotic analysis for elastic structures of composites materials with only including entirely basic configuration, Acta Math. Appl. Sin., 22 (1999), 3846 (in Chinese). 
[21] 
W. Allegretta, L. Q. Cao and Y. P. Lin, Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients, Discret Contin. Dyn. S., 20 (2008), 543576. 
[22] 
L. Q. Cao, Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains, Numer. Math., 103 (2006), 1145. doi: 10.1007/s0021100506684. 
show all references
References:
[1] 
S. T. Liu and Y. C. Zhang, Multiscale analysis method for thermal conductivity of composite material with radiation, Multidiscipline Modeling in Mat. and Str., 2 (2006), 327344. 
[2] 
G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model.Sim., 7 (2008), 11481170. doi: 10.1137/080714737. 
[3] 
N. S. Bakhvalov, Averaging of the heat transfer process in periodic media with radiative, Differ. Uraun., 17 (1981), 17651773. 
[4] 
T. Tiihonen, StefanBoltzmann radiation on nonconvex surfaces, Math. Method. Appl. Sci., 20 (1997), 4757. doi: 10.1002/(SICI)10991476(19970110)20:1<47::AIDMMA847>3.0.CO;2B. 
[5] 
N. Qatanani, Analysis of the heat equation with nonlocal radiation terms in a nonconvex diffuse and grey surfaces, Eur. J. Sci. Res., 15 (2006), 245254. 
[6] 
K. Daryabeigi, Analysis and testing of high temperature fibrous insulation for reusable launch vehicles, 37th AIAA Aerospace Sciences Meeting and Exhibit, January 1114, (1999), Reno, NV. doi: 10.2514/6.19991044. 
[7] 
L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties, second edition, Cambridge University Press, 1997. 
[8] 
K. El Ganaoui, Homogénéisation de Modéles de Transferts Thermiques et Radiatifs: Application au Coeur des Réacteurs A Caloporteur Gaz, Ph.D thesis, Ecole Polytechnique, 2006. 
[9] 
K. Terada, M. Kurumatani, T. Ushida and N. Kikuchi, A method of twoscale thermomechanical analysis for porous solids with microscale heat transfer, Comp. Mech., 46 (2010), 269285. doi: 10.1007/s0046600904009. 
[10] 
F. Su, J. Z. Cui and Z. Xu, A twoorder and twoscale computation method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients, Appl. Math. MechEngl., 30 (2009), 15791588. doi: 10.1007/s104830091209z. 
[11] 
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, NorthHolland, Amsterdam, 1978. 
[12] 
O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, NorthHolland, Amsterdam, 1992. 
[13] 
L. Q. Cao, J. Z. Cui and D. C. Zhu, Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equation with oscillating coefficients over general convex domains, SIAM J.Numer.Anal., 40 (2002), 543577. doi: 10.1137/S0036142900376110. 
[14] 
Z. Q. Yang, J. Z. Cui, Y. F. Nie and Q. Ma, The secondorder twoscale method for heat transfer performances of periodic porous materials with interior surface radiation, CMES: Comp. Model. Eng., 88 (2012), 419442. 
[15] 
J. Z. Cui, T. M. Shin and Y. L. Wang, Twoscale analysis method for bodies with small periodic configurations, Struct. Eng. Mech., 7 (1999), 601614. doi: 10.12989/sem.1999.7.6.601. 
[16] 
A. A. Amosov, Semidiscrete and asymptotic approximations for the nonstationary radiativeconductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 176 (2011), 361408. doi: 10.1007/s1095801103992. 
[17] 
A. A. Amosov, Nonstationary radiativeconductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 169 (2010), 145. doi: 10.1007/s1095801000374. 
[18] 
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications II, SpringerVerlag, Berlin, 1972. 
[19] 
G. Allaire and Z. Habibi, Homogenization of a conductive, convective and radiative heat transfer problem in a heterogeneous domain, SIAM J. Math. Anal., 45 (2013), 11361178. doi: 10.1137/110849821. 
[20] 
L. Q. Cao and J. Z. Cui, The twoscale asymptotic analysis for elastic structures of composites materials with only including entirely basic configuration, Acta Math. Appl. Sin., 22 (1999), 3846 (in Chinese). 
[21] 
W. Allegretta, L. Q. Cao and Y. P. Lin, Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients, Discret Contin. Dyn. S., 20 (2008), 543576. 
[22] 
L. Q. Cao, Multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains, Numer. Math., 103 (2006), 1145. doi: 10.1007/s0021100506684. 
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