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May  2014, 19(3): 827-848. doi: 10.3934/dcdsb.2014.19.827

The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials

Zhiqiang Yang 1, , Junzhi Cui 2, and  Qiang Ma 2,

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

Received  March 2013 Revised  August 2013 Published  February 2014

In this paper, a kind of second-order two-scale (SOTS) computation is developed for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials, where the convection part is composed of long thin parallel pipes with periodic distribution, the conduction part occupied by solid materials and the radiation part is on the pipe's walls and the surfaces of cavities. First of all, by asymptotic expansion of the temperature field, the homogenization problem, first-order correctors and second-order correctors are obtained successively. Then, the error estimation of the second-order two-scale approximate solution is derived on some regularity hypothesis. Finally, the corresponding finite element algorithms are proposed and some numerical results are presented. The numerical tests indicate that the developed method can be successfully used for solving the integrated heat transfer problem, which can reduce the computational efforts greatly.
Keywords: Second-order two-scale computation, periodic porous materials., integrated heat transfer problem, error estimation, homogenization.
Mathematics Subject Classification: Primary: 35B27, 80M40; Secondary: 80A2.
Citation: Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827
References:
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S. T. Liu and Y. C. Zhang, Multi-scale analysis method for thermal conductivity of composite material with radiation, Multidiscipline Modeling in Mat. and Str., 2 (2006), 327-344. Google Scholar

[2]

G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model.Sim., 7 (2008), 1148-1170. doi: 10.1137/080714737.  Google Scholar

[3]

N. S. Bakhvalov, Averaging of the heat transfer process in periodic media with radiative, Differ. Uraun., 17 (1981), 1765-1773.  Google Scholar

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T. Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces, Math. Method. Appl. Sci., 20 (1997), 47-57. doi: 10.1002/(SICI)1099-1476(19970110)20:1<47::AID-MMA847>3.0.CO;2-B.  Google Scholar

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N. Qatanani, Analysis of the heat equation with non-local radiation terms in a non-convex diffuse and grey surfaces, Eur. J. Sci. Res., 15 (2006), 245-254. Google Scholar

[6]

K. Daryabeigi, Analysis and testing of high temperature fibrous insulation for reusable launch vehicles, 37th AIAA Aerospace Sciences Meeting and Exhibit, January 11-14, (1999), Reno, NV. doi: 10.2514/6.1999-1044.  Google Scholar

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L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties, second edition, Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[9]

K. Terada, M. Kurumatani, T. Ushida and N. Kikuchi, A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer, Comp. Mech., 46 (2010), 269-285. doi: 10.1007/s00466-009-0400-9.  Google Scholar

[10]

F. Su, J. Z. Cui and Z. Xu, A two-order and two-scale computation method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients, Appl. Math. Mech-Engl., 30 (2009), 1579-1588. doi: 10.1007/s10483-009-1209-z.  Google Scholar

[11]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam, 1978.  Google Scholar

[12]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.  Google Scholar

[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), 543-577. doi: 10.1137/S0036142900376110.  Google Scholar

[14]

Z. Q. Yang, J. Z. Cui, Y. F. Nie and Q. Ma, The second-order two-scale method for heat transfer performances of periodic porous materials with interior surface radiation, CMES: Comp. Model. Eng., 88 (2012), 419-442.  Google Scholar

[15]

J. Z. Cui, T. M. Shin and Y. L. Wang, Two-scale analysis method for bodies with small periodic configurations, Struct. Eng. Mech., 7 (1999), 601-614. doi: 10.12989/sem.1999.7.6.601.  Google Scholar

[16]

A. A. Amosov, Semidiscrete and asymptotic approximations for the nonstationary radiative-conductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 176 (2011), 361-408. doi: 10.1007/s10958-011-0399-2.  Google Scholar

[17]

A. A. Amosov, Nonstationary radiative-conductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 169 (2010), 1-45. doi: 10.1007/s10958-010-0037-4.  Google Scholar

[18]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications II, Springer-Verlag, Berlin, 1972.  Google Scholar

[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), 1136-1178. doi: 10.1137/110849821.  Google Scholar

[20]

L. Q. Cao and J. Z. Cui, The two-scale asymptotic analysis for elastic structures of composites materials with only including entirely basic configuration, Acta Math. Appl. Sin., 22 (1999), 38-46 (in Chinese).  Google Scholar

[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), 543-576.  Google Scholar

[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), 11-45. doi: 10.1007/s00211-005-0668-4.  Google Scholar

show all references

References:
[1]

S. T. Liu and Y. C. Zhang, Multi-scale analysis method for thermal conductivity of composite material with radiation, Multidiscipline Modeling in Mat. and Str., 2 (2006), 327-344. Google Scholar

[2]

G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model.Sim., 7 (2008), 1148-1170. doi: 10.1137/080714737.  Google Scholar

[3]

N. S. Bakhvalov, Averaging of the heat transfer process in periodic media with radiative, Differ. Uraun., 17 (1981), 1765-1773.  Google Scholar

[4]

T. Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces, Math. Method. Appl. Sci., 20 (1997), 47-57. doi: 10.1002/(SICI)1099-1476(19970110)20:1<47::AID-MMA847>3.0.CO;2-B.  Google Scholar

[5]

N. Qatanani, Analysis of the heat equation with non-local radiation terms in a non-convex diffuse and grey surfaces, Eur. J. Sci. Res., 15 (2006), 245-254. Google Scholar

[6]

K. Daryabeigi, Analysis and testing of high temperature fibrous insulation for reusable launch vehicles, 37th AIAA Aerospace Sciences Meeting and Exhibit, January 11-14, (1999), Reno, NV. doi: 10.2514/6.1999-1044.  Google Scholar

[7]

L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties, second edition, Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[9]

K. Terada, M. Kurumatani, T. Ushida and N. Kikuchi, A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer, Comp. Mech., 46 (2010), 269-285. doi: 10.1007/s00466-009-0400-9.  Google Scholar

[10]

F. Su, J. Z. Cui and Z. Xu, A two-order and two-scale computation method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients, Appl. Math. Mech-Engl., 30 (2009), 1579-1588. doi: 10.1007/s10483-009-1209-z.  Google Scholar

[11]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam, 1978.  Google Scholar

[12]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.  Google Scholar

[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), 543-577. doi: 10.1137/S0036142900376110.  Google Scholar

[14]

Z. Q. Yang, J. Z. Cui, Y. F. Nie and Q. Ma, The second-order two-scale method for heat transfer performances of periodic porous materials with interior surface radiation, CMES: Comp. Model. Eng., 88 (2012), 419-442.  Google Scholar

[15]

J. Z. Cui, T. M. Shin and Y. L. Wang, Two-scale analysis method for bodies with small periodic configurations, Struct. Eng. Mech., 7 (1999), 601-614. doi: 10.12989/sem.1999.7.6.601.  Google Scholar

[16]

A. A. Amosov, Semidiscrete and asymptotic approximations for the nonstationary radiative-conductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 176 (2011), 361-408. doi: 10.1007/s10958-011-0399-2.  Google Scholar

[17]

A. A. Amosov, Nonstationary radiative-conductive heat transfer problem in a periodic system of grey heat shields, J. Math. Sci., 169 (2010), 1-45. doi: 10.1007/s10958-010-0037-4.  Google Scholar

[18]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications II, Springer-Verlag, Berlin, 1972.  Google Scholar

[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), 1136-1178. doi: 10.1137/110849821.  Google Scholar

[20]

L. Q. Cao and J. Z. Cui, The two-scale asymptotic analysis for elastic structures of composites materials with only including entirely basic configuration, Acta Math. Appl. Sin., 22 (1999), 38-46 (in Chinese).  Google Scholar

[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), 543-576.  Google Scholar

[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), 11-45. doi: 10.1007/s00211-005-0668-4.  Google Scholar

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