American Institute of Mathematical Sciences

Advanced
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:
[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.   Google Scholar

[2]

G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem,, Multiscale Model.Sim., 7 (2008), 1148.  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.   Google Scholar

[4]

T. Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces,, Math. Method. Appl. Sci., 20 (1997), 47.  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.   Google Scholar

[6]

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

[7]

L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties,, second edition, (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, (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.  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.  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, (1978).   Google Scholar

[12]

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

[2]

G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem,, Multiscale Model.Sim., 7 (2008), 1148.  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.   Google Scholar

[4]

T. Tiihonen, Stefan-Boltzmann radiation on non-convex surfaces,, Math. Method. Appl. Sci., 20 (1997), 47.  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.   Google Scholar

[6]

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

[7]

L. J. Gibson and M. F. Ashby, Cellular Solids:Structure and Properties,, second edition, (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, (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.  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.  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, (1978).   Google Scholar

[12]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization,, North-Holland, (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.  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.   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.  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.  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.  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, (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.  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.   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.   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.  doi: 10.1007/s00211-005-0668-4.  Google Scholar

[1]

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
[2]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[3]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[4]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[5]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[6]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053
[7]

Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022
[8]

Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083
[9]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
[10]

Jiyu Zhong, Shengfu Deng. Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1581-1600. doi: 10.3934/dcdsb.2018062
[11]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[12]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[13]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[14]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[15]

Xi-De Zhu, Li-Ping Pang, Gui-Hua Lin. Two approaches for solving mathematical programs with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 951-968. doi: 10.3934/jimo.2015.11.951
[16]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687
[17]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[18]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
[19]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339
[20]

Jiann-Sheng Jiang, Kung-Hwang Kuo, Chi-Kun Lin. Homogenization of second order equation with spatial dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 303-313. doi: 10.3934/dcds.2005.12.303

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences