March  2016, 36(3): 1415-1430. doi: 10.3934/dcds.2016.36.1415

Effective boundary conditions of the heat equation on a body coated by functionally graded material

1. 

Center for Partial Di erential Equations, East China Normal University, Minhang, Shanghai, 200241, China

Received  December 2014 Revised  May 2015 Published  August 2015

We consider the linear heat equation on a bounded domain, which has two components with a thin coating surrounding a body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The mixed part is considered to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The diffusion tensor is isotropic on the body, and allowed to be anisotropic on the coating; and the size of diffusion tensor may differ significantly in these components. We find effective boundary conditions (EBCs) that are approximately satisfied by the solution of the heat equation on the boundary of the body. A concrete example is considered to study the effect of FGM coating. We also provide numerical simulations to verify our theoretical results.
Citation: Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415
References:
[1]

J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steady-state heat transfer in an exponentially graded anisotropic material,, Z. angew. Math. Phys., 56 (2005), 293. doi: 10.1007/s00033-004-1131-6. Google Scholar

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J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body,, Proc. Amer. Math. Soc., 137 (2009), 1711. doi: 10.1090/S0002-9939-08-09766-9. Google Scholar

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J. Li, X. Wang, G. Zhang and K. Zhang, Asymptotic behavior of Robin problem for heat equation on a coated body,, Rocky Mountain J. Math., 42 (2012), 937. doi: 10.1216/RMJ-2012-42-3-937. Google Scholar

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J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer,, Math. Models Methods Appl. Sci., 21 (2012), 1153. doi: 10.1142/S0218202511005283. Google Scholar

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M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview,, Proceedings of the World Congress on Engineering, (2012). Google Scholar

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S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body,, SIAM J. Appl. Math., 66 (2006), 1895. doi: 10.1137/040621181. Google Scholar

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J. Wessel, Handbook of Advanced Materials: Enabling New Design,, J. Wiley, (2004). doi: 10.1002/0471465186. Google Scholar

show all references

References:
[1]

J. Berger, P. Martin, V. Mantic and L. Gray, Fundamental solution for steady-state heat transfer in an exponentially graded anisotropic material,, Z. angew. Math. Phys., 56 (2005), 293. doi: 10.1007/s00033-004-1131-6. Google Scholar

[2]

H. Brezis, L. A. Caffarelli and A. Friedman, Reinforcement problem for elliptic equations and variational inequalities,, Ann. Mat. Pura Appl., 123 (1980), 219. doi: 10.1007/BF01796546. Google Scholar

[3]

H. Carslaw and J. Jaeger, Conduction of Heat in Solids,, Reprint of the second edition, (1988). Google Scholar

[4]

X, Chen, C. Pond and X. Wang, Effective boundary conditions resulting from anisotropic and optimally aligned coatings: The two dimensional case,, Arch. Ration. Mech. Anal., 206 (2012), 911. doi: 10.1007/s00205-012-0547-y. Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998). Google Scholar

[6]

H. Li, X. Wang and Y. Wu, The effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body,, J. Differential Equations, 257 (2014), 3640. doi: 10.1016/j.jde.2014.07.004. Google Scholar

[7]

J. Li, Asymptotic behavior of solutions to elliptic equations in a coated body,, Comm. Pure Appl. Anal., 8 (2009), 1251. doi: 10.3934/cpaa.2009.8.1251. Google Scholar

[8]

J. Li, S. Rosencrans, X. Wang and K. Zhang, Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body,, Proc. Amer. Math. Soc., 137 (2009), 1711. doi: 10.1090/S0002-9939-08-09766-9. Google Scholar

[9]

J. Li, X. Wang, G. Zhang and K. Zhang, Asymptotic behavior of Robin problem for heat equation on a coated body,, Rocky Mountain J. Math., 42 (2012), 937. doi: 10.1216/RMJ-2012-42-3-937. Google Scholar

[10]

J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer,, Math. Models Methods Appl. Sci., 21 (2012), 1153. doi: 10.1142/S0218202511005283. Google Scholar

[11]

M. Mahamood, T. Akinlabi, M. Shukal and S. Pityana, Functionally Graded Material: An Overview,, Proceedings of the World Congress on Engineering, (2012). Google Scholar

[12]

S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body,, SIAM J. Appl. Math., 66 (2006), 1895. doi: 10.1137/040621181. Google Scholar

[13]

J. Wessel, Handbook of Advanced Materials: Enabling New Design,, J. Wiley, (2004). doi: 10.1002/0471465186. Google Scholar

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