# American Institute of Mathematical Sciences

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 [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

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
 [1] Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071 [2] Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147 [3] Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627 [4] Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209 [5] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [6] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [7] María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure & Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313 [8] María Teresa González Montesinos, Francisco Ortegón Gallego. The thermistor problem with degenerate thermal conductivity and metallic conduction. Conference Publications, 2007, 2007 (Special) : 446-455. doi: 10.3934/proc.2007.2007.446 [9] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [10] Michel Cristofol, Shumin Li, Eric Soccorsi. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Mathematical Control & Related Fields, 2016, 6 (3) : 407-427. doi: 10.3934/mcrf.2016009 [11] Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041 [12] Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073 [13] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [14] Wei Lv, Ruirui Sui. Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 47-57. doi: 10.3934/naco.2015.5.47 [15] Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343 [16] Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861 [17] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [18] Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 [19] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 [20] Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190

2018 Impact Factor: 1.143