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 and Continuous Dynamical Systems, 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), 293-303. doi: 10.1007/s00033-004-1131-6.

[2]

H. Brezis, L. A. Caffarelli and A. Friedman, Reinforcement problem for elliptic equations and variational inequalities, Ann. Mat. Pura Appl., 123 (1980), 219-246. doi: 10.1007/BF01796546.

[3]

H. Carslaw and J. Jaeger, Conduction of Heat in Solids, Reprint of the second edition, New York, 1988.

[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), 911-951. doi: 10.1007/s00205-012-0547-y.

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[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), 3640-3668. doi: 10.1016/j.jde.2014.07.004.

[7]

J. Li, Asymptotic behavior of solutions to elliptic equations in a coated body, Comm. Pure Appl. Anal., 8 (2009), 1251-1267. doi: 10.3934/cpaa.2009.8.1251.

[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), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9.

[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), 937-958. doi: 10.1216/RMJ-2012-42-3-937.

[10]

J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer, Math. Models Methods Appl. Sci., 21 (2012), 1153-1192. doi: 10.1142/S0218202511005283.

[11]

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

[12]

S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math., 66 (2006), 1895-1916; Corrigendum, SIAM J. Appl. Math., 68 (2008), p1202. doi: 10.1137/040621181.

[13]

J. Wessel, Handbook of Advanced Materials: Enabling New Design, J. Wiley, New Jersey, 2004. doi: 10.1002/0471465186.

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), 293-303. doi: 10.1007/s00033-004-1131-6.

[2]

H. Brezis, L. A. Caffarelli and A. Friedman, Reinforcement problem for elliptic equations and variational inequalities, Ann. Mat. Pura Appl., 123 (1980), 219-246. doi: 10.1007/BF01796546.

[3]

H. Carslaw and J. Jaeger, Conduction of Heat in Solids, Reprint of the second edition, New York, 1988.

[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), 911-951. doi: 10.1007/s00205-012-0547-y.

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[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), 3640-3668. doi: 10.1016/j.jde.2014.07.004.

[7]

J. Li, Asymptotic behavior of solutions to elliptic equations in a coated body, Comm. Pure Appl. Anal., 8 (2009), 1251-1267. doi: 10.3934/cpaa.2009.8.1251.

[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), 1711-1721. doi: 10.1090/S0002-9939-08-09766-9.

[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), 937-958. doi: 10.1216/RMJ-2012-42-3-937.

[10]

J. Li and K. Zhang, Reinforcement of the Poisson equation by a thin layer, Math. Models Methods Appl. Sci., 21 (2012), 1153-1192. doi: 10.1142/S0218202511005283.

[11]

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

[12]

S. Rosencrans and X. Wang, Suppression of the Dirichlet eigenvalues of a coated body, SIAM J. Appl. Math., 66 (2006), 1895-1916; Corrigendum, SIAM J. Appl. Math., 68 (2008), p1202. doi: 10.1137/040621181.

[13]

J. Wessel, Handbook of Advanced Materials: Enabling New Design, J. Wiley, New Jersey, 2004. doi: 10.1002/0471465186.

[1]

Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[6]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, Johannes Lankeit. The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6529-6546. doi: 10.3934/dcds.2020289

[7]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[8]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078

[9]

Paolo Luzzini, Paolo Musolino. Perturbation analysis of the effective conductivity of a periodic composite. Networks and Heterogeneous Media, 2020, 15 (4) : 581-603. doi: 10.3934/nhm.2020015

[10]

María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure and Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313

[11]

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

[12]

Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201

[13]

Michel Cristofol, Shumin Li, Eric Soccorsi. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Mathematical Control and Related Fields, 2016, 6 (3) : 407-427. doi: 10.3934/mcrf.2016009

[14]

Yaguang Wang, Shiyong Zhu. Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3233-3244. doi: 10.3934/cpaa.2020141

[15]

Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041

[16]

Felipe Ponce-Vanegas. Reconstruction of the derivative of the conductivity at the boundary. Inverse Problems and Imaging, 2020, 14 (4) : 701-718. doi: 10.3934/ipi.2020032

[17]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems and Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073

[18]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[19]

Wei Lv, Ruirui Sui. Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 47-57. doi: 10.3934/naco.2015.5.47

[20]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks and Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]