Figure 1.
Left: the periodic cell $Y$ . ${E_{{\rm{int}}}}$ is the shaded region and ${E_{{\rm{out}}}}$ is the white region. Right: the region Ω
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June
2018, 23(4): 1739-1756.
doi: 10.3934/dcdsb.2018078
Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices
| 1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy |
| 2. | Dipartimento di Matematica ed Informatica, Università di Firenze, Via Santa Marta 3, 50139 Firenze, Italy |
In this paper we study a model for the heat conduction in a composite having a microscopic structure arranged in a periodic array. We obtain the macroscopic behaviour of the material and specifically the overall conductivity via an homogenization procedure, providing the equation satisfied by the effective temperature.
Keywords:
Homogenization,
unfolding periodic method,
Laplace-Beltrami operator,
tangential derivatives,
partial differential equations,
polymer encapsulation.
Mathematics Subject Classification:
Primary: 35B27; Secondary: 35Q79, 35K20.
Citation:
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
![]()
References:
| [1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
| [2] |
G. Allaire, A. Damlamian and U. Hornung,
Two-scale convergence on periodic surfaces and applications, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, (1995), 15-25.
|
| [3] |
G. Allaire and H. Hutridurga,
Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-815.
doi: 10.1093/imamat/hxs049. |
| [4] |
G. Allaire and F. Murat,
Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95.
|
| [5] |
M. Amar, D. Andreucci and D. Bellaveglia,
Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 56-77.
doi: 10.1016/j.na.2016.05.018. |
| [6] |
M. Amar, D. Andreucci and D. Bellaveglia,
The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700.
doi: 10.4171/RLM/781. |
| [7] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1261-1295.
doi: 10.1142/S0218202504003623. |
| [8] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.
doi: 10.1002/mma.709. |
| [9] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues, Euro. Jnl. of Applied Mathematics, 20 (2009), 431-459.
doi: 10.1017/S0956792509990052. |
| [10] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues, J. Mechanics of Material and Structures, 4 (2009), 211-223.
doi: 10.2140/jomms.2009.4.211. |
| [11] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range, Communications on Pure and Applied Analysis, 9 (2010), 1131-1160.
doi: 10.3934/cpaa.2010.9.1131. |
| [12] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential and Integral Equations, 26 (2013), 885-912.
|
| [13] |
M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator,
(2018), submitted. |
| [14] |
M. Amar and R. Gianni, Error estimate for a homogenization problem involving the Laplace-Beltrami operator,
Mathematics and Mechanics of Complex Systems (2018), to appear, arXiv: 1705.04345v2. |
| [15] |
M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators,
(2018), work in progress. |
| [16] |
D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki,
The periodic unfolding method in domains with holes, SIAM Journal on Mathematical Analysis, 44 (2012), 718-760.
doi: 10.1137/100817942. |
| [17] |
D. Cioranescu, A. Damlamian and G. Griso,
Periodic unfolding and homogenization, Comptes Rendus Mathematique, 335 (2002), 99-104.
doi: 10.1016/S1631-073X(02)02429-9. |
| [18] |
D. Cioranescu, A. Damlamian and G. Griso,
The periodic unfolding method in homogenization, SIAM Journal on Mathematical Analysis, 40 (2008), 1585-1620.
doi: 10.1137/080713148. |
| [19] |
D. Cioranescu, P. Donato and R. Zaki,
Periodic unfolding and robin problems in perforated domains, Comptes Rendus Mathematique, 342 (2006), 469-474.
doi: 10.1016/j.crma.2006.01.028. |
| [20] |
D. Cioranescu, P. Donato and R. Zaki,
The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496.
|
| [21] |
P. Donato and Z. Yang,
The periodic unfolding method for the wave equation in domains with holes, Adv. Math.Sci. Appl., 22 (2012), 521-551.
|
| [22] |
P. Donato and Z. Yang,
The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906.
doi: 10.1007/s11425-015-5103-4. |
| [23] |
H. Ebadi-Dehaghani and M. Nazempour,
Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540.
|
| [24] |
L. Flodén, A. Holmbom, M. Olsson and J. Persson,
Very weak multiscale convergence, Applied Mathematics Letters, 23 (2010), 1170-1173.
doi: 10.1016/j.aml.2010.05.005. |
| [25] |
L. Flodén, A. Holmbom and M. Olsson Lindberg, A strange term in the homogenization of parabolic equations with two spatial and two temporal scales,
Journal of Function Spaces and Applications (2012), Art. ID 643458, 9 pp. |
| [26] |
A. Holmbom,
Homogenization of parabolic equations an alternative approach and some corrector-type results, Applications of Mathematics, 42 (1997), 321-343.
doi: 10.1023/A:1023049608047. |
| [27] |
S. Kemaloglu, G. Ozkoc and A. Aytac, Thermally conductive boron nitride/sebs/eva ternary
composites: processing and characterization, Polymer Composites (Published online on http://onlinelibrary.wiley.com/doi/10.1002/pc.20925/full, 2009, Society of Plastic Engineers), (2010), 1398–1408. |
| [28] |
G. Nguetseng,
A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [29] |
G. Nguetseng and J. Woukeng,
Σ-convergence of nonlinear parabolic operators, Nonlinear Analysis, 66 (2007), 968-1004.
doi: 10.1016/j.na.2005.12.035. |
| [30] |
W. Phromma, A. Pongpilaipruet and R. Macaraphan, Preparation and thermal properties of PLA filled with natural rubber-PMA core-shell/magnetite nanoparticles, European Conference; 3rd, Chemical Engineering, Recent Advances in Engineering. Paris, (2012). |
| [31] |
K. M. Shahil and A. A. Balandin,
Graphene-based nanocomposites as highly efficient thermal interface materials, Graphene Based Thermal Interface Materials, (2011), 1-18.
|
| [32] |
V. Zhikov,
On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.
|
show all references
References:
| [1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
| [2] |
G. Allaire, A. Damlamian and U. Hornung,
Two-scale convergence on periodic surfaces and applications, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, (1995), 15-25.
|
| [3] |
G. Allaire and H. Hutridurga,
Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-815.
doi: 10.1093/imamat/hxs049. |
| [4] |
G. Allaire and F. Murat,
Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95.
|
| [5] |
M. Amar, D. Andreucci and D. Bellaveglia,
Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 56-77.
doi: 10.1016/j.na.2016.05.018. |
| [6] |
M. Amar, D. Andreucci and D. Bellaveglia,
The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700.
doi: 10.4171/RLM/781. |
| [7] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1261-1295.
doi: 10.1142/S0218202504003623. |
| [8] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.
doi: 10.1002/mma.709. |
| [9] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues, Euro. Jnl. of Applied Mathematics, 20 (2009), 431-459.
doi: 10.1017/S0956792509990052. |
| [10] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues, J. Mechanics of Material and Structures, 4 (2009), 211-223.
doi: 10.2140/jomms.2009.4.211. |
| [11] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range, Communications on Pure and Applied Analysis, 9 (2010), 1131-1160.
doi: 10.3934/cpaa.2010.9.1131. |
| [12] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni,
A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential and Integral Equations, 26 (2013), 885-912.
|
| [13] |
M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator,
(2018), submitted. |
| [14] |
M. Amar and R. Gianni, Error estimate for a homogenization problem involving the Laplace-Beltrami operator,
Mathematics and Mechanics of Complex Systems (2018), to appear, arXiv: 1705.04345v2. |
| [15] |
M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators,
(2018), work in progress. |
| [16] |
D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki,
The periodic unfolding method in domains with holes, SIAM Journal on Mathematical Analysis, 44 (2012), 718-760.
doi: 10.1137/100817942. |
| [17] |
D. Cioranescu, A. Damlamian and G. Griso,
Periodic unfolding and homogenization, Comptes Rendus Mathematique, 335 (2002), 99-104.
doi: 10.1016/S1631-073X(02)02429-9. |
| [18] |
D. Cioranescu, A. Damlamian and G. Griso,
The periodic unfolding method in homogenization, SIAM Journal on Mathematical Analysis, 40 (2008), 1585-1620.
doi: 10.1137/080713148. |
| [19] |
D. Cioranescu, P. Donato and R. Zaki,
Periodic unfolding and robin problems in perforated domains, Comptes Rendus Mathematique, 342 (2006), 469-474.
doi: 10.1016/j.crma.2006.01.028. |
| [20] |
D. Cioranescu, P. Donato and R. Zaki,
The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496.
|
| [21] |
P. Donato and Z. Yang,
The periodic unfolding method for the wave equation in domains with holes, Adv. Math.Sci. Appl., 22 (2012), 521-551.
|
| [22] |
P. Donato and Z. Yang,
The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906.
doi: 10.1007/s11425-015-5103-4. |
| [23] |
H. Ebadi-Dehaghani and M. Nazempour,
Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540.
|
| [24] |
L. Flodén, A. Holmbom, M. Olsson and J. Persson,
Very weak multiscale convergence, Applied Mathematics Letters, 23 (2010), 1170-1173.
doi: 10.1016/j.aml.2010.05.005. |
| [25] |
L. Flodén, A. Holmbom and M. Olsson Lindberg, A strange term in the homogenization of parabolic equations with two spatial and two temporal scales,
Journal of Function Spaces and Applications (2012), Art. ID 643458, 9 pp. |
| [26] |
A. Holmbom,
Homogenization of parabolic equations an alternative approach and some corrector-type results, Applications of Mathematics, 42 (1997), 321-343.
doi: 10.1023/A:1023049608047. |
| [27] |
S. Kemaloglu, G. Ozkoc and A. Aytac, Thermally conductive boron nitride/sebs/eva ternary
composites: processing and characterization, Polymer Composites (Published online on http://onlinelibrary.wiley.com/doi/10.1002/pc.20925/full, 2009, Society of Plastic Engineers), (2010), 1398–1408. |
| [28] |
G. Nguetseng,
A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
| [29] |
G. Nguetseng and J. Woukeng,
Σ-convergence of nonlinear parabolic operators, Nonlinear Analysis, 66 (2007), 968-1004.
doi: 10.1016/j.na.2005.12.035. |
| [30] |
W. Phromma, A. Pongpilaipruet and R. Macaraphan, Preparation and thermal properties of PLA filled with natural rubber-PMA core-shell/magnetite nanoparticles, European Conference; 3rd, Chemical Engineering, Recent Advances in Engineering. Paris, (2012). |
| [31] |
K. M. Shahil and A. A. Balandin,
Graphene-based nanocomposites as highly efficient thermal interface materials, Graphene Based Thermal Interface Materials, (2011), 1-18.
|
| [32] |
V. Zhikov,
On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.
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