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February  2018, 11(1): 35-57. doi: 10.3934/dcdss.2018003

Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium

1. 

Université de La Rochelle, Laboratoire MIA, 23 Avenue A. Einstein, BP 33060,17031 La Rochelle, France

2. 

Université Moulay Ismaïl, FST Errachidia, Laboratoire M2I, Equipe MAMCS, BP: 509 Boutalamine, 52000 Errachidia, Maroc

* Corresponding author: Catherine Choquet

Received  September 2016 Revised  January 2017 Published  January 2018

Fund Project: This work was partially supported by the Volubilis project MA/14/301

We study the Landau-Lifshitz-Gilbert equation in a composite ferromagnetic medium made of two different materials with highly contrasted properties. Over the so-called matrix domain, the effective field, the demagnetizing field and the bulk anisotropy field are scaled with regard to a parameter $ε$ representing the size of the matrix blocks. This scaling preserves the physics of the magnetization as $ε$ tends to zero. Using homogenization theory, we derive the corresponding effective model. To this aim we use the concept of two-scale convergence together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry. We prove that the less magnetic part of the medium contributes through additional memory terms in the effective field.

Citation: Catherine Choquet, Mohammed Moumni, Mouhcine Tilioua. Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 35-57. doi: 10.3934/dcdss.2018003
References:
[1]

E. AcerbiV. ChiadòG. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. Google Scholar

[2]

A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford University Press, London, 1996.Google Scholar

[3]

B. Aktas, L. Tagirov and F. Mikailov (Eds), Nanostructured Magnetic Materials and their Applications Vol. Ⅱ/143, NATO Sciences Series, Kluwer Academic Publishers, Boston, 2004. doi: 10.1007/978-1-4020-2200-5. Google Scholar

[4]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

[5]

H. AmmariL. Halpern and K. Hamdache, Thin ferromagnetic films, Asympt. Anal., 24 (2000), 277-294. Google Scholar

[6]

T. ArbogastJ. DouglasJr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836. doi: 10.1137/0521046. Google Scholar

[7]

L. BerlyandD. Cioranescu and D. Golaty, Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions, J. Math. Pures et Appl., 84 (2005), 97-136. doi: 10.1016/j.matpur.2004.09.013. Google Scholar

[8]

A. BourgeatS. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543. doi: 10.1137/S0036141094276457. Google Scholar

[9]

H. Brézis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. Google Scholar

[10]

C. Choquet, Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, Appl. Anal., 83 (2004), 477-499. doi: 10.1080/00036810310001643194. Google Scholar

[11]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding method in homogenization, C. r. Acad. Sci., sér. 1, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar

[12]

W. Deng and B. Yan, Quasi-stationary limit and a degenerate Landau-Lifshitz equation of ferromagnetism, Appl. Math. Res. Express. AMRX, (2013), 277-296. Google Scholar

[13]

W. Deng and B. Yan, On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics, Evol. Equ. Control Theory, 2 (2013), 599-620. doi: 10.3934/eect.2013.2.599. Google Scholar

[14]

K. Hamdache, Homogenization of layered ferromagnetic media, Preprint 495, CMAP Polytechnique, UMR CNRS 7641,91128 Palaiseau cedex (France), December 2002.Google Scholar

[15]

K. Hamdache and M. Tilioua, On the zero thickness limit of thin ferromagnetic films with surface anisotropy energy, Math. Models Methods Appl. Sci., 11 (2001), 1469-1490. doi: 10.1142/S0218202501001422. Google Scholar

[16]

K. Hamdache and M. Tilioua, Interlayer exchange coupling for ferromagnets through spacers, SIAM J. Appl. Math., 64 (2004), 1077-1097. doi: 10.1137/S0036139901398916. Google Scholar

[17]

J. L. JolyG. Métivier and J. Rauch, Global solution to Maxwell equation in a ferromagnetic medium, Ann. Inst. H. Poincaré, 1 (2000), 307-340. doi: 10.1007/PL00001007. Google Scholar

[18]

E. Ya. Khruslov and L. Pankratov, Homogenization of boundary problems for GinzburgLandau equation in weakly connected domains, In Spectral Operator Theory and Related Topics, (ed. V. A. Marchenko), AMS Providence, 19 (1994), 233-268 Google Scholar

[19]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269. Google Scholar

[20]

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. Google Scholar

[21]

L. Pankratov, Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Appl. Anal., 69 (1998), 31-45. doi: 10.1080/00036819808840644. Google Scholar

[22]

G. A. Pavliotis and A. Stuart, Multiscale Methods. Averaging and Homogenization Springer-Verlag, New York, 2008. Google Scholar

[23]

K. Santugini-Repiquet, Homogenization of the demagnetization field operator in periodically perforated domains, J. Math. Anal. Appl., 334 (2007), 502-516. doi: 10.1016/j.jmaa.2007.01.001. Google Scholar

[24]

J. Starynkévitch, Local energy estimates for the Maxwell--Landau--Lifshitz system and applications, J. Hyperbolic Differ. Equ., 2 (2005), 565-594. doi: 10.1142/S0219891605000555. Google Scholar

[25]

M. Valadier, Admissible functions in two-scale convergence, Port. Math., 54 (1997), 147-164. Google Scholar

[26]

A. Visintin, Towards a two-scale calculus, ESAIM: COCV, 12 (2006), 371-397. doi: 10.1051/cocv:2006012. Google Scholar

show all references

References:
[1]

E. AcerbiV. ChiadòG. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. Google Scholar

[2]

A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford University Press, London, 1996.Google Scholar

[3]

B. Aktas, L. Tagirov and F. Mikailov (Eds), Nanostructured Magnetic Materials and their Applications Vol. Ⅱ/143, NATO Sciences Series, Kluwer Academic Publishers, Boston, 2004. doi: 10.1007/978-1-4020-2200-5. Google Scholar

[4]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

[5]

H. AmmariL. Halpern and K. Hamdache, Thin ferromagnetic films, Asympt. Anal., 24 (2000), 277-294. Google Scholar

[6]

T. ArbogastJ. DouglasJr and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21 (1990), 823-836. doi: 10.1137/0521046. Google Scholar

[7]

L. BerlyandD. Cioranescu and D. Golaty, Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions, J. Math. Pures et Appl., 84 (2005), 97-136. doi: 10.1016/j.matpur.2004.09.013. Google Scholar

[8]

A. BourgeatS. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 27 (1996), 1520-1543. doi: 10.1137/S0036141094276457. Google Scholar

[9]

H. Brézis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. Google Scholar

[10]

C. Choquet, Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, Appl. Anal., 83 (2004), 477-499. doi: 10.1080/00036810310001643194. Google Scholar

[11]

D. CioranescuA. Damlamian and G. Griso, Periodic unfolding method in homogenization, C. r. Acad. Sci., sér. 1, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar

[12]

W. Deng and B. Yan, Quasi-stationary limit and a degenerate Landau-Lifshitz equation of ferromagnetism, Appl. Math. Res. Express. AMRX, (2013), 277-296. Google Scholar

[13]

W. Deng and B. Yan, On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics, Evol. Equ. Control Theory, 2 (2013), 599-620. doi: 10.3934/eect.2013.2.599. Google Scholar

[14]

K. Hamdache, Homogenization of layered ferromagnetic media, Preprint 495, CMAP Polytechnique, UMR CNRS 7641,91128 Palaiseau cedex (France), December 2002.Google Scholar

[15]

K. Hamdache and M. Tilioua, On the zero thickness limit of thin ferromagnetic films with surface anisotropy energy, Math. Models Methods Appl. Sci., 11 (2001), 1469-1490. doi: 10.1142/S0218202501001422. Google Scholar

[16]

K. Hamdache and M. Tilioua, Interlayer exchange coupling for ferromagnets through spacers, SIAM J. Appl. Math., 64 (2004), 1077-1097. doi: 10.1137/S0036139901398916. Google Scholar

[17]

J. L. JolyG. Métivier and J. Rauch, Global solution to Maxwell equation in a ferromagnetic medium, Ann. Inst. H. Poincaré, 1 (2000), 307-340. doi: 10.1007/PL00001007. Google Scholar

[18]

E. Ya. Khruslov and L. Pankratov, Homogenization of boundary problems for GinzburgLandau equation in weakly connected domains, In Spectral Operator Theory and Related Topics, (ed. V. A. Marchenko), AMS Providence, 19 (1994), 233-268 Google Scholar

[19]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269. Google Scholar

[20]

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. Google Scholar

[21]

L. Pankratov, Homogenization of Ginzburg-Landau heat flow equation in a porous medium, Appl. Anal., 69 (1998), 31-45. doi: 10.1080/00036819808840644. Google Scholar

[22]

G. A. Pavliotis and A. Stuart, Multiscale Methods. Averaging and Homogenization Springer-Verlag, New York, 2008. Google Scholar

[23]

K. Santugini-Repiquet, Homogenization of the demagnetization field operator in periodically perforated domains, J. Math. Anal. Appl., 334 (2007), 502-516. doi: 10.1016/j.jmaa.2007.01.001. Google Scholar

[24]

J. Starynkévitch, Local energy estimates for the Maxwell--Landau--Lifshitz system and applications, J. Hyperbolic Differ. Equ., 2 (2005), 565-594. doi: 10.1142/S0219891605000555. Google Scholar

[25]

M. Valadier, Admissible functions in two-scale convergence, Port. Math., 54 (1997), 147-164. Google Scholar

[26]

A. Visintin, Towards a two-scale calculus, ESAIM: COCV, 12 (2006), 371-397. doi: 10.1051/cocv:2006012. Google Scholar

Figure 1.  An example of periodic structure for the domain and the standard cell
Figure 2.  A simple setting, $\overline{\Omega} =[-1/2;3/2]^3$. Representation of $\Omega^{1}$, $\Omega^{1/2}$ and $\Omega^{1/3}$ with the corresponding points belonging to $\mathcal{C}$
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