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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
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An example of periodic structure for the domain and the standard cell
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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