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

  • * Corresponding author: Catherine Choquet

    * Corresponding author: Catherine Choquet 

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

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  • 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.

    Mathematics Subject Classification: Primary: 35Q60, 35B27; Secondary: 35K55, 82D40, 78A25.

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  • 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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