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**Micromagnetic simulations with finite elements**

We have developed a robust, efficient and versatile micromagnetic code in finite elements : feeLLGood

**Micromagnetic softwares based on finite difference (FD) approximation are commonly used in the community, meaning that the magnetic system is divided into regular orthorhombic cells**. In these the implementation of FD solvers in time and in space for the Landau-Lifshitz-Gilbert (LLG) equation is relatively easy, however the geometry is suitable only for prismatic geometries. **An alternative approach for describing complex shape systems is based on finite elements**. However, we have shown that the finite element formulations proposed in the literature could not ensure the reliability of the results obtained, due to an issue with the conservation of the norm of magnetisation.

Aware of this, we have lead a **joint research program with F. Alouges from CMAP-Polytechnique for developing novel time-dependent finite element formulations ensuring stability and convergence**. The main idea was to project the LLG equation with vector test functions belonging to the plane tangent to magnetisation. A set of robust numerical schemes for 3D micromagnetic simulations based on the LLG equation has then been produced and validated on test cases, ending in a code named feeLLGood (Finite Element Equations for LLG with Object-Oriented Development).

**The software has been designed for continuous improvement of its technical features**. For example, an almost-second-order-in-time solver was recently developed, implementing slope limiter techniques used in the treatment of shock waves equations. The resulting advanced midpoint-like rule equation allows for much bigger time steps to be used than in first order schemes aiming at the same precision, therefore reducing the computational time. Additional physical torques may be easily implemented in FeeLLGood. For example, for studying Spin Torque Oscillators developed by Spintec (coll. A. Vaysset, L. Prejbeanu), the simplified expression of spin transfer torque obtained by Slonczewski has been implemented. The frequency jumps observed in experiments could be reproduced and explained by finite elements simulations with feeLLGood.

**Links and related pages**

[1] S. Da Col, S. Jamet, N. Rougemaille, A. Locatelli, T. O. Mentes, B. Santos Burgos, R. Afid, M. Darques, L. Cagnon, J. C. Toussaint, and O. Fruchart, *Observation of Bloch-point domain walls in cylindrical magnetic nanowires*, Phys. Rev. B 89, 180405(R) (2014).

[2] Growth and micromagnetism of self-assembled epitaxial fcc(111) cobalt dots, O. Fruchart, A. Masseboeuf, J. C. Toussaint, P. Bayle-Guillemaud, J. Phys. : Condens. Matter, 25, 496002 (2013).

[3] A convergent finite element approximation for Landau-Lifschitz-Gilbert equation, F. Alouges, E. Kritsikis, J. C. Toussaint, Physica B, 407, 1345 (2012)

**Contact**

Jean-Christophe Toussaint, Institut Néel, Departement for Nanosciences, Team Micro- and Nano-Magnétisme

- - Micromagnetic simulations with finite elements
- - Generalizing demagnetizing factors to synthetic antiferromagnets
- - Fast computation of magnetostatic fields by nonuniform fast Fourier transforms

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