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University of Wales, Bangor - Department of Mathematics
Computational applied mathematics preprints 2002 |
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The Gilbert equation is introduced and a discussion of damped gyro-magnetic precession is given. We then consider the dynamics of a single-spin system and its response in an applied magnetic field. A brief description of our variational finite element model of magnetisation dynamics is then given. The reversal of an isolated Voronoi grain is investigated and its response to the applied magnetic field is shown to vary with the damping parameter in a manner that is somewhat different to the single-spin system, thus highlighting the need for sub-grain discretization. Implicit periodic boundary conditions are then employed to simulate the effects of damping in a polycrystalline thin film. This reveals that damping influences not only the speed, but the mode of magnetisation reversal in such films. The minimum of magnetisation reversal time is shown to occur at the same value of the damping parameter in all three systems.
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A variational finite element model of magnetisation dynamics is described. The implementation of implicit periodic boundary conditions is then discussed. Convergence estimates of the magnetostatic calculation are given as well as applications of the dynamic model to polycrystalline thin film media.
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Geometric Integration refers to numerical methods which aim to preserve the qualitative and geometric features of a differential equation after discretization. In micromagnetics the magnetisation vector M represents a statistical average of magnetic moments, the magnitude of which should be conserved in time. It can be shown that an implicit midpoint scheme preserves the modulus of solutions on the sphere due to intrinsic quadratic invariance. This method has been implemented to solve the Landau-Lifshitz and Landau-Lifshitz-Gilbert equations within a finite difference formulation by various. In this paper it is shown that an explicit Euler method will over-estimate |M| while an implicit Euler method will make an under-estimate, whereas the midpoint method conserves |M| up to round-off error. The midpoint scheme is then utilised within a variational finite element formulation of the Gilbert equation. A posteriori error estimators are considered using the reversal of a cobalt nano-particle as an example calculation. Comparison is made with standard methods highlighting the improved numerical stability of the scheme.
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A numerical scheme for the modelling of micromagnetics dynamics must solve for the magnetisation within the regions of magnetic material. A crucial component of the effective magnetic field \bsy H which drives the dynamics is the exchange field which is defined inside the magnetic region and is proportional to the Laplacian of the magnetization.
Most schemes have solved for the magnetization in a pointwise fashion which is fast but necessitates an approximation to the exchange field as a sum of inverse square terms which leads to results that are mesh dependent.
We describe a method which allows the proper variational formulation of all terms which leads to a finite element discretisation that converges with mesh refinement.
A numerical scheme for the modelling of micromagnetics dynamics must solve for the magnetisation within the regions of magnetic material. A crucial component of the effective magnetic field \bsy H which drives the dynamics is the demagnetising contribution which is defined both inside and outside the magnetic region but only its values inside affect the magnetization.
Using standard finite element discretizations for the associated Poisson problem for the demagnetization potential entails meshing the whole domain, including non-magnetic regions, and solving throughout the mesh. This leads to significant use of computer storage and processor resources to calculate nodal values of the potential which are not used for the magnetization dynamics.
We describe a hybrid formulation which uses a boundary integral formulation on the interfaces to remove the calculation of the potential in the non-magnetic region. The finite element method is used to calculate a potential field within the magnetic region only which is then corrected by coupling with the discretisation of the boundary integral via an influence matrix.
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