# Computational applied mathematics preprints 2002

#### 02.11 : SPARGO, A.W., RIDLEY, P.H.W. & ROBERTS, G.W.

### Gilbert damping in polycrystalline thin films

This was presented at the Moscow International Symposium on Magnetism, Moscow State University, June 2002.

#### Summary:

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.

#### Published in:

J. Magn. Magn. Mater. (2003) 258-259.#### Download:

gzipped postscript: 02_11.ps.gz

#### 02.12 : SPARGO, A.W., RIDLEY, P.H.W. & ROBERTS, G.W.

### Periodic finite element simulation of magnetisation dynamics

This was presented at the 10th Biennial IEEE Conference on Electromagnetic Field Computation, Perugia, June 2002.

#### Summary:

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.

#### Download:

gzipped postscript: 02_12.ps.gz and 02_12fig.ps.gz

#### 02.13 : SPARGO, A.W., RIDLEY, P.H.W. & ROBERTS, G.W.

### Geometric Integration of the Gilbert Equation

This has been submitted to the*47th Conference on Magnetism and Magnetic Materials*

#### Summary:

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.

#### Published in:

J. Appl. Phys. 93(10) (2003).#### Download:

gzipped postscript:

#### 02.19 : LAMBE, L.A., LUCZAK, R. & NEHRBASS, J.W.

### Symbolic computation in electromagnetic modeling

#### Abstract:

A standard computational tool for approximating solutions to systems of partial differential equations with boundary conditions is the finite difference method.In the case of the Helmholtz equation

*nabla^2 u = - kappa^2 u*, it was shown in Nehrbass' PhD thesis [15, Ch.4], that numerical errors that can occur in the central difference quotients cound be corrected

*without*increasing the order by optimally adjusting "weights" for these quotients.

The problem of accurately interpolating when a mesh is refined in some subregion of interest for a given initial mesh for the Helmholtz equation was also addressed in [15, Ch.5] and this is reviewed in section 4. The calculations necessary for this work involved working out some rather complicated integral formulas and manipulating complex algebraic expressions, all of which can be tedious and error prone. In this paper, it will be shown how computer aided symbolic computation (SC) can be used to relieve the tedium and eliminate the inevitable typographical errors involved in hand calculations.

#### Published in:

#### Download:

gzipped postscript (revised version 21/12/02): lln.ps.gz

#### 02.29 : RIDLEY, P.H.W. & ROBERTS, G.W.

### Variational approach to micromagnetics

#### Summary:

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.

#### Published in:

#### Download:

pdf file: 02_29.pdf

#### 02.30 : RIDLEY, P.H.W. & ROBERTS, G.W.

### Hybrid finite element/boundary integral methods for micromagnetics

#### Summary:

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.

#### Published in:

#### Download:

pdf file: 02_30.pdf