[Back]

G. Hrkac, M. Page, D. Praetorius, D. Süss:
"Numerical Integrator for the LLG equation with Magnetostriction";
Talk: MATHMOD 2012 - 7th Vienna Conference on Mathematical Modelling, Wien (invited); 02-14-2012 - 02-17-2012.

The theoretical understanding and practical prediction of micromagnetic
phenomena is of utmost importance for the improvement
of existing and development of future magnetic based devices
like e.g. storage devices, sensors, or magnetic RAM. However,
certain aspects do not need the practical development of
prototypes, but can also be well understood by means of numerical
simulations. This relies on the mathematical modelling of micromagnetics.
In physics, it is well-accepted that the dynamics
of micromagnetics is described best by the nonlinear Landau-
Lifshitz-Gilbert equation (LLG), where time evolution is driven by
the so-called effective field heff .
In [Alouges, 2008], a numerical integrator is proposed for a simplified effective
field, where only the so-called exchange energy is reflected.
In our generalization of [Alouges, 2008], we further include the cristalline anisotropy
energy, the magnetostatic energy, the exterior Zeeman
energy, as well as the magnetostrictive energy. The latter couples
LLG with the conservation of momentum equation (CM) and
includes an additional nonlinearity. This coupling was first analyzed
in [Banas/Slodicka, 2006], where a different algorithm was proposed. In our
work, we combine the approaches of [Alouges, 2008] and [Banas/Slodicka, 2006].
Besides the nonlinarities of LLG and CM, numerical difficulties arise from a
non-convex side constraint |m| = 1 in space-time for the magnetization
and from a certain non-local, but linear integral operator
P involved for the computation of the demagnetization field.
The developed numerical integrator is linear implicit and treats
the known nonlinearities in an effective manner. The key features
of our integrator read as follows:

(o) First, the implicit part only deals with the higher-order
term stemming from the exchange energy, whereas the
remaining lower-order terms are treated explicitly. In particular,
this includes the numerical computation of the
demagnetization field which is the most time and memory
consuming part of the simulation.

(o) Second, the integrator decouples LLG and CM. Overall
and besides the demagnetization field, this results in the
fact that only two linear systems per timestep have to be
solved.

(o) Finally and from utmost importance for reliable simulations,
we prove that our integrator is unconditionally convergent
as time step-size k and mesh-size h tend to zero.

http://publik.tuwien.ac.at/files/PubDat_206976.pdf