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

Created from the Publication Database of the Vienna University of Technology.