Effective Simulation of a Macroscopic Model in Micromagnetics
Dirk Praetorius (University Wien)
The large body limit in the LandauLifshitz equations of
micromagnetics [1] yields a macroscopic model without
exchange energy and convexified side conditions for the macroscopic
magnetisation vectors. Its Euler Lagrange equations (P) read:
Given a magnetic body , d=2,3,
an exterior field and the convexified anisotropy
density , find a magnetization and a Lagrange multiplier such that
a.e. in
The potential solves the (Maxwell) equations
in the entire space. It therefore appears natural to recast
the associated far field energy into an integral operator
which maps to the corresponding stray field
.
The proposed numerical scheme involves the operator
and replaces pointwise sidecondition by
a penalization strategy. Given a triangulation , the induced
space of piecewise constant functions on , and a
penalization parameter , the discrete penalized problem
reads: Find such that for all
with .
Numerical aspects addressed in the presentation include the
integration of the matrices with quadrature rules and hierarchical
matrices as well as a priori and a posteriori error control
and adaptive meshdesign.
[1] A. DESIMONE:
Energy Minimizers for Large Ferromagnetic Bodies,
Arch. Rational Mech. Anal. 125 (1993), 99143.
[2] D. PRAETORIUS:
Analysis, Numerik und Simulation eines relaxierten Modellproblems zum
Mikromagnetismus,
Doctorial thesis, Vienna University of Technology 2003.
[3] C. CARSTENSEN, D. PRAETORIUS:
Numerical Analysis for a Macroscopic Model in Micromagnetics,
submitted 2003.
This is a joint work with C. Carstensen (Vienna University of Technology).
