Martin Kruzik: Mesoscopic and macroscopic models of hysteresis in micromagnetics and their numerical treatment
The energy functional describing equilibrium configurations in steady state
micromagnetics usually does not posses any minimizer due to
faster and faster oscillations of a minimizing sequence of magnetizations.
Therefore a suitable extension (relaxation) of the functional is needed.
This can be done by means of Young measures which record the oscillations
into magnetic microstructure.
The model, which we propose, combines tendency for minimization of
relaxed micromagnetic energy with the rate-independent dissipation
mechanism that reflects the macroscopic quantum of energy required
to change one pole of a magnet to another.
We analyze and discretize this model and we also derive optimality
conditions for the discrete model. Those optimality conditions stated in
the form of Weierstrass's maximum principle
are then used for an effective numerical solution.
On the other hand, the model can be equivalently formulated in terms of the
magnetization. After a time discretization it leads to a sequence of
convex minimization problems which can be solved by means of corresponding
Euler-Lagrange equations. The uniqueness and periodicity of a time-discrete
solution can be proved.
Computational examples for uniaxial magnets will be shown.