Workshop on

numerical methods for multiscale problems

 

     
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  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 macroscopic 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.
Impressum
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