Mathematical and computational modelling of bulk ferromagnets Tom - PDF document

✬ ✩ March 14-16, 2005: MULTIMAT, Paris Mathematical and computational modelling of bulk ferromagnets Tom´ aˇ s Roub´ ıˇ cek and Martin Kruˇ z´ ık Charles University & Academy of Sci., Prague http://www.karlin.mff.cuni.cz/ ˜ roubicek/multimat.htm Steady-state problem, a microscopical level: (Landau and Lifshitz (1935), Brown (1962-66))  �  minimize E ε ( m, u ) − h · m d x     Ω   � � ϕ ( m ) + ε |∇ m | 2 �    where E ε ( m, u ) := d x     Ω   �  +1  R n |∇ u | 2 d x ,  2 I    subject to | m | = M s on Ω ,        R n ,  div( ∇ u − χ Ω m ) = 0 in I        R n ) , R n ) , u ∈ W 1 , 2 (I m ∈ L ∞ (Ω; I  i.e. minimization of anisotropy + exchange + magnetostatic + interaction energy; R n = magnetization, m : Ω → I R n → I u : I R=magnetostatic potential, ∇ u = demagnetizing field, ✫ ✪ M s = saturation magnetization (given). 1

✬ ✩ Meso-scopical level (DeSimone (1993), Pedregal (1994)...): zero-exchange energy limit: ε → 0 and “ m ε ⇀ ν ” ∗ �  minimize E ( ν, u ) − h · (id • ν ) d x,     Ω  � �  ϕ • ν d x + 1  R 3 |∇ u | 2 d x,  where E ( ν, u ) :=  2 Ω I � �  R 3 ,  subject to div ∇ u − χ Ω (id • ν ) = 0 on I        R n ) , u ∈ W 1 , 2 (I ν ∈ Y (Ω; S M s ) , where ν : Ω → rca( S M s ) is a Young measure, thus ν x ≡ ν ( x ) describes volume fractions of m at x , � [ f • ν ]( x ) := R 3 f ( m ) ν x (d m ), I R 3 → I R 3 =the identity id : I then id • ν =the macroscopical magnetization M , w (Ω; rca( S M s )) ∼ = L 1 (Ω; C ( S M s )) ∗ Y (Ω; S M s ) ⊂ L ∞ the set of all Young measures, R n of the radius M s . S M s =the ball in I A macro-scopical level (DeSimone (1993)): � �  ϕ eff ( M ) − h · M d x + 1 R 3 |∇ u | 2 d x, minimize   2 Ω I  R 3 ,  subject to div( ∇ u − χ Ω M ) = 0 on I � � ∗∗ , M =macroscopical magnetization. ✫ ϕ eff := ϕ + δ S M s ✪ 2

✬ ✩ Evolution on the microscopical level: Gilbert-Landau-Lifshitz model: ∂m ∂t = λ 1 m × h eff − λ 2 m × ( m × h eff ) , h eff := h − ϕ ′ ( m ) + ε ∆ m − 1 2 ∇ u , u again determined from div( ∇ u − χ Ω m ) = 0, ϕ ′ =the derivative of ϕ . The balance of magnetic energy E ε (test by h eff ): � � d E ε ( m, u ) h eff · ∂m | m × h eff | 2 d x ≤ 0 , = − ∂t d x = − λ 2 d t Ω Ω which expresses Clausius-Duhem’s inequality; the “precession” λ 1 -term does not dissipate energy, the λ 2 -term: a phenomenological “viscous” damping. The multiwell structure of ϕ | S M s : a nearly rate-independent hysteretic response. The width of the hysteresis loop in the m/h -diagram can thus be indirectly controlled by a shape of ϕ . Evolution on the macroscopical level: Rayleigh, Prandtl and Ishlinski˘ ı model (1887) or Preisach’s (1935) model (a continuum of activation thresholds) ✫ ✪ Visintin (2000) (a one-threshold dry-friction) 3

✬ ✩ Evolution on the mesoscopical level: • rate-independent dissipation (independent of frequency of h ) Assumption: the amount of dissipated energy within the phase transformation from one pole to the other = a single, phenomenologically given number (of the dimension J/m 3 =Pa) depending on the coercive force H c . identification of poles through a vectorial order parameter : L : S M s → △ L R L ; ξ i ≥ 0 , i = 1 , ..., L, � L △ L := { ξ ∈ I i =1 ξ i = 1 } . L i ( s ) is equal 1 if s is in i -th pole, i.e. s ∈ S M s is in a neighborhood of i -th easy-magnetization direction. λ = Λ ν := L • ν : mesoscopic order parameter � [ L • ν ]( x ) := S M s L ( s ) ν x (d s ) R L → I R + ̺ : I 0 ̺ ( ˙ λ ) = H c | ˙ λ | L : specific dissipation potential R L | · | L : a norm on I set of admissible configurations: � q = ( ν, λ ) ∈Y (Ω; S M s ) × L ∞ (Ω; I R L ) ; Q : = � λ ( x ) ∈ △ L , Λ ν = λ for a.a. x ∈ Ω ✫ ✪ 4

✬ ✩ Mielke’s dissipation distance: � � 1 ̺ (d λ λ ∈ C 1 � R L � δ ( λ 1 , λ 2 ) := inf d t ) d t ; [0 , 1]; I , 0 � λ ( t ) ∈ co L ( S M s ) , λ (0) = λ 1 , λ (1) = λ 2 . in our case: δ ( λ 1 , λ 2 ) = H c | λ 1 − λ 2 | L total dissipation distance: � D ( q 1 , q 2 ) := δ ( λ 1 , λ 2 ) d x, q i = ( ν i , λ i ) . Ω energy regularization (with α, ρ > 0):  ρ || λ || 2 R L ) , if λ ∈ W α, 2 (Ω; I  W α, 2 (Ω;I R L ) E ρ ( ν, λ ) := E ( ν ) + + ∞ otherwise,  Zeeman’s (external field) energy: � H ( t ) , q � = � ν, h ( · , t ) ⊗ id � ; Gibbs’ energy: G ( t, q ) := E ρ ( q ) − � H ( t ) , q � ✫ ✪ 5

✬ ✩ Mielke & Theil’s definition of an energetic solution : A process q = q ( t ) is stable if ∀ t ∈ [0 , T ]: ∀ ˜ q ∈ Q : G ( t, q ( t )) ≤ G ( t, ˜ q ) + D ( q ( t ) , ˜ q ) . A process q = q ( t ) satisfies the energy equality if ∀ t, s ∈ [0 , T ] , s ≤ t , G ( t, q ( t )) + Var( D , q ; s, t ) � �� � � �� � Gibbs’ ener- dissipated energy gy at time t � t � d H � = G ( s, q ( s )) − d t , q ( θ ) d θ , s � �� � � �� � Gibbs’ ener- reduced work of gy at time s external field q = q ( t ) ≡ ( ν ( t ) , λ ( t )) is an energetic solution if • ν ( t ) ∈ Y (Ω; S M s ) for all t ∈ [0 , T ], R L )), λ ∈ BV([0 , T ]; L 1 (Ω; I q ( t ) ∈ Q for all t ∈ [0 , T ], • it is stable and satisfies the energy equality, • q (0) = q 0 . ✫ ✪ 6

✬ ✩ The existence of an energetic solution: a semi-discretization in time by the implicit Euler scheme with a time step τ > 0, assuming T/τ an integer, and a sequence of τ ’s converging to zero, and such that, τ i /τ i +1 is integer. Then we put q 0 τ = q 0 , a given initial condition, and, for k = 1 , ..., T/τ we define q k τ recursively as a solution of the minimization problem  I ( q ) := G ( kτ, q ) + D ( q k − 1 Minimize , q )  τ  subject to q ≡ ( ν, λ ) ∈Q , If a solution (i.e. a global minimizer) is not unique, we just take an arbitrary one for q k τ . Then we define the piecewise constant interpolation:  q k  for t ∈ (( k − 1) τ, kτ ] , τ q τ ( t ) = q 0 for t = 0 .  A-priori estimates: λ τ ∈ L ∞ (0 , T ; H α (Ω; I R L ) ∩ L ∞ (Ω; I R L )) ∩ BV([0 , T ]; L 1 (Ω; I R L ) , ν τ ∈ L ∞ (0 , T ; L ∞ w (Ω; rca( S M s )) . G τ ∈ BV([0 , T ]) where G τ ( t ) := G ( t, q τ ( t )). ✫ ✪ 7

✬ ✩ q k τ minimizes I & triangle inequality for D q ) + D ( q k − 1 q ) − D ( q k − 1 ⇒ G ( kτ, q k , q k τ ) ≤ G ( kτ, ˜ , ˜ τ ) τ τ q ) + D ( q k ≤ G ( kτ, ˜ τ , ˜ q ) ⇒ stability of q τ : ∀ ˜ q ∈ Q : G ( t, q τ ( t )) ≤ G ( t, ˜ q ) + D ( q τ ( t ) , ˜ q ) . 1) stability of q k − 1 q := q k vs. ˜ τ τ τ minimizes I in comparison with q k − 1 2) q k τ ⇒ a two-sided energy inequality: � t � d H � − d t , q τ ( θ ) d θ s ≤ G ( t, q τ ( t )) + Var( D , q τ ; s, t ) − G ( s, q τ ( s )) � t � d H � ≤ − d t , q τ ( θ − τ ) d θ s Convergence for τ → 0 (Mielke-Francfort scheme): Step 1a : Selection of a subsequence (Helly’s theorem): G ∈ BV([0 , T ]) : ∀ t ∈ [0 , T ] : G τ ( t ) → G ( t ) R L )) : λ ∈ BV([0 , T ]; L 1 (Ω; I R L ) ∀ t ∈ [0 , T ] : λ τ ( t ) → λ ( t ) weakly in L 1 (Ω; I � d H � → P ∗ weakly in L 1 (0 , T ) and P τ := − d t , q τ P ( t ) := lim sup τ → 0 P τ ( t ). Step 1b : Selection of a finer net (Tikhonov theorem): ∀ t ∈ [0 , T ] ∃ a Young measure ν ( t ) ∈ Y (Ω; S M s ) ∃ { q τ ξ } ξ ∈ Ξ finer than the (sub)sequence { q τ } : ν τ ξ ⇀ ν t ∗ ✫ ✪ 8

✬ ✩ Step 2 : Stability of the limit process q : closedness of the graph of the stable-set mapping � � t �→ S := q ∈Q : ∀ ˜ q ∈Q : G ( t, q ) ≤ G ( t, ˜ q ) + D ( q, ˜ q ) . Step 3 : (Moore-Smith’) convergence of the stored energy: lim ξ ∈ Ξ G ( t, q τ ξ ( t )) = G ( t, q ( t )) for any t ∈ [0 , T ] so that G τ ξ ( t ) = G ( t, q ( t )) Step 4 : Upper energy estimate: limit passage in the 2 nd double-sided energy inequality � t ⇒ G ( t, q ( t )) + Var( D , q ; 0 , t ) ≤ G (0 , q 0 ) + P ∗ ( s )d s 0 � t ≤ G (0 , q 0 ) + P ( s ) d s 0 Step 5 : Lower energy estimate: a suitable partition 0 ≤ t ε 1 < t ε 2 < ...t ε k ε ≤ T , stability of q ( t ε q := q ( t ε i − 1 ) vs. ˜ i ) approximation of a Lebesgue integral by Rieman’s sums � t ⇒ G ( t, q ( t )) + Var( D , q ; 0 , t ) ≥ G (0 , q 0 ) + P ( s ) d s. ✷ 0 Remark: 1) P = P ∗ , 2) t �→ ν ( t ) weakly* measurable ⇐ a suitable a-posteriori selection (A.Mainik, PhD-thesis 2004) ✫ ✪ 9