Micromagnetism - PeopleMicromagnetism - Philippe Corboz, 29.10.2002 – p.6/15 Discretization LLG...
Transcript of Micromagnetism - PeopleMicromagnetism - Philippe Corboz, 29.10.2002 – p.6/15 Discretization LLG...
Micromagnetism• Equations• Discretization FEM/Euler• Newton Algorithm• Results
Micromagnetism - Philippe Corboz, 29.10.2002 – p.1/15
EquationsLandau-Livshits-Gilbert equation:
dM
dt= −γM × Heff −
γα
Ms
M × (M × Heff) .
Effective magnetic field:
Heff = −∂E(m)
∂m
Micromagnetism - Philippe Corboz, 29.10.2002 – p.2/15
EquationsGibbs free energy:
E(M) =
∫
Ω
A
M2s
|gradM|2 + Φ(M/Ms) − 2µ0He · M dx + 1
2µ0
∫
R3
|gradψ|2 dx .
Potential equation:
µ0∆ψ = div M inR3
M satisfies Neumann boundary conditions
Micromagnetism - Philippe Corboz, 29.10.2002 – p.3/15
Discretization: TimeTimestepping:
δmn+ 1
2 = −mn+ 1
2 × hn+ 1
2
eff − αmn+ 1
2 × (mn+ 1
2 × hn+ 1
2
eff )
hn+ 1
2
eff := η∆mn+ 1
2 +Qd (d · mn+ 1
2 ) − gradψn+ 1
2 + he(tn+ 1
2)
with
dm
dt(t
n+ 1
2
) ≈ δmn+ 1
2 :=mn+1 − mn
τ, m(t
n+ 1
2
) ≈ mn+ 1
2 :=mn+1 + mn
2
Micromagnetism - Philippe Corboz, 29.10.2002 – p.4/15
Discretization: Spatial
Micromagnetism - Philippe Corboz, 29.10.2002 – p.5/15
Discretization: Weak formWeak form of LLG and potential equation:
1
1 + α2
∫
Ω
(δmn+ 1
2 − αmn+ 1
2 × δmn+ 1
2 ) · v dx =
=
∫
Ω
η grad(mn+ 1
2 × v) · gradmn+ 1
2 −Q(mn+ 1
2 × d)(d · mn+ 1
2 )v+
+ (mn+ 1
2 × gradψ) · v − (mn+ 1
2 × he) · v dx
∫
D
gradψ · gradϕ dx =
∫
D
mn+ 1
2 · gradϕ dx .
for all v ∈ (H1(Ω))3, and ϕ ∈ H10 (D)
Micromagnetism - Philippe Corboz, 29.10.2002 – p.6/15
DiscretizationLLG → |m| is invariant in timeNot the case for our discrete equations.Need to introduce a reduced integration for coupling term:
∫
Ω
η grad(mn+ 1
2 × v) · gradmn+ 1
2 dx
Micromagnetism - Philippe Corboz, 29.10.2002 – p.7/15
DiscretizationLLG → |m| is invariant in timeNot the case for our discrete equations.Need to introduce a reduced integration for coupling term:
∫
Ω
η grad(mn+ 1
2 × v) · gradmn+ 1
2 dx
↓↓↓
∫
Ω
η grad(I(mn+ 1
2 × v)) · gradmn+ 1
2 dx
Micromagnetism - Philippe Corboz, 29.10.2002 – p.7/15
Newton-Algorithm
fl(m) =kA1(ml − ml + α(ml+2ml+1 − ml+1ml+2))
+ η(A3l+2ml+1 − A3
l+1ml+2) − Al+1ml+2 + Al+2ml+1 = 0
Start with m = m
Do while norm(f(m)) > tol
• mit+1 = mit − (Df)−1f(mit)
• if it > itmax, break and error
Micromagnetism - Philippe Corboz, 29.10.2002 – p.8/15
Results: ExamplesStrong coupling vs weak coupling
Micromagnetism - Philippe Corboz, 29.10.2002 – p.9/15
Results: Energy contributions
0 10 20 30 40 50 60 70 80 90 100−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
time
ener
gy
exchange energyanisotropy energyexternal fieldmagnetic energytotal energy
Micromagnetism - Philippe Corboz, 29.10.2002 – p.10/15
Results: Different mesh widths
0 20 40 60 80 100 1200.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
time
ener
gy
4*ho2*ho3/2*hoho
Micromagnetism - Philippe Corboz, 29.10.2002 – p.11/15
Results: Different timesteps
0 1 2 3 4 5 6 7 8 9 100.28
0.3
0.32
0.34
0.36
0.38
0.4
time
ener
gy
0.10.050.01
Micromagnetism - Philippe Corboz, 29.10.2002 – p.12/15
Results: Varying Size of D
0 10 20 30 40 50 60 70 80 90 1000.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
time
ener
gy
D=4D=8
Micromagnetism - Philippe Corboz, 29.10.2002 – p.13/15
Results: Convergence
0 100 200 300 400 500 6001.5
2
2.5
3
timestep
# ite
ratio
ns
Micromagnetism - Philippe Corboz, 29.10.2002 – p.14/15
Results: Movie example
Dynamical evolution of the magnetization
Micromagnetism - Philippe Corboz, 29.10.2002 – p.15/15