Some control considerations for ferromagnetic materials › trelat › GDT › confs ›...

EDSFerromagnetismMicromagnetism

Control

Γ-convergenceHysteresisAssymptotic

Temperature

EDP

Multi-scalesMagnetostaticMonte-Carlo

Ito

Some control considerations forferromagnetic materials

Stephane Labbe

Joseph Fourier University, Jean Kuntzmann Laboratory.

Groupe de Travail Controle, LJLL – 13 fevrier 2015

Stephane Labbe (Joseph Fourier University), Ferromagnetism,

PlanPlan Modeling of ferromagnetic materials

Physical principles

• Modeling of ferromagnetic materials• Physical principles• The microscopic scale• The micromagnetic model• Link between scales

• One particle• The stochastic model• Results• Illustrations•Which noise for which model?

• Net of particles• The deterministic model• Stable configurations• Control• The sotchastic case


Ferromagnetic materialsFerromagnetic materials Modeling of ferromagnetic materials

Physical principles

Characterization

Remanent magnetization.

A critical temperatures separating linear andnon-linear behavior.

Microstructures and domain formation.



Physical principles

Aimantation rémanente

Température

Comportement non linéaire Comportement linéaire

Température de Curie

Characterization






Physical principles

Characterization





Several scalesSeveral scales Modeling of ferromagnetic materials

Physical principles


Analogical and numerical experimentsAnalogical and numerical experiments Modeling of ferromagnetic materials

Physical principles

0 2 4 6 8 10 12

1

2

3

4

5

6

7

8

9

10

11


Not so easy to control...Not so easy to control... Modeling of ferromagnetic materials

Physical principles



Physical principles

atomic scale: every thing visible, modeling

via DFT.

microscopic scale: atom kernels are assimilated to punctualcharges bearing a magnetic moment

Mescopic scale: matter is modeled as continuous,ferromagnetic behavior are observable.

Macroscopic scale: ferromagnetic behaviors

are no more observable.



Physical principles

atomic scale: characteristic time ofthe spin-orbit interactions

Larmor scale: characteristic of themagnetic moment precession

dynamic scale: characteristic timeof the equilibrium relaxation

adiabatic scale: slow dynamic of equilibrium states



The microscopic scale





Microscopic modelMicroscopic model Modeling of ferromagnetic materials


GivenR the periodic net Z3, we setRε = εR andRε,Ω = Rε ∩ Ω.

Static

Atom kernels are seted in a cristalline configuration : Rε,Ω,

Magnetic moment beared by dirach measures : (µx )x∈Rε,Ω , avec µ =∑

x∈Rε,Ω

µx δx ,

Electromagnetic interactions: Maxwell system in void.

Heisenberg interaction energy: EHeis(µ) = −A∑

x∈Rε,Ω,y∈V(x)

µx · µy .


Microscopic modelMicroscopic model Modeling of ferromagnetic materials


Dynamic

Quasi instantaneous minimization of the local Heisenberg energy.

Larmor precesion :

∀x ∈ Rε,Ω,dµx

dt= −γµx ∧ H(µ)(x),

where H(µ)(x) is the electromagnetic files at point x generated by µ augmented by the externalfield.

Heisenberg interactions can be modeled by introduction of an heuristic dissipation term.



The micromagnetic model





Basis of micromagnetism (1)Basis of micromagnetism (1) Modeling of ferromagnetic materials


Thermodynamical description of ferromagnetic materials: Micromagnetism, W.F. BROWN, in the60’s.

Notations

Magnetic domain: Ω, open in R3

Unit sphere: S2

Magnetization: m, vector fields of Ω, whose values are in S2

Energy functional: E defined on H1(Ω,R3) with values in R

Equilibrium State: element of H1(Ω,S2) which minimizes E


Basis of micromagnetism (2)Basis of micromagnetism (2) Modeling of ferromagnetic materials


Energy functional E

E : H1(Ω,S2) −→ R is defined by

E(m)=A2

∫Ω|∇m|2 +

12

∫R3|Hd (m)|2 −

∫Ω

m · Hext

Hd (m): Demagnetizing field solution of the following equation (where m is an extension of m by 0in R3):

curl(Hd ) = 0 in D′(R3,R3)div(Hd ) = −div(m) in D′(R3,R3).

Hext : Zeeman, models the action of an external field (independant of m).


Basis of micro micromagnetismBasis of micro micromagnetism Modeling of ferromagnetic materials


Dynamic: the Landau-Lifchitz system.

Landau et Lifchitz

∂m∂t

= −m ∧ H(m)− αm ∧ (m ∧ H(m)),

H(m) is the effective field.

H(m) = −dedm

, with E(m) =

∫Ω

e(m)dx .

Hypothesis: ϕ(m) = K2 (|m|2 − (m · u)2)

where u is in L∞(R3,S2).

Champ effectif

H(m) = A4m+Hd (m)+K (m.u)u+Hext ,

Some remarks:

Equilibrium states verify ‖H(m) ∧m‖0,Ω = 0.

For an autonomous external field, the energy of system’s solutions is decreasing.

The magnetization modulus is preserved overall the domain.



Link between scales





The static caseThe static case Modeling of ferromagnetic materials

Link between scales

µ, minimizer of the energy under constraint

EHeis(µ) +∫R3 |Hd (µ)|2 dx

m, minimizer of the energy under constraint

A∫

Ω |∇m|2 dx +∫R3 |Hd (m)|2 dx

ε

Ω

Γ-converge,H1(Ω,R3)

ε→ 0

Rε,Ω = εZd ⋂Ω

m ∈ H1(Ω; S2).

B. Bidegaray, Q. Jouet and S. Labbe, Static ferromagnetic materials: from the microscopic to the mesoscopic scale,Communications in Contemporary Mathematics, 2013.


The dynamic caseThe dynamic case Modeling of ferromagnetic materials

Link between scales

Development of a particular notion of Gamma Convergence: convergence of trajectories.

Static results induces dynamical results: notion of gradient flows (see for example works by E. Sandierand S. Serfaty in the Ginzburg Landau context).

Work in progress with H. Pajot and B. Rufini.


PlanPlan One particle

The stochastic model





ProblemProblem One particle


Goal: to point out temperature like effects.

Tool: introduction of a stochastic perturbation.

What we want to emphazise :

quantifiable thermal effects,

hysteric phenomenas.

The deterministic model: Find µ(t), de R+ in S2, such that

dµdt = −µ ∧ Hext − αµ ∧ (µ ∧ Hext ),

The question: how to introduce the stochastic perturbation in the system preserving its structuralproperties?P. Etore, S. Labbe et J. Lelong, Hysteretic behavior of a stochastic nano particle, Journal of Differential Equations,2014.


The model: the studied EDSThe model: the studied EDS One particle


Given (Ω,F , (Ft )t>0,P) a filtered probabilist space and W standard Brownian process adapted to thefiltration (Ft )t>0 and valued in IR3. We set

dYt = −µt ∧ (b dt + ε dWt )− αµt ∧ µt ∧ (b dt + ε dWt )

µt = Yt|Yt |

Y0 = y ∈ S2,

We remark that the process µt preserves the structural properties of the deterministic model:

preservation of the modulus in time,

combination of to movements who, in the determinist case, are deriving from an hamiltoniandynamic of one and purely dissipative one for the other.



Results





Some propertiesSome properties One particle

Results

Proposition

When the pari of processes (Y , µ) is solution of the EDS we have

d |Yt |2 = 2ε2(α2 + 1)dt

so, |Yt | is deterministic.

Comes from the Ito formula.

We introduce h(t) = |Yt |h(t) =

√2ε2(α2 + 1)t + 1.

Then, we are able to define an EDS for µt · b

d(µt · b) = −(µt · b)h′(t)h(t)

dt −α

h(t)

((µt · b)2 − |b|2

)dt

−ε

h(t)

(− L(µt )b + α((µt · b)µt − b)

)· dWt

where L(x) is the edge by x operator.


Long time behaviorLong time behavior One particle

Results

The EDS for µt .b allows us to determine the long time behavior of the main EDS.

Lemma

supt

∫ t

0

1h(u)

(− µu ∧ b + α((µu · b)µu − b)

)· dWu <∞ a.s.

This lemma is based on the Doob inequality for martingales and gives access to the proof of thetheorem on long time behavior of the solutions

Theorem

µt · b −−−−→t→∞

|b| p.s.


Convergence speedConvergence speed One particle

Results

We obtain a convergence rate for the L1 norm

Theorem

limt−→∞

E(h(t) ||b| − µt · b|) =ε2(1 + α2)

2α.

We also prove the following corollary based upon the Markov inequality

Corollary

For all 0 < β < 1/2 and η > 0, P(tβ(|b| − µt · b) > η) −→ 0.


The hysteresis modelThe hysteresis model One particle

Results

Applied external field obeying to a slow dynamic

bη(t) = (1− 2t η) b ∀t 6 1/η

b(t) = (1− 2t) b ∀t 6 1

EDS for the processes Yt and µt :dYηt = −µηt ∧ (bη(t) dt + ε dWt )− αµηt ∧ µ

ηt ∧ (bη(t) dt + ε dWt )

µηt =Yηt|Yηt |

Yη0 = b

Then, we focus on the process Zηt = Yηtη

which is re-scaled in time. We set ληt =Zηt|Zηt |

.


The hysteresis phenomenaThe hysteresis phenomena One particle

Results

We prove that

Theorem

∀t ∈ [0,12

], E(ληt · b) >1√

1 + ε2(1+α2)η

This means that the trajectory of the process ληt · b in expectation goes over 0 strictly when b isvanishing, which is an hysteric behavior.



Illustrations





Almost sure convergence of µt · b for µ0 = −b, |b| = 1, ε = 0.1

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0.5

1

0 5 10 15 20

alpha=1.0alpha=0.7alpha=0.5


Convergence of 2α√

2ε√

(1+α2)

√t E(|b| − µt .b) pour µ0 = −b, |b| = 1 et ε = 0.1. dashed line: level at 1.

The expectation is computed via a Monte-Carlo method using 100 samples.

0

10

20

30

40

50

0 5 10 15 20

alpha=1.0alpha=0.7alpha=0.5


For α = 1, ε = 0.01 et η = 3.1 10−5

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1

0 0.2 0.4 0.6 0.8 1


Zoom near t = 12

-1

-0.5

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0.5

1

0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52


For α = 1, ε = 0.005 et η = 0.01

-1

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0.5

1

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Which noise for which model?





Stratonovich versus Ito?Stratonovich versus Ito? One particle


In the case of the Stratonovich integral, equal to the It one up to a finite variation process , the system iswritten as follows

∂µt = −µt ∧ (b∂t + ε∂Wt )− αµt ∧ µt ∧ (b∂t + ε∂Wt ), µ0 ∈ S(R3).

Then, this Stratonovich integral is transformed into an It integral1:

dµt = −A(µt )dt + εA(µt )dWt +12ε2

3∑q=1

3∑j=1

(AjqDj (Aiq))(µt ) dt

Then, µt is solution of the Ito EDS:

dµt = −(A(µt ) + ε2(α2 + 1)µt )dt + εA(µt )dWt .

who implies

d(µt · b)|µt = b|b|

= −ε2(α2 + 1)|b|dt,

1voir Rogers and Williams, V.30


Stratonovich versus Ito?Stratonovich versus Ito? One particle


Then, we have

E[µt · b]′ = α(|b|2 − E[(µt · b)2])− ε2(α2 + 1)E[µt · b]

E[µt · b]− E[µ0 · b] e−ε2(α2+1)t = e−ε

2(α2+1)t∫ t

0α(|b|2 − E[(µs · b)2]) e−ε

2(α2+1)s ds

lim supt→+∞

E[µt · b] = − lim inft→+∞

e−ε2(α2+1)t

∫ t

0αE[(µs · b)2]) e−ε

2(α2+1)s ds

lim supt→+∞

E[µt · b] 6 −α

ε2(α2 + 1)lim inft→+∞

E[(µt · b)2]

and similarly, we can show that

lim inft→+∞

E[µt · b] > −α

ε2(α2 + 1)lim supt→+∞

E[(µt · b)2].

So far, the process µt stays on the lower half sphere.


Differences: stability point of vueDifferences: stability point of vue One particle


ItoAsymptotically stable states remain asymptotically stables.

µt · b → 1, p.s. when t goes to infinity.

StratanovichAsymptotically stable states become stables.∫ t+σ

t(µs · b)ds → β < 1, p.s. when t, σ go to infinity.


PlanPlan Net of particles

The deterministic model





A net of par tiles and a lecture head

x

y

z

l

δl

diple magntique

G.Carbou, S. Labbe, C. Prieur and S. Agarwall Control of a network of magnetic ellipsoidal samples. MathematicalControl and Related Fields, 2011.



Stable configurations





Theorem

There exists γ0 > 0 (form parameter of the net) such that

Vl3≤ γ0 , (3.1)

There exists ν0 > 0, et c > 0 such that, for every pertinent configuration m0 associated to ε (m0i = εi ~e2

pour tout i), for all minit ∈ Vε(ν0), the solution m of the system with M ≡ 0 associated to the initialcondition m(0) = minit satisfies:

∀ t ≥ 0,m(t) ∈ Vε(ν0e−ct ).


Unstable configuration

x

y

z

x

y

z


Stable configuration

x

y

z

x

y

z



Control





ControlControl Net of particles

Control

Given m[ and m] two pertinent configurations. We choose the initial state in the neighborhood of m[,then the configuration m] is the goal.

∀t ∈ [ilv, i

lv

+ δlv

],M(t) =

+M, si m[i = −m]i = ~e2,

0 si m[i = m]i ,−M, si m[i = −m]i = −~e2,

∀t /∈N⋃

i=0

[ilv, i

lv

+ δlv

], M(t) = 0.


ControlThe magnetic dipole induces a magnetic field Ωi0 given by

Happ(t,M)(i0) =µ0M4π

1r3

(2 cos(θ)ur + sin(θ) uθ),

where

r is the distance between the dipole and the considered particle:

r = [(x0 + vt − i0l)2 + δ2l2]12 ,

ur et uθ are given by

ur =1r

x0 + vt − i0l−δl

0

, uθ =1r

δlx0 + vt − i0l

0

θ is the angle (−~e2, ur ).

M is the control.


Theorem

Given γ0 and c chosen in the stability zone. There exists γ1 > 0 with γ1 < γ0, there exists ν1 > 0,M > 0 and δ > 0 such that

Vl3≤ γ1,

then we can claim the following stability result:given m[ and m] two pertinent configurations t 7→ M(t) the control obtain auctioning the dipole. Ifminit ∈ Vε[ (ν1), The solution m of the system associated to the initial condition minit satisfies:

∀ t ≥ Tf , m(t) ∈ Vε] (ν1e−c(t−Tf )).



The sotchastic case





The studied systemThe studied system Net of particles

The sotchastic case

Similarly to the study performed for one particle, where are now interested in a net of particles∀σ ∈ Σl,n :dYσ,t = −µσ,t ∧ (H(µt )(σ)dt + εdWσ,t )− α µσ,t ∧ (µσ,t ∧ (H(µt )(σ)dt + εdWσ,t )) ,

µσ,t =Yσ,t|Yσ,t |

,

Y0 = y ∈ (S2)Σl,n .

Goal: to retrieve a phase transition behavior for configurations.

Work in progress with J. Lelong and A. Kritoglou.


Examples of dynamicsExamples of dynamics Net of particles

The sotchastic case

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Thank you for your attention.


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