Parameter-Dependence

We have investigated the influence of the interfacial energy parameter γ on the domain morphology and the scaling of the asymptotic energy. Qualitatively, a decrease in the typical domain wall energyγtriggers the influence of the dipolar interaction, and therefore tends to favour oscillating solutions and eventually microstructure. Consequently, as displayed in Figure 7.3, we observe a decrease of the typical size of the domains. In the opposite regime, for increasing values of the interfacial energy parameter, the dipolar interaction has a declining influence. For sufficiently large γ, the problem almost reduces to the minimal interface problem. In that case, the absolute minimizer is a single domain state taking the value either+1or −1everywhere. This statement is illustrated in Figure 7.5. The pattern-evolution was obtained for a random initial condition (7.1), but with the parameter value γ = 1000.

In a more quantitative study we have investigated the precise scaling law for the en-ergy asγ varies. As pointed out earlier, real micromagnetic applications involve a second parameterδthat corresponds to the film thickness relative to the dimension of the period-icity cell. Formal results have been obtained in the physics literature: the theory of Kooy and Enz [38] predicts an algebraic energy scaling of order(γ/δ)^1/2 for γ δ, whereas for γ δ the approach of Gehring and Kaplan [31] predicts an exponential dependence of type C1δ(1−C2exp[−C₃(γ/δ)]) with positive constantsC1,C2 and C3.

In our model problem we bypass the additional complexity arising from the presence of

7.4. Parameter-Dependence

time=0

(a)

time=132.2633

(b)

time=1076.05

(c)

time=1130.6633

(d)

time=13246.29

(e)

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10^−1.5

10^−1.4

Time

Energy

(f)

Figure 7.2.: (a)–(e) Formation of a checkerboard-pattern and subsequent evolution to-wards a stripe-pattern. (f) The corresponding energy profile.

(a) (b) (c)

(d) (e) (f)

Figure 7.3.: Typical domain sizes observed for increasing values of the interfacial energy parameter, respectively (a) γ = 0.00125, (b) γ = 0.002, (c) γ = 0.005, (d) γ = 0.008, (e) γ = 0.0175and (f) γ= 0.035.

a thickness parameter δ by setting it to1 for convenience. Accordingly, we have studied the evolution of the asymptotic energy as a function of the interfacial energy parameterγ only. Taking a random initial condition andε= 1/20, we ran the algorithm for values ofγ ranging from 1/800to 1/2in the first instance, and subsequently for values ranging from 10to1000. For the small values of the interfacial energy parameter, we obtain a scaling of the asymptotic energy as an algebraic power ofγ; cf. Figure 7.4(a). For larger values ofγ, the solutions converge towards a single domain state. The energy fluctuations are much less prominent than in the previous case, nevertheless the plot of the asymptotic energy as a function of γ suggests a scaling law of exponential type, as predicted by Kaplan and Gehring; cf. Figure 7.4(b). These scaling laws are consistent with the formal results predicted in [38] and [31], and a cross-over in the energy scaling can be observed.

7.4. Parameter-Dependence

10⁻³ 10⁻² 10⁻¹ 10⁰

10⁻² 10⁻¹ 10⁰

Interfacial Parameter γ

Asymptotic Energy Algebraic Scaling γ^1/2

(a)

10 100 300 500 750 1000

0.4990 0.4994 0.4998 0.5

Interfacial Parameter γ

Asymptotic Energy

(b)

Figure 7.4.: Evolution of the asymptotic discrete energy as a function of the interfacial energy parameterγ. (a) Logarithmic plot for values ofγ ranging from1/800 to 1/2 and comparison with the algebraic scaling γ^1/2 predicted by [38]. (b) Plot of the asymptotic discrete energy for values ofγ ranging from10to1000.

time=0.0005

(a)

time=0.00145

(b)

time=0.0053833

(c)

time=0.0116

(d)

time=0.026167

(e)

time=0.036783

(f)

Figure 7.5.: Temporal evolution towards a single-domain state. The patterns were ob-tained on a 512×512 grid for random initial condition and the parameter valueγ = 1000.

A. Norm Equivalence for Trigonometric Polynomials

We shall show here that on the space SN, the norms k · k_N and k · k_L2(Ω) coincide. Let thenN ∈N>0 be fixed and let u, v∈S_N be two trigonometric polynomials given by

u(x) = X

k∈Z²N

a(k) e^{2iπ k·x} and v(x) = X

k⁰∈Z²N

b(k⁰) e^{2iπ k0·x}

for all x∈Ω. On considering the discrete inner-product h·,·i_N defined in Section 6.3, we have of the right-hand side of the previous equation yields

1

so that finally we obtain

hu, vi_N = X

On proceeding as in the discrete case and developing the last term of the previous equation withx= (x1, x2), we obtain

ˆ

Ω

e^{2iπ(k−k0)·x}dx= hˆ ₁

0

e^{2iπ(k1−k01)x1}dx1

ihˆ ₁

0

e^{2iπ(k2−k20)x2}dx2

i

=

(1 if k1=k₁⁰ andk2=k₂⁰

0 else ,

so that as expected we have

(u, v)_L2(Ω;C) = X

k∈Z²N

a(k)b(k) =hu, vi_N.

Consequently, there holds as well

kuk_N =kuk_L2(Ω) for allu∈SN.

B. Implementation Code (MATLAB)

The following programm and functions realize the implementation in MATLAB of the numerical scheme (6.1) as presented in Chapter 7.

Implicit Scheme Implementation

% C U S T O M SET UP --g r i d s i z e =1;

N = 5 1 2 ; % N u m b e r of G r i d p o i n t s dt = 1 / 3 0 0 0 ; % Time step

c0 = 9 / 3 2 ; % N o r m a l i z a t i o n c o n s t a n t for d o u b l e well e p s i l o n = 1 / 2 0 ;

g a m m a = 1 / 1 0 0 ;

th =1; % T h i c k n e s s p a r a m e t e r

Nmax =40; % M a x i m u m i t e r a t i o n s for f i n d i n g f i x e d p o i n t tol =10^( -8) ; % T o l e r a n c e for f i n d i n g f i x e d p o i n t

s t o p _ c r i t =10^( -8) ; % S t o p p i n g c r i t e r i o n for time i t e r a t i o n m a x _ i t = 5 0 0 0 0 ;

% G E N E R I C SET UP --x = g r i d s i z e / N *(0: N -1) ;

k =[0: N /2 - N /2+1: -1] ’; % Wave n u m b e r s 1 D [ xi , eta ]= n d g r i d ( k , k ) ; % 2 D Wave n u m b e r s m o d k 2 = xi .^2+ eta .^2;

modk = sqrt ( m o d k 2 ) ;

L =(2* pi ) ^2* g a m m a * e p s i l o n *( m o d k 2 ) + s i g m a ( th * modk ) ; % L i n e a r o p e r a t o r in F o u r i e r S p a c e

Mult = ones ( N ) - dt /2* L ; % M u l t i p l i c a t i n g o p e r a t o r in F o u r i e r S p a c e (1 - dt /2* L )

Inv = ones ( N ) + dt /2* L ; % I n v e r t o p e r a t o r in F o u r i e r S p a c e (1+ dt /2*

L )

% TIME S T E P P I N G LOOP I N I T I A L I Z A T I O N --n_it =1;

time =0;

i =1;

time = time + dt ; u _ i n t = u0 ; Du =1;

D e n e r g y = 1 0 0 0 ;

% Time V e c t o r V a l u e s

--t i m e _ v e c --t o r = z e r o s ( max_i--t ,1) ; t i m e _ v e c t o r (1) =0;

% E N E R G Y C O M P U T A T I O N - I N I T I A L I Z A T I O N

--[ E n e r g y (1 , i ) ]= e n e r g y _ v a l u e ( gamma , epsilon , N , u0 , th , modk , modk2 , c0 ) ;

i = i +1;

% TIME S T E P P I N G LOOP --w h i l e ( Denergy > s t o p _ c r i t )

% C o m p u t a t i o n of f i x e d p o i n t

[ k , u , err , conv ]= f i x p t ( u_int , L , dt , N , epsilon , gamma , Nmax , tol , c0 ) ;

if ( conv ==1)

% E n e r g y C o m p u t a t i o n

--[ E n e r g y (1 , i ) ]= e n e r g y _ v a l u e ( gamma , epsilon , N , u , th , modk , modk2 , c1 ) ;

% E r r o r in L - i n f i n i t y norm --Du = max ( max ( abs ( u - u _ i n t ) ) ) ;

% -- E n e r g y D e c a y

D e n e r g y = E n e r g y (1 , i -1) - E n e r g y (1 , i ) ;

% Time V e c t o r

--t i m e _ v e c --t o r ( i ) = --t i m e _ v e c --t o r ( i -1) + d--t ;

% P a t t e r n v i s u a l i z a t i o n --i m a g e s c ( x , x , u ) ; c o l o r m a p ( gray ) ; t i t l e ([ ’ time = ’ n u m 2 s t r ( time ) ]) ; F r a m e s (: , i -1) = g e t f r a m e ;

% I n c r e m e n t a t i o n --u _ i n t = --u ;

time = time + dt ; n_it = n_it +1;

i = i +1;

e l s e i f ( conv ==0)

dt = dt /4; % try with r e d u c e d time step end

end

Computation of Fixed Point

The following function computes the solution of the numerical scheme (6.1) using the procedure explained in Section 7.2. Arguments of the algorithm are the initial guess for the fixed point, the linear operatorL_ε, the time step∆tand the gridsizeN, the parameters γ andεas well as the normalizing factorc0for the double-well potential and the tolerance for finding the fixed point. It returns the number of iterations needed to reach the fixed

point, the fixed point itself, the error in the approximation as well as a boolean taking the value 0 if a fixed point has not been reached within a chosen number of iterations.

f u n c t i o n [ k , u_n , err , conv ]= f i x p t ( u0 , L , dt , N , epsilon , gamma , Nmax , tol , c0 )

% k = n u m b e r of i t e r a t i o n s n e e d e d to o b t a i n f i x e d p o i n t

% u_n = a p p r o x i m a t i o n to f i x e d p o i n t

% err = e r r o r in a p p r o x i m a t i o n ( L - inf norm )

% u0 = i n i t i a l g u e s s for f i x e d p o i n t

% L = L i n e a r o p e r a t o r of g r a d i e n t flow e q u a t i o n ( in F o u r i e r S p a c e )

% dt = time step

% N = grid size

% eps = e p s i l o n

% g a m m a

% Nmax = m a x i m u m n u m b e r of i t e r a t i o n s

% tol = e r r o r t o l e r a n c e

% N o n l i n e a r f u n c t i o n

--NL = i n l i n e ( ’ 2* g a m m a * c0 /( e p s i l o n ) *( u + v ) .*(1 -( abs ( u ) .^2+ abs ( v ) .^2) /2) ’ , ’ u ’ , ’ v ’ , ’ e p s i l o n ’ , ’ g a m m a ’ , ’ c0 ’ ) ;

% -- C a l c u l a t i o n of c o n s t a n t term [(1 - dt /2 L ) /(1+ dt /2 L ) U_n ] CT = real ( i f f t 2 (( ones ( N ) - dt /2* L ) ./( ones ( N ) + dt /2* L ) .* fft2 ( u0 ) ) ) ;

% I n i t i a l i z a t i o n of I t e r a t i o n step --j =1;

u _ i n t 2 = N * ones ( N , N ) ; u _ i n t = u0 ;

e r r o r =10;

c o n v e r g e n c e =0;

% F i x e d P o i n t I t e r a t i o n Loop --w h i l e ( j < Nmax ) && ( error > tol )

test = NL ( u_int , u0 , epsilon , gamma , c0 ) ;

u _ i n t 2 = real (( i f f t 2 ( fft2 ( dt * test ) ./( ones ( N ) + dt /2* L ) ) + CT ) ) ; e r r o r = max ( max ( abs ( u_int2 - u _ i n t ) ) ) ;

u _ i n t = u _ i n t 2 ; j = j +1;

end

if ( error < tol ) c o n v e r g e n c e =1;

end k = j ;

err = e r r o r ; u_n = u _ i n t 2 ;

conv = c o n v e r g e n c e ;

Evaluation of Discrete Energy

The following function computes the value of the discrete energy EN(u) = γ

εh1, W(u)i_N +1

2hu,L_εui_N

for a given solution u ∈ X_N. Parameters taken in entry are γ, ε, the gridsize N, the solution u, the thickness parameter th, the normalizing constant c₀ for the double well and the matricesmodk andmodk2composed of the values of|k|, respectively|k|², for all k∈Z²_N.

f u n c t i o n [ d i s c r e t e _ e n e r g y _ v a l u e ]= e n e r g y _ v a l u e ( gamma , epsilon , N , u , th , modk , modk2 , c0 )

W = d o u b l e _ w e l l ( u , c0 ) ; ftu =1/ N ^2*( fft2 ( u ) ) ;

d i s c r e t e _ e n e r g y _ v a l u e =( g a m m a / e p s i l o n ) *1/ N ^2*( sum ( sum ( W ) ) ) + 1 / 2 * sum ( sum (( s i g m a ( th * modk ) + g a m m a * e p s i l o n *( m o d k 2 ) ) .*(( abs ( ftu ) ) .^2) ) ) ;

Double Well

The function double_well evaluates in this case the quantityW(u) =c₀(1−u²)² for a given solution u. It is needed for the evaluation of the discrete energy. Included in the entries are the solution u and the normalizing parameter for the double wellc0.

f u n c t i o n W = d o u b l e _ w e l l ( u , c0 ) u2 =1 -( abs ( u ) ) .^2;

W = c0 *(( u2 ) .^2) ;

Fourier Multiplier σ

The following function computes the quantities σ(k) = 1−exp(−2π|k|)

2π|k| for frequencies k given in entry.

f u n c t i o n sig = s i g m a ( A ) if ( abs ( A ) <1.0 e -12)

B =1 -( pi * abs ( A ) ) ; else

B =(1 - exp ( -2* pi * abs ( A ) ) ) . / ( 2 * pi * abs ( A ) ) ; end

sig = B ;

Bibliography

[1] R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.

[2] G. Alberti. Variational models for phase transitions, an approach viaΓ-convergence.

In Calculus of variations and partial differential equations (Pisa, 1996), 7, pages 95–114. Springer, Berlin, 2000.

[3] F. Alouges and A. Soyeur. On global weak solutions for Landau-Lifshitz equations:

existence and nonuniqueness. Nonlinear Anal., 18(11):1071–1084, 1992. ISSN 0362-546X.

[4] H. W. Alt. Lineare Funktionalanalysis. Springer, fourth edition, 2002.

[5] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. ISBN 0-19-850245-1.

[6] H. Attouch. Variational convergence for functions and operators. Applicable Math-ematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984. ISBN 0-273-08583-2.

[7] M. Bahiana and Y. Oono. Cell dynamical system approach to block copolymers.

Phys. Rev. A, 41(12):6763–6771, Jun 1990. doi: 10.1103/PhysRevA.41.6763.

[8] C. Bernardi and Y. Maday. Approximations spectrales de problèmes aux limites el-liptiques, volume 10 of Mathématiques & Applications (Berlin) [Mathematics & Ap-plications]. Springer-Verlag, Paris, 1992. ISBN 2-287-00387-8.

[9] J. P. Boyd. Chebyshev and Fourier spectral methods. Dover Publications Inc., Mine-ola, NY, second edition, 2001. ISBN 0-486-41183-4.

[10] A. Braides. Γ-convergence for beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002. ISBN 0-19-850784-4.

[11] H. Brezis. Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983. ISBN 2-225-77198-7. Théorie et applications. [Theory and applications].

[12] I. N. Bronstein and K. A. Semendjajew.Taschenbuch der Mathematik. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, fifth edition, 1991. ISBN 3-8154-2000-8. Trans-lated from the Russian, Translation edited and with a foreword by G. Grosche, V.

Ziegler and D. Ziegler.

[13] C. Canuto and A. Quarteroni. Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp., 38(157):67–86, 1982. ISSN 0025-5718.

[14] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods in fluid dynamics. Springer Series in Computational Physics. Springer-Verlag, New York, 1988. ISBN 0-387-17371-4.

[15] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods. Sci-entific Computation. Springer-Verlag, Berlin, 2006. ISBN 978-30725-9; 3-540-30725-7. Fundamentals in single domains.

[16] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods. Sci-entific Computation. Springer, Berlin, 2007. ISBN 978-3-540-30727-3. Evolution to complex geometries and applications to fluid dynamics.

[17] R. Choksi. Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci., 11(3):223–236, 2001. ISSN 0938-8974.

[18] R. Choksi and X. Ren. On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Statist. Phys., 113(1-2):151–176, 2003. ISSN 0022-4715.

[19] N. Condette, C. Melcher, and E. Süli. Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth.Math.

Comp., 80:205–223, 2011.

[20] J. W. Cooley and J. W. Tukey. An algorithm for the machine calculation of complex Fourier series. Math. Comp., 19:297–301, 1965. ISSN 0025-5718.

[21] G. Dal Maso. An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston Inc., Boston, MA, 1993.

ISBN 0-8176-3679-X.

[22] E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad.

Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58(6):842–850, 1975.

[23] A. Desimone, R. V. Kohn, S. Müller, and F. Otto. A reduced theory for thin-film micromagnetics.Comm. Pure Appl. Math., 55(11):1408–1460, 2002. ISSN 0010-3640.

[24] A. DeSimone, R. V. Kohn, S. Müller, and F. Otto. Recent analytical developments in micromagnetics. In G. Bertotti and I. Mayergoyz, editors,The Science of Hysteresis, volume 2, chapter 4, pages 269–381. Elsevier Academic Press, 2005. ISBN 0-12-480874-3.

[25] Q. Du and R. A. Nicolaides. Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal., 28(5):1310–1322, 1991. ISSN 0036-1429.

[26] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathe-matics. American Mathematical Society, Providence, RI, 1998. ISBN 0-8218-0772-2.

[27] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions.

Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. ISBN 0-8493-7157-0.

BIBLIOGRAPHY

[28] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wis-senschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[29] A. Finlayson and L. Scriven. The method of weighted residuals: a review. Applied Mechanics Review, 19:735–748, 1966.

[30] B. Fornberg. A practical guide to pseudospectral methods, volume 1 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 1996. ISBN 0-521-49582-2.

[31] G. Gehring and B. Kaplan. The domain structure in ultrathin magnetic films. J.

Magn. Magn. Mat., 128:111–116, 1993.

[32] G. Gioia and R. James. Micromagnetics of very thin films. Proc. R. Soc. Lond. A, 453:213–223, 1997.

[33] D. Gottlieb and S. A. Orszag. Numerical analysis of spectral methods: theory and applications. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26.

[34] B.-Y. Guo. Spectral methods and their applications. World Scientific Publishing Co.

Inc., River Edge, NJ, 1998. ISBN 981-02-3333-7.

[35] A. Hubert. Statics and dynamics of domain walls in bubble materials.J. Appl. Phys., 46(5):2276–2287, 1975.

[36] A. Hubert and R. Schäfer. Magnetic Domains: The Analysis of Magnetic Microstruc-tures. Springer Verlag, 1998.

[37] L. Kantorovic. On a new method of approximate solution of partial differential equation. Dokl. Akad. Nauk SSSR, 2:532–536, 1934.

[38] C. Kooy and U. Enz. Kooy-enz theory. Philips Res. Rep., 15(7), 1960.

[39] H.-O. Kreiss and J. Oliger. Comparison of accurate methods for the integration of hyperbolic equations. Tellus, 24:199–215, 1972. ISSN 0040-2826.

[40] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural⁰ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967.

[41] C. Lanczos. Trigonometric interpolation of empirical and analytical functions. J.

Math. Phys., 17:123–199, 1938.

[42] L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion, 8:153–169, 1935.

[43] C. Melcher. The logarithmic tail of Néel walls. Arch. Ration. Mech. Anal., 168(2):

83–113, 2003. ISSN 0003-9527.

[44] B. Mercier. An introduction to the numerical analysis of spectral methods, volume 318 ofLecture Notes in Physics. Springer-Verlag, Berlin, 1989. ISBN 3-540-51106-7.

[45] L. Modica. The gradient theory of phase transitions and the minimal interface crite-rion. Arch. Rational Mech. Anal., 98(2):123–142, 1987. ISSN 0003-9527.

[46] Y. Nishiura and I. Ohnishi. Some mathematical aspects of the micro-phase separation in diblock copolymers. Phys. D, 84(1-2):31–39, 1995. ISSN 0167-2789.

[47] T. Ohta and K. Kawasaki. Equilibrium morphology of block copolymer melts. Macro-molecules, 19:2621–2632, 1986.

[48] E. Schlömann. Domain walls in bubble films. i. general theory of static properties.

J. Appl. Phys., 44(4):1837–1849, 1973.

[49] J. Slater. Electronic energy bands in metal. Phys. Rev., 45:794–801, 1934.

[50] J. Slonczewski. Theory of domain wall motion in magnetic films and platelets. J.

Appl. Phys., 44:1759–1770, 1973.

[51] L. N. Trefethen.Spectral methods in MATLAB, volume 10 ofSoftware, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. ISBN 0-89871-465-6.

Selbständigkeitserklärung

Hiermit versichere Ich, meine Dissertation Pattern Formation in Magnetic Thin Films:

Analysis and Numerics selbstständig und ohne unerlaubte Hilfsmittel erstellt zu haben.

Die verwendte Literatur und die Hilfsmittel sind in der Arbeit angegeben. Die Promo-tionsordnung der Mathematisch-Naturwissenschaftlichen Fakultät II ist mir bekannt. Ich habe mich nicht anderwärts um einen Doktorgrad beworben und ich besitze noch keinen Doktorgrad im Promotionsfach.

Berlin, den 29.01.2010 Nicolas Condette

Im Dokument Pattern formation in magnetic thin films (Seite 84-99)