A finite element approximation for the quasi-static Maxwell Landau Lifshitz Gilbert equations

Transcription

1 ANZIAM J. 54 (CTAC2012) pp.c681 C698, 2013 C681 A finite element approximation for te quasi-static Maxwell Landau Lifsitz Gilbert equations Kim-Ngan Le 1 T. Tran 2 (Received 31 October 2012; revised 29 July 2013) Abstract Te quasi-static Maxwell Landau Lifsitz Gilbert equations wic describe te electromagnetic beaviour of a ferromagnetic material are igly nonlinear. Sopisticated numerical scemes are required to solve te equations, given teir nonlinearity and te constraint tat te solution stays on a spere. We propose an implicit finite element solution to te problem. Te resulting system of algebraic equations is linear wic facilitates te solution process compared to nonlinear metods. We present numerical results to sow te efficacy of te proposed metod. ttp://journal.austms.org.au/ojs/index.pp/anziamj/article/view/6318 gives tis article, c Austral. Matematical Soc Publised December 24, 2013, as part of te Proceedings of te 16t Biennial Computational Tecniques and Applications Conference. issn (Print two pages per seet of paper.) Copies of tis article must not be made oterwise available on te internet; instead link directly to tis url for tis article.

2 Contents C682 Contents 1 Introduction C682 2 A variational formulation of te MLLG equations C685 3 Te finite element sceme C A basis for W (j) C A basis for Y C691 4 Numerical experiments C692 References C695 1 Introduction Te Maxwell Landau Lifsitz Gilbert (mllg) equations describe te electromagnetic beaviour of a ferromagnetic material. For simplicity, we assume tat tere is a bounded cavity D R 3 (wit perfectly conducting outer surface D) into wic a ferromagnet D R 3 is embeded. We furter assume tat D\ D is a vacuum. Over time period (0, T) we let D T := (0, T) D and D T := (0, T) D, and let S 2 be te unit spere. We denote te unit vector of te magnetisation by m(t, x) : D T S 2, and te magnetic field by H(t, x) : DT R 3 over time t and space x. Te quasi-static mllg system is m t = λ 1 m H eff λ 2 m (m H eff ) in D T, (1a) µ 0 m t = µ 0 H t + σ ( H) in D T, (1b) in wic te subscript t indicates a partial derivative wit respect to time, λ 1 0, λ 2 > 0, σ 0 and µ 0 > 0 are constants and H eff is te effective magnetic field wic is dependent on bot H and m. Here m : DT R 3 is

3 1 Introduction C683 te zero extension of m onto D T, tat is { m(t, x) (t, x) D T, m(t, x) = 0 (t, x) D T \D T. Te system (1a) (1b) is supplemented wit initial conditions and boundary conditions m(0,.) = m 0 in D and H(0,.) = H 0 in D, (2) m n = 0 on D T and ( H) n = 0 on D T, (3) were n is te outward normal vector to te relevant surface. Equation (1a) is te first dynamical model for te precessional motion of te magnetisation, suggested by Landau and Lifsitz [8] in In tis model, te time derivative of te magnetisation m is a combination of te precessional movement m H eff and te dissipative movement m (m H eff ) ; see Figure 1. Cimrák [4] sowed existence and uniqueness of a local strong solution of (1a) (3). He also proposed a finite element metod to approximate tis local solution and provided an error estimation [3]. Baňas, Bartels and Prol [2] proposed an implicit nonlinear sceme using te finite element metod, and proved tat te approximate solution converges to a weak global solution. Teir metod required te condition k = O( 2 ) on te time step k and space step for te convergence of te nonlinear system of equations resulting from te discretisation. We propose an implicit linear finite element sceme to find a weak global solution to (1a) (3). Tis approac was initially developed by Alouges and Jaison [1] for te single Landau Lifsitz equation (1a). We extend teir approac to te system (1a) (1b). Te advantage of tis approac is tat tere is no condition imposed on te time step and te space step. For simplicity we coose te effective field H eff = m + H. We focus on implementation issues of te metod. In particular, we sow ow te

4 1 Introduction C684 Figure 1: Te combination of te precessional movement m H eff and te dissipative movement m (m H eff ). finite element spaces and teir bases are constructed. In anoter article we conducted a convergence analysis [9]. In Section 2 we rewrite (1a) in a form wic is more suitable for our approac; see (4). We ten introduce a variational formulation of te mllg system. Section 3 is devoted to te presentation of te implicit linear finite element sceme. Numerical experiments are presented in te last section.

5 2 A variational formulation of te MLLG equations C685 2 A variational formulation of te MLLG equations Before presenting a variational formulation for te mllg equations, we sow ow to rewrite (1a) in te form introduced by Gilbert [6] in wic µ = λ λ2 2. λ 1 m t + λ 2 m m t = µm H eff, (4) Lemma 1. Equation (1a) and equation (4) are equivalent. Proof: By using te elementary identity a (b c) = (a c)b (a b)c, for all a, b, c R 3, and te property m = 1 we obtain m [m (m H eff )] = m H eff. (5) Assume tat m is a solution to (1a). Multiplying bot side of (1a) by λ 2 m using te vector product and using (5) we obtain λ 2 m m t = λ 1 λ 2 m (m H eff ) + λ 2 2m H eff. Multiplying bot sides of (1a) by λ 1 and adding te resulting equation to te above equation, we deduce tat m satisfies (4). Now assume tat m is a solution to (4). On multiplying bot sides of tis equation by λ 2 m using te vector product and noting m (m m t ) = (m m t )m m 2 m t = m t, (6) we deduce λ 1 λ 2 m m t λ 2 2m t = λ 2 µm (m H eff ).

6 3 Te finite element sceme C686 Subtracting tis equation from equation (4) times λ 1 and dividing bot sides of te resulting equation by µ, we obtain (1a). Tis proves te lemma. Before presenting te variational form of tis problem, it is necessary to introduce te function spaces { } H 1 (D, R 3 ) = u L 2 (D, R 3 u ) L 2 (D, R 3 ) for i = 1, 2, 3, x i H(curl; D) = { u L 2 ( D, R 3 ) u L 2 ( D, R 3 ) }. Here L 2 (Ω, R 3 ) is te usual space of Lebesgue integrable functions defined on Ω and taking values in R 3. Following Lemma 1, instead of solving (1a) (3) we solve (1b) (4). A variational form of tis problem is as follows. For all φ C (D T, R 3 ) and ζ C ( D T, R 3 ), find m H 1 (D T ) and H L 2 ( D T ) suc tat H t L 2 ( D T ) and H L 2 ( D T ) to satisfy te Landau Lifsitz Gilbert (llg) equation λ 1 m t φ dx dt + λ 2 (m m t ) φ dx dt D T D T = µ m (m φ) dx dt + µ (m H) φ dx dt, (7) D T D T and Maxwell s equation µ 0 H t ζ dx dt + σ H ζ dx dt = µ 0 m t ζ dx dt. (8) D T D T D T In te following section we introduce a finite element sceme to approximate te solution (m, H) of (7) (8). 3 Te finite element sceme Let T be a regular tetraedrisation of te domain D into a tetraedra of maximal mes size, and let T D be te restriction of T to te domain D.

7 3 Te finite element sceme C687 Te set of N vertices is N := {x 1,..., x N } and te set of M edges is M := {e 1,..., e M }. To discretise te llg equation (7) we introduce te finite element space V of all continuous piecewise linear functions on T D, wic is a subspace of H 1 (D, R 3 ). A basis for V is cosen to be (φ n ) 1 n N, were φ n (x m ) = δ n,m, and δ n,m is te Kronecker delta. Te interpolation operator from C 0 (D, R 3 ) onto V is N I V (v) = v(x n )φ n for all v C 0 (D, R 3 ). n=1 To discretise Maxwell s equation (8), we use te space Y of lowest order edge elements of Nedelec s first family [10] wic is a subspace of H(curl; D). For a function u wic is Lebesgue integrable on all edges in M, we define [10] te interpolation I Y onto Y as I Y (u) = M u q ψ q for all u C 0 ( D, R 3 ), q=1 were u q = u τ q ds, e q in wic τ q is te unit vector in te direction of edge e q. Fixing a positive integer J, we coose te time step k = T/J, and define t j = jk for j = 0,, J. Te functions m(t j, ) and H(t j, ) are approximated by m (j) V and H (j) Y, respectively, for j = 1, 2,..., J. Since m t (t j, ) m(t j+1, ) m(t j, ) k m(j+1) k m (j),

8 3 Te finite element sceme C688 we evaluate m (j+1) from m (j) using were v (j) is an approximation of m t (t j, ). However, to maintain te con- = 1, we normalise te rigt and side of (9) and terefore belonging to V by dition m (j+1) define m (j+1) m (j+1) := m (j) + kv(j), (9) ( m (j+1) m (j) = I + kv(j) V m (j) + kv(j) ) = N m (j) n=1 m (j) Hence it suffices to propose a sceme to compute v (j). Motivated by m t m = 0, we find v (j) Given m (j) (x n) + kv (j) (x n) (x n) + kv (j) (x φ n. n) in te space W(j) defined by }. (10) W (j) := {w V w(x n ) m j (x n) = 0, n = 1,..., N V, if (7) is used to compute v (j), an approximation to m t(t j, ), ten a different test space from V is required because te test function φ in (7) is not perpendicular to m, unlike m t. To circumvent tis difficulty, we use (6) to rewrite (7) as λ 2 m t w dx dt λ 1 (m m t ) w dx dt D T D T = µ m w dx dt + µ H w dx dt (11) D T D T were w = m φ. Now bot m t and w are perpendicular to m for all (t, x) D T. Hence, given m (j) V and H (j) Y, we compute te approximations v (j) and H (j+1) of m t (t j, ) and H(t j+1, ), respectively, as follows. Find v (j) W(j) and H(j+1) Y satisfying, for all w (j) W(j) and

9 3 Te finite element sceme C689 ζ Y, λ 2 and Here, v (j) D = µ D µ 0 w(j) dx λ 1 ( m (j) D + θkv(j) (m (j) ) v(j) ) w(j) dx w (j) dx + µ H (j+1/2) D w (j) dx dt, (12) d t H (j+1) ξ dx dt + σ H (j+1/2) ξ dx dt D D = µ 0 H (j+1/2) v (j) ξ dx dt. (13) D := H(j+1) 2 + H (j) and d t H (j+1) := k 1 (H (j+1) H (j) ). Te parameter θ is arbitrarily cosen to be in [0, 1]. Te metod is explicit wen θ = 0 and fully implicit wen θ = 1. Te algoritm for te numerical approximation of te mllg system is summerised in Algoritm 1. By te Lax Milgram teorem, for eac j > 0 tere exists a unique solution (v (j), Hj+1 ) W(j) Y of equations (12) (13). Since m (0) (x n) = 1 and v (j) (x n) m (j) (x n) = 0 for all n = 1,..., N, and j = 0,..., J, by induction, m (j) (x n) + kv (j) (x n) 1 and m (j) (x n) = 1. Terefore, te algoritm is well defined. We now comment on te construction of basis functions for W (j) wic are necessary in solving (12) (13) in Step 5 of Algoritm 1. and Y

10 3 Te finite element sceme C690 Algoritm 1: Numerical approximation of te mllg system 1 begin 2 Set j = 0 ; 3 Coose m 0 = I V m 0 and H 0 = I Y H 0 ; 4 for j = 0, 1, 2,..., J 1 do 5 Solve (12) and (13) to obtain (v (j), H(j+1) ) W (j) Y ; 6 Define 7 end 8 end m (j+1) (x) := N m (j) n=1 m (j) (x n) + kv (j) (x n) (x n) + kv (j) (x φ n (x) ; n) 3.1 A basis for W (j) From (10), te basis functions of W (j) at eac iteration depend on te solution m (j) computed in te previous iteration. Terefore tey must be computed again for eac iteration of j in Algoritm 1. For eac w W (j), let α n = w(x n ) n = 1,, N, and let α (1) n and α (2) n be two basis vectors of te plane tangential to te vector m (j) (x n) = (u 1, u 2, u 3 ) T R 3. It follows from (10) tat α n = β n α (1) n + γ n α (2) n, for some real numbers β n and γ n. In our computation we take α (1) n = Am (j) (x n) and α (2) n = m (j) (x n) α (1) n,

11 3 Te finite element sceme C691 were u 2 u A = 1 u 2 u u 2 u 3 A basis for W (j) is defined from bases of te 2D planes wic are perpendicular to m (j) (x n) for all n = 1,..., N. Terefore dim(w (j) ) = 2N and w is expressed in terms of α (1) n and α (2) n by It can be sown tat space W (j). w(x) = N n=1 ( ) βn α (1) n + γ n α (2) n φn (x). (14) { } (α (1) n φ n, α (2) n φ n ) 1 n N is a basis for te vector 3.2 A basis for Y A basis {ψ 1,..., ψ M } of Y is defined as follows [10, Section 5.5.1]. Consider an edge e q, q = 1,..., M, and let K be te tetraedron aving e q as an edge. Let λ (1) q and λ (2) q be te barycentric coordinate functions corresponding to te endpoints of e q. We define Te relation ψ q K := λ (1) q λ (2) q λ (2) q λ (1) q. ( ψ q K ) = 2 λ (1) q λ (2) q, is useful in te computation of ( ψ q K ).

12 4 Numerical experiments C692 4 Numerical experiments In order to carry out pysically relevant experiments, te initial conditions of te mllg equations sould be cosen to satisfy te divergence-free constraint [7] div(h 0 + χ D m 0 ) = 0 in D, were χ D is te caracteristic function of D. Tis is acieved by taking H 0 = H 0 χ D m 0, (15) were H 0 is some function defined on D satisfying div H 0 = 0. In our experiment, for simplicity, we coose H 0 to be a constant. We solve te standard problem #4 proposed by te Micromagnetic Modeling Activity Group at te National Institute of Standards and Tecnology [5]. In tis model, te initial conditions m 0 and H 0, and te effective field H eff are m 0 = (1, 0, 0) in D, H 0 = (0.01, 0.01, 0.01) in D, H eff = 2A µ 0 M m+h. s Te parameters λ 1 = γ and λ 1 + α 2 2 = γα 1 + α, 2 were te positive pysical constants are te damping parameter α, te gyromagnetic ratio γ, te vacuum permeability µ 0, te excange constant A, and te magnitude of magnetisation M s. Te values of te pysical constants are α = 1, σ = S 1 m s 1, µ 0 = H m 1 s 1, A = J m 1, µ 0 = H m 1, γ = m A 1 s 1, M s = A m 1. Te domains D and D are cosen to be D = (0, 0.5) (0, 0.125) (0, 0.003),

13 4 Numerical experiments C Figure 2: Te magnetisation domain D (in red) is a tin film. Te magnetic domain D is in blue. and D = ( 0.583, 1.083) ( 0.146, 0.271) ( 0.767, 0.770), wit distances measured in µm; see Figure 2. Te domain D is uniformly partitioned into cubes of dimensions µm 3, were eac cube consists of six tetraedra. We generate a nonuniform mes for te magnetic domain D in suc a way tat it is identical to te mes for D in te region near D, and te mes size gradually increases away from D. A cross section of te mes at x 3 = 0 is displayed in Figure 3. At eac iteration of Algoritm 1, te system to be solved is linear and of

14 4 Numerical experiments C694 Table 1: A comparision between Baňas, Bartels and Prol s (bbp) metod and our metod. bbp Our metod Discrete system nonlinear linear (uses fixed-point iteration) (solved directly) Degrees of freedom 3N + M 2N + M Required condition k = O( 2 ) None Basis functions same different of solution space for all iterations in eac iteration size 2N + M. A comparision of our metod and te metod proposed by Baňas, Bartels and Prol [2] is presented in Table 1. For j = 0, 1, 2,..., let E (j) T E (j) T := 2σA M s D := E (j) ex + E (j) H + E(j) E, be te total energy at time t j = jk defined by H (j) 2 dx + λ 2 H (j) D µ 2 dx D m (j) 2 dx + 2σµ 0 were E (j) ex, E (j) H and E(j) E are te excange energy, magnetic field energy and electric field energy, respectively. Our computation sows tat te total energy decreases, tat is E (j+1) T E (j) T for all j 0 ; (16) see Figure 4. Te decrease of te discrete energy E (j) T, j = 1,..., J, suggests te gradient stability of te mllg solutions. In Figure 4 we plot different scaled versions of te different energies versus log t. Te cange in E (j) T dominated by te cange in E (j) H. Figure 4 also sows tat te sequence { m (j) } j 0 is bounded in L 2 (D), and te sequences {H (j) } j 0 and { H (j) } j 0 are bounded in L 2 ( D). In a fortcoming paper, we will sow tat our numerical solution converges to is

15 References C695 Figure 3: Mes for te domain D at x 3 = 0. a weak solution of te problem (1b) (4) by proving tat te inequality (16) olds. Acknowledgements Te autors acknowledge financial support troug te Australian ARC project DP References [1] F. Alouges. A new finite element sceme for Landau Lifcitz equations. Discrete Contin. Dyn. Syst. Ser. S, 1: (2008). doi: /dcdss C683 [2] L. Baňas, S. Bartels, and A. Prol. A convergent implicit finite element discretization of te Maxwell Landau Lifsitz Gilbert equation. SIAM

16 References C E energy H Energy excange energy total energy 1 Energy (J) log(t) (ns) Figure 4: Evolution of te energies, log(t) E E (t)/600, E H (t)/10 6, 10 4 E ex (t), E T (t)/10 6.

17 References C697 J. Numer. Anal., 46: (2008). doi: / C683, C694 [3] I. Cimrák. Error analysis of a numerical sceme for 3D Maxwell Landau Lifsitz system. Mat. Metods Appl. Sci., 30: (2008). doi: /mma.863 C683 [4] I. Cimrák. Existence, regularity and local uniqueness of te solutions to te Maxwell Landau Lifsitz system in tree dimensions. J. Mat. Anal. Appl., 329: (2007). doi: /j.jmaa C683 [5] Micromagnetic Modeling Activity Group, Center for Teoretical and Computational Materials Science, National Institute of Standards and Tecnology, USA. ttp:// C692 [6] T. L. Gilbert. A Lagrangian formulation of te gyromagnetic equation of te magnetic field. Pys Rev, 100: (1955). Abstract only; full report, Armor Researc Foundation Project No. A059, Supplementary Report, May 1, 1956 (unpublised). C685 [7] B. Guo and S. Ding. Landau Lifsitz Equations, volume 1 of Frontiers of Researc wit te Cinese Academy of Sciences. World Scientific, Hackensack NJ (2008). doi: / C692 [8] L. Landau and E. Lifscitz. On te teory of te dispersion of magnetic permeability in ferromagnetic bodies. Pys Z Sowjetunion, 8: (1935); Perspectives in Teoretical Pysics, pp , Pergamon, Amsterdam (1992). doi: /b C683 [9] K.-N. Le and T. Tran. A convergent finite element approximation for te quasi-static Maxwell Landau Lifsitz Gilbert equations. Researc Report, arxiv: , 2012, submitted to Computers & Matematics wit Applications. ttp://arxiv.org/abs/ C684

18 References C698 [10] P. Monk. Finite Element Metods for Maxwell s equations. Numerical Matematics and Scientific Computation. Oxford University Press, New York (2003). C687, C691 Autor addresses 1. Kim-Ngan Le, Scool of Matematics and Statistics, Te University of New Sout Wales, Sydney 2052, Australia. mailto:n.le-kim@student.unsw.edu.au 2. T. Tran, Scool of Matematics and Statistics, Te University of New Sout Wales, Sydney 2052, Australia. mailto:tan.tran@unsw.edu.au