An Effective Integrator for the Landau-Lifshitz-Gilbert Equation

Transcription

1 ASC Report No. 02/2012 An Effective Integrator for te Landau-Lifsitz-Gilbert Equation P. Goldenits, G. Hrkac, D. Praetorius, D. Suess Institute for Analysis and Scientific Computing Vienna University of Tecnology TU Wien ISBN

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3 An Effective Integrator for te Landau-Lifsitz-Gilbert Equation P. Goldenits G. Hrkac D. Praetorius D. Suess Institute for Analysis and Scientific Computing, Vienna University of Tecnology, 1040 Wien, Wiedner Hauptstraße 8-10, Austria, ( Institute for Analysis and Scientific Computing, Vienna University of Tecnology, 1040 Wien, Wiedner Hauptstraße 8-10, Austria and Department of Materials Science and Engineering, University of Seffield, S1 3JD Seffield, Mappin Street, UK, ( Institute for Analysis and Scientific Computing, Vienna University of Tecnology, 1040 Wien, Wiedner Hauptstraße 8-10, Austria, ( Institute of Solid State Pysics, Vienna University of Tecnology, 1040 Wien, Wiedner Hauptstraße 8-10, Austria, ( Abstract: We consider a lowest-order finite element sceme for te Landau-Lifsitz-Gilbert equation (LLG) wic describes te dynamics of micromagnetism. In contrast to previous works from te matematics literature, we examine LLG including te total magnetic field induced by pysical penomena described in terms of excange energy, anisotropy energy, magnetostatic energy, as well as Zeeman energy. Besides a strong non-linearity and a non-convex side constraint, te non-local dependence of te demagnetization field from te magnetization represents a callenging task for te numerical integrator. In our numerical sceme, only te igest order term, namely te excange contribution, is treated implicitly, wereas te remaining contributions are computed explicitly. Tis is, in particular, advantageous for te computation of te demagnetization field by means of te popular approac of Fredkin et al. (1990). Furtermore, our sceme requires to solve only one linear system per time-step and allows a simplified computation of te arising system matrices by mass-lumping. Finally, te proposed integrator is matematically reliable in te sense tat we prove unconditional convergence for te approximation of a weak solution. Keywords: finite elements, micromagnetism, Landau-Lifsitz-Gilbert equation, time-integration, non-convex, non-local, non-linear, demagnetization field 1. INTRODUCTION Te understanding of te dynamic beaviour of a micromagnetic body under te influence of certain micromagnetic penomena is essential and of utter relevance for te development of magnetic materials. In te literature, te Landau-Lifsitz-Gilbert equation (LLG) is well-accepted to model te dynamics of micromagnetism. A variety of applications suc as for example te development of magneto-resistive storage devices as well as te amount of numerical issues makes LLG of interest for bot, pysicists and matematicians. In our contribution we consider a polyedral bounded Lipscitz domain R 3 and a fixed time interval (0,τ). Let m 0 : S 2 = {x R 3 x = 1} be some given initial state. Ten, te non-dimensional formulation of LLG reads m t = α 1+α 2m (m eff(m,f)) 1 1+α 2m eff(m,f), (1) m(0) = m 0 in, (2) ν m = 0 on (0,τ), (3) were te (unknown) magnetization is denoted by a vector-valued function m : (0,τ) S 2 wic satisfies te non-convex side constraint m = 1 a.e. in (0,τ). Here,α > 0referstoteGilbertdampingparameterwic depends only on te material. Moreover, m t is te time derivative of m, and eff (m,f) denotes te total magnetic field and is given by te negative variation of te Gibbs Free energy eff (m,f) = δe(m) δm. (4) In tis work, te bulk energy e( ) consists of excange energy, anisotropy energy, magnetostatic energy, as well

4 as Zeeman energy and tus reads e(m) = C ex m 2 +C ani Φ(m) (5) m u f m. 2 Here, Φ refers to te anisotropy density, f denotes an applied external field, and u is te demagnetization field. Te latter is obtained from te magnetostatic Maxwell s equations wit u being te solution u = (u int,u ext ) of te full space transmission problem u int = divm in, u ext = 0 in R 3 \, [u] = 0 on Γ, [ ν u] = m ν on Γ, u ext (x) = O ( 1/ x ) as x. Here, [u] and [ ν u] denote te jumps of u and its normal derivative across te boundary Γ of. Te contribution of te magnetostatic potential, tus involves certain integral operators. Terefore, te computation of te demagnetization field is te most time and memory consuming part in numerical simulations and as to be realized effectively. In Goldenits et al. (2012), we discuss several approaces from te literature to solve (6) numerically. In te present work, we restrict ourselves to te ybrid FEM-BEM approac proposed in Fredkin et al. (1990), wic is mostly used in te pysics literature: Let u 1 be te (up to an additive constant) unique solution of te Neumann problem u 1 = divm in, ν u 1 = m ν on Γ, and extend u 1 by zero to te entire space R 3. Wit u te magnetostatic potential determined by (6), te remainder u 2 = u u 1 satisfies u 2 = 0 in R 3 \Γ, [u 2 ] = u 1 Γ on Γ, [ ν u 2 ] = 0 on Γ, u 2 (x) = O(1/ x ) as x. Here and in te following, u 1 Γ denotes te interior trace u 1 Γ of u 1. As is known from potential teory, te unique solution of (8) is te double-layer potential u 2 = Ku 1 Γ were (Ku 1 Γ )(x) := 1 (y x) ν(y) 4π Γ y x 3 u 1 (y) (9) for all x R 3 \Γ and wit ν te exterior unit normal vector on Γ. According to te jump of K across Γ, one can sow tat u 2 on is caracterized by te inomogeneous Diriclet problem u 2 = 0 in, u 2 = (K 1/2)u 1 Γ on Γ, and we ave u = u 1 + u 2 in. (6) (7) (8) (10) Remark. We stress tat te factor c = 1/2 for te trace jump (K c)u 1 Γ of te double-layer potential Ku 1 Γ in olds only almost everywere on Γ, were Γ is flat. At corners or on edges of Γ, te factor c depends on te interior angle of. In order to provide a numerical sceme, we obtain tat supplemented by te same initial and boundary conditions (2) and (3), te classical formulation of LLG, cf. (1), can equivalently be stated as αm t +m m t = eff (m,f) (m eff (m,f))m, (11) cf. e.g. Goldenits (2012) for a detailed proof. We empasis, tat tis alternative formulation still is non-linear in consideration of te magnetization m but is linear in consideration of its time derivative m t. Formulation (11) will serve as te basis for our finite element (FE) sceme to solve LLG numerically,wereweapproximatem (t, ) m(t, ) and v (t, ) m t (t, ) for all times t (0,τ). Note tat, due to te non-convex constraint m = 1 a.e., te time derivative m t belongs to te tangential space of m, i.e. m m t = 0 a.e. in τ. 2. NUMERICAL ALGORITHM Let T denote a quasi-uniform and regular triangulation of te domain into tetraedra and N be te set of its nodes. To discretize te magnetization m in te spatial variable, we use te vector-valued Courant FE space V = S 1 (T ) 3 of piecewise linear and globally continuous functions. To discretize te time interval, we consider a uniform partition 0 = τ 0 < τ 1 < < τ J = τ wit timestep size k = k j := τ j+1 τ j for j = 0,...,J 1. For eac (discrete) function ϕ, ϕ j denotes te evaluation ϕ(τ j ) at time τ j. Now, let M = { } φ V φ (z) = 1 for all z N (12) be te restricted finite element set, were our solution m j m(τ j) is sougt due to te non-convex constraint m = 1. Furtermore, for φ M, let K φ = { } ψ V ψ (z) φ (z) = 0 for all z N (13) be te discrete tangential space associated wit φ M, were te discrete time derivative v j m t(τ j ) is sougt. To obtain a numerical integrator for LLG, we follow te idea of Alouges (2008), were te small particle limit eff (m) = m is considered: We proceed by setting v = m t in (11) and by discretizing te weak form of it according to our framework.we treat te term of igest order implicitly, namely te excange contribution, wereas te remaining tree terms of te effective magnetic field eff (m,f) = C ex m+ low (m,f) wit low (m,f) := C ani DΦ(m) u+f (14) are computed explicitly. For te numerical integrator, we set low (m j,fj ) := C anidφ(m j ) uj +fj. (15) Here, u j := uj 1 + uj 2 S1 (T ) denotes an FE solution of te superposition ansatz of Fredkin et al. (1990), were we proceed as follows: First, let u 1 S 1 (T ) wit e.g. u 1 = 0 be te unique FE solution of u 1 v = m v (16) for all v S 1 (T ) wit v = 0. Second, let u 2 S 1 (T ) be te unique solution of te Diriclet problem u 2 v = 0 (17)

5 for all v S 1 (T ) wit v Γ = 0, wic additionally satisfies te inomogeneous discrete Diriclet condition u 2 Γ = S (K 1/2)u 1 Γ. Here, S maps te continuous boundary data (K 1/2)u 1 Γ onto some discrete boundary data in S 1 (T Γ ), were T Γ denotes te induced triangulation of te boundary Γ into flat surface triangles. A possible coice for te approximation operator S is given by te Clément-Operator J v := z N Γ v z β z wit v z := 1 γ z Here, γ z denotes te surface node-patc γ z := {T Γ : T T wit z T} Γ γ z v. of a boundary node z N Γ. Moreover, β z S 1 (T ) is te corresponding at function. Finally, te discrete demagnetization field is defined by u = u 1 + u 2. Considering te last term in (15), we stress tat f j f(τ j) approximates te given applied field f. If f is continuous in time, a valid coice is te evaluation f j := f(τ j) of f at time τ j. If f is continuous in space and time, f j V can be cosen to be te nodal interpolant of f(τ j ) wic furter simplifies te implementation. Finally, wit te nodal interpolation operator I : C() S 1 (T ), we include te so-called mass-lumping of te L 2 -scalar-product to compute te arising massmatrices only approximately. Tis results in corresponding matrix blocks wic are diagonal instead of sparse only. Te proposed time-splitting sceme now reads as follows: Algoritm. Input: Discretized initial data m 0 M, damping parameter α > 0, parameter 0 < θ 1, counter j = 0. (i) Compute v j K m j by solving te (regular) linear system α I (v j ψ )+ I ((m j vj ) ψ ) = C ex (m j +θkvj ) ψ (18) + low (m j,fj ) ψ for all test functions ψ K m j tangential space. (ii) Define m j+1 M by setting from te discrete m j+1 (z) = mj (z)+kvj (z) m j (z)+kvj (z) (19) for all nodes z N and go to (i). Output: Sequence of functions v j K m j as well as m j+1 M for j 0. Some remarks are in order to comment on te wellposedness and te advantages of te proposed algoritm: According to te Lemma of Lax-Milgram,(18) admits a unique solution v j K m j. Asaconsequenceofteortogonalityrelationm j (z) v j (z) = 0, one as mj (z) + kvj (z) 2 = m j (z) 2 + k 2 v j (z) 2 1. Terefore, te discretized magnetization m j+1 M defined in step (ii) of our Algoritm is well-defined. We stress tat only one (sparse) linear system (18) as to be solved per time-step and te non-convex side constraint m = 1 is fulfilled node-wise. Te assembly of tis system is te topic of te subsequent Section 3. Altoug, one may also drop te nodal interpolation I in te linear system (18), we put empasis on te fact tat te use of mass-lumping for te cross product contribution of (18) is implementationally very attractive. An explicit treatment of te non-local contribution stemming from te magnetostatic potential u is included, i.e. te computation of m j+1 only requires te approximate field u j from te previous timestep. Put differently, te approac by Fredkin et al. (1990) is only used once per time-step. Tis results in te solution of two additional (sparse) linear systems per time-step. Formally, te Crank-Nicolson type sceme θ = 1/2 is of second order in time, wereas θ = 1 corresponds to an implicit Euler sceme. Instead of te approac of Fredkin et al. (1990) to approximate te demagnetization field u, also oter approaces can be used. Te same applies for te assumptions on and te discretization of te exterior field f. We refer to te remarksin Section 4 and Goldenits et al. (2012) for furter details. 3. IMPLEMENTATION In tis section, we focus on te computation of v j K m j in step (i) of our algoritm. Empasis is put on te assembly and structure of te matrices of te linear system(18),wicisposedontesubspacek m j oftefe space V = S 1 (T ) 3. Let β i S 1 (T ) denote te canonical at function associated wit te node z i N. We define β i+(l 1)n := β i e l wit te l-t unit vector e l R 3 and note tat β 1,...,β 3N is te canonical basis of V. In a first step, we rewrite te variational form (18) as follows: Find v j K m j suc tat a(v j,ψ )+b j (v j,ψ ) = L j (ψ ) (20) for all ψ K m j, were we use te abbreviate notation

6 a(φ,ψ ) = α I (φ ψ ) +θkc ex φ ψ, b j (φ,ψ ) = (21) I ((m j φ ) ψ ), (22) L j (ψ ) = C ex m j ψ C ani DΦ(m j ) ψ u j ψ + f j ψ. (23) Due to te coice of te basis functions β i+(l 1)n = β i e l, te bilinear form a(, ) from (21) corresponds to a symmetric block diagonal matrix A = A0 A 0 R 3N 3N A 0 sym (24) wit symmetric blocks A 0 R N N sym, were A 0 ii = α I (β i β i )+θkc ex β i β i = α ω i δ ii +θkc ex β i β i, (25) wit ω i = {T T : z i T} te volume patc associated wit te node z i N. Here, δ ii denotes Kronecker s delta wit δ ii = 1 for i = i and δ ii = 0 oterwise. Te occurring integrals can be computed by closed formulae. Note tat A 0 is te (positively weigted) sum of te standard stiffness and an approximated diagonal mass matrix. Consequently, A 0 and ence A are positive definite sparse matrices and do not depend on te time step τ j. For te bilinear form b j (, ) from (22), we use te identity a (b c) = (a b) c to see b j (φ,ψ ) = I ((m j φ ) ψ ) = I ((φ ψ ) m j ). To derive te corresponding matrix B R 3N 3N, note tat β i e l β i e l = β i β i e l e l and e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2, and e l e l = 0. By coice of te basis functions β i+(l 1)n = β i e l, B terefore as some block structure of te type B = 0 +B3 B 2 B 3 0 +B 1 R 3N 3N (26) +B 2 B 1 0 wit diagonal blocks B l R N N, were B l ii = I (β i β i m j e l) = ω i m j (z i) e l δ ii. Clearly, B is a sparse matrix wic is skew-symmetric and positive semidefinite, since b j (φ,ψ ) = b j (ψ,φ ) and b j (φ,φ ) = 0. Altogeter, te system matrix M = A + B R 3N 3N is positive definite and ence regular. Terefore te variational formulation (20) as even a unique solution if posed on te entire space V = S 1 (T ) 3. However, v j is determined by solving (20) in te subspace K m j. We realize te linear constraints m j (z) vj (z) = 0 for all z N (27) by a Lagrange multiplier ansatz, i.e. we obtain te unknown coefficients ν R 3N of 3N v j = ( )( ) ( M Λ ν m β m by solution of T ν b =. Λ 0 λ 0) m=1 Here, λ R N is te Lagrange multiplier and Λν = 0 realizes te constraints (27). Consequently, te Lagrange matrix reads Λ = ( Λ 1 Λ 2 Λ 3) R N 3N wit Λ l R N N, Λ l ii = k2 m j (z i) e l δ ii, i.e. te matricesλ l arediagonaland scaledby k 2 in order to stabilize te sceme. 4. CONVERGENCE RESULT Te definition of a weak solution to LLG is based on te idea of Alouges et al. (1992) and reads as follows: Definition. Let m 0 H 1 (;S 2 ) be a given initial magnetization. Ten, m is called a weak solution to LLG, if tere olds for all times τ > 0: (i) m H 1 ( τ ;S 2 ) wit m(x,0) = m 0 (x) in te sense of traces; (ii) for all φ C0 ( τ ;R 3 ), tere olds m t φ α (m m t ) φ τ τ (28) = ( eff (m,f) m) φ; τ (iii) for almost all t (0,τ), tere olds 1 m(t) 2 +C 1 m t 2 2 τ 1 (29) m 0 2 +C 2, 2 wit positive constants C 1,C 2 > 0 wic depends only on f, and α > 0. We interpolate te discrete solution m j for j = 1,...,J of te numerical integrator from Section 2 as a continuous, piecewise affine function in time: For all x and all times t (0,τ) wit j = {0,...,J 1} suc tat t [jk,(j +1)k), we define m k (t,x) := t jk m j+1 (j +1)k t (x)+ m j (x). (30) k k Te following convergence teorem generalizes te result of Alouges (2008), yields reliability of te proposed algoritm, and even proves existence of global weak solutions. We stress tat no coupling of te time-step size k and te space-mes size is imposed.

7 Convergence Teorem. Let θ (1/2,1] be a fixed parameter and let T be a family of sape-regular triangulations of te magnetic domain wit mes-size 0. Let te at functions {β i } associated wit T fulfill β i β i 0 for all i i. (31) For te discrete initial magnetization m 0 M, we assume m 0 m 0 in H 1 () as 0. Ten, tere olds weak convergence in H 1 ( τ ) of a subsequence of te in time interpolated discrete magnetization m k from (30) to a weak solution m of LLG as (,k) 0. Remark. Due to Bartels (2005), condition (31) implies te following energy estimate for te excange energy in step (ii) of te algoritm: I ( φ φ ) 2 φ 2 (32) for all φ V wit φ (z) 1 and for all nodes z N. Remark. Under certain assumptions one may approximate te demagnetization field u as well as te applied external field f differently witout canging te result of te convergence teorem: We assume tat te approximation P m j related to te demagnetization field Pm = u as its discrete counterpart, fulfills te following properties P m j L 2 () C P m j L 2 () (33) as well as Pm P m L2 ( τ) 0 as 0 (34) wit a positive constant C P > 0. We note tat te approximate stray-field operator P m j = u,wicisgivenbyteapproacoffredkin et al. (1990) and discussed above, fulfills property (33) as well as (34). A detailed proof is given in Goldenits et al. (2012). We consider te approximation f k in space and time of an applied external field f, wic is given by f k (t,x) = f j (x) for all t j t t j+1,x. Wenotetatanydiscretizationf k off wicsatisfies f f k L2 ( τ) 0 as (,k) 0 (35) will provide an admissible coice in te sense tat te convergence teorem olds. In particular, it is also possible to deal wit discontinuous applied fields. Te discretization of continuous f by nodal interpolation in time or in space-time, as proposed in Section 2, guarantees (35). 5. SKETCH OF PROOF In tis section we briefly comment on te essential arguments to prove te convergence teorem. A rigorously elaborated proof is given in Goldenits (2012). In a first step, we aim for convergence properties of te output v j and mj for j > 0 of our algoritm. To tis end, we also interpret te output function v j as a piecewise constant function in time: For all x and all times t (0,τ) wit j = {0,...,J} suc tat t [jk,(j +1)k), we define v k (t,x) := v j (x). On te one and, te definition of m k in (30) as well as te non-convex property m j (x) 1 for j 0 and all x, wic olds due to step (ii) of our time-splitting sceme, implies m k (t,x) 1. On te oter and and by use of te test function ψ = v j in (18), one may deduce some stability estimate J 1 1 m J 2 2 +kc v j 2 j=0 1 m (36) + k J 1 low (m j C,fj ) vj. ex j=0 Tis yields uniform boundedness of m k in H 1 ( τ ) as well as of v k in L 2 ( τ ). Terefore, we may extract a subsequence of v k and m k, respectively, suc tat m k m weakly in H 1 ( τ ) as (,k) 0, (37) m k m strongly in L 2 ( τ ) as (,k) 0, (38) v k v weakly in L 2 ( τ ) as (,k) 0, (39) wit certain v L 2 ( τ ;R 3 ) and m H 1 ( τ ;R 3 ). Te remaining part of te proof is dedicated to te stepwise verification tat te limit m is a weak solution to LLG, see Definition 4. We remark tat tis also involves te proof of v = m t to link te limits (37) (39). ACKNOWLEDGEMENTS Te autors acknowledge support of te Viennese Science and Tecnology Fund (WWTF) troug te researc project MA Micromagnetic Simulation and Computational Design of Future Devices. REFERENCES F. Alouges. A new finite element sceme for Landau- Lifcitz equations. Discrete and Continuous Dynamical Systems Series S, Vol. 1, 2008, F. Alouges, and A. Soyeur. On global weak solutions for Landau-Lifsitz equations: existence and nonuniqueness. Nonlinear Analysis, 18: , S. Bartels. Stability and convergence of finite-element approximation scemes for armonic maps. SIAM J. Numer. Anal., 43: , D. Fredkin, and T. Koeler. Hybrid metod for computing demagnetizing fields. IEEE Transactions on Magnetics, 26: , P. Goldenits, G. Hrkac, M. Page, D. Praetorius, T. Screfl, and D. Süss. Effective Simulation of te Dynamics of Ferromagnetism. Work in progress, P. Goldenits. Pd-Tesis. Institute for Analysis and Scientific Computing, Vienna University of Tecnology.