An Implicit Finite Element Method for the Landau-Lifshitz-Gilbert Equation with Exchange and Magnetostriction

Transcription

1 SMA An Implicit Finite Element Metod for te Landau-Lifsitz-Gilbert Equation wit Excange and Magnetostriction Jonatan ROCHAT under te supervision of L. Banas and A. Abdulle

2 2 ACKNOWLEDGEMENTS I would like to tank Assyr Abdulle, wo gives me te opportunity to come ere in Edinburg and work on tis fascinating subject. My tanks goes to Lubomir Banas, wo follows me during all work. I tank Lubomir for is support, is motivation and all time we pass in is office discussing on te subject. I also tank im to let me freedom to try tings on my own. At last, I would like to tank all te PHD and Erasmus students from Heriot-Watt for te good moments we ad togeter during te work and out te university.

3 CONTENTS 3 Contents 1 Introduction 4 2 Te Landau-Lifsitz equation Maxwell s equations and magnetization Te Landau-Lifsitz equation Conservation of magnitude Gilbert form of te Landau-Lifsit equation Magnetostriction LLG equation wit excange and magnetostriction An implicit metod wit magnetostriction Preliminaries Convergence of te Algoritm Solving te non-linear system Numerical Experiments LLG wit exact solution A Blow Up Example Introduction Initial Data LLG wit excange LLG wit magnetostriction LLG wit excange and magnetostriction A problem wit magnetrostriction Conclusion 49 6 Appendix Notations and Preliminaries Functional Analysis Cross Product Alternative Energy Inequality Implementation wit FreeFem FreeFem Reduced Integration Discrete Laplacian L 2 -projector of te magnetostriction FreeFem and Paraview Code References 69

4 1 INTRODUCTION 4 Abstract Te Landau-Lifsitz-Gilbert Equation describes te dynamics of ferromagnetism, were strong nonlinearity and non-convexity are ard to tackle. Based on te work of S.Bartels and A.Prol "Convergence of an implicit finite element metod for te Landau-Lifsitz-Gilbert equation" ([4]), we present in tis report a fully implicit finite element sceme wit excange and magnetostriction. We verify unconditional convergence and present numerical examples 1 Introduction Te Landau-Lifsitz-Gilbert equation wit excange and magnetostriction describes te magnetization of non-linear magneto-elastic materials, suc as ferromagnetism; let α denote te damping factor and Ω T = Ω (, T ), wit a Lipscitz boundary continuous Γ, wic consists in two non-overlapping parts Γ D and Γ N. Ten te magnetization m : Ω T S 2, were S 2 = {x R 3 x = 1}, and te displacement u : Ω T R 3 solve on Ω T for all T > : { mt + αm m t = (1 + α 2 )(m ( m + σ )) u tt σ = (1) were σ = σ(u, m) is a tensor from magneto-dynamic and σ is te magnetostriction vector. Te system is supplemented by initial conditions m() = m H 1 (Ω, S 2 ), u() = u H 1 (Ω), u t () = v H 1 (Ω) and boundary conditions m ν = on Ω, u = on Γ D, σ ν = on Γ N. Te construction of convergent sceme for tis system is a non-trivial task due to te non-convex side-contraint m = 1 a.e in (, T ) Ω, wic is difficult to realize in numerical approximation scemes. Te autors in [4] ave developed an implicit finite element sceme wit tis property to solve te Landau-Lifsitz- Equation equation wit excange and tey sow it was better tan te standard explicit sceme. In tis work, we extended tis algoritm wit te magnetostriction: we take te lowest order finite element space V H 1 (Ω, R 3 ) subordinate to a triangulation T of Ω and a time-step size k >. We note V, = {ϕ V ϕ = on Γ D }. Let m V, u, v V,. Given a time step k >, j and m j V, u j, d tu j V,, our approximative sceme reads as follows: Algoritm Determine m j+1 V from (d t m j+1, φ ) + α(m j d tm j+1, φ ) 2. Determine u j+1 = (1 + α 2 )( m j+ 1 2 ( m j P V j+ 1 2 σ, φ ) φ V. V, from (d 2 t u j+1, ϕ ) + (σ j+1, (ϕ )) = ϕ V,.

5 1 INTRODUCTION 5 3. set j = j + 1 and return to 1. Here, (, ) denote te reduced integration and te terms, P V σ are respectively te discrete implementation of te Laplacian and te L 2 projection of te magnetostriction. We use d t ψ j+1 = k 1 (ψ j+1 ψ j ) for j 1, d 2 t ψ j+1 = k 2 (d t ψ j+1 d t ψ j ) for j 2 and ψ j+ 1 2 = 1/2(ψ j+1 + ψ j ) for j and a sequence ψ j. Te report is organized as follows. We first present te Landau-Lifsitz equation, some of its properties and preliminaries about finite element space (2). Ten we prove tat te algoritm (1.1) conserves te lengt of te magnetization m and verifies a discrete energy inequality: N mn 2 L 2 + α 1 + α 2 d t m j+1 2 k j= d tu N 2 L ε(un ) 2 L 2 C(λ e, λ m ) m 2 L v 2 L ε(u ) 2 L 2 were λ e, λ m are symmetric tensors wic describe te properties of te material. Our main result is teorem 3.2, wic verifies unconditional convergence for te algoritm (1.1). Fixed-point iteration 3.3 is used to solve te non-linearity, wose convergence is establised for k = O( 2 ). We discuss some numerical examples implemented in FreeFem++. An exact solution problem in one dimension (4.1) allows to look at te convergence of te algoritm. Te program is also motivated by possible finite time blow-up beaviour of te solution from (17) and te impact of te magnetostriction on te excange field, (4.2) Finally, te last problem (4.3) sows te deformation of a ferromagnetic material under te effect of magnetostriction.

6 2 THE LANDAU-LIFSHITZ EQUATION 6 2 Te Landau-Lifsitz equation In solid state pysics, te Landau-Lifsitz equation, named after Lev Landau ( ) and Evgeny Lifsitz ( ), is a partial differential equation describing time evolution of magnetization in solids. Depending on te time variable and generally two or tree space variables, te model first appears for te precessional motion of te magnetization in We introduce ere te Landau-Lifsitz equation wit some of its properties, te effect of te magnetostriction and te definition of te problem tat will be studied trougout tis report. 2.1 Maxwell s equations and magnetization Te electromagnetic field on te macroscopic scale is described by te four Maxwell equations: t D H = σe J, t B + E =, <, B > =, D = ρ. Te four space and time dependent vector functions E, D, H, B are respectively te electric field, electric displacement, magnetic field and magnetic induction. Moreover, ρ is te scalar carge density function, J is te current density vector and σ is te conductivity. In vacuum or in free space, relations exist beween te electric and magnetic variables: D = ε E, B = µ H were ε, µ are respectively te permittivity and magnetic permeability. For linear material, we ave te following correspondances: D = εe, B = µh were ε, µ are 3 3 dependent tensor. In tis work, we are interesting in non-linear materials, suc as ferromagnets. In tis case, te precedent relation is usually a bit more complicated: D = ε E, B = µ (H + m), were m is te magnetization of te material, governed by te Landau-Lifsitz equation below. 2.2 Te Landau-Lifsitz equation We searc te magnetization m : (, T ) Ω R 3 suc tat m t = m T αm (m T ) (2) wit te initial condition m() = m H 1 (Ω; S 2 ) and n m = on (, T ) Ω. Te term m T is te precession part and αm (m T )

7 2 THE LANDAU-LIFSHITZ EQUATION 7 is te damping part, were α denote te damping parameter. Te vector T is te total effective field and consist of were T = + anis + exc + σ, 1. is te magnetic field wic comes from Maxwell s equation and is supposing to be known; 2. Magnetic properties depend of te direction. Tis is given by te anisotropy field anis wic takes te form anis = K < p, m > p, were p is a direction of te uniaxial anisotropy and K a constant tat depends of tat direction; 3. exc = m is te excange field wic is due to te interactions between te magnetic spins; 4. σ is te magnetostriction tat will be described later. A basic model to understand te effect of te effective and damping part is to consider one constant effective field and te following Landau-Lifsitz equation: m t = m. (3) Tis is te precession part and te solution m of (3) rotates around and te angle between m and remains contant. If we add a damping part, te equation is written m t = m αm (m ). (4) Te magnetization m of (4) rotates around until teir mutual angle becomes zero, wic means tat te magnetization becomes parralel to, te steady state. 2.3 Conservation of magnitude In te Landau-Lifsitz Equation, te magnitude of te magnetization m is conserved. Indeed, a scalar multiplication of (2) and using te relation < a b, a >= gives < t m, m >= 1 2 t m 2 =. Consequently, m = C. Tis is an important conservation property of te Landau- Lifsitz Equation. In general, te magnetization will be ten usually been in (, T ) Ω S 2, were S 2 = {x R 3 x = 1}.

8 2 THE LANDAU-LIFSHITZ EQUATION Gilbert form of te Landau-Lifsit equation Te construction of te numerical sceme of tis report is based on a reformulation of (2) by T.L.Gilbert. Taking te cross product of (2) wit m, we obtain m t m = [ αm (m T )] m + (m T ) m, Using tat m 2 = 1 and te proposition (6.1) we ave Consequently (m T ) m =< m, m > T < m, T > m = T < m, T > m. m t = m T + α T α < m, T > m, and taking again te cross product wit m, te last equation reduces to We finally ave m t m = (m T ) m + α T m. m (m T ) = m t m αm T, and inserting tis term in te equation (2) yields m t = m T αm (m T ) = m T α( m t m αm T ) = αm t m + (1 + α 2 )m T. Tis last assertation gives te following version of equation (2), known as te Landau-Lifsitz-Gilbert equation (written LLG equation) : Tis form will be used trougout te report. 2.5 Magnetostriction m t + αm m t = (1 + α 2 )m T. (5) Magnetostriction is a property of ferromagnetic materials tat cause tem to cange teir sape or dimension during te process of magnetization. To include magnetostriction effects in te LLG equation, we need some expression of magnetomecanical energy. We define te stress tensor by σ = λ e ε e (u, m), σ ij = kl λ e ijklε e kl, (6) were ε e (u, m) = ε(u) ε m (m). Te vector u is te displacement and te total strain tensor ε is written ε(u) = 1 2 ( u + u T ). Te magnetic part of te total strain is given by ε m (m) = λ m mm T.

9 2 THE LANDAU-LIFSHITZ EQUATION 9 Te tensor λ e and λ m, wic describes te properties of te materials, are supposed to be symmetric: λ ijkl = λ jikl = λ ijlk = λ klij and positive definite ijkl λ ijkl ξ ij ξ kl λ ij wit bounded elements. We can use all te precedent relation to write te contribution of te magnetostriction σ to te effective field in te LLG equation. Te magnetostrictive field σ as te form σ = λ m σm. wic means for eac s-component s = 1, 2, 3: σ,s = klr λ m klrsσ kl m r. ξ 2 ij, Te stress tensor σ and displacement vector u obey te equation of elastodynamics: ϱu tt σ = in Ω (, T ), were ϱ is te constant mass density independent of te deformation. Te initial conditions are: u(, x) = u (x) in Ω, (7) u t (, x) = v (x) in Ω, and te boundary conditions: u = on Γ D, σ ν = on Γ N. (8) 2.6 LLG equation wit excange and magnetostriction Te coupled problem is ten written: let Ω T = Ω (, T ), wit a Lipscitz boundary continuous Γ, wic consists in two non-overlapping parts Γ D and Γ N. For any T >, we searc for m : Ω T S 2 and u : Ω T R 3 suc tat { mt + αm m t = (1 + α 2 )(m ( m + σ )) u tt σ = (9) wit te initial conditions: m(, x) = m (x) in Ω, u(, x) = u (x) in Ω, u t (, x) = v (x) in Ω, (1) and te boundary conditions: m = on Ω, ν u = on Γ D, σ ν = on Γ N. (11)

10 2 THE LANDAU-LIFSHITZ EQUATION 1 We coose V = H 1 (Ω T, R 3 ) and V = {ϕ V, ϕ = on Γ D }. Using te proposition (6.15), te LLG equation from (17) can be written: m t + αm m t = (1 + α 2 )( (m m) + m σ ). We multiply ten by a function φ C (Ω T ; R 3 ) and we integrate over Ω T : < m t, φ > dxdt + α < m m t, φ > dxdt Ω T Ω T = (1 + α 2 ) < (m m) + m σ, φ > dxdt. Ω T Using te integration by part for te term on te rigt, we obtain te weak formulation for te LLG equation: < m t, φ > dxdt + α < m m t, φ > dxdt Ω T Ω T = (1 + α 2 ) < m m, φ > dxdt Ω (1 + α 2 ) < m σ, φ > dxdt φ C (Ω T ; R 3 ). Ω T For te mecanical part, we multiply te elastodynamics equation by a vector field in te space C T (Ω T ; R 3 ) = {ϕ C (Ω T ; R 3 ) ϕ(t ) = }. We will see tat tis condition is necessary to pass te limit for te convergence of te algoritm. Ten we integrate over Ω to obtain: (u tt, ϕ) = ( σ, ϕ). Integrated te left term by part and using te boundary conditions (11) we obtain (u tt, ϕ) = (σ, ϕ). We develop ten, using te symmetry of σ and ε: < σ, ϕ > = < σ ij, ϕ j > x ij i = 1 < σ ij + σ ji, ϕ j > + < σ ji + σ ij, ϕ i > 4 x ij i x ij j = 1 < σ ij + σ ji, ε ij (ϕ) > 2 = 1 2 ij ij kl < (λ e ijkl + λ e jikl)ε e kl, ε ij (ϕ) > = < λ e ijkl(ε kl ε m kl), ε ij (ϕ) > ij kl = < λ e (ε m ε), ε(ϕ) >.

11 2 THE LANDAU-LIFSHITZ EQUATION 11 T ϕ C (Ω ; R 3 ). We integrate over Ω T te precedent relation: Ω < u tt, ϕ > dxdt Ω + < λ e ε(u), ε(ϕ) > dxdt = < λ e ε m (m), ε(ϕ) > dxdt Ω T T T Using te integration by part, we obtain te weak formulation of te elastodynamics part: < λ e ε(u), ε(ϕ) > dxdt < u t, ϕ t > dxdt Ω T + Ω = < λ e ε m (m), ε(ϕ) > dxdt Ω + < u t (), ϕ t () > dx ϕ CT (Ω T ; R 3 ). Ω Te variational formulation of te problem (17) motivates te following definition: Definition 2.1 (Weak Solution of te LLG equation wit magnetostriction) Let m V suc tat m = 1, u V and v L 2 (Ω T ). A couple of functions (m, u) is called a weak solution of te LLG equation wit magnetostriction if for all T > it verifies: 1. m V, u V and m = 1 a.e in Ω T ; 2. for all φ C (Ω T ; R 3 ), ϕ CT (Ω T ; R 3 ) tere olds < m t, φ > dxdt + α < m m t, φ > dxdt Ω T = (1 + α 2 ) Ω T < m m, φ > dxdt Ω T (1 + α 2 ) < m σ, φ > dxdt Ω T and < u t, ϕ t > dxdt + Ω T 3. m() = m in te sense of te trace; = + Ω T Ω T Ω < λ e ε(u), ε(ϕ) > dxdt < λ e ε m (m), ε(ϕ) > dxdt < u t (), ϕ t () > dx; 4. for almost all T (, T ) we ave te following inegality of energy: 1 m(t ) 2 dx + 1 ε(u(t )) 2 dx + α 2 Ω 2 Ω 1 + α 2 m t 2 dxdt Ω T + 1 u t (T ) 2 dx C(m, u, v ). 2 Ω Remark 2.2 Te teorem 3.2 below will prove te existence of a couple of weak solution (m,u) tat solve te LLG equation wit excange and magnetrostriction (17).

12 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 12 3 An implicit metod wit magnetostriction From te difficult task of te conservation of magnitude and te ard non-linearity in te LLG equation, a lot of different numerical metods were developed during years. To preserve te lengt of te magnetization, projection metods are currently used, wit explicit Euler approximation in time ([1], [9], [7]). In [5] is presented an explicit metod wit energy inequality, good for small two dimensional problems, but wic fails to be useful for more complicated simulation due to very ard timestep restriction. In [4] a norm conservative implicit finite element sceme is presented for te LLG equation wit excange field. Te metod from tis paper is better and more stable tan te explicit sceme of [5]. Te autors of [4] motive also te use of reduced integration (mass-lumped integral) in te numerical sceme, tat allows better stability of te algoritm wen te damping factor α. Te goal of tis capter is to extend te algoritm developed in [4] to te case of te LLG equation wit te excange and magnetostriction fields. 3.1 Preliminaries In tis section we consider a regular and quasi-uniform triangulation T. We define te lowest order finite element space V H 1 (Ω; R 3 ) by: V = {φ C( Ω; R 3 ) : φ K P 1 (K; R 3 ) K T }, were P 1 (K; R 3 ) is te space of polynome of degree less or equal to one. We consider moroever in addition V, = {ϕ V ; ϕ = on Γ D }. Definition 3.1 (Interpolation Operator) For any l L, we denote by ϕ l C( Ω) te basis function wic are affine and satisfies ϕ l (x m ) = 1 if m = l and oterwise. Given te set of node of te triangulation {x l : l L}., we note te local interpolate operator I : C( Ω; R 3 ) V, suc tat I (φ (x l ) = φ (x l ). Tis can be written: I (φ) = l L φ(x l )ϕ l. Definition 3.2 (Discrete inner product of functions) We define te discrete inner product (, ) : H 1 (Ω; R 3 ) H 1 (Ω; R 3 ) V, by (φ, ξ) = I (< φ, ξ >) = < φ(x l ), ξ(x j ) > I (ϕ l ϕ j ) Ω Ω j,l L = < φ(x l ), ξ(x j ) > δ jl ϕ j Ω j,l L = B l < φ(x l ), ξ(x l ) >, l L wit β l = Ω ϕ ldx. Definition 3.3 (Discrete Laplacian) Te discrete version of te Laplacian : H 1 (Ω; R 3 ) V is given by ( φ, ϕ ) = ( φ, ϕ ) ϕ V. (12)

13 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 13 Definition 3.4 (L 2 projection on V ) We define te L 2 projector P V : L 2 (Ω T ) V by (P V u, φ ) = (u, φ ) φ V. Te next standard results are useful to sow te convergence of te algoritm: Proposition 3.5 For all φ V, tere olds (Id I )φ L 2 C 2 φ 2 L 2 (K) K T Proposition 3.6 For a regular and quasi-uniform triangulation T, tere exists a constant c 1 suc tat for all φ V te following estimation old: φ L 2 c 1 1 φ L 2, were is te maximal mes side, = max{diam(k) : K T }. Proposition 3.7 (Equivalence of norms) For all φ V, φ 2 L 2 φ 2 5 φ 2 L 2. Proposition 3.8 For all φ V, φ c 1 1 φ L 2. Proof. We coose ϕ = φ in (12). Using ten te Holder inequality, te proposition (3.6) and (3.7), we obtain: φ 2 = ( φ, φ ) φ L 2 φ ) L 2 c 1 1 φ L 2 φ Proposition 3.9 For all φ V, φ (x l ) c 2 2 φ L l L. Proof. From te definition of te discrete inner product, we ave ( φ, ϕ l φ (x l )) = k L β l < φ (x k ), ϕ l (x k ) φ (x l ) >= β l φ (x l ) 2. Consequently, φ (x l ) 2 = βl 1 < φ, ϕ l (x k ) φ (x l ) > = βl 1 < φ, (ϕ l φ (x l )) > = βl 1 < φ (x m ), φ (x l ) > ( ϕ m, ϕ l ) m L K T,x m,x l K β 1 l φ (x l ) m L K T,x m,x l K φ (x m ) ( ϕ m, ϕ l ).

14 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 14 On te oter and, ϕ m L 2 cβ 1 2 l for all m L, wic gives: ( ϕ m, ϕ l ) ϕ m L 2 ϕ l L 2 c 2 β l. Finally, using tat te cardinality of te set {m L : K T, x m, x l T } is bounded independently, we can conclude tat φ (x l ) 2 C 2 φ φ (x l ). Definition 3.1 For any T >, te time is divided [, T ] into N equidistant subintervals [t j, t j+1 ] wit t j = jk, i =,..., N 1 and wit te time step k = T N. For an approximative solution f, we denote f j+1 = f (t j+1 ), d t f j+1 = fj+1 f j, d 2 t f j+1 = d tf j+1 d t f j. k k Algoritm 3.11 Let m V, u, v V,. Given a time step k >, j, m j V, u j, d tu j V,. 1. Determine m j+1 V from (d t m j+1, φ ) + α(m j d tm j+1, φ ) = (1 + α 2 )( m j+ 1 2 ( m j P V j+ 1 2 σ, φ ) φ V, wit j+ 1 2 σ = λ m λ e (ε(u j ) εm (m j )) mj+ 1 2 ; 2. Determine u j+1 V, from (d 2 t u j+1, ϕ ) + (λ e ε(u j+1 ), ε(ϕ )) = (λ e ε m (m j+1 ), ε(ϕ )) ϕ V, ; (13) 3. set j = j + 1 and return to 1. Remark 3.12 Te second term in te LLG equation is motivated by te relation (m j d tm j+1, φ) = 3.2 Convergence of te Algoritm (( ) m j+ 1 k 2 2 d tm j+1 = ( m j+ 1 2 dt m j+1, φ). ) d t m j+1, φ Proposition 3.13 Suppose tat m (x l) = 1 for all l L. Ten te sequences (m j, uj ) j obtained from algoritm (22) satisfies for all j : 1. m j+1 (x l ) = 1 l L;

15 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION d t m j+1 2 L 2 + α 1 + α 2 d tm j+1 2 = (d t m j+1, j+ 1 2 σ ). Proof. For te proof of te first assertion, we coose φ = ϕ t m j+ 1 2 (x l ) V in te LLG equation, for all l L. Using ten te relation (3.12) : = (d t m j+1, ϕ t m j+ 1 2 (x l )) = = 1 ( ) m j+1 m j 2k, ϕ tm j+1 (x l ) + ϕ t m j (x l) = β l 2k < mj+1 (x l ) m j (x l), m j+1 (x l ) + m j (x l) > = β ( l m j+1 (x l ) 2 m j 2k (x l) 2), and tus m j+1 (x l ) 2 = m j (x l) 2 = 1. In order to verify te second assertion, we first coose φ = m j+ 1 2 P V j+ 1 2 σ V in te LLG part of te algoritm (3.11). We obtain wit (3.12) and te definition of : (d t m j+1, m j+ 1 2 P j+ 1 2 V σ ) + α(m j d tm j+1, m j+ 1 2 P j+ 1 2 V = ( d t m j+1, m j+ 1 2 ) (d t m j+1, P j+ 1 2 V + α( m j+ 1 2 d t m j+1, m j+ 1 2 P j+ 1 2 V = 1 2 d t m j+1 2 L 2 (d t m j+1, P V j+ 1 2 σ ) α( m j+ 1 2 d t m j+1, m j P V j+ 1 2 σ ) =. On te oter and, taking φ = d t m j+1 α 1 + α 2 d tm j+1 2 = σ σ ) ) in (3.11) yields α 1 + α 2 (d tm j+1, d t m j+1 = α( m j+ 1 2 ( m j P V j+ 1 2 σ ), d t m j+1 ). Finally, te relation (a b, c) = (a c, b) and te definition of te projection operator P V conclude tat 1 2 d t m j+1 2 L 2 + α 1 + α 2 d tm j+1 2 = (d t m j+1, P j+ 1 2 V σ ) = (d t m j+1, j+ 1 2 σ ). ) σ )

16 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 16 Proposition 3.14 Te following estimate olds for te solution u : 1 2 d tu N 2 L ε(un ) 2 L j=1 ( ε(u j+1 ) ε(u j ) 2 L 2 + d tu j+1 d t u j 2 L 2 ) C + (λ e ε m (m N ), ε(u N )) (λ e ε m (m 1 ), ε(u )) k j= Proof. From te elastodynamics equation: (λ e d t ε m (m j+1 ), ε(u j )). (d 2 t u j+1, ϕ ) + (λ e ε(u j+1 ), ε(ϕ )) = (λ e ε m (m j+1 ), ε(ϕ )), we coose φ = u j+1 u j and sum over j: j= (d 2 t u j+1, u j+1 u j N 1 ) + (λ e ε(u j+1 ), ε(u j+1 u j )) = j= j= (λ e ε m (m j ), ε(uj+1 u j )). We develop te first identity wit te Abel summation and we obtain : j= (d 2 t u j+1, u j+1 u j N 1 ) = (d t u j+1 d t u j, d tu j+1 j= = 1 2 d tu N 2 L d tu 2 L We do te same for te second sum j= j=1 (λ e ε(u j+1 ), ε(u j+1 u j )). ) d t u j+1 d t u j 2 L 2 and combining te two precedent identities wit te boundeness and positive definition of λ e, we finally ave 1 2 d tu N 2 L ε(un ) 2 L ( ) ε(u j+1 ) ε(u j 2 ) 2 L 2 + (d tu j+1 d t u j ) 2 L 2 j=1 1 2 d tu 2 L λe ε(u ) 2 L 2 }{{} =C + j= (λ e ε m (m j+1 ), ε(u j+1 u j )).

17 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 17 Moroever, te rigt and side can be written: j= (λ e ε m (m j+1 ), ε(u j+1 u j )) = (λ e ε m (m N ), ε(u N )) (λ e ε m (m 1 ), ε(u )) j= (λ e (ε m (m j+1 ) ε m (m j )), ε(uj )) = (λ e ε m (m N ), ε(u N )) (λ e ε m (m 1 ), ε(u )) wic concludes te proof. k j= (λ e d t ε m (m j+1 ), ε(u j )), Lemma 3.15 Tere exists η i >, i = 1,..., N and η > sufficiently small suc tat ( 1 α 2 mn 2 L 2 + ) 1 + α 2 C max η i }{{} > + 1 ( ) 1 2 d tu N 2 L η } {{ } > j=1 d t m j+1 2 k ε(u N ) 2 L 2 C m 2 L v 2 L ε(u ) 2 L 2. Proof. We sum te assertions from (3.13) and (3.14) and we integrate over time. Wit we ave : A = 1 2 j=1 d t m j+1 2 L 2k + α 1 + α d tm N 2 L ε(mn ) 2 L j=1 j=1 d t m j+1 2 k ( ε(u j+1 ) ε(u j ) 2 L 2 + d tu j+1 d t u j 2 L 2 ), A C + (d t m j+1, j+ 1 2 σ )k k j= j=1 (λ e d t ε m (m j+1 ), ε(u j )) + (λ e ε m (m N ), ε(u N )) (λ e ε m (m 1 ), ε(u )). Using ten te Caucy s and Young inequalities, we obtain for any positive η and

18 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 18 η j, j = 1,..., N: A C j= j=1 η N d t m j+1 2 L 2k j=1 η j (λ e dt(ε m (m j+1 )) 2 L 2k C ηn j+ 1 2 σ 2 L 2k j= C ηj ε(u j )) 2 L 2k η ε(un ) 2 L C η (λ e ε m (m N ) 2 L 2 + C λe ε m (m 1 ) 2 L 2 + C ε(u ) 2 L 2. Coosing η sufficiently small, using te boundeness of te coefficients λ e and te conservation of te magnitude we ave A 1 2 η ε(un ) 2 L 2 C N 1 2 C m i = 1 i =,..., N, (14) j= j=1 η N d t m j+1 2 L 2k j=1 η j d t (ε m (m j+1 )) 2 L 2k C η1 j+ 1 2 σ 2 L 2k j= C ηj ε(u j ) 2 L 2k. From te definition of ε m, te boundeness of te tensor λ m and te proposition (3.13) we ave on one and and on te oter and Consequently, dt(ε m (m j+1 )) 2 L 2 C dtmj+1 2 L 2 j+ 1 2 σ 2 L 2 = λm λ e (ε(u j ) εm (m j )) mj L 2 C + C ε(uj ) 2 L 2. N 1 A 1 2 η ε(un ) 2 L 2 C + C + C 1 2 max η i }{{} i=,n j=1 ε(u j N 1 ) 2 L 2k + j=1 d t m j+1 j= ε(u j ) 2 L 2k N 1 2 L 2k + C j= d t m j+1 2 L 2k. Rassembling te precedent terms, we obtain A 1 N 1 2 η ε(un ) 2 L 2 C + ε(u ) 2 L 2 + 2C + 2C 1 2 max η i }{{} i=,n j=1 j=1 d t m j+1 2 L 2k, ε(u j ) 2 L 2k

19 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 19 and ten A 1 2 η ε(un ) 2 L 2 2C 1 N 1 2 max η i d }{{} t m j+1 N 1 2 L 2k C + C ε(u j ) 2 L 2k. j=1 j= i=,n Finally, from te fact tat N 1 ε(u N ) 2 L 2 C + te discrete Gronwall inequality give j= N 1 ε(u N ) 2 L 2 C + exp Nk ε(u j ) 2 L 2k, j= Nk C. To conclude, if we coose η >, max η i > sufficiently small suc tat ten we ave C max η i < α 1 + α 2, η < 1 2, N j=1 ( ) d t m j+1 α 2 L 2k α 2 C max η i j=1 d t m j+1 2 k d tu N 2 L 2 + ( 1 2 η ) ε(u N ) 2 L 2 C. Moroever, we remark tat we can develop te first integral: j=1 to finally obtain te relation: N 1 d t M + 2 L 2k = j= m j+1 2 L 2 m j 2 L 2 k = m N 2 L 2 m 2 L 2, ( ) 1 α 2 mn 2 L α 2 C max η i d t m j+1 2 k j= d tu N 2 L 2 + ( 1 2 η ) ε(u N ) 2 L 2 C m 2 L v 2 L ε(u ) 2 L 2. k

20 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 2 Definition 3.16 For x Ω, t [t j, t j+1 ) we define M(t, x) := t t j k mj+1 (x) + t j+1 t m j k (x); M (t, x) := m j (x), M+ (t, x) := m j+1 j+ (x), M(t, x) := m 1 2 (x); U(t, x) := t t j k uj+1 (x) + t j+1 t u j k (x); V(t, x) := t t j k d tu j+1 (x) + t j+1 t d t u j k (x); U (t, x) := u j (x), U+ (t, x) := u j+1 (x), Ū(t, x) := ū j+ 1 2 (x); V (t, x) := d t u j (x), V+ (t, x) := d t u j+1 (x), V(t, x) := [dt ū ] j+ 1 2 (x) and H σ (t, x) = λ m λ e (ε(u ) ε m (M )) M. All tese functions ave te property to be very close eac oter wen k becomes small. Tis is illustrated by te following proposition: Proposition 3.17 For all t [t j, t j+1 ) we ave wen k, M M L 2, U U L 2, V V L 2. Similar results are available for M +, U +, V +. Proof. By te lemma (6.19) and (6.18), tere exists a constant C > suc tat M t L 2 C, U t L 2 C, N V + V L 2 C. j= Consequently, using te definition of M and M yields M M 2 L 2 On te oter and, we ave for V: Te proof is similar for U. V V 2 L 2 k 2 M + M k 2 L 2 N k 3 M t 2 L 2 Ck3. j= k 2 V + V k 2 L 2 N k V + V L 2 Ck. j= Te next tool is an important result from Sobolev Spaces teory:

21 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 21 Proposition 3.18 Let M ν be a sequence of vector functions. If tere exists a constant C > suc tat M ν H 1 (Ω T ;R 3 ) C, ten tere exists a subsequence M νi and a limit m H 1 (Ω T ; R 3 ) suc tat M νi m in H 1 (Ω T ; R 3 ). Proposition 3.19 Let u, v be two sequences in V. Let u, v V. Suppose tat Ten 1. lim (u, v ) (u, v ) = ; 2. u u in L 2 (Ω T ); 3. v v in L 2 (Ω T ). lim (u, v ) dt = (u, v)dt. k, Proof. A triangle inequality gives, (u, v ) (u, v)dt (u, v ) (u, v ) dt + (u, v ) (u, v) dt By te dominated convergence teorem (u, v ) (u, v ) dt wen k,. For te second integral, we use te relation ab cd = 1 ((a c)(b + d) + (a + c)(b d)) 2 to obtain: (u, v ) (u, v) dt (u u, v + v) dt + (u + u, v v) dt (u u, v + v) dt + u + u L 2 v v) L 2. }{{} }{{}, because u u in L 2 Teorem 3.2 Suppose tat m (x l) = 1 for all l L, and let (m j, uj ) j solve algoritm (3.11). Suppose tat m m in H 1 (Ω, R 3 ) and u u in V for. For k, tere exists m H 1 (Ω T ; R 3 ), u V suc tat M subconverges to m in H 1 (Ω t, R 3 ) and U subconverges to u in V. Moreover, m and u are weak solutions of te LLG equation wit magnetostriction.

22 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 22 Proof. From te lemma (6.19), we obtain N mn 2 L 2 + d t m j+1 2 k j= d tu N 2 L 2 + ε(un ) 2 L 2 C( m 2 L v 2 L ε(u )) 2 L 2. Consequently, using te definition (3.16) tis inequality can be written for all T > 1 T 2 M+ (T ) 2 L 2 + M t U t(t ) 2 L 2 + ε(u+ (T ) 2 L 2 C m 2 L v 2 L ε(u 2 L 2. Tis bound te proposition (3.18) and (3.17) and te Korn s inequality yields existence of a couple (m, u) [L (, T, H 1 (Ω, S 2 )) H 1 (Ω T ; R 3 )] V, wic is te weak limit (as k, ) of a subsequence {(M, U)} k, suc tat (M; U) (m; u) in (H 1 (Ω T ; R 3 ); V ); ( M, M +, M, M; U, U +, U, Ū) ( m, u) in L2 (Ω T, R 3 ). Moreover, by te teorem of Rellic Kondracov, H 1 (Ω T ; R 3 ) L 2 (Ω T, R 3 ) wit compact embedding. It means tat tere exists subsequence suc tat (M, M +, M, M) m in L 2 (Ω T, R 3 ), (U, U +, U, Ū) u in L2 (Ω T, R 3 ). On te oter and, te proposition (3.14) gives te estimate and ten 1 2 d tu N 2 L 2 C 1 2 V+ (T ) 2 L 2 C. Tis bound yields te existence of a limit u t in L 2 (Ω T ; R 3 ) and a subsequence V suc tat V u t in L 2 (Ω T, R 3 ). We now verify tat te limit (m, u) is a weak solution of te LLG equation wit magnetostriction; we verify tat it satisfies te four points of te definition (2.1):

23 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION From te proposition (3.13) M (t, x l ) = 1 for every l L and almost all t (, T ). Consequently, using standard finite element and Caucy-Scwarz inequalities, tere olds for every K T M 2 1 L 2 (K) C ( M 2 1) L 2 (K) = C 2( M )M L 2 (K) 2C M L 2(K), wic means tat M 1 in L 2 (K), and consequently m = 1 a.e in Ω T. 2. We recall te algoritm (3.11) (a) Determine m j+1 V from (d t m j+1, φ ) + α(m j d tm j+1, φ ) wit (b) Determine u j+1 = (1 + α 2 )( m j+ 1 2 ( m j P V j+ 1 2 σ, φ ) φ V j+ 1 2 σ = λ m λ e (ε(u j ) εm (m j )) mj+ 1 2 ; V, from (d 2 t u j+1, ϕ ) + (λ e ε(u j+1 ), ε(ϕ )) = (λ e ε m (m j+1 ), ε(ϕ )) ϕ V, ; (c) set j = j + 1 and return to 1. For te LLG equation, using te definition of te subsequences, integrate on (, T ) and taking φ = I φ(t, ) for φ C (Ω T ; R 3 ) goes to (M t, φ ) + α (M M t, φ ) = (1 + α 2 ) ( M M, φ ) We need to sow tat wen, k, we ave + (1 + α 2 ) ( M P V H, φ ). (m t, φ)dt + α (m m t, φ) = (1 + α 2 ) (m m, φ) + (1 + α 2 ) Te effect of te reduced integral is controlled by te relation (ξ, ν ) (ξ, ν ) C ξ L 2 ν L 2. Consequently, for almost all t (, T ) we ave (M t, φ ) (M t, φ ) C M t L 2 φ L 2. (m σ, φ). Moroever, since M m in H 1 (Ω T ), we ave tat M t m t in L 2 (Ω T ). On te oter and φ φ L 2 (Ω T ) = (I I )φ(t, ) L 2 C φ(t, ) L 2,

24 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 24 wic means tat φ φ in L 2 (Ω t, R 3 ). We use te proposition (3.19) to conclude tat lim (M t, φ ) dt = (m t, φ)dt. k, For te second integral, we ave by standard estimate (M t, M φ ) (M t, M φ ) C M t L 2 ( M φ ) L 2. Using ten tat M φ m φ in L 2 (Ω T, R 3 ), tis yields again wit te precedent argument lim ( M M t, φ ) dt = (m m t, φ)dt. k, For te tird term of te equation, we write: ( M M, φ ) = ( M φ, M) = (Id I )( M φ, M) ) (I Id)( M φ, M) ( M φ, M) = (Id I )( M φ, M) + ( ((I Id)( M φ )), M) + ( ( M φ ), M) = I + II + III. We use for I te bound from te proposition (3.8) I = (Id I )( M φ, M) (Id I )( M φ L 2 M) L 2 C 1 (Id I )( M φ ) L 2 M L 2. On te oter and, te proposition (3.9) gives (Id I )( M φ ) L 2 C 2 We can ten easily conclude tat K T D 2 ( M φ ) L 2(K) C 2 M L 2 φ ) L. lim I =.,k Te same argument allows to sow tat For III, te proposition (6.13) gives II C M 2 L 2 φ ) L. III = ( ( M φ ), M) = ( M φ, M). Using one more time te proposition (3.19) we obtain lim ( M M, φ ) = (m φ, m)dt = (m m, φ)dt. k,

25 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 25 For te last term, we ave by definition of P V, ( M P V H, φ ) = ( M φ, P V H) = (I ( M φ ), P V H) = (I ( M φ ), H σ ) = (I I )( M φ, H σ ) ( M φ, H σ ). Linearity of te limit sows tat ε(u ) ε(u) in L 2 (Ω T, R 3 ) and ten Finally, using H(t, x) = λ m λ e (ε(u ) ε m (M )) M (I I )( M φ, H σ ) L 2 C we obtain lim k, λ m λ e (ε(u) ε m (m))m = σ. ( M P V H, φ ) dt = lim K T ( M φ ) L 2 (K) H σ L 2 (K), = = k, ( M φ, H σ ) ( m φ, σ ) (m σ, φ). For te elastodynamics equation, we coose ϕ = I ϕ(t, ) for ϕ C T (Ω T ; R 3 ). Using te definition of te subsequences and integrate on (, T ) goes to ( t V, ϕ ) + (λ e ε(u + ), ε(ϕ )) = (λ m ε m (M + ), ε(ϕ )). (15) We need to prove tat (15) converge wen k, to (u t, ϕ t ) + = (λ e ε(u), ε(ϕ)) (λ e ε e (m), ε(ϕ)) + (u t (), ϕ t ()). We integrate by part te first integral of (15), and using ϕ (T ) = gives: ( t V, ϕ ) = (V, ϕ ) T (V, t ϕ ) = (V(), ϕ ()) (V, t ϕ ) We ave also seen tat tere exists a subsequence V suc tat V u t in L 2 (Ω T, R 3 ). By ypotesis, u u in V and ten lim (V(), ϕ ()) = (u t (), ϕ()) k,

26 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 26 Te proposition (3.19) also concludes tat lim k, (V, t ϕ ) = (u t, t ϕ). From U + u in L 2 (Ω T, R 3 ) we ave by linearity of te limit On te oter and, ε(u + ) ε(u) in L 2 (Ω T, R 3 ). ε(ϕ ) ε(ϕ) L 2 C 1 ϕ ϕ L 2 = C 1 (I I)ϕ L 2 C 1 2 K T D 2 (ϕ) L 2 (K). and consequently ε(ϕ ) ε(ϕ) in L 2 (Ω T, R 3 ). We apply again te proposition (3.19) to obtain Te fact tat lim k, allows to sow tat Following te same argument: lim k, (λ e ε(u + ), ε(ϕ )) = M m in L 2 (Ω T, R 3 ) (λ e ε(u), ε(ϕ)). ε m (M ) ε m (m) in L 2 (Ω T, R 3 ). (λ m ε m (M ), ε(ϕ )) = (λ m ε m (m), ε(ϕ)). Altogeter, we obtain tat te couple (m, u) verifies te weak LLG equation wit magnetostriction. 3. By ypotesis, m m in H 1 (Ω T ; R 3 ). By te continuity of te trace operator, we ave m() = m in te sense of traces. 4. To prove te energy inequality, remember tat te norm in H 1 (Ω T ; R 3 ) is lower semi-continuous, wic means tat for all sequence suc tat a ν a in H 1 (Ω T ; R 3 ), we ave lim a ν ν H 1 a H 1. Consequently, we ave lim 1 k, 2 M+ (T ) 2 L 2 α 1 + α 2 M t U t(t ) 2 L 2 + ε(u+ (T ) 2 L 2 + C + m 2 L v 2 L ε(u 2 L m(t ) 2 L 2 α T 1 + α 2 m t u t(t ) 2 L 2 + ( 1 2 η ) ε(u(t ) 2 L 2 + C(1 + m 2 L v 2 L ε(u 2 L2),

27 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 27 wic concludes te proof. 3.3 Solving te non-linear system We employ a fixed-point iteration to solve te non linear system in te step 1 of te algoritm (3.11). Given m j V and u j V, for j, we compute m j+1 by intermediate m j+1,l+1 for some l N. We coose a condition suc tat after a certain number of iteration, m j+1,l+1 m j+1 sufficiently precisely. To simplify te algoritm, instead of computing m j+1,l+1 we compute w j+1,l+1 = mj+1,l+1 + m j m j It means tat m j+1,l+1 = 2w j+1,l+1 m j. We write also l σ = λ m λ e (ε(u j ) εm (m j ))wl Algoritm 3.21 We set m j+1, := m j and l =. Given uj V 1. compute (w l+1 V suc tat for all φ V tere olds 2 k (wl+1 (1 + α 2 )(w l+1, φ ) + 2α k (mj wl+1, φ ) = 2 k (mj, φ ) φ V ; ( w l + P V l σ ), φ ) 2. If (w l+1 w l )+P V ( l σ l 1 σ ) ε, ten stop and set m j+1 := 2w l+1 m j ; 3. Set l := l + 1 and go to 1. Te following proposition sows tat te iteration converge, provided tat ε = and k = O( 2 ). Proposition 3.22 Suppose tere exists a constant C > suc tat m j (x m) C, u j (x l) C for all m L. Ten for all l, tere exists a unique solution w j+1,l+1 to (3.21). For all l 1 tere olds w l+1 w l Θ w l w l 1, provided tat Θ = k(1 + α 2 )C(c C σ ) < 1. Moreover, for all l and all φ V, we ave (d t m j+1, φ ) + α(m j d tm j+1, φ ) (1 + α 2 )( m j+ 1 2 ( m j P V j+ 1 2 σ, φ ) (1 + α 2 )( m j+ 1 2 R j, φ ),

28 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 28 wit R j ε. By te Banac fixed-point teorem, te above contraction property implies for m j (x m) = 1 and for all m L te existence of a unique m j+1, V wic solves algoritm (3.11) and satisfies lemma (3.13). Proof. We coose φ = w l+1 V. We obtain for te left and-side: 1 k wl+1 2, wic means tat te bilinear form formed by te left and side is positive definite. Tus, te algoritm admits a unique solution. We control now te L norm : coose m L suc tat w l+1 L = wl+1 (x m). We take ten φ = ϕ m w l+1 (x m) in (3.21) and it gives 1 k wl+1 (x m) 2 = 2 k (mj, ϕ mw l+1 Consequently, (x m)) 2 k mj ϕ mw l+1 w l+1 L C. (x m) C k wl+1 (x m) Substraction of two subsequent equations in te fixed-point iteration yields 2 k (wl+1 w l, φ ) + 2α k (mj (wl+1 (1 + α 2 )((w l+1 ( w l + P V l σ, φ ) + (1 + α 2 )((w l ( w l 1 + P V l 1 σ, φ ) = 2 k (wl+1 w l, φ ) + 2α k (mj (wl+1 (1 + α 2 )((w l+1 ( w l + P V l σ, φ ) + (1 + α 2 )(w l ( w l + P V l σ, φ ) (1 + α 2 )(w l ( w l + P V l σ, φ ) + (1 + α 2 )((w l ( w l 1 = 2 k (wl+1 (1 + α 2 )((w l+1 =. + P V l 1 σ, φ ) w l, φ ) + 2α k (mj (wl+1 (1 + α 2 )(w l ( (w l w l 1 w l ), φ ) w l ), φ ) w l ), φ ) w l ) ( w l + P V l σ, φ ) ) + P V l σ P V l 1 σ, φ ) Coosing φ = w l+1 w l in te precedent identity gives: w l k(1 + α 2 )C( (w l w l 1 ) + P V ( l σ l 1 σ ) ) k(1 + α 2 )C(c w l w l 1 + l σ l 1 σ ) k(1 + α 2 )C(c C σ ) w l w l 1, w l+1 wen we use tat l σ l 1 σ λ m λ e (ε(u j ) εm (m j )) wl w l 1 C σ w l w l 1.

29 3 AN IMPLICIT METHOD WITH MAGNETOSTRICTION 29 Consequently if we coose k < 1 (1 + α 2 )C(c C σ ), tere exists a unique solution of te algoritm (3.21) by te Banac fixed-point teorem. To connect te algoritms (3.11) and (3.21), we use te fact tat for all l Finally, were 2 k (wl+1 (1 + α 2 )(w l+1 = 2 k (mj, φ )., φ ) + 2α k (mj wl+1, φ ) (d t m j+1, φ ) + α(m j d tm j+1, φ ) ( w l + P V l σ, φ ) (1 + α 2 )( m j+ 1 2 ( m j P V j+ 1 2 σ, φ ) = (1 + α 2 )( m j+ 1 2 R j, φ ), R j = (w l+1 w l ) + P V ( l σ l 1 σ ). Ten, if R j L 2 < ε for ε sufficiently small, we ave also te convergence of te algoritm (3.11), wic acieves te proof. At eac iteration l of te algoritm (3.21), we need to solve a linear system to find w l+1. Since te matrix of te system is sparse, we can coose a direct metod suc UMFPACK (from LU decomposition). On more complicated problem in tree dimensions or wen as small values, we can use iterative metods like GMRES. For te elastodynamics equation, we generally took direct solvers. Remark 3.23 Wen we consider te LLG equation only wit magnetostriction, te time step restriction becomes: 1 k < (1 + α 2 )C(C σ ). In tis case te restriction is less important provided te tensors λ are reasonably small. It allows an important gain of time in te computation. Remark 3.24 We can easily extend te algoritm (3.11), te proof of te convergence and te nonlinear system wen we consider te magnetic field and te anisotropy. Indeed, te term (w l+1 ( + C < p, w l > p, φ ) present no difficulties in te precedent algoritm and te convergence is obvious. At worst, we sould maybe coose k a bit smaller, depending of te value of p and C.

30 4 NUMERICAL EXPERIMENTS 3 4 Numerical Experiments 4.1 LLG wit exact solution We test te accuracy of our implicit algoritm on a one dimensional example wit excange, magnetostriction and magnetic field. We coose te exact solutions sin(3t) m(x, t) = cos(x) cos(3t) sin(x) cos(3t) and u(x, t) = cos(x) cos(3t). Te oters parameters used were λ e = λ e 3333 = λm 3333 (, 1), T = 1 and (t) = (sin(.9t), cos(.9t), ). Consequently, te magnetostriction takes te form σ (x, t) =. u x m 3 m 3 3 = 1, α = 1, η = 1, Ω = Suc tat te above functions are te exact solutions, we need to add appropriate rigt-and sides, and te problem becomes in tis case: m t + αm m t = (1 + α 2 )(m ( m + σ + )) (1 + α 2 )(m (f + f σ + )) + f mt + α(m f mt ) u tt σ = f u (16) wit Diriclet boundary conditions and te rigt and-sides: 3 cos(3t) f = cos(x) cos(3t) f mt = 3 cos(x) sin(3t), sin(x) cos(3t) 3 sin(x) sin(3t) f σ =, sin(x) 2 cos(3t) 2 sin(x) 3 cos(3t) 3 f u = 8 cos(x) cos(3t) + 2 sin(x) cos(x) cos(3t) 2. Remark 4.1 Wit tese rigt and-sides, te algoritm will no longer conserve te norm of te magnetization because of te term ( m j+ 1 2 f mt ). Neverteless, for k sufficiently small, te magnitude stays very close to one.

31 4 NUMERICAL EXPERIMENTS 31 and In tables 1 3 below, we present te following errors for differents problem: Err(m) = max j Err(u) = max j m j m L 2 + max m j j m L 2, u j u L 2 + max u j j u L 2. In table 1, we can see te H 1 error wit different value of k and N. Te algoritm (3.11) is expected to ave at least a convergence of O() + O(k), but tis rate is not observable in tat example. Indeed, for a space discretization N fixed, te functions Err(m) and Err(u) do not decrease wen we diminis te time step k. For example, if N = 5, te error will stay around C 5, because of te relation m m j H1 C(k + ). Tis restriction comes from te small value of k we need to coose to solve te non-linear system (k = O( 2 ) from te proposition (3.22)). Tus, te convergence rate in tat example can only depend of te space discretization: wic is ten visible in table 1. m m j H 1 C(O(2 ) + ) C. In table 2, we solve only te linear wave equation so we do not ave restriction on k because te system is linear. Consequently, for a sufficiently small, te function Err(u) diminis wen we decrease k. Te functions Err(m) and Err(u) are sown in table 3 for te case wen we solve te LLG equation only wit te magnetostriction field. In tis case, te time step restriction does not depend of, consequently we can observe a linear decreasing of tese functions for sufficiently small k. Indeed, taking for example k =.1 and = 1/1 we ave Err(m)=.31 and Err(u)=.25. Coosing ten k =.1 and = 1/1, te values of Err(m) and Err(u) are divided by ten. Tese computations ave been done on my personal computer, resulting sometimes in a large CPU time. For N = 1 and k =.1, te time of computation is almost 6 [s]. For more complicated problem, development of parallel solver and computation on a cluster is ten igly recommended.

32 4 NUMERICAL EXPERIMENTS 32 k N u m u m u m N.A.N N.A.N N.A.N N.A.N N.A.N N.A.N.59.8 Table 1: Error for te LLG equation wit excange and magnetostriction N \ k Table 2: Error for te Wave equation k N m u m u m u m u Table 3: Error for te LLG equation wit magnetostriction

33 4 NUMERICAL EXPERIMENTS A Blow Up Example Introduction In tis section we discuss numerical experiments wic motivate finite-time blowup. Tis example as been developed in [4] and [5] for respectively te previous implicit metod witout magnetostriction and an explicit sceme. We extend te experiment wit te magnetostriction and study te impact of te tensor λ and te parameter α on te blow-up time Initial Data In [5], te initial data is cosen to be an armonic map tat solves m = m 2 m. Tis function is te static solution of te LLG equation and is crucial to te formation of singularity in te case of te LLG equation wit excange field. Te initial condition is given by: Ω = (.5,.5) 2 and m : Ω S 2 defined by were A := (1 2 x )4 s. We solve te following LLG problem: (,, 1) for x.5 m (x) = (2xA,A 2 x 2 (A 2 + x 2 for x.5, { mt + αm m t = γ(1 + α 2 )(m ( m + σ )), u tt σ =. (17) were γ is a given constant. For te magnetostriction, we took λ e and λ m 2 2 tensors suc tat λ e = λ e 1111 = λ e 2222 = C 1, λ m = λ m 1111 = λ m 2222 = C 2. for given constant C 1, C 2. Te initial condition for te elastodynamics equation are u =, v = and omogeneous Diriclet boundary conditions. Te triangulation T l used in te numerical simulation are defined troug a positive integer l and consists of 2 2l+1 alved square wit edge lengt = 2 l. Te restriction on k by te non-linear algoritm (3.12) motives to coose k = α 2. We set ε = 2 16 in (3.12) and te non-linear system terminates after at most 13 iterations. As discrete intial data we employ te nodal interpolation m = I Tl m.

34 4 NUMERICAL EXPERIMENTS LLG wit excange In tis example we solve te LLG equation wit only te excange field T = m: m t + αm m t = (1 + α 2 )(m ( m). (18) In figure 1 and 2 we display snapsots of te numerical approximation provided by algoritm (3.11) witout magnetostriction and wit α = 1, s = 1 and l = 4. Te plots display te first two components at various time. A zoom towards te central nodes is seen in figure 2. We observe tat at time t.529 te vector at te origin points in anoter direction tan all surrounding vectors, resulting in a large W 1, norm. Tis reveals tan in tis experiment, regularity of te exact solution cannot be expected. Figure 3 sows similar snapsots for α = 1/64, s = 1 and l = 4. In tis case te numerical solution is even less regular tan in te previous experiment and at time t =.3 te solution is still unsteady. Figure 1: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange (18), wit s = 1, α = 1, l = 4 at time t =,.119,.297,.529,.588,.6. Figure 2: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange (18), wit s = 1, α = 1, l = 4, at time t =,..529,.6.

35 4 NUMERICAL EXPERIMENTS 35 Figure 3: Snapsots of te Numerical Approximation of te magnetization m for te problem (18), wit s = 1, α = 1/64, l = 4, at time t =,.12,.297,.492,.687,.178,.1371,.1664,.254,.2347,.2738,.2999.

36 4 NUMERICAL EXPERIMENTS 36 For T = m and witout displacement, te lemma (6.19) is reduced to te following relation: 1 m(t, ) 2 α t + 2 Ω γ(1 + α 2 d t m j+1 2 = 1 m 2. ) 2 Ω Consequently, we expect tat te energy E(m, u, t) = 1 2 is a decreasing function of time because p(t) = Ω α t γ(1 + α 2 ) m(t, ) 2. d t m j+1 2 is a strictly non-decreasing function. We define also te W 1, semi-norm: m(t) 1, = m(t) L In figure 4 we sow te energy and te W 1, semi-norm for different values of. We observe tat te maximum value of m L is and wen becomes small, te blow-up time (wen te m L is maximal) seems to approac t =.3. Figure 4: Energy and W 1, semi-norm for te magnetization m from te problem (18), wit α = 1, s = 4 and for different values of. Beaviour of te blow-up time wen α as small value is displayed in figure 5. Te blow-up time seems to converge at t =.85 wen α. For α 1/256, te W 1, semi-norm oscillates and te energy decreases very slowly, meaning possible unstability. Finally, figure 6 sows tat te Blow-Up time decreases for larger value of s.

37 4 NUMERICAL EXPERIMENTS 37 Figure 5: Energy and W 1, semi-norm for te magnetization m from te problem (18), wit = 1/32, s = 4 and for different values of α. Figure 6: Energy and W 1, semi-norm for te magnetization m from te problem (18), wit = 1/32, α = 1 and for different values of s.

38 4 NUMERICAL EXPERIMENTS LLG wit magnetostriction We solve now te LLG equation wit magnetostriction: { mt + αm m t = γ(1 + α 2 )(m ( σ )), u tt σ =. (19) were γ = 4 and s = 1. For te magnetostriction, we took λ e and λ m 2 2 tensors suc tat λ e = λ e 1111 = λ e 2222 = 5, λ m = λ m 1111 = λ m 2222 = 5. Te initial conditions for te elastodynamics equation are u =, v = and omogeneous Diriclet boundary conditions. Snapsots of te magnetization m wit α = 1 are sown in figure. After t.5[s], we observe tat te vectors near te center converge to (,, 1) and te ones near te boundary to (,, 1). Figure 7: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit magnetostriction (19), wit α = 1, = 1/16, for time t =,.2, 1. Te displacement, sown in figure 8, spreads rapidly on te diagonal of te square, wit stronger component in u 1. It progressively vanises wen te system becomes steady. Figure 8: Isoclines of te numerical approximation of te displacement u 1 (t, ), u 2 (t, ) for te elastodynamic equation from (19), wit α = 1, = 1/16 and at time t =.25.

39 4 NUMERICAL EXPERIMENTS 39 In figure 9 and 1 we sow te evolution of te magnetization for α = 1/16. Te vector turns around teir nodes, diagonaly symmetric, and te steady state is only reaced after t = 5[s]. Te spread of te displacement (figure 11) is stronger due to te small value of α, and te vectors on te diagonal start to turn after t =.2[s] until te steady state is reaced. Figure 9: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit magnetostriction (19) wit α = 1/16, = 1/16, for time t =,.15,.1,.4, 1, 3. Figure 1: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit magnetostriction (19) wit α = 1/16, = 1/16, for time t =, 1, 3.

40 4 NUMERICAL EXPERIMENTS 4 Figure 11: Isoclines for te numerical approximation of te displacement u 1 (t, ), u 2 (t, ) for te elastodynamic equation from (19), wit α = 1/16, = 1/16 and at time t = LLG wit excange and magnetostriction We solve te LLG equation wit excange and magnetostriction: { mt + αm m t = γ(1 + α 2 )(m ( m + σ )), u tt σ =. (2) were γ = 4 and s = 1. We took λ e and λ m 2 2 diagonal tensors suc tat λ e ij = 4δ ij, λ m ij = 1δ ij. Tese tensors ave been cosen wit ig value in order to ave an impact on te blow-up time. On figure 12 we display snapsots of te numerical magnetization for α = 1. Te vectors turn around teir nodes wit a diagonal symmetry. In figure 13, we observe tat at time t.74, tat te vector at te origin point in anoter direction tan all surrounding vectors. It means tat te blow-up appened again wit te magnetostriction. Neverteless, te first component of te displacement is strong (figure 14) and consequently te vectors move after te blow-up time, and so te solution fails to become stationary for t.1.

41 4 NUMERICAL EXPERIMENTS 41 Figure 12: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange and magnetostriction (2), wit α = 1, = 1/16, for time t =,.15,.1,.6,.725,.1. Figure 13: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange and magnetostriction (2), wit α = 1, = 1/16, for time t =,.65,.7,.8.

42 4 NUMERICAL EXPERIMENTS 42 Figure 14: Isoclines of te numerical approximation of te displacement u 1 (t, ), u 2 (t, ) for te elastodynamic equation from (2), wit = 1/16, α = 1 and at time t =.1. In figure 15 we sow snapsots for te magnetization wit α = 1/4. Te solution is less regular tan for α = 1 and te blow-up time is smaller (figure 16). Te displacement as te same order and consequently te steady state is still not reaced for t.1. We define te following energies: E T (m, u, t) = 1 2 m (t) 2 L 2 + α t γ(1 + α 2 ) d tu (t) 2 L ε(u (t)) 2 L 2; d t m j+1 2 E(m, u, t) = 1 2 m (t) 2 L d tu (t) 2 L ε(u (t)) 2 L 2; = E m (u, t) + E ut (u, t) + E ux (u, t). By te lemma (6.19), E T (m, u, t) = E(m, u, t) + α t γ(1 + α 2 ) d t m j+1 2 C(λ e, λ m ) m 2 L v 2 L ε(u ) 2 L 2 We cannot expect tat te function E(m, u, t) σ decreases, because te constant C(λ e, λ m ) only depend of te tensors λ, wic ave ig value in our experiment. We sow tese different energies for λ e = 4, λ m = 1 in figure 17 (after rescaling). We note tat te total energy increases quickly and ten stabilizes around a constant of order 1 2 C(λ e, λ m ).

43 4 NUMERICAL EXPERIMENTS 43 Figure 15: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange and magnetostriction (2), wit α = 1/4, = 1/16, for time t =,.2,.5,.1,.2,.3,.1,.24,.41,.5,.6.

44 4 NUMERICAL EXPERIMENTS 44 Figure 16: Snapsots of te numerical approximation of te magnetization m for te LLG equation wit excange and magnetostriction (2), wit α = 1/4, = 1/16, for time t =,.2874,.4869, Figure 17: Te different energies for te magnetization m and te displacement u from te problem (2), wit = 1/16, λ e = 4, λ m = 1 and α = 1.

45 4 NUMERICAL EXPERIMENTS 45 Impact of te tensors λ on te Blow-Up time : On figure 18 and 19 we plot te energy and te W 1, semi-norm for different values of λ m respectively λ e. Te magnetostriction ave an impact on te excange field only for λ m λ e 3. For iger values, te Blow-Up time increases quickly and its existence is terefore not guarantee. Impact of α on te Blow-Up time : Finally, we sow on figure 2 te energy and te W 1, semi-norm for different values of α. Te more we decrease α, te more te Blow-Up time decreases. Te oscillations of te energy and W 1, semi-norm for α 1/2 mean tat te algoritm start to become unstable for too small value of α. Figure 18: Energy and W 1, semi-norm for te magnetization m and te displacement u from te problem (2), wit = 1/16, λ m = 1, α = 1 and for different values of λ e.

46 4 NUMERICAL EXPERIMENTS 46 Figure 19: Energy and W 1, semi-norm for te magnetization m and te displacement u from te problem (2), wit = 1/16, λ e = 4, α = 1 and for different values of λ m. Figure 2: Energy and W 1, semi-norm for te magnetization m and te displacement u from te problem (2), wit = 1/16, λ m = 1, λ e = 4 and different values of α.

47 4 NUMERICAL EXPERIMENTS A problem wit magnetrostriction We take Ω = (, 1) 2 and m : Ω 2S 2 be defined by exp 2 x x 2 m (x) = 2 (exp 2 x x 2 ) 2 wit x = (.5,.5). We solve te LLG equation wit magnetostriction: { mt + αm m t = m γ(1 + α 2 )m σ, u tt σ =. (21) wit γ = 4 and m = 2. For te magnetostriction, we took λ e and λ m 2 2 tensors suc tat λ e = λ e 1111 = λ e 2222 = 1, λ m = λ m 1111 = λ m 2222 = 1. For te elastodynamics equation we take omogeneous Diriclet condition wit u = and v =. Te triangulation T l used in te numerical simulation are defined troug a positive integer l and consists of 2 6l+1 alved square wit edge lengt = 2 6l. Te center of te triangulation as been refined. Te oters parameters used were α = 1, η = 4, ε = 1 16, k = 1 2 (weakly dependence wit ). Te initial condition of te magnetization is radially symmetric. Wen we compute te LLG equation wit te magnetostriction, te displacement evoluate from around te center and te symmetry is still conserved. Te magnetization, te displacement and te mes can be seen on figure 21 and 22 at time t = 1 and on figure 23 for t = and t =.5. Figure 21: Isoclines of te numerical approximation of te magnetization m 1 (t, ), m 2 (t, ) and m 3 (t, ) for te LLG equation wit magnetostriction (21), wit = 1/3, α = 1 and at time t = 1.

48 4 NUMERICAL EXPERIMENTS 48 Figure 22: Isocline of te numerical approximation of te displacement u 1 (t, ), u 2 (t, ) for te LLG equation wit magnetostriction (21), and te mes T, wit = 1/3, α = 1 and at time t = 1. Figure 23: Numerical approximation of te magnetization m for te LLG equation wit magnetostriction (21), wit = 1/3, α = 1 and at time t =,.5.