Adaptive mesh refinement for micromagnetics simulations

1648 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 6, JUNE 2006

Adaptive Mesh Refinement for MicromagneticsSimulations

Carlos J. García-Cervera1 and Alexandre M. Roma2

Mathematics Department, University of California, Santa Barbara, CA 93106 USADepartamento de Matemática Aplicada, Universidade de São Paulo, Instituto de Matemática e Estatística,

CEP 05311-970 São Paulo, Brazil

We present a methodology for efficient micromagnetics simulation. The method combines an unconditionally stable, finite differencesscheme with an adaptive mesh refinement technique. It enhances accuracy by covering locally special regions of the domain with asequence of nested, progressively finer rectangular grid patches that dynamically follow sharp transitions of the magnetization field(e.g., walls and vortices). To illustrate our approach, we consider a rectangular sample of infinite thickness with strong anisotropy in theout-of-plane direction.

Index Terms—Adaptive mesh refinement, finite differences, Landau–Lifshitz–Gilbert, multilevel-multigrid.

I. INTRODUCTION

UNDERSTANDING the mechanisms of magnetization re-versal in ferromagnetic samples of nano-scale size is of

interest in the study of the magnetic recording process, in par-ticular in computer disks and in computer memory cells, suchas magnetic random access memories (MRAMs) [1]–[3].

The relaxation process of the magnetization distributionin a ferromagnetic material is described by the Landau–Lif-shitz–Gilbert (LLG) equation [4], [5]

(1)

where is the saturation magnetization which, farfrom the Curie temperature, is usually set to be a constant; thefirst and second terms are the gyromagnetic and the dampingterms, respectively, with being the gyromagnetic ratio, andthe dimensionless damping coefficient; is the effective field

(2)

In (2), represents the anisotropy energy per unit volume.The parameters and are the exchange constant and per-meability of vacuum, respectively. is the stray field, andis the external field.

Typically, the magnetization profile in a ferromagneticsample displays large domains where the magnetization isslowly varying. These domains are usually of the order of a fewhundreds of nanometers in size, and are separated by magneticwalls, and magnetic vortices. The core size of these sharptransition regions is of the order of a few nanometers. Most ex-perimental studies coincide that the presence of magnetizationvortices inside a ferromagnetic sample has a dramatic effect in

Digital Object Identifier 10.1109/TMAG.2006.872199

the magnetization reversal process [6]–[10]. In addition, the re-versal process occurs as a consequence of domain wall motion.In order to carry out realistic micromagnetics simulations, itis therefore necessary to resolve numerically a broad range oflength scales, spanning from the nanometer size magnetic wallsand vortices to the macroscopic size of magnetic memoriesand hard drives [11]. Moreover, the overall accuracy of thenumerical simulation depends strongly on how well these localphenomena are resolved [12].

Fully resolved three-dimensional simulations that use a uni-form grid may be too costly for currently available computer re-sources. To increase the computational efficiency, and to allowfor complex problems to be tackled, reductions in the processingtime and in the computer memory consumption are desirable.These reductions can be achieved by adaptively refining thespatial mesh locally around walls and vortices, while resolvingmagnetic domains with a coarser grid.

In micromagnetics simulations, the adaptivity has usuallybeen achieved by employing adaptive refinement finite-elementmethods [13]–[15], or by employing methods based on amoving mesh [16]–[18].

Adaptive finite elements started to be applied to several prob-lems in micromagnetics in the 1990s. Among those are the simu-lations of longitudinal thin film media [13], domain structures insoft magnetic thin films [14], and domain wall motion in perma-nent magnets [15], to name a few. Arbitrarily shaped domainscan be handled naturally by finite element discretizations. Dueto the way information is stored and retrieved at nodal points,the mesh adaptivity may impose a relatively heavy computa-tional overhead when compared to the processing time taken bythe overall method, especially if remeshing must be performedfrequently.

In the moving mesh method, a mapping between a uniformlogical mesh and the physical mesh is defined. The physicalmesh deforms from one time step to the next in order to ac-commodate the underlying structure of the solution. The newmapping may be obtained either by an optimization procedure[19], or dynamically, as in [20]. This methodology has been em-ployed in the study of singularity formation in different contexts

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GARCÍA-CERVERA AND ROMA: ADAPTIVE MESH REFINEMENT FOR MICROMAGNETICS SIMULATIONS 1649

[21], [20]. In this procedure, the equations on the physical gridare rewritten on the logical grid. This gives rise to a differen-tial equation with nonconstant coefficients. In the context of mi-cromagnetics, this may considerably complicate the stray fieldcomputation.

Here, we introduce another strategy for adaptive micromag-netics simulations which combines the Gauss–Seidel projectionmethod (GSPM) [12], [22] with an adaptive mesh refinement(AMR) technique. The GSPM is an unconditionally stablescheme for micromagnetics simulations whose complexity iscomparable to that of solving the linear heat equation withthe backward Euler method. The AMR technique employedis based on the works of Berger and Collela [23], for solvinghyperbolic equations on rectangular grids, and Berger andRigoutsos [24], for point clustering and mesh generation.Enhanced accuracy is attained by covering certain regionsof the domain with a sequence of nested, progressively finerrectangular grid patches which dynamically follow specialfeatures of the solution (e.g., sharp property transitions). Sincethe convergence properties of the GSPM are well understoodon rectangular grids, this refinement technique seems to beparticularly appealing. Moreover, rectangular grids have asimple user interface and, by separating the integration methodfrom the adaptive strategy, we can use the GSPM on fine andcoarse grids without modification.

By combining the GSPM with the AMR, we obtain an un-conditionally stable, adaptive method for the simulation of theLLG equation on rectangular domains, with optimal asymptoticcomplexity. Second-order finite differences for spatial approx-imations are employed and only the solution of Poisson-typeequations with constant coefficients are required. Although weonly consider a rectangular sample, arbitrarily shaped domainscan be handled efficiently by carefully adding boundary correc-tions to the exchange and stray fields [25].

For the reasons above, the AMR-GSPM is an attractive al-ternative approach to both adaptive finite elements and movingmesh methods. We illustrate our approach by finding the steadystate of the magnetization in a rectangular sample of infinitethickness, with strong anisotropy in the out-of-plane direction,and for which a large number of complex, transient structuresdevelop.

II. MICROMAGNETICS SIMULATIONS

The Landau–Lifshitz–Gilbert equation (1) is solved em-ploying the GSPM [12], [22]. Only linear systems of the form

(3)

must be solved. By carefully introducing the nonlinearity a pos-teriori, an unconditionally stable finite differences method forthe LLG equation is obtained, and we may employ time stepson the order of 1 ps, even in the presence of thermal agitations[22].

The stray field can be written as , where is themagnetostatic potential. solves the magnetostatic equation

(4)

where is the volume occupied by the sample, and repre-sents the jump at the material/vacuum interface. Equation (4)must be solved in the whole space. This requires the introduc-tion of far field boundary conditions. In order to avoid this, wedecompose the potential into two parts: . The func-tion satisfies equation

(5)

and it contains the bulk contribution of to the stray field.The function is extended to be equal to zero outside .

The boundary contributions are included in , which satisfiesequation

(6)

The solution to (6) is

(7)

where , and is the Newtonianpotential in free space. Formula (7) can be used to evaluate onthe boundary of the domain, and therefore can be determinedinside the domain solving (6) with Dirichlet boundary data. Thisapproach is similar to the hybrid method introduced by Fredkinand Koehler [26]. Our approach differs in that we use Dirichletboundary conditions instead of the Neumann boundary condi-tions considered in [26], which changes the integral representa-tion of the boundary contribution from a double-layer potential,to the single-layer potential (7). The single-layer potential is lesssingular than the double-layer potential, and thus can be handlednumerically more easily.

We approximate integral (7) on the boundary of the domainby approximating using piecewise polynomial interpolation.The corresponding moments of the Newtonian potential can beevaluated analytically. In two dimensions, the resulting sum canbe evaluated in operations by direct summation, whereis the total number of grid points in the domain, if a uniform gridwas used. In three dimensions, however, the evaluation of theboundary values by direct summation is an operation.


Fig. 1. Ghost cells near the interface.

Solving Poisson’s equation with Multigrid [27] is an op-eration. Therefore, in two dimensions our procedure has optimalcomplexity. In three dimensions, the evaluation of the boundaryvalues by direct summation dominates the CPU time. The com-putational time can be further reduced using a fast summationtechnique [28], [29]. Results in this direction will be presentedelsewhere.

III. ADAPTIVE MESH REFINEMENT

In our approach, the ferromagnetic sample is covered by com-posite grids, that is, by block-structured grids, defined as a hi-erarchical sequence of nested, progressively finer levels. Eachlevel is formed by a set of disjoint rectangular grids and the re-finement ratio between a level and the next finer level is two.The magnetization is defined at the center of the computationalcells. Ghost cells are employed around each grid, for all levels,and underneath fine grid patches to formally prevent the finitedifferences operators from being redefined at grid borders andat interior regions which are covered by finer levels. We usesecond- or third-order accuracy interpolation schemes to pro-vide values at these cells. The description of the composite gridsis given in [23] in greater detail. In Fig. 1, we show an interfacebetween two successive refinement levels, and the location ofcoarse and fine variables.

The use of an unconditionally stable time stepping proceduresuch as the GSPM allows us to evolve the solution on all grids,and in all levels, with the same time step. The complexity of theGSPM is comparable to that of solving a linear heat equationusing backward Euler. Equations (3), (5), and (6) are solvedusing a multilevel-multigrid method [30], [31].

A. Composite Grid Generation and Remeshing

Composite grid generation depends on the flagging step, thatis, determining first the cells whose collection gives the regionwhere refinement is to be applied. We mark for refinement thecells at which the absolute value of the in-plane divergence ofthe magnetization field is at least 10% of its global maximum,that is, if

(8)

Also, it is convenient to mark for refinement a layer of cellsclose to the boundary of the domain. This way, we are able tocompute the boundary integral given by (7) with the accuracyof the finest level.

Other criteria that can be employed for marking cells for re-finement, either simultaneously or separately, are the norm ofthe gradient of the angle formed by the two in-plane compo-nents, and the norm of the rotational of the field, .

Once the collection of flagged cells is obtained, grids in eachlevel are generated by applying the algorithm for point clus-tering proposed by Berger and Rigoutsos [24]. We require that,at least, from 70% to 85% of the cells in each grid patch wereflagged (grid efficiency). The rest of the cells are included so thegrid patch is rectangular.

Remeshing is performed in two situations: 1) whenever highin-plane divergence values escape from the region covered bythe finest level and 2) at every certain number of fixed time steps(e.g., at every 50 time steps) in order to refresh the compositegrid; otherwise, composite grids generated with finest levelscovering a large portion of the domain would tend to stay per-manently in use, and the integration would become inefficient.

IV. RESULTS

To illustrate the proposed methodology, we consider a rect-angular cylinder of infinite thickness parallel to the -axis,with strong anisotropy in the out-of-plane direction. TheLandau–Lifshitz energy per unit length becomes

(9)

where is the anisotropy constant, and is a rectangle in the-plane. The effective field reduces to

(10)

where . The stray field is computed solving (4) intwo dimensions.

The rectangular domain has dimensions 1 m 250 nm. Wechoose the exchange constant and saturation magnetization tomimic Permalloy ( J/m, A/m).The anisotropy constant is chosen such that , inorder to impose a high penalty on the out-of-plane componentof the magnetization. The damping parameter was set to

.Our motivation to study this problem is twofold. On the one

hand, the energy (9) resembles the energy of a thin ferromag-netic film, where the shape anisotropy penalizes out-of-planeexcursions of the magnetization, and the leading term in thein-plane components of the stray field is given by a logarithmicconvolution kernel [32]–[34]. On the other hand, the geometricsetting and the parameters considered here allow us to reducethe problem to a two dimensional computation, which simpli-fies the evaluation of the stray field. Despite this reduction, the


Fig. 2. Diamond domain structure. We present a (color) plot of the angle thatthe in-plane components of the magnetization form with the OX-axis. (Colorversion available online at http://ieeexplore.ieee.org.)

Fig. 3. Sketch of the diamond domain structure. The arrows indicate the direc-tion of the in-plane components of the magnetization. (Color version availableonline at http://ieeexplore.ieee.org.)

minimizers of energy (9) show complex domain structures, withmagnetic walls and vortices, making this an ideal test problemfor the numerical method presented here

The domain is discretized employing four levels. Level 1 (thebase level) is a uniform grid given by 128 32 cells, on top ofwhich we add three refinement levels. A uniform grid wouldrequire 1024 256 cells (that is, 262 144 cells) to have its res-olution equivalent to the finest level.

The solution is evolved in time until no visible differencein the magnetization field between successive time steps is no-ticeable. The final time is 1.5 ns, reached in 1500 time steps.At steady state, the magnetization distribution develops the di-amond domain structure depicted in Fig. 2. A sketch of thisstructure is presented in Fig. 3. The arrows show the directionof the in-plane components of the magnetization. Lengths aremeasured in units of m.

Employing the divergence of the in-plane components (seeSection III-A), and the distance to the boundary of the domainas the flagging criteria, 118 composite grids are generated in the1500 time steps. The number of cells in the composite grids,summing over all the levels, ranged from about 51 10 toabout 213 10 , the finest level covering from about 11% toabout 53% of the domain. In average, the finest level coveredonly 25% of the domain (6% standard deviation).

Typical grid patches and the corresponding grid cells em-ployed close to the steady-state solution are shown in Figs. 4and 5, respectively.

Only the region near one of the vortices is plotted. The com-posite grid dynamically adapts itself to the underlying structureof the magnetization and concentrates grid points on the domainwalls, vortices, and on the boundary of the domain.

To illustrate how the grid patches adapt to the underlyingstructure of the magnetization, we present different stages in theevolution of the magnetization in Figs. 6 and 7. Both the mag-netization and the corresponding grid patches are plotted. The

Fig. 4. Grid patches near a vortex and in a neighborhood of the north andsouth domain boundaries. (Color version available online at http://ieeexplore.ieee.org.)

Fig. 5. Grid cells near the vortex shown in Fig. 4. (Color version availableonline at http://ieeexplore.ieee.org.)

large number of small structures that arise during the time evolu-tion makes this a good test problem for the procedure presentedhere.

In Fig. 8, we show the third component of the magnetiza-tion at steady state, computed using 128 32 cells. This corre-sponds to the coarse grid used in Level 1. The exchange lengthis nm, and therefore vortices cannotbe resolved with such a coarse grid. This can produce erroneousresults, as shown in [12], [35]. For the parameters used in thisexample, at least 512 128 cells are necessary to resolve thecore of the vortex. In Fig. 9, we plot the out-of-plane compo-nent of the magnetization computed on the composite grid. The


Fig. 6. Evolution of the magnetization toward the diamond structure, with thecorresponding grid patches. Darker areas (blue) indicate finer grid patches,while lighter areas (yellow) indicate coarser grid patches. Time is measuredin precession time units (T = ( � M ) ). (a) Initial condition. Grid pointsaccumulate at the interface and the domain boundary. The arrows indicate thedirection of the magnetization. (b) As finer structures develop, the grid adaptsdynamically. (c) Additional patches are added wherever necessary. (Colorversion available online at http://ieeexplore.ieee.org.)

resolution of the composite grid, in a vicinity of the vortex, isequivalent to that of a 1024 256 uniform grid.

Fig. 7. (Continuation of Fig. 6.) (a) As the fine structures disappear, fine gridpatches are replaced by coarser ones. (b) and (c) The diamond structure is almostformed, and the grid patch distribution becomes more stable. (Color versionavailable online at http://ieeexplore.ieee.org.)

To perform the visualization of any function defined on acomposite grid, we first interpolate its values to a series of pro-gressively finer uniform grids whose mesh widths range fromthe mesh width of the base level to the mesh width of the finest


Fig. 8. Out-of-plane component of the magnetization at steady state, computedwith a uniform grid, with 128� 32 grid points. The vortices cannot be resolvedin such a coarse grid. (Color version available online at http://ieeexplore.ieee.org.)

Fig. 9. Out-of-plane component of the magnetization at steady state, on theadaptive grid. The grid around the vortex is equivalent to a uniform grid of1024� 256 grid points. The vortex is fully resolved. (Color version availableonline at http://ieeexplore.ieee.org.)

level. Remembering that level 1, the base level, is given by auniform grid, we:

1) interpolate values from the uniform grid with meshwidth to a uniform grid with mesh width

;2) overwrite the interpolated values with the computed func-

tion values coming from level of the composite grid.We execute this loop for all the levels from 1 to finest level-1.For the number of levels considered in this example, the

steady-state solution is obtained on the composite grid spendingless than 65% of the time needed for the equivalent 1024 256

uniform grid, solving in average for about 60% fewer un-knowns. For this example, remeshing represents less than 1%of the total processing time.

V. CONCLUSION

We have presented a finite-difference methodology to per-form micromagnetics simulations employing adaptive mesh re-finement. The spatial grid dynamically adapts itself accordingto both the size of the divergence of the in-plane components ofthe magnetization, and the distance from the computational cellto the boundary of the domain.

An unconditionally stable time-stepping procedure is em-ployed, allowing for the same time step to be taken for allthe grids in all the refinement levels. In particular, we can uselarge time steps for micromagnetics simulations, and a reducednumber of grid points, depending only on the intricate structureof the magnetization.

The Poisson and Helmholtz-type equations are solved usinga multilevel-multigrid method, which has asymptotically op-timal complexity. The numerical results indicate that greater ef-ficiency in both the processing time and in the memory con-sumption can be achieved, compared to results obtained with auniform grid of the same accuracy. Also, remeshing imposes anegligible computational overhead.

Ferromagnetic layers are an integral part of larger structures,such as hard drives, and MRAMs. An adaptive procedure likethe one described here can find applications not only in the studyof large ferromagnetic samples, with a variety of small scales,but also in the study of macroscopic devices, such as the onesmentioned before. Although this goal has not been achieved yet,we believe this is a good step in that direction.

As future research, the methodology introduced in this papercan be combined with a fast summation technique on adaptivegrids to compute the stray field, relaxing therefore the require-ment of flagging the cells close to the boundary of the domainfor refinement. Also, to deal with arbitrarily shaped domains,boundary corrections can be added to the exchange and strayfields. Results in these directions are natural extensions of thisadaptive methodology and will be presented elsewhere.

ACKNOWLEDGMENT

The computations presented in this paper were carried outon a Sun Sparc and a Beowulf cluster purchased by the Mathe-matics department at UCSB with funds provided by a NationalScience Foundation (NSF) SCREMS Grant DMS-0112388. Theauthors would like to thank N. Rogers for his assistance withthe computational facilities at the Mathematics Department atUCSB. The work of C. J. García-Cervera was supported in partby NSF Grant DMS-0505738, and a Faculty Career Develop-ment Award from UCSB.

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Manuscript received October 5, 2005; revised February 16, 2006. Corre-sponding author: C. J. García-Cervera (e-mail: [email protected]).

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Adaptive mesh refinement for micromagnetics simulations