Finite element micromagnetic simulations with adaptive mesh refinement

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method based on an adaptive refinement scheme is clearly desirable. Error control is essential if ... improve accuracy, adaptive mesh refinement techniques can.
Finite element micromagnetic simulations with adaptive mesh refinement K. M. Tako Computational Magnetism Group, School of Electronic Engineering and Computer Systems, University of Wales, Bangor, Dean Street, Bangor, Gwynedd LL57 1UT, United Kingdom

T. Schrefl Institute of Applied and Technical Physics, T.U., Vienna, Wiedner Hauptstrasse 8, A-1040 Vienna, Austria

M. A. Wongsam and R. W. Chantrell Computational Magnetism Group, School of Electronic Engineering and Computer Systems, University of Wales, Bangor, Dean Street, Bangor, Gwynedd LL57 1UT, United Kingdom

The finite element method has recently been employed in investigating magnetization processes in materials with an irregular microstructure. In order to improve the accuracy of the solution, a method based on an adaptive refinement scheme is clearly desirable. Error control is essential if accuracy of results is to be optimal. We have applied an adaptive scheme to a system of interacting two-dimensional polycrystalline grains. The error estimator is based on the divergence and curl of both the magnetization and demagnetizing field. It is shown that increasing levels of discretization improve the micromagnetic solution, and the spatial localization of the additional nodes demonstrates the importance of the adaptive meshing technique. © 1997 American Institute of Physics. @S0021-8979~97!28608-1#

I. INTRODUCTION

II. SIMULATION MODEL

The use of the finite element method ~FEM! in modeling magnetization processes in micromagnetics demands a great deal of intuition. The FEM is ideally suited for handling regions of arbitrary geometry, hence its use in modeling materials with an irregular microstructure. The demagnetizing field within the particles is generally nonuniform giving rise to nonuniform magnetization states.1 The reliability of all finite element solutions requires that the error be kept to a minimum. This entails that the accuracy must be optimal. To improve accuracy, adaptive mesh refinement techniques can be employed. The objective of any adaptive procedure is to increase the accuracy of the finite element solution by refining the finite element mesh and reducing the discretization error until a desired accuracy is achieved. On the basis of error estimates that indicate the regions of the mesh with the largest errors, the procedure continually improves the accuracy by refining these regions. Adaptive meshing is attractive because mesh generation is automatic and hence user friendly, nodes are added only to regions which are in need and error estimates always accompany the finite element solution. Over the years, adaptive meshing techniques specific to certain problems have been very useful for analysis and design, especially in the areas of fluid flow and thermal analysis. The application of adaptive meshing in micromagnetics is relatively new.2 The main objective of this work was to develop and implement an adaptive scheme for a two-dimensional system of interacting polycrystalline grains. To this end, an error estimator that takes into account the combined effects of the exchange and demagnetizing field gradients on the discretization error was successfully implemented. We give a brief description of the micromagnetic model in Sec. II, the implementation of the adaptive technique in Sec. III and we present the results of simulations of a two-dimensional platelet in Sec. IV.

The micromagnetic model employed in our study is here outlined briefly since the details are given elsewhere.3 The two-dimensional system is modeled as an array of interacting polycrystalline grains produced using the Voronoi construction. The grains are all single crystals and the uniaxial anisotropy is random in-plane. The initial triangular discretization involves joining the nodes of each grain to its center point. The magnetization varies linearly within each triangle and there are contributions to the total magnetic field at each point from external, anisotropy, exchange, and magnetostatic forces. The formulation of all field terms has been presented elsewhere.3 All the magnetization processes are driven by the dynamic gyromagnetic Landau–Lifshitz equation.3 The resulting system of differential equations are solved using a fourth order predictor-corrector scheme3 with step size adjustment and error control. The vorticity function4 has been used to characterize the magnetic structures obtained.

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III. ADAPTIVE PROCEDURE

The implementation of adaptive refinement demands computer power and storage facility. In simulating a hysteresis curve, we have restricted the adaptive process to the zero field configuration. The main errors in numerical micromagnetics arise from exchange and demagnetizing discretization. The refinement procedure reduces these errors. The most important aspects of any adaptive process are to identify an error estimator which can adequately represent the local error in each finite element and a robust method of adding finite elements in the regions of largest errors. Another important aspect is the method of dividing the finite elements into smaller elements. For every adaptive process, an appropriate quantity must be used as a basis for refinement. In micromagnetic problems, this could be an error in

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© 1997 American Institute of Physics

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the energy, field, or magnetization. Knowing the error in every finite element for example, we could define an error norm i e i , a global quantity, such that N

i e i 25

( e iA i ,

~1!

i51

where N is the total number of elements, e i the local error in the generic element i, and A i its corresponding area. Since an exact solution is not available, it is not simple to calculate e i . The refinement algorithm is based on the evaluation of two parameters:5 the refinement indicator h and the convergence parameter c. h, which is based on error estimates,5 tells us where to refine the mesh without necessarily telling us what the errors are, while c tells us when to stop the adaptive process.5 The exchange energy6 can be expressed in terms of ~¹3m!21~¹•m!2. A modified form of h has been used to estimate the exchange and demagnetizing discretization errors, hex and h d , respectively. The refinement indicator h i d over element i is defined as h i 5 Ah ex i 1 h i , where

h ex i 5

@~ ¹3m! 2 1 ~ ¹•m! 2 # 3A i

~2!

( Nj51 i mi 2

and

h di 5

@~ ¹3hd ! 2 1 ~ ¹•hd ! 2 # 3A i ( Nj51 i hs i 2

,

~3!

where imi and ihs i are computed using nodal averages for the magnetization and demagnetizing field components. h i is thus computed as a mean value for contributions from exchange and demagnetizing effects. An element i is chosen for refinement when h i . s h max, where hmax is the maximum value of h in the problem domain and 0,s,1. In our simulations, s50.2 and c5 A( Ni h 2i . Refinement is terminated when c, e , for some user defined tolerance limit e; e50.05. We have also calculated the total energy of the system E, and if c or E were to change by an insignificant amount after each refinement step, then there is convergence of the algorithm. An algorithm based on regular division,7 to be described elsewhere, divides the elements with the largest errors. E52 ( Ni51 @mn •hn ]•A i /9, where mn and hn denote the total nodal magnetization and field components averaged over each element.

FIG. 1. Variation of 2E, c, and hmax with the number of elements N.

mum after around 4000 elements. The total energy E decreases rapidly at first due to an initial rapid increase in the number of elements. As we refined the mesh further, the energy is virtually constant indicating convergence of the algorithm. Additional elements would have no significant effect on the system energy, therefore the adaptive process can be terminated. Figure 2 shows the mesh after refinement. The refinement is concentrated at about the center of the vortex structure. This is due to a rapid variation in the magnetization and demagnetizing field in these regions. Rapid variations result in large errors hence smaller finite elements are concentrated in the region. Figure 3 shows the magnetization pattern before refinement. The arrows represent the vector magnetizations in each grain. There is evidence of nucleation of a vortex close to the particle edge. However, there is still a remanent magnetization of 0.471. Figure 4 shows the magnetization pattern after refinement. The refined mesh allows the vortex to propagate into

IV. RESULTS AND DISCUSSION

There are free boundary conditions and the grains are assumed to be magnetically decoupled. A51026 ergs/cm, K54.23106 ergs/cm3 and M s 51.43103 emu/cm3. The constants have the usual definitions. The film thickness and grain size are 200 Å, respectively. In this article we present a study of the remanent state of a two-dimensional platelet of size 1000 Å containing 25 grains. The convergence of the technique is demonstrated in Fig. 1 which shows the variation of c, hmax , and the energy E with the number of elements N. In Fig. 1, the general trend is a reduction in c and hmax with increasing number of elements indicating a reduction in the discretization errors tending asymptotically to a miniJ. Appl. Phys., Vol. 81, No. 8, 15 April 1997

FIG. 2. Refined mesh. N56822, 4440 nodes. Tako et al.

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FIG. 5. Variation of Vorticity G with radial distance r. ~a! Before refinement, ~b! after refinement. FIG. 3. Before refinement. N5150, 175 nodes.

the particle forming a more stable structure. The refinement thus leads to a more stable magnetization configuration, giving in this case a lower remanence of 0.195. The concentration of elements in Fig. 2 is clearly evident from the structure in Fig. 4. Refinement occurs where there are large dispersions in the spatial variation of the magnetization. In a previous paper,4 we have used the radial variation of the average vorticity function to characterize the magnetization microstructure. Figure 5 shows the vorticity of the magnetisation G as a function of radial distance r before and after

refinement. Vortex structures are characterized by a pronounced peak in the vorticity function. Clearly, there is a more pronounced peak after refinement, which acts to quantify the better defined vortex structure after refinement which is evident from the structure in Fig. 4. V. CONCLUSION

An adaptive FEM approach has been developed, using an error estimator based on the spatial variation of both the exchange and magnetostatic energies. The method has been applied to the study of remanent states in a two-dimensional platelet. The error estimator has been shown to correctly identify the areas requiring improvement. The results indicate a significant improvement in the calculated magnetization structure after refinement, specifically the development of a well-defined solenoidal structure. In order to achieve the same improvement with a uniform refinement, a factor of 50 increase in the number of elements would be required, which highlights the importance of adaptive meshing in allowing detailed micromagnetic studies within reasonable computational requirements. ACKNOWLEDGMENTS

This work was supported by EPSRC and carried out within the EU CAMST project. M. E. Schabes, J. Magn. Magn. Mater. 95, 249 ~1991!. T. Schrefl, H. F. Schmidts, J. Fidler, and H. Kronmu¨ller, J. Magn. Magn. Mater. 124, 251 ~1993!. 3 K. M. Tako, M. A. Wongsam, and R. W. Chantrell, J. Magn. Magn. Mater. 155, 40 ~1996!. 4 K. M. Tako, M. A. Wongsam, and R. W. Chantrell, J. Appl. Phys. 79, 8 ~1996!. 5 P. Fernandes, P. Girdino, P. Molfino, G. Molinari, and M. Repetto, IEEE Trans. Magn. 26, 795 ~1990!. 6 A. S. Arrott and B. Heinrich, J. Magn. Magn. Mater. 93, 571 ~1991!. 7 W. F. Mitchell, ACM Trans. Mathematical Software 15, 326 ~1989!. 1 2

FIG. 4. After refinement. N56822, 4440 nodes.

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