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2124 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 3, JULY 1993

A New Finite Element Method in Micromagnetics Wenjie Chen, D. R. Fredkin, and T. R. Koehler

Abstract-We present a new finite element method in micromagnetics. The magnetization is linearly interpolated within each tetrahedral element and the magnetostatic interaction is accurately obtained by integration. The equilibrium magnetizations at the nodal points can be found by minimizing the total energy of the system. Two different minimization schemes are compared.

I. INTRODUCTION N recent years there have been continuing efforts [ 1 ] - I [SI to numerically model magnetization processes in

magnetic materials. Finite element methods are attractive because complex geometries can be handled easily. In the algorithm developed by Fredkin and Koehler, [2] the magnetization is assumed to be uniform in each tetrahedral element and therefore is a piece-wise constant function. The gradient of magnetization, and therefore the exchange energy, is ill-defined, and some reasonable but ad hoc method must be used to obtain the exchange energy. The magnetic scalar potential is found by solving Pois- son’s equation [ l ] .

Different schemes for representing the magnetization M and calculating the magnetostatic energy are proposed. M is linearly interpolated within tetrahedral elements from its values at the nodal points. The magnetic scalar potential is exactly calculated using Collie’s technique [6] and the magnetostatic energy is accurately obtained by numerical integration using Gauss-Jacobi quadrature formulas [7 ] , [8]. Note that in this method M is continuous and surface poles appear only on the boundary surface of the sample. A relaxation dynamics method and a conjugate gradient method have been used to minimize the system energy. Both give the same results. Their perfor- mances are compared.

11. THE METHOD The total energy of a ferromagnetic system is the sum

of exchange energy E,,, magnetostatic interaction energy Emag, crystalline anisotropy energy Eonis and Zeeman energy in an applied field Ezeem. This is written for uniaxial

Manuscript received July 31, 1992; revised July 31, 1992. This work has been supported by the National Science Foundation under the grant No. DMR-90-10908 and by the Center for Magnetic Recording Research at the University of California, San Diego.

W. Chen and D. R. Fredkin are with the Department of Physics, Uni- versity of California, San Diego, 9500 Gillman Drive, La Jolla, CA 92093.

T. R. Koehler is with the IBM Research Division, Almaden Research Center, 650 Harry Road (K66/803), San Jose, CA 95120.

IEEE Log Number 9209187.

anisotropy, as

1 - M - Hex, - M - ; K ( M e)2’\ dV (2) L 1

where A is the exchange constant, M, is the saturation magnetization density, Hd is the demagnetizing field, Hex, is the external magnetic field, K is the uniaxial anisotropy constant and e is the anisotropy axis.

Consider a tetrahedral element e. Let V‘ denote the volume, M: the magnetization at vertex a of this element, and q: the linear shape function that equals 1 at vertex a and 0 at all other vertices. The magnetization in each tetrahedral element is linearly .interpolated,

4

M(r) = C M:q:. (3) e a = l

Calculations of the exchange energy, the Zeeman energy and the anisotropy energy are straightforward.

A d

4 4

where the volume integral depends upon whether a and 0 are equal,

The Zeeman energy of the system in a uniform external field is given by

Ezeem = -E 1 Hex, - M z q Z d V (7) e a = l e

4

= - C C Ha, * M : V e / 4 . (8)

A nonuniform external field may be approximated by a piece-wise polynomial function, but in most of our calculations the external field is assumed to be uniform, as reflected in (7) and (8). The magnetostatic interaction

e a = l

0018-9464/93$03.00 0 1993 IEEE

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CHEN er a[.: NEW FINITE ELEMENT METHOD IN MICROMAGNETICS 2125

needs to be treated more elaborately: . P P 4

( r - r' l3 - 3(r - r ' ) ( r - r ' )

( r - r' l5

C

- The total energy can be expressed as a quadratic poly- Fig. 1 . Division of tetrahedron ABCD into eight sub-elements. E , F, G ,

H , I , and J are the m 9 i n t s of the edges. The eightsub-elements are: m, W F , W F , DHJI, GHFI, CHEF, JEHF, and JIHF. nomial with the constraint that (MI is a constant.

The K matrix contains all information about the magnetostatic interaction, the anisotropy energy and the exchange interaction. Zi only involves the external field. After the K and 2 matrices are obtained, neither element geometries nor the connectivity matrix is needed for the iterative calculation. The magnetostatic part consumes al- most all of the CPU time to calculate the K matrix. How- ever this part depends only upon the mesh, and therefore it needs to be done only once for each mesh. The total magnetostatic energy is always finite, but the demagnetizing field is divergent at the comers of the polyhedral mesh, which makes a direct use of (9) difficult. There- fore, we utilize the magnetic scalar potential which is a linear combination of magnetizations on the nodes:

especially when the two are the same element, the scalar potential is a rapidly varying function in space. Merely increasing the order of the polynomial does not guarantee that the numerical integral will approach the true value. In this case subdivision of integration volume is needed to achieve faster convergence. A tetrahedron is cut into eight smaller ones with equal volumes by connecting the midpoints of its six edges (Fig. l) , and this subdivision procedure can be repeated as needed. This subdivision scheme requires that the order of the polynomial approximation be greater than 3.

A. Newton-Cotes Method in the Barycentric Coordinate System

An integrand is approximated by a polynomial of degree N . For a volume integral on T3, the (N + 1)(N + 2) (N + 3)/6 values needed to uniquely determine this polynomial are chosen to be the values of the integral at

(13) i l r l + i2r2 + i3r3 + i4r4

N The magnetostatic energy is a sum of contributions from surface poles and volume poles: r { i l , i2 , i3 , i4} =

- se +(r )Vv , * Ma dV] (12)

The integral in (1 1) can be calculated analytically [6]. + (r) is a continuous function, hence the surface integral in (12) needs to be evaluated only on the true boundary surfaces of the sample. Two methods of numerical integration have been tested: the Newton-Cotes method in the barycentric coordinate system and application of Gauss-Jacobi quadrature formulas [7], [8]. Since a volume integral on any tetrahedron can be converted to one on the standard simplex T3: {(OOO), (loo), (OlO), (OOl)), and a surface integral on any triangle can be converted to one on the standard simplex T2: {(00), (lo), (Ol)}, all

where r l , r2, r3 and r4 are the four vertices of the tetrahedron T3 and the non-negative integers i,, i2, i3 , and i4 satisfy i l , i2 , i3 , and i4 = N. For convenience, the values will be denoted by rA where the subscript A labels a qua- druple { i l , i2 , i3 , i 4 } . The polynomial shape function hA of order N at point rA is defined as a polynomial of degree N that equals 1 at rh and 0 at rA, for A' # A. It can be expressed in terms of the four linear shape functions q l , q2, q3, and q4. If we define

if i, = 0

the polynomial shape function can be written as:

h = Cf'~(r, i l)P2(r, i2)P3(r, i3)P4(r, id, (15) integrals are considered on T2 or T3. where C is the normalization constant such that

111. INTEGRALS FOR MAGNETOSTATIC ENERGY h , ( a , ) = 8 ~ ~ r . (16) We now discuss in detail the computation of the indi-

vidual integrals in (12). In the two integration methods given below, the scalar potential is approximated by a

With this shape function, one has

1 +(r) dV = jT +@A)hA(r) dV = c A VAA+(~A) polynomial function. This is sufficient when the potential varies slowly. When two elements are near one another,

T (17)

2126 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 3 , JULY 1993

where

A - l h,(r) dV. " v T

When calculating the weights Ah, the following formula is useful:

i, ! i2! i3 ! i4 ! (19)

The method can be formulated the same way for two

ii i2 i3 i4 V I 9 2 7?3 7?4 dV = 6

(il + i2 + i3 + i4 + 3)! s, dimensional integrals on a triangle.

B. Gauss-Jacobi Quadrature Hillion [7], [8] has given integration formulas of arbi-

trary order for numerical integrations on triangles and tet- rahedra as products of one-dimensional Gauss-Jacobi formulas. The quadrature formula which is exact for polynomials of order d is denoted by I d .

For the standard triangle, P P n

(20 ) where (t i , wi) are the roots and the weights of the Gauss- Legendre method and ( T ~ , aj) those of the Gauss-Jacobi method. If both methods have order 2n - 1, this formula also has order 2n - 1, and n 2 function evaluations are needed.

For the standard tetrahedron,

n 2 0 1 0 0 0 2 0 I O 0 0 2.0 1.0 = W i ; n w j ; n W k : n f ( t i , ' n tj:n t k : n , t i , n t ; ,n

i , j , k = 1

(1 - t g ) , t y ( 1 - t;;:)),

where t:;," is the i 'th root in the interval (0,l) of the Jacobi polynomial H;' of degree n and w;is is the weight associated with this root. This quadrature formula is exact for a polynomial integrand of degree 2n - 1, and n3 function evaluations are needed.

C. Comparison of the Two Methods If the scalar potential is approximated by a linear func-

tion, the Newton-Cotes method in barycentric coordinate system requires function evaluations at all the four vertices, while Gauss-Jacobi quadrature formula for volume integral requires function evaluation at only one interior point. Since the total number of nodes is much less than the total number of elements, the Newton-Cotes method is faster. But as will be shown below, the linear approximation is not sufficient and a higher order approximation must be applied.

To know which order of quadrature formula is neces- sary to achieve a desired accuracv. a comnarison has been

made between accurate results and the approximations of the self interaction matrix of the standard tetrahedron, which is the strongest among all possible pair interac- tions. Here accurate results are obtained by recursively calling a one dimensional integration subroutine implementing an adaptive Gauss-Legendre algorithm, with the relative error being controlled to be less than Some typical matrix elements of the self interaction matrix obtained with the two methods of various orders are listed in Table I . It shows that Gauss-Jacobi quadrature con- verges to the accurate value faster than the Newton-Cotes method does.

When two elements are far apart, Gauss-Jacobi quadrature formulas of low orders are sufficiently good. In our production code, the order of quadrature formula is chosen according to the following rule:

(1) self interaction and nearest neighbors: d = 5 with one subdivision,

( 2 ) D < 15: d = 3 , ( 3 ) D > 15: d = 1,

where D is the ratio of distance between the centroids of two elements to their average dimension.

I v . MINIMIZATION OF T H E S Y S T E M ENERGY In quasistatic micromagnetics we seek a local minimum

of the energy , which is approximated by (10). One method for minimizing (10) is to write and approximately solve a relaxation dynamics

This is not the discretized equation of motion that would be derived from the Landau-Lifshitz-Gilbert equation, and there is no need to solve it with precision; we need only get the correct values for the Mi after a long time. Accordingly, we adopt the following procedure: We set y = 1 since merely rescales the time variable, which does not have physical significance here. For each site, define Hi* = -aE,oral/Mi. If, at some time t, the angle between Mi and H: is 8 ( t ) , then rotate Mi in the plane of Mi and HF so that the angle between Mi and HT is re- duced to

O(t + 6t) = 2 arctan

where H,,,, = maxi IHTI. The step size 6t is automatically adjusted to be approximately as large as possible without allowing the energy to increase in a single step.

The conjugate gradient method [9] is an altemative approach for the search for a local minimum of (10) which is especially appropriate here where the number of vari- ables is large. The constraints /Mil = M, can be enforced by representing each Mi by its stereographic coordinates, which can be generated by projection from any pole. As the magnetization reverses. it is necessarv to change poles.

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CHEN et al . : NEW FINITE ELEMENT METHOD IN MICROMAGNETICS 2127

gauss8 bary 3 bary 7 barylO bary20 GJ Odiv GJ ldiv

(W, M3

34.9 133 27.6 167 34.1960 34.6304 34.8562 35.1492 35.0034

3.20455 2.04910 3.18313 3.21351 3.20495 3.24343 3.20824

7.08195 1.25764 7.90101 0.07650 7.08953 1.22081 7.07488 1.25087 7.08011 1.25614 7.11132 1.33964 7.08460 1.26445

22.6411 20.8402 22.4589 22.5571 22.6232 22.7575 22.6754

22.9 150 13.4733 22.5566 22.9337 22.9274 23.4961 22.9661

Some typical elements (multiplied by 1OOO) of magnetostatic interaction matrix of tetrahedron {(OW), (loo), (OlO), (001)) interacting with itself, obtained with different approximations. Accurate results are obtained by recursively calling a one dimensional integration subroutine implementing adaptive Gauss-Legendre algorithm. Bary3 is for the 3rd order Newton- Cotes method in the barycentric coordinate system, bary7 is for the 7th order, barylO for the 10th order and bary20 is for the 20th order. GJOdiv is for the 5th order Gauss-Jacobi quadrature formulas with no subdivision. And GJldiv is also for the 5th order Gauss-Jacobi quadrature formulas, but with one subdivision.

Fig. 2. The magnetization pattem of a 9721 A x 4860.5 A X 972 A permalloy particle at its remanent state.

This complication can be avoided by replacing (10) with

We have compared the relaxation and conjugate gradient methods. In all but one case, the two methods gave the same results; in one case the conjugate gradient method gave a slightly lower switching field, which prob- ably indicates that it was using too large a step size and therefore left the neighborhood of the previous local minimum prematurely. For platelike particles, both methods took the same processor time near reversal, but the conjugate gradient method was substantially faster near saturation. For ellipsoidal particles, on the other hand, the conjugate gradient method did not seem to have any ad- vantage over the relaxation dynamics method. We use the relaxation dynamics method in our production code.

V. DISCUSSION A potential difficulty with the present method is that the

magnitude of magnetization density is in general not pre- served within elements. This problem becomes serious

when M varies rapidly with respect to the grid size; how- ever, any discretization method will break down in this case. Fig. 4 shows th? seven domain pattern of a 9721 A X 4860.5 A X 972 A permalloy particle at its remanent state as obtained by our micromagnetic code. Fig. 3 shows the relative deviation of magnetization density from its saturation value. For this particular magnetization config- uration, the overwhelming majority (85 %) of the elements have errors less than 10%.

The approximation of M by a piece-wise constant function preserves the magnitude of M , but the evaluation of the exchange energy is ambiguous. In particular, ambi- guities in the treatment of surface elements do affect quan- titative results significantly. Further, the number of nodal points in a finite element mesh is much less than the number of elements so that the number of unknowns is re- duced with the present method. For example, in the mesh we used to model a platelike particle, there are 3818 elements but only 834 nodes. This makes our new method very efficient and fast.

We have utilized this new micromagnetic code to study the magnetization processes in platelike permalloy particles [lo], [ l l ] . Good agreement with experimental data was found.

2128 IEEE TRANSACTIONS ON MAGNETICS. VOL. 29, NO. 3. JULY 1993

0 20 40 80 80 100 Relative &or ( X )

Fig. 3. Departure of the magnitude of the magnetization from a constant. The vertical axis is the fraction of elements in which the relative error of the mean magnetization magnitude exceeds that given by the horizontal axis.

REFERENCES 111 D. R. Fredkin and T. R. Koehler, “Hybrid method for computing

demagnetizing fields,” IEEE Trans. Mugn., vol. 26, pp. 415-417, 1990.

[2] D. R. Fredkin and T. R. Koehler, “Ab Initio Micromagnetic Calcu- lations for Particles,” J. Appl. Phys., vol. 67, pp. 5544-5548, 1990.

131 D. R. Fredkin, T. R. Koehler, J. F. Smyth and S. Schultz, “Mag- netization reversal in permalloy particles: Micromagnetic computa- tions,” J . Appl. Phys., vol. 69, pp. 5276-5278, 1991.

[4] M. E. Schabes, H. N. Bertram, “Magnetization processes in feno- magnetic cubes,” J. Appl. Phys., vol. 64, pp. 1349-1359, 1988.

151 J. F. Smyth, S. Schultz, D. R. Fredkin, T. R. Koehler, 1. R. Mc- Fayden, D. P. Kern and S. A. Rishton, “Hysteresis in lithographic arrays of permalloy particles: experiment and theory,’’ J. Appl. Phys.,

[6] C. J. Collie, “Magnetic Fields and Potentials of Linearly Varying Current or Magnetization in a Plane Bounded Region,” in Proceed- ings of COMPUMAG, Oxford, England, pp. 86-95, 1976.

[7] P. Hillion, “Numerical integration on a triangle,” IJNME, vol. 11,

[8] P. Hillion, “Numerical integration on a tetrahedron,” Culcolo, vol.

191 Philip E. Gill, Walter Murray, Margaret H. Wright, Prucricul opri- mizurion, London; New York: Academic Press, 1981.

[lo] Wenjie Chen, D. R. Fredkin, T. R. Koehler, “Micromagnetic studies of interacting permalloy particles,” IEEE Trans. Magn., vol. 28, pp. 3168-3170, Sept. 1992.

[ I l l Wenjie Chen, D. R. Fredkin, T. R. Koehler, “Micromagnetics of imperfect permalloy particles,” presented at the Magnetism and Mag- netic Materials Conference, Houston, December 1992.

vOI. 69, pp. 5262-5266, 1991.

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18, pp. 117-130, 1981.

Wenjle Chen was born in January 1965, in China. He graduated with the B.S. in physics from the University of Science and Technology of China in Hefei, China, in 1986. He received the M.S. and Ph.D. in physics from the University of California, San Diego in I988 and 1992, respectively.

He was a postdoctoral researcher at the Physics Department, U.C. San Diego, from February to October, 1992. His research interest has been magnetism and magnetic materials, and their applications in magnetic recording. Currently he is with Maxtor Corporation in San lose, California.

D. R. Fredkin was bom in New York on Sept. 28, 1935. He received the A.B. degree in mathematics in 1956 from New York University and the Ph.D. degree in mathematical physics in 1961 from Princeton University.

Since 1961 he has been at the University of Califomia, San Diego where he is professor of physics. His research deals with theoretical condensed matter physics, biophysics, and statistical problems arising in neurophys- iology. He has been associated with the Center for Magnetic Recording Research since 1984. Dr. Fredkin is a member of the American Physical Society and the Society for Industrial and Applied Mathematics.

At various times he has also been associated with AT&T Bell Labora- tories, the Aerospace Corporation, C.E.N. Saclay, A.E.R.E. Harwell, and the West Los Angeles Medical Center of the Veterans’ Administration.

T. R. Kwhler was bom in Toledo, Ohio on Aug. 8, 1932. He received the B.S. degree in physics in 1954 from Seattle University and the Ph.D. degree in physics in 1960 from the California Institute of Technology.

Since 1960 he has been a Research Staff Member of the IBM Research Division and has been associated with the San Jose Research Center prior to 1987 and with the Almaden Research Center after that. His work mostly consists of computer simulation applied to condensed matter Physics.

Dr. Koehler is a Fellow of the American Physical Society.

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