Direct vector solution of threedimensional magnetic

Direct vector solution of threedimensional magnetic field problems S. Ratnajeevan H. Hoole and Z. J. Cendes Citation: J. Appl. Phys. 57, 3835 (1985); doi: 10.1063/1.334940 View online: http://dx.doi.org/10.1063/1.334940 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v57/i8 Published by the American Institute of Physics.

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Direct vector solution of three-dimensiona~ magnetic field problems s. Ratnajeevan H. Hoolea) and Z. J. Cendes Magnetics Technology Center. Carnegie-Mel/on University. Pittsburgh. Pennsylvania 15213

A new finite-element formulation of magnetic field problems is presented in which the vector field is solved for directly. Unlike solutions obtained by using scalar and vector potentials, a continuous solution in terms of the magnetic field intensity vector is obtained. The formulation is based on a matrix representation of the curl and divergence operators and provides a least-squares solution of the overdetermined system of equations. The procedure is used to solve both two- and threedimensional problems in magnetic recording and in electric machine design.

INTRODUCTiON

THE FORMULATION

Recent years have witnessed a flurry of activity in procedures to solve three-dimensional magnetic field problems. I - 7 However, researchers thus far have followed the trend set by the pioneers in two-dimensional magnetic field analysis and have consistently solved not for the magnetic field vector, but for an associated potential function. While the advantage of solving for the vector potential in two dimensions is easily seen, that of a vector potential formulation in three dimensions is less clear. In two-dimensional problems, the two-component transverse magnetic field vector is replaced by a single-component vector potential at a great saving in complexity. The single-component vector potential function is automatically nondivergent and variational expressions for the vector potential are easily derived. In three dimensions, however, both the vector potential and the magnetic field intensity have three components so that apparent computer memory requirements are the same for both variables. Furthermore, neither the vector potential nor the magnetic field intensity vector is automatically nondivergent in three dimensions, and determining the proper variational expression for the vector potential in three dimensions is a complicated matter. While no one has attempted it, there is a great advantage in solving for the magnetic field directly instead of for a potential function. Since numerical differentiation is an inaccurate process, the accuracy of the magnetic field intensity vector obtained by taking the curl or gradient potential solution is less than that of the potential itself. Indeed, differentiating a continuous nth-order finite-element solution of the vector potential results in a discontinuous (n - 1)st-order approximation for the magnetic field intensity; alternatively, if we solve for the magnetic field directly in terms of a continuous set of nth-order finite functions, then the corresponding vector potential function, obtained by integration, will be of order n + 1 and have derivative continuity. In view of this, it is proposed here to solve directly for the magnetic vector field intensity. In this way, the equivalent of a derivative continuous vector potentia] solution is obtained without the burden of having to impose derivative continuity, and the numerically inaccurate process of having to differentiate a potential function to obtain the magnetic field is eliminated. -'Presently with Drexel University. Philadelphia. Pennsylvania 19104 3835

J. Appl. Phys. 57 (1). 15 April 1985

In linear homogeneous media, the magnetic flux intensity vector B is governed by the equations div B = 0,

(1)

curl B = J,

(2)

together with appropriate boundary conditions. To make a finite-element approximation of the field and obtain a matrix version ofEqs. (1) and (2), let he components ofB be approximated by nth-polynomial finite-element basis functions aln) .8

In general, if a scalar function/is expressed in terms of values of/at the interpolation nodes and an interpolation polynomial a as

/=a[,

(3)

then the derivative of/with respect to the coordinates will be another polynomial of one lower order. Since differentiation is a linear process lx, the derivative of/with respect to x, may be expressed in terms of the coefficients of the polynomial which is being differentiated. Thuslx becomes I'

_

Jx -

a/ _ -aa-In)/_ -a-In-I)Dx/ '

ax

ax

-

-

(4)

where Dx is called the differentiation matrix in the x direction. 9 To approximate the three-component vector B, we may write (5)

B = (an).!!, where

la"I~G"

0 an

t)

and

B~(~)

(6)

0 Substituting Eq. (5) into Eq. (1), and employing Eq. (4), then gives

aln -

I)Md

.!! = 0,

(7)

where Md is the matrix operator equivalent of divergence, given in terms of differentiation matrices as (8)

In a similar way, recognizing that J must be (n - 1)st order if B is nth order, Eq. (2) yields

0021.-8979/85/013835-03$02.40

© 1985 American Institute of Physics


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.......,,,

......

,, ..-.

...... "-

..--.

""'-

.---.

---.

.......... ..:

\

""'~

,

\

\

,

\

~ ___ ~L. __LJ ________LJ ____ L_J FIG. I. Magnetic field intensity around a current-carrying cylinder.

(9)

where Me is the matrix operator equivalent of curl and is given as I

Me

=~l f-l

0

-Dz

Dz

0

-Dy

(10)

Dx

It should be noted that the only approximation in either Eq.

, , r--

II

(7) or (9) is the fact that B is set equal to an nth-order polynomial; Md and Me are exact divergence and curl operators over nth-order polynomial fields.

1I

!

I ]

I

'

I

1 :

I

I i

I

II

U

11

I t.J

_ .-

I

I

~

!

LEGEND r. ~ X-CCM"C,'·JENT VECTOR C - ,{-[;C'j"(";ENT VEe TOR I. ~ X-CC:'l"G!JENT St~~P.R .. - ,(-CC:i"WENT SGFlLP.R

I

1

1 .

~J

r---l J

I

UNIQUENESS AND RANK

Equations (7) and (9) need to be solved simultaneously to compute the approximate solution B. To determine the nature of this system of equations, one needs to know the number of approximation nodes in a 3-D mesh. It can be

l~

.~

i

._... J

II

FIG. 3. Flux distribution in the (a) x-y plane, (b) at Xo in the parallel to the y-z plane for one quadrant of a three-pole magnetic recording head. The figure is symmetric about the front and left sides.

proved by induction that the number of tetrahedrons ntet in a 3-D finite element mesh is (11)

where ne is the number of edges in the mesh, nv is the number of vertices, and nb is the number of boundary nodes, The number of degrees offreedom y(n) within a single nth order tetrahedron is given by y(n) = 'r.>

"~. ::::::~::::rr~:l :::::::::::::::::::::::~~:::::i:::::::::f::::::t:::r: -!MID'

10'

20 11.9TR!X SIZE

30

FIG. 2. Error versus matrix size for direct vector and potential solutions of the problem in Fig. I.

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J. Appl. Phys., Vol. 57, No.1, 15 April 1985

g(n + l)(n + 2)(n + 3).

(12)

Thus the ranks of Me and Md are, respectively, 3y(n - 2) - y(n - 3) and y(n - 1). It follows that the rank of the combined system is R = 4y(n - 2) - y(n - 3), Consequentl.y, a vector field B with 3y(n) d.egrees of freedom inside a tetrahedron needs only 3y(n) - R components to be specified by boundary conditions to ensure uniqueness within the tetrahedron. Continuity of flux will in general provide more S. R. H. Hoole and Z. J. Cendas

3836


than this number of auxilary conditions and hence implies that the system of Eqs. (7) and (9) is overdetermined. BOUNDARY CONDITIONS

Equation (2) is valid only in homogeneous media. To solve problems in inhomogeneous media, we must enforce the correct discontinuity in the magnetic flux which occurs at material boundaries. This is accomplished by deriving a matrix relationship between the magnetic flux components on one side of a boundary and the magnetic flux components on the other side: (13)

Using Eq. (11) we may combine the equations governing the field in homogeneous regions into a single equation for the entire homogeneous problem. VARIATIONAL FORM

Since the system of Eqs. (7) and (9) is in general overdetermined, some method of computing the best possible approximate solution must be devised. This is accomplished in either one of two different ways: (1) Define a least-squares error function corresponding to Eqs. (7) and (9) as F(B) =

f

(M~Mc + M~Md)-# = M~.[. (17) Both of the above procedures give good results. However, the second one is somewhat easier to program and, in general, more accurate. The examples presented below were produced by using Eq. (17).

QUANTIFICATION OF THE ERROR

To determine how the error from the direct method compares with the conventional potential method, we decided to solve for the magnetic field within and about a currentcarrying cylinder of relative permeability of 2. The problem has a well-known solution: the field rises linearly from zero to the edge of the cylinder and then decays as r- I to infinity. 2 The problem was solved by the direct method in Eq. (17) and the flux density is shown by arrows in Fig. 1. The same problem was solved by using the vector potential method and the variation of errors with matrix size is shown in Fig. 2. Several complex three-dimensional problems have been solved by using the method described above. One such problem is a magnetic recording head, one half of which is shown in Fig. 3. A plot of the flux in the two main axes is presented in Fig. 4.

((an - l)Mc# - (an - I).[] 2dR

+

J

(a1n-I)Md

-# )2dR.

(14)

Minimizing this with respect to B gives (M~[T]Mc +M~[T]Md)-#=M~

(T]!,

(15)

where [T] is the metric tensor. (2) Note that Eq. (7) and (9) must be valid independent of a and write them as the coupled system (16)

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Evaluating the least-squares vector solution of this overdetermined system of equations gives

J. Appl. Phys., Vol. 57, No.1, 15 April 1985

'I. Simkin and C. W. Trowbridge, Proc. IEEE 127 (1980). 2M. V. K. Chari, Z. 1. Cendes, P. P. Silvester, A. Konrad and M. A. Palmo, IEEE Trans. Power Appar. Syst. PAS-lOO, 4007 (1981). 3N. A. Demerdash, T. W. Nehl, F.A. Fouad, andO. A. Mohammed, IEEE Trans. Power Appar. Syst. PAS-IOO (1981). 41. Coulomb, IEEE Trans. Magn. MAG-I7, 3241 (1981). 5Z. 1. Cendes, J. Weiss, and S. R. H. Hoole, IEEE Trans. Magn. MAG-IS, 367 (1982). 6p. R. Kotiuga and P. P. Silvester, 1. AppJ. Phys. 54, 8399 (1982). 70. A. Mohammed, W. A. Davis, B. D. Popovic, T. W. Nehl, and N. A. Dernerdash, 1. AppJ. Phys. 54, 8422 (1982). "P. P. Silvester and R. L. Ferrari, Finite Elements/or Electrical Engineers (Cambridge University, Cambridge, England, 1983). "P. P. Silvester, F. U. Minhas, and Z. 1. Cendes, Comput. Phys. Commun. 24,173 (1981).

S. R. H. Hoole and Z. J. Candes


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