E E E TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5 , SEPTEMBER 1991 3808

VECTOR POTENTIAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSJONAL MAGNETOSTATIC

J . P . Adriaens - F . Delince' * - P . Dular * - A . Genon - W . Legros - A . Nicolet

University ofLi2ge - Dept of Elecrricnl Engineering Insriritl Montefiore - Sari Tilman BLit. B28 - B-4000 Likge (Belgium)

Abstract - A three-dimensional magnetostatic field computation method is presented. It uses the vector potential formulation which is valid for any topology in opposition to the scalar formulation needing special cuts which are not always obvious. We give an attention to the calculation of strongly singular integrals using the Cauchy principal value. Finally, two examples show the good accuracy of this method.

I NT RO DUCTI0 N

The boundary element method is well suited to study linear materials, and particularly the exterior homogeneous region. Indeed, i t leads to less unknowns than the finite element method and it allows to model open boudary problems 11 J.

Magnetic vector potential formulation is valid for any topology where the scalar fomiulation needs special cuts which are not ;ilw;iys obvious. A magnetostatic program using vector potential can also be used as a starting point for an eddy curl-ent computation program.

This paper deals with a direct vector formulation of the boundary e!ement method for three-dimensional magnetostatic. I t describes a method to calculate strongly singular integrals in the Cauchy principal value sense. It gives some computation results for structures whose boundaries are meshed with triangular linear elements.

BOUNDARY 1NTEGRAL. EQUATION

By applying the Green's vector theorem to the equations of the vector magnetostatic model (1)

curl €1 = j div B = 0 B = p H B = curl A div A = 0

where H is the magnetic field, B is the magnetic induction and A is the magnetic vector potential, we obtain the following integral equation (2) [2,3,4]

where V is the concerned domain and S is its boundary (fig. 1 ), P is the computation point and Q is the source point,

G = L is the Green's function, 4 7 r r 1 ~

rrQ.n dsQ is the solid a~igle from which S is seen from P,

n i r the un i t normal vector of S . r = rQp = -rpQ.

Fig1 Doinain V

The integral equation (2) shows that the vector potential A inside V is given by summing two integrals which represent respectively the effect of the internal sources and the external elements. This equation can be written in the form (3)

I-

INTEGRALS

When the computation point P is near the surface S, it is necessary to take some precautions since P and Q may be identical in the integration : singularities appear and have to carefully be processed .

Only the last term of the surface integral of (3) is not integrable i n the coninion sense. Yet, the problem of the singularity l/r2 can be solved by using the surface integration in the sense of the Cauchy principal value. This one is the result of the integration on S except mi an infinitesimal section in the vicinity of P.

Indeed. the integration of (nxr)/r3 on ;i plane disc centered in P is nul and WK write

where the stiu represents the integration in the Cauchy principal value sense. When the surface is curved, the radius of the disc has to be infinitesimal.

I t can be shown that this disc notion is also valid when P is on an edge or a corner (geometrical discontinuity). Then, we define as many disc portions as there are adjacent faces to P (fig.2).

* Thcse authors are Rcrcarch Assisunts \wlh the B c l p a n Sattonal Fund Tor ScicnLlllL Rcrc,rrch.

(x)18-9464/91$01,00 0 1991 IEEE

EEE TRANSACTIONS ON MAGNETICS. VOL. 27, NO. 5, SEPTEMBER 1991 3809

NUMERICAL SOLUTION

We are interested in the integration in the Cauchy principal value sense. We consider a surface mesh of triangular linear elements and a point P located on one of its nodes. The surface around P is then made with pieces of planes (fig.4).

' Ponanedge P on a comer

Fig. 2 Geometrical discontinuities

Let's consider the case where P is on an edge at the intersection of two planes (fig.3).

Fig. 4

On each of them, we can break up A into two parts, a constant vector (A in P) and a variable vector (variation of A around P) :

Fig. 3 P is on an edge

Let's integrate the function ( n x r d / r 3 on the surface built with two half-discs centered in P, each of them is located in one of the planes. If we write

r = rl+ rll

where rl and r I I are respectively perpendicular and parallel to the edge and are included in the same plane as r , then we have

where a,(r,O) = r a(@) because A varies linearly on an element;

a@) depends on the nodal values of A on the concerned element

(A).

Then, we have

There is no problem with the first term of (4) as Ap is constant; we have to make an integration on the mangle A except on a portion of the disc of a certain radius, called the limit radius, which can be finite. Here we consider separately each element (A) including P, but i t is clear the cancelling of the integration on a piece of disc is only meaningful when all the elements are gathered together with a unique limit radius. The second term is regular.

+ [ (m7 ds + ds

= (0) + (0) + (0) = 0 (Q.E.D.)

A similar reasoning can be perform for a general corner. Thus the integration of the function can be made on S except on portions of discs around P.

COMPUTATION RESULTS

1. A magnetic sphere in a uniform external magnetic field

We consider a magnetic sphere (radius R=O.l m; center O=(O,O,O); kl=lOO) in a uniform extemal magnetic field Bo=(O,O,l) [tesla]. The computation is made with 266 nodes and 588 triangular linear elements and the variations of B, along a radial axis in the plane z=O are compared to the analytical ones (fig.5).

3810 IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5, SEPTEMBER 1991

REFERENCES

[ l ] S.J. SALON & J. D'ANGELO, "Applications of the hybrid finite element - boundary element method in electromagnetics", IEEE Transactions on Magnetics, vo1.24, No.1, January 1988,

J.A. STRATTON. "Electromagnetic theory", Mc Graw Hill, pp.80-85.

r21

SPHERE pr= 100

Fig.5 A magnetic sphere in a uniform extemal magnetic field

2. A magnetic cylinder in the field of a current loop

We consider a magnetic cylinder (radius R=0.05 m; height =

0.1 m; p,=1000) in the field of a half-way current loop (radius R1=0.075 m; current I=1000 A). The computation is made with 242 nodes and 480 triangular linear elements and the variations of Bz along a radial axis at a height of 0.07 m are compared to those computed with an axisymmetric program (fig.6).

L 1

New York, 1941. LIXIN LI & J . LUOMI, "On three-dimensional boundary element methods for magnetostatics in vector variables", IEEE Transactions on Magnetics, vo1.24, No. 1, January 1988,

[3]

pp. 19-22. [4] H. TSUBOI & M. TANAKA, "External conditions for the

vector potential in the boundary element method", IEEE Transactions on Magnetics, ~01 .25 , No.5, September 1989, pp.4138-4140.

C n I N D E R pr = 1000

Fig.6 A magnetic cylinder in the field of a current loop