Axisymmetric formulation for boundary integral equation methods in scalar potential problems

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Axisymmetric formulation for boundary integral equation methods in scalar potential problems

Laurent Krähenbühl, Alain Nicolas

To cite this version:

Laurent Krähenbühl, Alain Nicolas. Axisymmetric formulation for boundary integral equation meth-

ods in scalar potential problems. IEEE Transactions on Magnetics, Institute of Electrical and Elec-

tronics Engineers, 1983, 19 (6), pp.2364 - 2366. �10.1109/TMAG.1983.1062872�. �hal-00414086�

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IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-19, NO. 6 , NOVEMBER 1983 AXISYUMETRIC FORMULATION FOR BOUNDARY INTEGRAL

EQUATION METHODS IN SCALAR POTENTIAL PROBLEMS by L. KRAHENBUHL and A. NICOLAS

Abstract : Axisymmetric 'geometries often appaear in electromagnetic device studies. The authors pre- sent an original formulation for Boundary Integral Equation methods in scalar potential problems. This technique requires only 2D boundary in the r-z plane and evaluation of the equations only on those boun- dari es.

INTRODUCTION

Scalar potential problems are described by Laplace's equation :

in the general 3D space.

When the geometry is an axisymmetric there are two possibilities :

-

use cylindrical coordinates, express equation (1) in this set of coordinates and note that each quan- tity is invariant in the azili~uthal direction Laplace equation becomes :

In axisymmetric geometry physical quantities keep constant values on circular lines : Green's function G and derivative function bG/ bn have to be integrated on circles.

By this way a 2D Boundary Integral Equation can be generated in the r-z plane, similar to those in x-y plane (41.

It can be noticed that G gives the potential of a uniform charge disgibution on a circle.

Classical development 121 can be applied to obtain function G . :

ax

where :

Boundary Integral Equations have to developed : complete elliptic integral of the first kind with operator :

13

1

b2 b 2 l b

z+

T f 7 . r

This gives complicated expressions.

p

,

R, D : as shovn on Fig. 1

Function G I a x is obtained by a similar development :

-

the second method we propose to develop is to

express the BIE in 3D space and to integrate all G i x = $=.dl .dG ( 9 ) the invariant quantities analytically before

solving the equations. =

-

S a ~ ( k 2

+

&.[~.coao

-

l . l . e o a ( a - ~ ) ] .e(k21 271 D

AXISYMMETRIC FORMULATION

In a 3D space Laplace equation is transformed with

into BIE equation : E : complete elliptic integral of the second kind.

a , ) . ' , D' : as shown on Fig. 1

. . . .

, : ~ = ! ! ~ , i v."G

A ,

I

-

ti G I ( 3 )

XG - / I , , '>/l. On

as shown in previous publications 111.

When p , cjl and H,, are constant along one part of the boundary 6&

,

partial integration of functions G and 6G/Sn along that direction can be made.

As an example it can be shown that partial integration of 3D Green's function

c = -

1 ⁽⁴⁾

47rr

over a stralght llne glves 2D Green's functlon :

CZU =

- -

I

.

^lop^r. ^{( 5 )}

2 7

- - -

'I'hrt authors are with

U~.partement dlElectrotechnique

-

^{ERA 908}

Ecole Centrale d e Lyon

-

BP 163

69131 ECULLY CEDEX

-

^FRANCE

4

Figure 1 0018-9464/83/1100-2364$01.00 O 1983 IEEE

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FLUX DENSITY CALCULATION

The induced magnetic field in a point P of the region HI :

can be computed directly by integration of cp and B on the boundary 6 @I:

with

d z (13)

cosa dG,,

dc:,

2.R dp (14)

dp

+ D.cosa

-

2.R.cos(a-Y) (dE dk + 2. (R-P) .E) 2.n.D" dk dp Dl2

cosa dG,, dGa'x = -,

-

dz 2.R dz (15)

+ ~ . c o s a

-

2.~.cos(a-Y).~~&

2.x.D" dk dz D 7

As these integrals have to be computed a large number of times, the third solution appears to be the most convenient.

Function E is always regular, but K becomes sin- gular when r pq-O. This singularity is similar to Log(r) singularity, and Gaussian quadrature formulae with weight function Log(t) are well matched for evaluation of the integral (6) of function K.

This development is now applied in PHIAX program, using classical techniques 111, 141.

NUMERICAL RESULT

The analytical solution is known for a ferroma- gnetic sphere in a constant field 151. We have choosen this example to test the method.

The sphere is discretised into 4 finite elements of second order (fig. 2). The accuracy of the solution on the boundary is better than 0.4 %O on the potential and 1. %O on the normal flux density

(fig. 3).

It must be noticed that these expressions depend on the same functions K and E as Gax and GIax.

When the point P is on the boundary, a more simple expression can be used :

an equivalent result is obtained in this way.

However computing time is less important in this case.

NUMERICAL DEVELOPMENT

Numerical calculation of integrals K and E can be done in several ways :

-

numerical integration ;

-

asymptotic development 131 ;

-

tabulation and interpolation.

finite element second order.

Fig. 2 : Finite element discretisation of the sphere.

Fig.3 : Solution on the boundary.

Point

1 2 3 4 5

Flux density Phiax Analit.

Potential Phiax Analit.

1.2.

.42.

.62.

.1L progr.

5.8381 5.4369 4.1628 2.2517

.4E-05 .4%

.3%., .3%.

.I%.

.042.

progr.

3.8808 3.5859 2.7441 1.4853 .1E-04

value 5.8823 5.4346 4.1595 2.2511 0.

value 3.8823 3.5868 2.7452 1.4857 0.

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The values of potential and flux density for 6 external points are also presented with comparison to analytical values (fig. 4). The points 6 and 7 are very near the boundary and the integration errors increase.

The same integrals give the angular factor "C"

(141 ; eq. ( 6 ) ) . SO this coefficient becomes a

"goodness factor" of the method and allows the defi- nition of a forbidden area around the boundary where potential and magnetic field cannot be computed.

Fig.4 : Potential and flux density on the straight line 8 .

Xpoints in the forbidden area.

CONCLUSION

The method we developed and exposed is the com- plementary to the BIE programs already existing in 2D and 3D. It solves a large set of problems at a low computing cost and needs a very short geometry description time. With PHI2D, PHI3D and now this PHIAX packages it is possible to solve with the BIE method all magnetostatic or electrostatic problems.

Point

3 6 7 8 9 10 1 1

Point

3 6 7 8 9 10 1 1

REFERENCES

111 ANCELLE B. : "Emploi de la mdthode des 6quations intdgrales de fontiere et mise en oeuvre de la CAO dans les calculs des systBmes dlectromagn6ti- ques" PH. D. Thesis - Grenoble 1979.

121 DURAND : "Electrostatique"

-

tome 1, Masson, Paris

( 3 1 ABRAMOUITZ M. and STEGUN I.A. : "Handbook of ma-

thematical functions". Dover Publication.

141 KRAHENBUHL L. and NICOLAS A. : "Efficient techniques for boundary equations method for potentla1 problems". Compumag 83.

(5 ( DURAND : "Magn6tosta~ique"

-

Masson, Paris.

Potential

Rel. error on ' c ' see eq. ( 6 )

7.52. X 11.X X

1 . 2

<I%.

/

Hz

Phiax Analit.

Phiax progr.

2.7441 2.5948 2.4736

1.8999 1.6938 1.3715

progr.

0.9731 -3.9932 1.3445 0.6744 0.5522 0.4691 0.3427 Analit.

value 2.7452 2.6114 2.4402 2.14342.1447 1.8998 1.6946 1.3726

value 0.9706 0.9005 0.8134 0.6702 0.5588 0.4707 0.3431 Phiax H~

progr.

2.9138 -2.0787 2.9622 2.0131 1.6701 1.4101 1.0284

Analit.

value 2.9118 2.7014 2.4402 2.0107 1.6763 1.4122 1.0295