Comparison of methods for modelling jumps in conductivity using

1295

IEEE TRANSACTIONS ON MAGNETICS. VOL. 33. NO. 2, MARCH 1997

Comparison of Met hods for Modelling Jumps in Conductivity using Magnetic Vector Potential based Formulations P.J.Leonard and D.Rodger University of Bath, Claverton Down, BATH BA2 7AY, UK

Abstract-This paper compares finite element formulations for modelling eddy current problems with jumps in conductivity. We study schemes which represent the eddy current regions using nodal or edge based magnetic vector potentials. The use of an additional electric scalar potential, V , in the eddy current regions is discussed. The nodal A scheme is shown to produce misleading results at low frequencies whilst the A V and edge A schenies are acceptable.

I . INTRODUCTION This work was motivated by the observation that the first order nodal magnetic vector potential produces erroneous current predictions at low frequencies/conductivities (skin depth much greater than element size). Such problems can arise in medical applications where an alternating magnetic field is used to induce currents to stimulate nerves. Non-destructive testing is another application where, for example, a corroded crack would be a region of low conductivity next to a higher conductivity region. Low frequencies must be used if the defect is say on the inside of a pipe.

In conducting regions the field is represented as ;

dA E=---QVT/ at

(1)

The consistency of Amperes Law is enforced by solving 1 VX-VXA-{VCXQ.A}+U P

The term in curly brackets is sometimes added to gauge these equations. Typically a = is chosen as the penalty weight. To make the solution unique A . ii = 0 can be enforced on the boundary of the A regions [l]. If V is kept as an unknown we typically solve V.u

($+w) =0

Manuscript received March 18,1996

111. TESTPROBLEM To highlight the problem we examine a very simple configuration that characterizes the observed problematic behaviour. A plate (16 x 16mm2 with a conductivity of 5 x lo7 S / m and ,U,. = 1) is placed between 2 infinitely permeable poles. The plate was discretized using a 21x21 regular element mesh. The magnetic scalar is fixed on the poles making the problem 2 dimensional with H x n^ set via H = -04. The plate was modelled with 4 low conductivity inserts ( r = 5 x lo4 S / m ) ,these should divert the current flow! We also looked at the case when the insert was removed leaving air. The formulations used 3D bricks with a single layer of elements. I apologise for the unlikely fields values, this is only a theoretical study.

A. Nodal A

11. THEORY

B =VXA

In this paper we look at schemes using first and second order nodal basis functions and first order edge variables [a] to represent A in eddy current regions. The magnetic scalar potential is typically used in non conductors, however in the examples in this paper it’s value is fixed and it acts only to provide the H x 6 boundary source term for the A region.

(3)

The results shown in Fig. 1 illustrate that first order nodal A does not predict the current correctly at 100Hz for the low conductivity slot case. This result is not completely surprising, it is well known that the divergence of current is violated at a conduciivity jump if a continuous nodally interpolated vector potential is used. Next we model the problem with an air filled slot (modelled with magnetic scalar). The result is shown in Fig. 2. This is quite disturbing, the weakly implied zero divergence of current has failed to keep the current from leaving the plate ! At higher frequencies the results are more acceptable. Fig. 3 shows the same problem at 1000Hz. Unfortunately the low conductivity insert case is not helped by increasing the frequency as shown in Fig. 4. One would think that improvement might be possible if we use more than one element to model the insert. Fig. 5 shows that this is not the case. The same test problem was modelled using a second order nodal A. The results in Fig. 6 show that the current

0018-9464/97$10.00 0 1997 IEEE

1296

Fig. 4. Current flow with iiisert, 1000 Hz,first order nodal A

Fig. 1. Current flow with low coiiductivityiiisert, 100 Hz,first order nodal A (1/4 symmetry)

l lo",

i

/m.11

i

I Tig. 5. Current flow with wide insert,1000 Hz,first order nodal A

Fig. 2. Current flow with air slot, 100 H z , first order nodal A

distribution for the air filled slot case is now reasonable. T h e use of second order elements did not significantly irriprove the results with the low conductivity insert.

B. Zero frequency limat As the frequency approaches zero the equations tends to LZ lwrrCsUlSl ~ o w m ~ I I I I u p 1 I B z

I

IW,

i

I

14.

,&.I,

Fig. 3 . Current flow with air slot, 1000 Hz,first order nodal A

Fig. 6. Current flow with air slot, 100 H z , second order iiodal A

1297

1 Vx-VxA PO

=0

(4)

This is clearly singular and leads to numerical problems. A gauge can be imposed on the nodal A equations by adding the penalty term,

1 V~-VXA-{VCXV.A}=O (5) P To rnake the solution unique the normal component of A on the A$ interface is also set to zero. Whilst this ensures that the system is well conditioned it can lead to inaccurate results. For example the problem with the air slot should yield uniform flux density with a frequency of 0 H z . Fig. 7 shows that the flux at the centre of the plate is 6.33 T compared the expected value of 12.57 T . Doubling the mesh density in fact makes the answer worse, 6.24 T ! Making the problem second order improves matters slightly 7.57 T . FILE : AJUO’KLPEN_O FIRST ORDER A IPENALTYI AIR OAP OH6

I

Fig. 8. Current flow with wide insert, 100 H z , using AV$

A t high frequencies the AV scheme can suffer from

ill conditioning due to the cancellation of and VV. Physically the induced field tries to cancel the source e.m.f. However, this problem is outside the scope of this I paper, which is concerned primarily with low frequency 13.747 problems. 12.301 10.854

9.4076

7.9610

D. Scheme with a discontinuous A

6.5144

5.0679 3.6213 2.1748 0.7282 -0.718 -2.164’ -3.6111

-6.0581 -6.5041

Fig. 7. Flux density at 0 H z using the nodal A with a divergence penalty scheme.

If there is a jump in conductivity the normal component of A should be discontinuous on the interface (if no V term is used). We have implemented a scheme that uses a distinct magnetic vector potential for each region of constant conductivity. At the interface between regions the magnetic vector potentials are coupled using a vector Lagrange multiplier that enforces the continuity of tangential H and normal B [3]. Fig. 9 shows the results using this scheme.

It should be noted that this error is associated with the “inside corner” that makes the magnetic vector potential perform a full 180 degree turn around the bottom of the slot. Problems without such features do not exhibit such drastic errors. Satisfactory results can be obtained without the imposed gauge but the matrix conditioning is bad and leads to convergence problems using ICCG.

6‘. Thf AV scheme

%

lJsing E = V V allows E to be discontinuous. The results of this scheme are shown in Fig. 8 Adding the divergence penalty on A is now valid even in the case of discontinuous conductivity. The VV takes care of the jump in electric field strength and V.A = 0 is a valid gauge. Theoretically we still have the overconstraint problem with the divergence penalty. However the “inside corners” can be filled with a region of A (without V).

Fig. 9. Current flow with wide insert, 1000 H z , discontinuous A

At lower frequencies the gauge of A becomes too weak and we get the unrealistic results shown in fig. 10.

1298

shown to be over-constrained. It was unable to model the uniform field for the example shown without introducing unacceptable errors. The best methods to model the current flow for the example problem with a jump in conductivity were the edge A and the AV$ schemes. The 2 directed electric field is shown in Fig. 12, the field is taken along a line through the centre of the insert to the plane of symmetry. It can be seen that there is excellent agreement between the two schemes. There was no significant difference when the penalty term was used in the AV scheme. FILE

’

LOW CONDUCTIVITY INSERT 100 Hz

Fig. 10. C’urrent flow with wide insert, 100 H z , discontinuous A

E. Edge A Edge variables maintain the tangential continuity of the variable but allow a normal discontinuity. This is exactly what is required for this case. We adopt the edge A representation within the eddy current regions leaving the nodal magnetic scalar potential to represent non-conductors. This was done to avoid problems associated with co-trees and ill conditioned matrices that can occur when edge A variables are used in non conductors. The results for the 100 H z problem with a low conductivity insert are shown in Fig. 11

2 W x

Fig. 12. Comparing E, predi at 100 H z

edge A methods

The efficiency of the two s Table I gives details of the nu convergence of the ICCG for the conductivity insert. It would a with the penalty is better conditioned.

ase with the low

TABLE 1 Comparison of efficiency

Scheme AV (no gauge)

,G”,

J

1

I

Iterations 139

1 1

Nonzeros 75328

I

1

Uiiknowns 3872

REFERENCES

mea,,

Fig. 11 Current flow with wide insert a t 100 H z Using edge A

IV. DISCUSSION We have looked at some very specific examples in order to highlight the problems in modelling jumps in conductivity and low frequencies. The nodal A with the penalty terms, the only scheme which claims to behave well even at zero frequency, was

[l] P.J.Leonasd D.Rodger and R J Hill-Cottiiighain “Calculation of AC losses in current forced conductors using 3D finite elements and the A I V method”. IEEE Tra September 1989. [Z] A. Kaineari. “Study on 3-D Eddy Current Anal Proceedangs of 4th Internatzonal IGTE Symp pean TEAM Workshop, pages pp91-99, October 1990. [3] D.Rodger and P.J.Leonard. “Alternative s modelling of rail guns a t speed using finite ele of the 3’rd European Symposzom on EML Techn England), April 1991.