Page 1

Abstract—A three-dimensional boundary integral method (3D

BIM) capable of analyzing the magnetic field at extremely low

frequency (ELF) next to thin conducting shields is presented.

This novel approach is formulated in terms of global variables

(loop currents), avoiding field discontinuities across the shield.

Using Non-orthogonal Dual Grids the unknowns can be defined

on nodes, thus greatly reducing computing requirements

compared to traditional Galerkin’s formulations. The procedure

is validated using an axisymmetric problem. It is shown that

analytical and numerical results are in good agreement even at

few millimeters from the shield surface.

I. BOUNDARY INTEGRAL FORMULATION

Numerical modeling of thin layers embedded in a large

non conductive domain (eg. air) may represent a demanding

task if traditional numerical methods based on Galerkin’s

integration scheme of Maxwell’s equations are employed. It

becomes thus mandatory to develop new formulations capable

of properly managing the intrinsic complexity of the problem.

An integral formulation based on the cell method (CM) has

been applied successfully to analyze ELF magnetic shields of

rectangular shape discretizing the conducting region only [1].

This formulation is extended here to non-structured grids in

order to model shields of arbitrary shape. Assuming the skin

depth of the conductive material to be larger than the shield

thickness, it is possible to discretize the conducting region by

a Delaunay triangulation. According to CM scheme, field

source variables –such as eddy currents i – are defined on dual

edges, whereas configuration variables – such as magnetic

fluxes b and electromotive forces e– are defined on primal

faces and edges respectively. The eddy current problem is

formulated in terms of loop currents io, exploting the similarity

between 2D tessellations and planar networks. The main

advantage is that loop currents form already an independent

set of DoFs, while a tree-cotree decomposition is required

using eddy currents.

The basic idea of this approach is that, thanks to the one-

to-one correspondence between primal and dual geometric

entities, loop currents can be defined on dual nodes rather than

primal faces, greatly reducing number of DoFs and, in turn,

the computational cost. Furthermore, loop currents are

continuous by definition across the layer, unlike the vector

variables of Galerkin’s methods. Stability and accuracy of

procedure are enforced by using div-conforming edge

elements to interpolate the eddy current density J inside each

simplex. In particular, it has been found that J results to be

uniform when expressed in terms of loop currents by

Kirchhoff’s law i=CTio, where C is the primal face-edge

incidence matrix. This allows the electric constitutive equation

e=Rio to be built from Ohm’s law in a straightforward way

similar to CM-based formulations for magnetostatics [2]. The

circulation of e along the boundary of primal faces is related to

the magnetic flux by Faraday’s law, written for time-harmonic

problems as Ce+jω ωb=0. In linear media, fluxes b can be

resolved into discrete curl-free, bs, and div-free, br.

components. bs is computed from source currents, br can be

expressed in terms of loop currents by the magnetic

constitutive equation br =Mio. Partial inductances assembled

in M are computed in a similar way as the moment method by

semi-analytical integration [3]. This calculation is made easier

compared to traditional edge-element based boundary element

methods such as [4] thanks to J uniformity, while the accuracy

of the solution is ensured by the first order interpolation.

Finally, the fundamental linear system {CR+jω ω M} io=−jω ω bs

is obtained by inserting both constitutive relations into

Faraday’s law. Once eddy currents have been computed from

loop currents via the continuity law, the magnetic flux density

in the air region can be computed via Biot-Savart’s law.

II. PROCEDURE VALIDATION AND DISCUSSION

In order to validate the developed method, an

axisymmetric problem with exact solution is considered. A

uniform magnetic flux density (1 µT, 50 Hz) is applied

orthogonally to a 2 mm thick conducting spherical shell of 1 m

diameter. Numerical and analytical values of magnetic flux

density are computed on the x-axis for different mesh sizes. It

has been found that the proposed procedure converges to the

exact solution even with very coarse meshes. Using triangular

elements of 10 cm size (252 nodes), the maximum error of

3D BIM is about 2.5% on the shell surface, decreasing for

x>0. For finer grids (5 cm size triangles, 1048 nodes) the

maximum error decreases up to 1%.

The excellent agreement with analytical data demonstrates

that the developed procedure is both efficient and reliable.

III. REFERENCES

[1] M. Guarnieri, F. Moro, and R. Turri, “An Integral Method for Extremely

Low Frequency Magnetic Shielding,” IEEE Trans. on Magnetics, Vol.

41, No. 5, pp. 1376-1379, May 2005.

[2] M. Repetto and F. Trevisan, “3-D Magnetostatic with the Finite

Formulation,” IEEE Trans. on Magnetics, Vol. 39, No. 3, pp. 1135 –

1138, May 2003.

[3] R.D. Graglia, “On the Numerical Integration of the Linear Shape

Functions Times the 3-D Green’s Function or its Gradient on a Plane

Triangle,” IEEE Trans. on Magnetics, Vol. 41, No. 10, pp. 1448-1454,

October 1993.

[4] H. Tsuboi, T. Asahara, F. Kobayashi, and T. Misaki, “Eddy Current

Analysis on Thin Conducting Plate by an Integral Equation Method

Using Edge Elements,” IEEE Trans. on Magnetics, Vol. 33, pp.1346-

1349, March 1997.

A Boundary Integral Formulation on Unstructured Dual Grids

for Eddy Current Analysis in Thin Shields

P. Alotto, M. Guarnieri, and F. Moro

Dipartimento di Ingegneria Elettrica, Università di Padova

Via Gradenigo 6/A, 35131 Padova, Italy

alotto/guarnieri/moro@die.unipd.it

9

PA1-6

1-4244-0320-0/06/$20.00 ©2006 IEEE