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ABSTRACT In this paper, we first re-examine the model of small apertures with equivalent dipoles and obtain, for each aperture, 3 basis functions that only need a local meshing of the aperture, the environment being replaced with an infinite

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A New Local Model for Small Apertures in a Multi-Domain Approach
L. Patier
(1)
, V. Gobin*
(1)
, P. Bonnet
(2)
and F. Paladian
(2)
(1) ONERA, B.P. 74025, Toulouse Cedex 4, 31055 FRANCE (2) Univ. Blaise Pascal, 24 av. des Landais, 63 177 Aubière, FRANCE E-mail: vincent.gobin@onera.fr
Introduction
The modeling of small apertures remains an interesting problem to study in the context of EM Compatibility when using Integral Equation techniques. Indeed, the difference of geometrical scale between the local details of the small apertures and the large shields materials lead to several specific difficulties: the small level of the internal EM fields into the structure is the final purpose of the computation and requires a methodology that keeps the expected precision of the result. In this paper, we first re-examine the model of small apertures with equivalent dipoles [1], [2] and obtain, for each aperture, 3 basis functions that only need a local meshing of the aperture, the environment being replaced with an infinite perfectly conducting plane. In a second step, we plan to introduce this representation of the fields close to the aperture in a full wave 3D modeling. Two strategies can be investigated. The first based upon the Equivalent Principle is divided into two steps: evaluation of the external problem, then internal problem (neglecting the feedback retroaction). The second one is based on the scattering matrix of each domain (outside and inside) and on application of final interconnection with boundary conditions. In this paper, we concentrate on the first step and modify the classical process by expending the field on the aperture on the first computed basis functions introducing the expression of the field on the aperture expended on the firstly computed basis functions.
Expending the fields on a small aperture
We know for years [1] that the behavior of the EM fields scattered by a small aperture can be expended on a development that two first terms happens to be interpreted respectively as a magnetic and an electric dipole. Defining the short-circuit field (Hsc and Esc) when the aperture is closed (see Fig.1), the dipole moments are respectively given by (1) where the parameters
m
α
and
e
α
summarize the shape and size of the aperture and are magnetic and electric polarizabilities, respectively.
Fig 1: E and H field for a short-circuited and an open small aperture
scm pm
H P
.
α
−=
(1)
sce pe
E P
.
α
−=
In (1), 'p' notation recalls that the dipoles radiate in the presence of the infinite PEC plane. One can apply image theory, doubling the dipole, if an expression of the free space fields is needed [2]. This approximation is valid with restrictive hypothesis: -
h1: small aperture (this meaning that its size is smaller than the wavelength of interest) -
h2: uniform local short-circuit fields, -
h3: point of evaluation of the fields "far" from the center of the hole (in practice, at a distance larger than its diameter). For a circular aperture of radius 'a', the polarisabilities are:
334
a
m
=
α
(2)
332
a
e
=
α
For any shape of aperture, one can evaluate the electric field (in practice
n E M
×=
) in the plan of the aperture, with an appropriate integral equation (3) which expression is very closed similar to the classical EFIE equation [1], [2].
[ ]
∫
+∇∇=
ssc
ds M k j H
ϕ π ε µ ωε
412
0200
(3) As an example, Fig 2 represents the electric field distribution in the aperture associated respectively to a unity short-circuit magnetic field (polarized along x and y axes) and electric field (normal to the plan of the aperture). These fields, solution of (3) for appropriate incident fields appears to be the basis functions that expend the electric tangential field at low frequency.
Fig 2: Tangential E field in a circular aperture corresponding to unity short-circuit field
As a validation, once
n E M
×=
has been obtained, we can evaluate the dipoles using (4) and finally deduce the polarizabilities values. For an aperture of radius a=1m, we obtain:
3
297.1
m
m
=
α
(1.333 in theory) and
3
646.0
m
e
=
α
(0.666 in theory)
dS nr E r P
dS nr E
jP
aperture peaperture pm
∫∫
××−=
×=
))'('('
2))'((
1
00
ε ωµ
(4)
Induction Principle Interpretation
The Induction Principle formalized by Harrington [3] will appears as a very useful technique for solving a general scattering problem in a two steps process. On Fig. 3a, we schematize the solution of an srcinal problem, using the well known EFIE, Z
EFIE
(J)=V
RHS
, where J is the current solution for a specified excitation (RHS written for Right Hand Side). The circles symbolize the fact that in this srcinal problem, the incident is evaluated on the whole surface of the scatterer. We now decide to divide this srcinal problem in a primary problem (Fig. 3b) and secondary problem (Fig. 3c).
(a)Original problem (b)Primary problem (c)Secondary problem Fig. 3 Scheme of the Induction Principle
In the first one, we only apply the incident field on an arbitrary part of the scatterer supposed to be truncated and obtain the partial solution J
primary
on this partial object. In this stage, we also evaluate the electric field E
primary
(symbolized with the cross) radiated by the primary current on the complementary part of the srcinal structure. It will contribute to the RHS for the secondary problem. This secondary problem corresponds to the whole object excited by the sum of srcinal incident field plus the primary field concentrated on the complementary part of the object. In [3], the demonstration of the validity of the approach is given.
Induction Principle applied the aperture problem
We now apply this methodology to the coupling of a field in a structure through an aperture. In srcinal problem (Fig. 4a), we look for the fields in a structure due to a distribution on magnetic currents on an aperture. We can modify the situation with an infinite plan prolonging the plan of the aperture without changing the internal field (Fig. 4b). We can now apply the Induction Principal on this modified problem. The primary problem appears to be the scattering of the magnetic currents on an infinite conducting plan, this been the case already solved at the beginning of this paper. Note that, in this stage, we have to calculate the E field on the complementary surface which is the structure without the plane of the aperture. With the hypothesis h3 described previously (we need the "far" fields from the source), we can now choose between a full computing of the fields from magnetic current in free space or an approximation deduce from equivalent dipoles. In the second and final step, where only the influence of the box is researched, the secondary current appears to be quite uniform in the vicinity of the aperture. Thus there is no need of a precise meshing of the geometry with detail (aperture) in this area. A keen idea consists in removing the aperture in the mesh, reducing deeply the process of generation of the mesh and reducing the size of the linear system.
(a) Original PB (b) Modified PB (c) Primary PB (d) Secondary PB Fig. 4 Application of the Induction Principle to the coupling through aperture in a structure
Validation
For the validation of our method, we consider a structure with a small aperture located on the top face. Fig. 5a represents a view of this upper face in the srcinal problem. Fig. 5b illustrates the benefit of our technique because no aperture is meshed anymore in the secondary problem. At the opposite, for the primary problem (not represented here) we can have a very detailed mesh of the aperture because it is the only present surface. Fig. 5c is a comparison of the final E fields obtained on the axes of the aperture compared to the srcinal field.
(a)Mesh for the srcinal PB (b)Mesh for the secondary PB (c)Final result for E Fig. 5 Application to the computing of the field along the axe of a box with a small aperture
Acknowledgment
This study has been supported by the
Délégation Générale pour l’Armement (DGA)
in the framework of thesis.
References
[1]
R.F. Harrington – Resonant Behavior of a Small Aperture Backed by a Conducting Body – IEEE transactions on antennas and propagation, Vol. AP-30, No. 2, p. 205–212, 1982. [2]
P. Degauque, A. Zeddam – Compatibilité électromagnétique : des concepts de base aux applications, Tome 1 – Éditions Hermès : Collection technique et scientifique des télécommunications, 2007. (English translation in progress) [3]
R.F. Harrington – Time-Harmonic Electromagnetic Fields – New York : Mc Graw Hill, 1961. [4]
L. Patier, V. Gobin, P. Bonnet, F. Paladian – « Détermination de la matrice admittance d’une ouverture sur un système faradisé » – NUMELEC 2008, 6e Conférence Européenne sur les Méthodes Numériques en Électromagnétisme, Liège (Belgique), p. 94 - 95, Décembre, 2008.

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