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Equivalent point-source modeling of small obstacles for electromagnetic waves
Justine Labat, Victor Péron, Sébastien Tordeux
To cite this version:
Justine Labat, Victor Péron, Sébastien Tordeux. Equivalent point-source modeling of small obsta-
cles for electromagnetic waves. WAVES 2019 - 14th International Conference on Mathematical and
Numerical Aspects of Wave Propagation, Aug 2019, Vienne, Austria. �hal-02278032�
WAVES 2019, Vienna, Austria 1
Equivalent point-source modeling of small obstacles for electromagnetic waves Justine Labat 1,∗ , Victor Péron 1 , Sébastien Tordeux 1
1 EPC Magique 3D, UPPA-E2S, Inria Bordeaux Sud-Ouest, LMAP UMR CNRS 5142, Pau, France
∗ Email: justine.labat@inria.fr Abstract
We develop reduced models to approximate the solution of scattering problem by electromag- netic obstacles that are small in comparison with the wavelength. Using the matched asymptotic expansions method, we investigate a meshless multi-scale approach where the scatterers are represented by equivalent point-sources. In the context of multiple scattering, we deduce from this a Foldy-Lax approximation whose accuracy and eciency are illustrated with numerical sim- ulations.
Keywords: Reduced models, Maxwell's equa- tions, Multiple scattering
Introduction
The propagation of time-harmonic electromag- netic waves of angular frequency ω > 0 in a ho- mogeneous and isotropic dielectric innite medium of electric permittivity ε > 0 and magnetic per- meability µ > 0 is described by an incident wave
Re E inc (x) exp(−iωt)
, x ∈ R 3 , t > 0.
In the presence of a small and smooth auto- similar obstacle ω δ = δω ⊂ R 3 centered at the origin, whose characteristic length δ is very small compared to the wavelength λ = ω 2π √ µε , the wave is scattered and gives birth to electro- magnetic elds E δ and H δ satisfying the time- harmonic Maxwell equations
( ∇ × E δ − iκH δ = 0,
∇ × H δ + iκE δ = 0,
where κ = 2π λ denotes the wave-number. For a perfectly conducting obstacle, the domain of propagation Ω δ is the exterior domain R 3 \ ω δ and the boundary condition reads as
n × E δ = −n × E inc on ∂ω δ .
The existence of a unique solution is guaranteed by the hypothesis of outgoing wave at innity given by the Silver-Müller radiation condition,
lim
|x|→∞ |x| (H δ × x b − E δ ) = 0 unif. in b x = x
|x| .
Numerical techniques based on a discrete ap- proximation of the geometry are limited and very expensive due to the smallness of the ob- stacle. To overcome these diculties, we investi- gate two meshless approaches involving approxi- mate solutions of the exterior Maxwell problem.
Equivalent point-source modeling
The rst one is a volumical approach based on the method of matched asymptotic expansions [1]. This method consists in constructing dis- tinct expansions of the solution in dierent re- gions of the domain of propagation with appro- priate scales, and matching them in an inter- mediate region called the matching area. Far from the obstacle, the obstacle is modeled like a dipolar source around the origin. As a result,
E δ ∼
δ→0 E elec [d e δ ] + E mag [d h δ ], (1) where E elec [d e δ ] (resp. E mag [d h δ ] ) is the electric eld generated by an electric (resp. magnetic) dipole of moment d e δ ∈ C 3 (resp. d h δ ∈ C 3 ), given by
E elec [d](x) = exp(iκr) r
"
2 r 2 − 2iκ
r
(d · x) b x b
+
− 1 r 2 + iκ
r + κ 2
( x b × d) × x b
# ,
E mag [d](x) = exp(iκr) r
1 r − iκ
(iκd) × x, b
where r = |x| . For the single-scattering case, the dipole moments will depend on the inci- dent eld, size and location of the scatterer.
In particular, for a spherical obstacle, we have d e δ = δ 3 E inc (0) and d h δ = − δ 2
3H inc (0). For the multiple-scattering case, each obstacle ω δ k = c k +δω ( k = 1, . . . , N obs ) is modeled as a dipolar source around its center c k . Following (1), the electric eld is approximated by
N
obsX
k=1
E elec [d e δ,k ](x − c k ) + E mag [d h δ,k ](x − c k ).
WAVES 2019, Vienna, Austria 2
According to Foldy-Lax theory, the dipole mo- ments depend not only on the incident elds, but also on all the other scattered elds, d e δ,k =δ 3 α(δ) n
E inc (c k )+
N
obsX
`=1, `6=k
E elec [d e δ,` ](c k − c ` ) + E mag [d h δ,` ](c k − c ` ) o
,
with a similar expression for the magnetic mo- ments d h δ,k . These expressions lead to a vecto- rial formulation involving 6N obs unknowns. The coecient α(δ) diers with dierent levels of ap- proximation. We dene
α(δ) =
1 : rst Foldy model,
1 + 3(κδ) 2
10 : collected Foldy model, 3i
2(κδ) 3
j 1 (κδ)
h (1) 1 (κδ) : modied Foldy model, where j 1 and h (1) 1 denote the bessel function and the hankel function of the rst kind, of order 1. The Born approximations are dened by ne- glecting the interactions between the obstacles.
Spectral method: the reference solution The electric eld has the integral representation
E δ (x) =
N
obsX
k=1
∇ × Z
∂ω
δkΦ(x, y)p k (y)ds y , x ∈ Ω δ , where Φ(x, y) = exp(iκ|x−y|)
4π|x−y| denotes the Green function associated to the Helmholtz equation.
The tangential densities p k solve the following boundary integral equations
N
obsX
`=1
M k` Γ p ` = −n × E inc on ∂ω k δ , (2) where the magnetic potentials M k` Γ λ are dened as an extension of
n(x Γ ) × lim
x→x
Γ∇ × Z
∂ω
`δΦ(x, y)λ(y) ds y
! , with x Γ ∈ ∂ω δ k . The spectral method [2] con- sists in discretizing (2) into a local spectral basis associated with the vectorial Laplace-Beltrami operator with N mod modes. For spherical obsta- cles, the basis is composed of the vector spheri- cal harmonics ∇ S Y n,m , curl S Y n,m ,
p ` =
N
modX
n=1 n
X
m=−n
p `,⊥ n,m ∇ S Y n,m ` + p `,× n,m curl S Y n,m ` ,
where Y n,m ` ( x) = b Y n,m ( x \ − c ` ) . This formula- tion leads to the linear system developed in [3]
with 2N mod (N mod + 2)N obs degrees of freedom.
The matrix becomes more ill-conditionned as the number of obstacles grows or the size of ob- stacles decreases. We make use of linear algebra tools, preconditionners and iterative solvers to perform simulations with thousands of spheres.
Numerical tests
The asymptotic models are validated with the spectral method, itself validated with nite el- ement solutions provided by Montjoie code, in spherical geometries. Figure 1 shows the per- formance of the reduced models. The incident wave is an electromagnetic plane wave of wave- length λ = 1.0 and the medium contains ve aligned spheres of radius δ varying between 10 −0.5 and 10 −2.75 . The reference solution is the spec- tral solution truncated at the order N mod = 10 .
0 1 2
−1
0
1
x
z
−1 0 1 2 3
(a) Reference (real part)
10−2 10−1
10−5 10−4 10−3 10−2 10−1 100 101
δ λ
1st Born 1st Foldy Col. Born Col. Foldy Mod. Foldy
(b) Distance ∝ λ
10−2 10−1
10−5 10−4 10−3 10−2 10−1 100 101
δ λ
(c) Distance ∝ √ δ
10−2 10−1
10−3 10−2 10−1 100 101
δ λ
