On the scattering of electromagnetic waves by small bodies

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Submitted on 8 Jan 2018

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On the scattering of electromagnetic waves by small bodies

Justine Labat, Victor Péron, Sébastien Tordeux

To cite this version:

Justine Labat, Victor Péron, Sébastien Tordeux. On the scattering of electromagnetic waves by small bodies. 2017. �hal-01677560�

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On the scattering of electromagnetic waves by small bodies

Justine LABAT

PhD student in Applied Mathematics at the University of Pau

EPC Magique 3D, INRIA Bordeaux Sud-Ouest, LMAP UMR CNRS 5142, Universit é de Pau et des Pays de l’Adour Advisors : Victor P ÉRONet S ébastien TORDEUX

Instituto de Matematicas, Pontificia Universidad Cat ´olica de Valpara´ıso, Chile May 5, 2017

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Scattering problems of electromagnetic waves by small inclusionsin 3D

incident field

δ Ωδ⊂R³

δ: small parameter O⁽¹⁾δ , . . . ,O^(N)δ : inclusions Ωδ=R³\SO^(j)_δ Γ^(j)_δ =∂O^(j)δ

Γδ=S Γ^(j)_δ

Reference shapeOb^(j)

1

O^(j)δ =cj+δbO^(j)

Ob^(j)Lipschitz-continuous Asymptotic assumption

δλ^inc

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Motivation−Applications and numerical difficulties

I Applications

I Medical imaging: detecting small tumours

I Non-destructive testing civil engineering: detecting impurities into concrete I Numerical approximation difficulties

I Infinite domain

I Mesh refinement near bodies (pictures from G. Vial)

I Our approach

I Reduced models derived thanks to asymptotic analysis: the radiusδis the small parameter

I Integration in a Boundary Element library (unbounded domain)

Figure:

https://people.eecs.ku.edu/∼

shontz/nsf career project.html

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Non-exhaustive references−Scattering wave problems by small bodies

I Electromagnetic waves

H. Ammari, M. S. Vogelius, D. Volkov (2001)

Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations

H. Ammari, H. Kang (2004)

Reconstruction of small inhomogeneities from boundary measurements

I Linear elasticity problems

V. Bonnaillie-No ¨el, D. Brancherie, M. Dambrine, F. H ´erau, S. Tordeux, G. Vial (2011) Multiscale expansion and numerical approximation for surface defects

I Acoustic waves

V. Mattesi (2014)

Propagation des ondes dans un domaine comportant des petites h ét érog én éit és : mod élisation asymptotique et calcul num érique

A. Bendali, P-H. Cocquet, S. Tordeux (2014)

Approximation by multipoles of the multiple acoustic scattering by small obstacles and application to the Foldy theory of isotropic scattering

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Scattering of electromagnetic waves byonesmall perfect condutor in3D

(E^inc,H^inc) Ωδ

n Oδ

Oδ=δOb: perfect conductor Ωδ=R³\Oδ: exterior domain Γδ=∂Ωδ

κ: wave number

E^inc,H^inc: incident fields (given) E^sca_δ ,H^sca_δ : scattered fields

Time-harmonic Maxwell equations











curl E^scaδ −iκH^scaδ =0 inΩδ

curl H^scaδ +iκE^scaδ =0 inΩδ

n×E^scaδ =−n×E^inc onΓδ

n·H^scaδ =−n·H^inc onΓδ

|x|→+∞lim |x|(H^scaδ ×er−E^scaδ ) =0







κ²=ω²µ

ε+iσ ω

Im(κ)≥0

ω >0: frequency

µ >0: magnetic permeability ε >0: electric permittivity σ≥0: electric conductivity

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Scattering of electromagnetic waves byonesmall perfect condutor in3D

(E^inc,H^inc) Ωδ

n Oδ

Oδ=δOb: perfect conductor Ωδ=R³\Oδ: exterior domain Γδ=∂Ωδ

κ: wave number

E^inc,H^inc: incident fields (given) E^sca_δ ,H^sca_δ : scattered fields

Time-harmonic Maxwell equations











curl E^scaδ −iκH^scaδ =0 inΩδ

curl H^scaδ +iκE^scaδ =0 inΩδ

n×E^scaδ =−n×E^inc onΓδ

n·H^scaδ =−n·H^inc onΓδ

|x|→+∞lim |x|(H^scaδ ×er−E^scaδ ) =0

(curl Eînc−iκHînc=0 inR³ curl Hînc+iκEînc=F inR³

I Fsmoothsource term I 0∈/suppF

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Guideline

Approximation method: Matched Asymptotic Expansions

“Formal” part Analysis part

I Postulate asymptotic expansions I Derive problems independent ofδ I Define a global approximationE^Nδ

I Modal representation of solutions I Validation of reduced models

E^Nδ −E^scaδ

X

=? O

δ→0(δ^N)

Numerical resolution method: Equivalent Source Problem

I Boundary obstacles considered as surface/ponctual source terms

I Implementation of Matched Asymptotic Expansions method

I Comparison with volumical methods

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Outline

1 Method of matched asymptotic expansions Asymptotic domain decomposition Far field expansion

Near field expansion Matching conditions Global approximation

2 Modal decomposition of the asymptotics Modal decomposition of the far field Modal decomposition of the near field Application of the matching conditions Case of the sphere: analytical solutions

3 Conclusion and perspectives

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Outline

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Asymptotic domain decomposition

Far field domain Near field domain

η_δ^F

η_δ^N

δ→0limηδ^F=0

δ→0lim η_δ^N

δ = +∞

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Asymptotic domain decomposition

Far field domain Matching area

η_δ^F

η^N_δ η^F_δ

δ→0limηδ^F=0

δ→0lim η_δ^N

δ = +∞

Asymptotic assumption δη^Fδ< ηδ^Nλ^inc

A.M. Il’in (1991)

Matching of Asymptotic Expansions of Solutions of Boundary Value Problems

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Asymptotic domain: far field

Far field domain

η^F_δ

δ−→0

Asymptoticfar field domain

R³\B(0, η_δ^F) Ω^?=R³\ {0}

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Asymptotic domain: near field

Near field domain Change of coordinates

Oδ

η^N_δ

Oδ=δOb bx=x

δ

δ→0

Ob

η^N_δ δ

Ob

Asymptoticnear field domain Ω =b R³\bO

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Far field expansion inΩ^? =R³\ {0}

1. ^Postulateformal series

E^scaδ (x) ∼

δ→0

∞

X

p=0

δ^pEep(x) H^scaδ (x) ∼

δ→0

∞

X

p=0

δ^pHep(x)

I Eep,Hepdefined inΩ^? I Eep,Hepindependent ofδ

I Eep,Heppotentially singular atx=0

2. Deriveformal problems: Forp≥0, findEep,Hep∈Hloc(curl,Ω^?)s.t.

(curlEep−iκHep=0 inΩ^? curlHep+iκeEp=0 inΩ^? + Silver-M ¨uller radiation condition

|x|→+∞lim |x|

Hep×er−Eep

=0

I Ill-posed problems

I Missing information matching conditions I Singular behavior atx=0

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E^scaδ (x) ∼

δ→0

∞

X

p=0

δ→0

∞

X

p=0

δ^pHep(x)

|x|→+∞lim |x|

Hep×er−Eep

=0

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E^scaδ (x) ∼

δ→0

∞

X

p=0

δ→0

∞

X

p=0

δ^pHep(x)

|x|→+∞lim |x|

Hep×er−Eep

=0

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Near field expansion inΩ =b R³\Ob 1. Postulateformal seriesin fast variablebx=^x_δ

E^scaδ (x) ∼

δ→0

∞

X

p=0

δ^pEbp(bx) H^scaδ (x) ∼

δ→0

∞

X

p=0

δ^pHbp(bx)

I Ebp,Hbpdefined inΩb I Ebp,Hbpindependent ofδ

I Ebp,Hbppotentially increasing at∞

E^inc(x) =

∞

X

p=0

δ^p X

|α|=p

1

α!d^αE^inc(0)·bx^α

| {z }

bE^inc_p(bx)

2. ^Deriveformal problems: Forp≥0findEbp,Hbp∈Hloc(curl,Ω)b s.t.

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

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



curlbEp−iκHbp−1=0 inΩb curlHbp+iκEbp−1=0 inΩb divEbp=divHbp=0 inΩb (bE−1=Hb−1=0)+ Boundary condition onbΓ =∂Ωb







n×Ebp=−n×bE^incp onΓb n·Hbp=−n·Hb^inc_p onΓb

I Missing information matching conditions I Increasing behavior at∞

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E^scaδ (x) ∼

δ→0

∞

X

p=0

δ→0

∞

X

p=0

δ^pHbp(bx)

E^inc(x)=

∞

X

p=0

δ^p X

|α|=p

1

| {z }

bE^inc_p(bx)

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











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E^scaδ (x) ∼

δ→0

∞

X

p=0

δ→0

∞

X

p=0

δ^pHbp(bx)

E^inc(x) =

∞

X

p=0

δ^p X

|α|=p

1

| {z }

bE^inc_p(bx)

















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Some notations

Far field decomposition

Eep(x) =Ee^regp (x) +Ee^singp (x) x∈Ω^? Hep(x) =He^regp (x) +He^singp (x) x∈Ω^? I Ee^reg_p ,He^reg_p : regular behavior atx=0(regular partof the far field) I Ee^singp ,He^singp : singular behavior atx=0(singular partof the far field) Near field decomposition

bEp(bx) =Eb^regp (bx) +bE^singp (bx) bx∈Ωb Hbp(bx) =Hb^regp (bx) +Hb^singp (bx) bx∈Ωb

I Eb^reg_p ,Hb^reg_p :“decreasing” behavior at∞(decreasing partof the near field) I Eb^singp ,Hb^singp : increasing behavior at∞(increasing partof the near field) Be careful. Weighted spaces X. Claeys, PhD thesis (2008)

Analyse asymptotique et num ´erique de la diffraction d’ondes

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Matching conditions

Find missing information

I Singular part of the far field ←−Decreasing part of the near field I Increasing part of the near field←−Regular part of the far field

Idea: developeEpclose tox=0andEbpnear∞

eEp(x) ∼

r→0

−1

X

`=−∞

r^`eE^(p)` (θ, ϕ) +

+∞

X

`=0

r^`eE^(p)` (θ, ϕ) x= (r, θ, ϕ)

bEp(bx) ∼

R→∞

−1

X

`=−∞

R^`Eb^(p)` (θ, ϕ) +

+∞

X

`=0

R^`bE^(p)` (θ, ϕ) bx= (R, θ, ϕ) =r δ, θ, ϕ

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Matching conditions

Idea

I Far field expansion inV(0)

E^scaδ (x) ∼

δ→0 +∞

X

p=−∞

δ^p

−1

X

`=−∞

r^`eE^(p)` (θ, ϕ) +

+∞

X

`=0

r^`Ee^(p)` (θ, ϕ)

!

I Near field expansion inV(∞)withR=_δ^r

E^scaδ (x) ∼

δ→0 +∞

X

p=−∞

δ^p

−1

X

`=−∞

R^`bE^(p)` (θ, ϕ) +

+∞

X

`=0

R^`Eb^(p)` (θ, ϕ)

!

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Matching conditions

Idea

I Far field expansion inV(0)

E^scaδ (x) ∼

δ→0 +∞

X

p=−∞

δ^p

+∞

X

`=−∞

r^`Ee^(p)` (θ, ϕ)

!

I Near field expansion inV(∞)withR=_δ^r

E^scaδ (x) ∼

δ→0 +∞

X

p=−∞

δ^p

+∞

X

`=−∞

R^`Eb^(p)` (θ, ϕ)

!

I Identificationof the two series







eE^(p)` =Eb^(p+`)` p≥0, `=−p, . . . ,−1 bE^(p)` =Ee^(p−`)` p≥0, `=0, . . . ,p







eE^(p)` =0 ` <−p bE^(p)` =0 ` >p

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Matching conditions

I Identificationof the two series







eE^(p)` =Eb^(p+`)` p≥0, `=−p, . . . ,−1 bE^(p)` =Ee^(p−`)` p≥0, `=0, . . . ,p







eE^(p)` =0 ` <−p bE^(p)` =0 ` >p

I Term by term Eep(x) ∼

r→0

−1

X

`=−p

r^`bE^(p+`)` (θ, ϕ)+

+∞

X

`=0

r^`eE^(p)` (θ, ϕ) (far field)

Ebp(bx) ∼

R→∞

−1

X

`=−∞

R^`bE^(p)` (θ, ϕ) +

p

X

`=0

R^`Ee^(p−`)` (θ, ϕ) (near field)

I Forp=0,eE0isonlyregular

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Global approximation Truncated series

Ee^Nδ(x) =

N

X

p=0

δ^peEp(x) He^Pδ(x) =

N

X

p=0

δ^pHep(x) (far fields)

Eb^Nδ(x) =

N

X

p=0

δ^pEbp(bx) Hb^Nδ(x) =

N

X

p=0

δ^pHbp(bx) (near fields)

Global approximation

E^Pδ(x) =χδ(|x|)bE^Pδ(x) + 1−χδ(|x|)

Ee^Pδ(x) x∈Ωδ

H^Pδ(x) =χδ(|x|)Hb^Pδ(x) + 1−χδ(|x|)

He^Pδ(x) x∈Ωδ

1

η_δ^F η_δ^N r

χδ(r)

Consistance and stabilityDetermineφs.t.

φ(N) −→

N→∞+∞and for anyN>0,ρ > ρ0

E^scaδ −E^Nδ

H(curl,Ω_δ∩B(0,ρ))≤C^EN,ρδ^φ(N)

H^scaδ −H^Nδ

H(curl,Ω_δ∩B(0,ρ))≤C^HN,ρδ^φ(N)

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Global approximation Truncated series

Ee^Nδ(x) =

N

X

p=0

δ^peEp(x) He^Pδ(x) =

N

X

p=0

δ^pHep(x) (far fields)

Eb^Nδ(x) =

N

X

p=0

δ^pEbp(bx) Hb^Nδ(x) =

N

X

p=0

δ^pHbp(bx) (near fields)

Global approximation

E^Pδ(x) =χδ(|x|)bE^Pδ(x) + 1−χδ(|x|)

Ee^Pδ(x) x∈Ωδ

H^Pδ(x) =χδ(|x|)Hb^Pδ(x) + 1−χδ(|x|)

He^Pδ(x) x∈Ωδ

1

η^F η^N r

χδ(r)

Consistance and stabilityDetermineφs.t.

φ(N) −→

N→∞+∞and for anyN>0,ρ > ρ0

E^scaδ −E^Nδ

H(curl,Ω_δ∩B(0,ρ))≤C^EN,ρδ^φ(N)

H^scaδ −H^Nδ

≤C^HN,ρδ^φ(N)

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Outline

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Modal decomposition of the far field of order 0

FindeE0andHe0∈Hloc(curl,R³)s.t.











curlEe0−iκHe0=0 inR³ curlHe0+iκeE0=0 inR³

|x|→∞lim |x|

He0×er−eE0

=0

I Ee0,He0areregular(no singularity atx=0)

I Ee0,He0satisfy S.M.=⇒eE0=0andHe0=0

I Far field expansion

E^scaδ (x)∼

∞

X

p=1

δ^pEep(x)

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Modal decomposition of the far field of orderp>0

FindeEpandHep∈Hloc(curl,Ω^?)s.t.











curlEep−iκHep=0 inΩ^? curlHep+iκEep=0 inΩ^? lim

|x|→∞|x|

Hep×er−Eep

=0

I Eep,Heparesingularatx=0

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Modal decomposition of the far field of orderp>0

FindeEpandHep∈Hloc(curl,Ω^?)s.t.











curlEep−iκHep=0 inΩ^? curlHep+iκEep=0 inΩ^? lim

|x|→∞|x|

Hep×er−Eep

=0

I Eep,Heparesingularatx=0 I Modal decomposition (singular at 0)

Eep(x) =

∞

X

n=1 n

X

m=−n

(

βe^(p),n,m^Ecurl

h⁽¹⁾n (κ|x|)Yn,m(bx)x

+βe^(p),n,m^H

iκ curl curl h

h⁽¹⁾_n (κ|x|)Yn,m(bx)xi )

I Finite numberofβe^(p),n,m^E,βe^(p),n,m^Hnon-equal to 0 I βen,m^(p),^E,βen,m^(p),^Hare given bymatching conditions I h⁽¹⁾: spherical Hankel function of first kind