Fast calculation of electromagnetic scattering in anisotropic multilayers and its inverse problem

(1)

HAL Id: hal-01101753

https://hal-supelec.archives-ouvertes.fr/hal-01101753

Submitted on 9 Jan 2015

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-

entific research documents, whether they are pub-

lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Fast calculation of electromagnetic scattering in

anisotropic multilayers and its inverse problem

Giacomo Rodeghiero, Ping-Ping Ding, Yu Zhong, Marc Lambert, Dominique

Lesselier

To cite this version:

Giacomo Rodeghiero, Ping-Ping Ding, Yu Zhong, Marc Lambert, Dominique Lesselier. Fast calcu-

lation of electromagnetic scattering in anisotropic multilayers and its inverse problem. ENDE 2014,

Jun 2014, Xi’an, China. �hal-01101753�

(2)

Fast Calculation of Electromagnetic Scattering in Anisotropic Multilayers

and its Inverse Problem

P.-P. Ding¹ G. Rodeghiero¹ Y. Zhong^2,3 M. Lambert¹ D. Lesselier¹ Cooperative works among

1Laboratoire des Signaux et Systèmes UMR 8506 (CNRS-SUPELEC-Univ. Paris-Sud)

2presently in Institute of High Performance Computing, Singapore

3formerly in National University of Singapore (NUS)¹

ENDE Xi’an June 2014

1. Special thanks to X. Chen from NUS

Ding, Rodeghiero, Zhong, Lambert, Lesselier (L2S)Direct and Inverse Problem in Anisotropic Multilayers ENDE Xi’an June 2014 1 / 24

(3)

Electromagnetic modeling of fiber-based composite structures

ñ

Motivation

Accurate computational models of complex anisotropic multilayered composite structures Robust, fast, end-user’s friendly imaging procedures

Forward modeling

¯¯r“diag “¯¯r11 ¯¯r22 ¯¯r22‰

Figure: Damaged structure with uniaxial dielectric

(glass-based) or conductive (graphite-based) multilayers Figure: The transformation between the local and global coordinates by rotation matrix

(4)

Forward modeling

EM response of anisotropic multilayers to distributed sources

The state equation d

dzϕpzq¯ =¯¯An¨ϕ pzq ` ¯¯ f pzq based on the 4-component vector :

¯

ϕ pkx, ky, zq “

»

—

— –

kxH˜xpkx, ky, zq ` kyH˜ypkx, ky, zq kyH˜xpkx, ky, zq ´ kxH˜ypkx, ky, zq kx˜Expkx, ky, zq ` ky˜Eypkx, ky, zq ky˜Expkx, ky, zq ´ kx˜Eypkx, ky, zq

fi ffi ffi ffi ffi ffi fl

The solution of the state equation

¯

ϕ pdn`1q “ e^¯¯Aⁿ^pδⁿ^q¨ϕ pd¯ nq ` ż_d_n`1

d_n

e^¯¯Aⁿp^δn¹q ¨ ¯fpz¹qdz¹

(5)

EM response of anisotropic multilayers to distributed sources

The new recurrence equation :

¯

ϕ pdnq “ ¯¯Ωn¨

„ αn

βn



New recurrence relations based on the propagator matrix method To efficiently calculate the spectral response of the laminate Capable of stably dealing with distributed source along z

More efficient compared to the traditional Green’s function method

To numerically solve the state equation containing the tangential components of the fields

(6)

Forward modeling

Two methods for calculating the scattered field

1^st method

Induced current integral equation (ICIE) windowing technique

Padua interpolation-integration technique

2^nd method

Lippman-Schwinger integral formula Polarization tensor

Padua interpolation-integration technique

Figure: Damaged structure with uniaxial dielectric (glass-based) or conductive (graphite-based) multilayers

(7)

Discretization of the integral equation - 1

^st

method

Induced current integral equation (ICIE)

¯¯χprq ¨ E^incprq “ Iprq

´i ω₀´ ¯χprq ¨ iωµ¯ ₀

ż ¯¯Gpr; r¹q ¨ Ipr¹qd r¹

By MoM

ř3

v “1χu,v;m,n,pE_v^inc_;m,n,p“ ´1 i ω₀

ř

m¹,n¹,p¹

I_m^puq1,n¹,p¹ψ^puq_m1,n¹,p¹prm,n,pq

´ ř3

v “1χu,v;m,n,p

ř3 κ“1

ř

p¹

ř

m¹,n¹

ηv κ;p;p¹;pm´m¹q,pn´n¹qI_m^pκq1,n¹,p¹

By fast Fourier transform (FFT)

ξ_{v κ}_;p;p1“ ÿ

m¹,n¹

η_{v κ}_;p;p1;pm´m¹q,pn´n¹qI_m^pκq₁_,n₁_,p₁“ IFFT

"

FFT

! η_{v κ}_;p;p1

) b FFT

"

p0qI^pκq_p1

**

(8)

Forward modeling

The techniques for calculating the impedance matrix - 1

^st

method

The impedance matrix :

η_{v κ}_;p;p1px , y q “ i ωµ₀ş

DGv κrpx ´ x¹q, py ´ y¹q; zp; z¹s ψ^pκq_0,0,p₁pr¹qd r¹

After detouring the integral path to avoid branch cuts or singularites for the integral involving rη_{v κ}_;p;p1pkx, kyq, the DFT of η¹v κ;p;p¹(the windowed version of ηv κ;p;p¹) can be constructed as

ˆη¹_{v κ;p;p}1;α,β “_4π₂_∆x∆y¹ exp´ i_M^2πα

t∆xxs` i_N^2πβ

t∆yys

¯

ť_`8

´8d σ_x^Rd σ^R_y ηr_{v κ}_;p;p1“σxpσ^R_x, σ_y^Rq, σypσ_x^R, σ^R_yq‰ ˇ ˇ ˇ ˇ ˇ

Bpσx, σyq Bpσ_x^R, σ_y^Rq

ˇ ˇ ˇ ˇ ˇ

ˆ

# `8

ř

l₁,l₂“´8

eⁱ^2πrl¹^px^s^{^∆xq`l²^py^s^{^∆yqswr

”2π

∆x

´α Mt` l₁

¯

´ σx`σ^R_x, σ^R_y˘ ,_∆y^2π´

β Nt` l₂

¯

´ σy`σ_x^R, σ_y^R˘ı +

2-D Hamming window

w px , y q “

"

0.54 ` 0.46 cos„ px ´∆x{2qπ a

*

ˆ

"

0.54 ` 0.46 cos„ py ´∆y{2qπ b

*

(9)

Padua interpolation-integration technique - calculating fast oscillating

integrals

The goal is to compute the I-FT of fast oscillating spectrum in the k^x´ ky plane G px , y q “ 1

4π²

8

ĳ

´8

G˜₀pkx, kyqep^ik^x^{x `ik}^y^yq dk^xdk^y

Interpolation of the non-oscillating part at the Padua points with Chebyshev’s polynomial interpolant

LnG˜₀pkx, kyq “

n

ÿ

k“0 k

ÿ

j“0

cj, k´jTˆjpkxq ˆTk´jpkyq ´1

2cn,0Tˆnpkxq T₀pkyq with weights cj, k´j computed using²

Fourier transform of Chebyshev polynomials given by ż₁

´1

Tˆnpkxqexpp´ik^xx qdk^x are managed using³

2. M. Caliari, S. De Marchi et al., Numer Algorithms, 56, 45-60, 2011 3. V. Dominguez, I. G. Graham et al., IMA J Numer Anal, 31, 1253-1280, 2011

(10)

Forward modeling

Padua interpolation-integration technique - calculating fast oscillating

integrals

Alternative representation as self intersections and boundary contacts of the generating curve γ ptq “ p´cos ppn ` 1q tq , ´ cos pntq , t P r0, πsq

5th European Congress of Mathematics - Amsterdam, July 14-18 2008

Near-optimal interpolation and quadrature

in two variables: the Padua points

M. C ÂLIARI â , S. D Ê M ÂRCHI â , A. S ÔMMARIVA ^b ÂND M. V ÎANELLO ^b

a

Department of Computer Science, University of Verona, ITALY

b

Department of Pure and Applied Mathematics, University of Padua, ITALY

Abstract

The Padua points, recently studied during an international

collaboration at the University of Padua, are the first known

near-optimal point set for bivariate polynomial interpolation

of total degree. Also the associate algebraic cubature formulas

are both, in some sense, near-optimal.

Which are the Padua points?

Chebyshev-Lobatto points in [ −1, 1]

C

_n+1

:= {cos(jπ/n), j = 0, . . . , n}

card(C

n+1

) = n + 1 = dim(P

¹n

)

Padua points in [ −1, 1] × [−1, 1]

Pad

n

:= (C

_n+1^odd

× C

n+2^even

) ∪ (C

n+1^even

× C

n+2^odd

) ⊂ C

ⁿ⁺¹

× C

ⁿ⁺²

card(Pad

n

) = (n + 1)(n + 2)

2 = dim(P

²n

)

Alternative representation as self-intersections and boundary

contacts of the generating curve

g(t) := ( − cos((n + 1)t), − cos(nt)) , t ∈ [0, π]

F

IGURE

1. The

Padua points

with their

generating curve

for n = 12 (left, 91

points) and n = 13 (right, 105 points), also as

union of two Chebyshev-Lobatto subgrids

(red and blue bullets).

-1 -0,5 0 0,5 1

There are 4 families of such points, corresponding to succes-

sive rotations of 90 degrees.

Interpolation at the Padua points

trigonometric quadrature on the generating curve

⇓

algebraic cubature at the Padua points ξ ∈ Pad

ⁿ

for the product Chebyshev measure

dµ = dx

1

dx

2

π

²

!

(1 − x

²1

)(1 − x

²2

)

with weights

w

ξ

:= 1

n(n + 1) ·

 



 

1/2 if ξ is a vertex point

1 if ξ is an edge point

2 if ξ is an interior point

near-exactness in P

²2n

namely, exactness in P

²2n

∩ (span{T

2n

(x

₁

) })

^⊥^dµ

⇓

Lagrange interpolation formula

L

Padn

f (x) = &

ξ∈Padn

f (ξ) L

ξ

(x) , L

ξ

(η) = δ

ξη

L

ξ

(x) = w

ξ

(K

n

(ξ, x) − T

n

(ξ

1

)T

n

(x

1

))

T

n

( ·) = cos (n arccos (·)) and K

ⁿ

(x, y) reproducing kernel of

the product Chebyshev orthonormal basis

Θ

_j

(x) = T

_j^∗₁

(x

₁

) T

_j^∗₂

(x

₂

) , j = (j

₁

, j

₂

) , 0 ≤ |j| = j

1

+ j

₂

≤ n

K

n

(x, y) = &

|j|≤n

Θ

j

(x) Θ

j

(y)

The Lebesgue constant

Unisolvence gives the projection operator

L

Padn

: C([ −1, 1]

²

, ) · )

∞

) → P

²n

([ −1, 1]

²

, ) · )

∞

)

Theorem (interpolation stability, cf. [1])

)L

Padn

) = max

x∈[−1,1]²

&

ξ∈Padn

|L

ξ

(x) | = O(log

²

(n))

i.e., the Lebesgue constant has optimal order of growth [9].

Conjecture

x∈[−1,1]

max

²

&

ξ∈Padn

|L

^ξ

(x) | ! 4

π

²

log

²

(n)

and the max is attained at one of the vertices of the square.

Classical convergence estimate

)L

^Padn

f −f)

∞

≤ (1+)L

^Padn

)) min

p∈P²n

)f − p)

∞

= o(n

^−p

log

²

(n))

for f ∈ C

^p

([ −1, 1]

²

), p > 0

Numerical implementation

Representation of the interpolant at the Padua points in the

product Chebyshev orthonormal basis

L

^Padn

f (x) = &

|j|≤n

c

^'_j

Θ

j

(x)

with coefficients

c

^'_j

= c

j

:= &

ξ∈Padn

w

ξ

f (ξ) Θ

j

(ξ) , c

^'_(n,0)

= c

(n,0)

2 which are approximate Fourier-Chebyshev coefficients

(with a correction for j = (n, 0))

Two different fast algorithms (cf. [3, 5]) using

• optimized matrix subroutines

• 2-dimensional Fast Fourier Transform

(competitive and more stable at large degrees)

Other features

• extension to interpolation over rectangles

• practical convergence estimate

)L

^Padn

f − f)

∞

"

&

n

|j|=n−2

|c

^'j

| )Θ

^j

)

∞

≤ 2

&

n

|j|=n−2

|c

^'j

|

Numerical results on interpolation

F

IGURE

2. Six test functions from the Franke-Renka-Brown test set [7];

top:F1, F2, F3;bottom:F5, F7, F8.

0.2 0 0.6 0.4 1 0.8 0

0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6

0.8 1

0 0.5 1 0 0.05 0.1 0.15 0.2 0.25

0 0.5

10 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

0.5

1

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0 0.2 0.4 0.6

0.8 1

0 0.5 1

−2

−1 0 1 2 3

0 0.2 0.4 0.6

0.8 1

0 0.5 1 0 0.5 1 1.5 2 2.5

T^ABLE1.

True and (estimated) interpolation errors for the test functions above.

n card(Padn) F1 F2 F3 F5 F7 F8

10 66 9E-2 4E-1 8E-3 4E-2 3E-1 1E-1

(2E-1) (6E-1) (6E-2) (2E-1) (1E+0) (4E-1)

20 231 7E-3 6E-2 1E-5 6E-5 8E-6 3E-3

(2E-2) (8E-2) (8E-5) (8E-4) (2E-4) (1E-2)

30 496 1E-4 1E-2 2E-8 1E-8 7E-13 2E-5

(8E-4) (1E-2) (1E-7) (2E-7) (2E-11) (1E-4)

40 861 3E-6 2E-3 2E-11 4E-13 4E-14 6E-8

(1E-5) (2E-3) (2E-10) (2E-11) (8E-15) (6E-7)

50 1326 1E-8 4E-4 1E-13 1E-15 7E-14 5E-11

(8E-8) (4E-4) (4E-13) (1E-15) (1E-14) (6E-10)

Nontensorial Clenshaw-Curtis cubature

• integration of the interpolant at the Chebyshev-Lobatto

points gives the 1D Clenshaw-Curtis quadrature formula

• integration of the interpolant at the Padua points gives a 2D

nontensorial Clenshaw-Curtis cubature formula

I(f ) :=

''

[−1,1]²

f (x) dx ≈ I

^Padn

(f ) :=

''

[−1,1]²

L

^Padn

f (x) dx

= &

|j|≤n

c

^'_j

m

j

= &

ξ∈Padn

λ

ξ

f (ξ) , λ

ξ

:= w

ξ

&

|j|≤n

m

^'_j

Θ

j

(ξ)

(exact for f ∈ P

²n

), via the Chebyshev moments

m

j

:=

''

[−1,1]²

Θ

j

(x) dx =

'

1

−1

T

_j^∗₁

(t) dt

'

1

−1

T

_j^∗₂

(t) dt

The cubature weights {λ

ξ

} are not all positive, nevertheless

Theorem (cubature stability and convergence, cf. [8])

n

lim

→∞

)I

^Padn

) = lim

_n

→∞

&

ξ∈Padn

|λ

^ξ

| = area of the square = 4

and I(f ) = I

Padn

(f ) + o(n

^−p

) for f ∈ C

^p

([ −1, 1]

²

), p ≥ 0.

Numerical results on cubature

• near-optimality: on non-entire integrands, I

Pad_n

performs

better than tensor-product Gauss-Legendre quadrature and

even than the few known minimal formulas, see Fig. 3 right

(similar to the well-known 1D phenomenon studied in [10])

• fast algorithm: computing the weights by a suitable matrix

formulation (cf. [3]) is very fast in Matlab/Octave (e.g. less

than 0.01 sec at n = 100 on a 2.4Ghz PC)

F

IGURE

3.

Cubature errors

of tensorial and nontensorial cubature formulas

versus the

number of evaluations: CC=Clenshaw-Curtis, GL=Gauss-Legendre,

MPX=Morrow-Patterson-Xu [11], OS=Omelyan-Solovyan (minimal) [6].

Left:f = (x1+ x2)²⁰

; f = exp (x

1+ x2); f = exp (−(x²1+ x²₂))

Right:f = 1/(1 + 16(x²₁+ x²₂)); f = exp (−1/(x²1+ x²₂)); f = (x²₁+ x²₂)^3/2

0 100 200 300 400 500 600

10⁻¹⁶ 10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

Number of points

Relative Cubature Error

OS pts. Non tens. CC MPX pts. Non tens. CC Padua pts. Tens. CC pts. Tens. GL pts.

0 100 200 300 400 500 600

10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Number of points

0 100 200 300 400 500 600

10⁻¹⁸ 10⁻¹⁶ 10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴

Number of points

0 100 200 300 400 500 600

10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

Number of points

0 100 200 300 400 500 600

10⁻¹⁶ 10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

Number of points

0 100 200 300 400 500 600

10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

Number of points

References

[1] L. BOS, M. CALIARI, S. DEMARCHI, M. VIANELLO ANDY. XU, Bivariate Lagrange in- terpolation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006). [2] L. BOS, S. DEMARCHI, M. VIANELLO ANDY. XU, Bivariate Lagrange interpolation at the

Padua points: the ideal theory approach, Numer. Math. 108 (2007).

[3] M. CALIARI, S. DEMARCHI, A. SOMMARIVA ANDM. VIANELLO, Fast interpolation and cubature at the Padua points in Matlab/Octave, in preparation.

[4] M. CALIARI, S. DEMARCHI ANDM. VIANELLO, Bivariate polynomial interpolation on the square at new nodal sets, Appl. Math. Comput. 162 (2005).

[5] M. CALIARI, S. DEMARCHI ANDM. VIANELLO, Padua2D: Lagrange Interpolation at Padua Points on Bivariate Domains, ACM Trans. Math. Software 35-3 (2008).

[6] I.P. OMELYAN ANDV.B. SOLOVYAN, Improved cubature formulae of high degree of exactness for the square, J. Comput. Appl. Math. 188 (2006).

[7] R.J. RENKA ANDR. BROWN, Algorithm 792: Accuracy tests of ACM algorithms for interpo- lation of scattered data in the plane, ACM Trans. Math. Software 25 (1999).

[8] A. SOMMARIVA, M. VIANELLO ANDR. ZANOVELLO, Nontensorial Clenshaw-Curtis cu- bature, Numer. Algorithms, published online 27 May 2008.

[9] L. SZILI ANDP. VERTESI, Some New Theorems on Multivariate Projection Operators, C.R. Acad. Bulgare Sci. 61 (2008).

[10] L.N. TREFETHEN, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev. 50 (2008). [11] Y. XU, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87

(1996). Figure: The Padua points with their generating curve for n “ 12 (left, 91 points) and n “ 13 (right, 105

points), also as union of two Chebyshev-Lobatto sub-grids (red and blue bullets). Image taken from⁵ M. Caliari, S. De Marchi et al., 5th European Congress of Mathematics - Amsterdam, July, 2008

(11)

Fast calculation of EM scattering problems - 2

^nd

method

The Lippman-Schwinger integral formulation E^scaprq “ i ωµ₀

ż ¯¯G^ee`r, r¹˘

¨ ¯χ¯`r¹˘

¨ E^tot`r¹˘ dr¹ where the contrast function

¯¯χ prq “ ´iω0¨`

¯¯i´ ¯¯b

˘

¯¯i is the permittivity tensor of an inclusion in the composite medium.

¯¯bis the permittivity tensor of the composite medium in the global coordinate system.

The incident field is defined as

E^incprq “ i ωµ₀

ż ¯¯G^ee`r, r¹˘

¨ J₀`r¹˘ dr¹

“ i ωµ₀¯¯G^ee`r, r¹˘

¨ βIl

(12)

Forward modeling

Asymptotic formulation with polarization tensor - 2

^nd

method

The Lippman-Schwinger integral formulation

When permittivity tensor of the inclusion ¯¯ⁱ“ ¯¯Ii and its size is small enough, the scattered field can be derived into the asymptotic formulation.

E^scaprq “ i ωµ₀¯¯G^eepr, rmq ¨ ¯¯% ¨ E^incprmq

The polarization tensor

The polarization tensor in the local coordinate for the inclusion with volume V is :

¯¯% “ ´iω0V ¨

» –

αl 0 0

0 αt 0

0 0 αt

fi fl

α_l“ ₁₁ i´ ₁₁

₁₁` Llpi´ ₁₁q αt“ ₂₂ i´ ₂₂

₂₂` Ltpi´ ₂₂q Ll and L^t are dependent on the shape of inclusion.

For a cubic inclusion, Ll“ carctan `c {?

1 ` 2c˘ and Lt“ p1 ´ Llq {2, where c “ 11{ ₂₂⁶.

6. W. S. Weiglhofer and A. Lakhtakia, Int. J. Infrared Millimeter Waves, 17, 1365–1376, 1996.

(13)

Comparison of the scattered field by MoM and COMSOL - 1

^st

Case

Figure: The Configuration of the 1^stCase⁷

freq=6 GHz

¯¯2“ ₀diag r3, 2, 2s, rotation angle θ₂“30^˝

¯¯3“ ₀diag r4, 2.5, 2.5s, rotation angle θ₃“60^˝

7. Y. Zhong, P.-P. Ding, M. Lambert, D. Lesselier, and X. Chen, Fast calculation of scattering by 3-D inhomogeneities in uniaxial anisotropic multilayers, submitted to IEEE Trans. Antennas Propagat., 2014.

(14)

Numerical examples

Comparison of the scattered field by MoM and asymptotic method - 2

^nd

Case

Figure: The Configuration of the 2^nd Case

freq=3 GHz

¯¯1“

₀diag r2 ` i0.3, 3 ` i0.1, 3 ` i0.1s rotation angle θ “ 60°

11 electric dipole sources with x/y/z polarization a cubic air scatterer with side length 0.1λ0

Figure: The scattered electric field with sources in x/y/z-polarization

(15)

Comparison of the scattered field by MoM and asymptotic method - 3

^rd

Case

Figure: The Configuration of the 3^rdCase freq “ 3 GHz

d₁“0.5λ0; d2“0.2λ0; d3“0.35λ0; d4“0.2λ0

layer 1 : ¯¯1“ ₀diag r2 ` i0.3, 3 ` i0.1, 3 ` i0.1s ; rotation angle θ1“60°

layer 2 :

¯¯2“ ₀diag r4.5 ` i0.2, 6 ` i0.05, 6 ` i0.05s ; rotation angle θ2“45°

11 electric dipole sources with x polarization

a cubic air scatterer with side length 0.1λ0 Figure: The scattered electric field with the x-polarization source located at p0, 0, 0.5λ0q

(16)

Inverse problem

MUSIC (MUltiple SIgnal Classification) imaging method

Singular value decomposition

K “ U S V^˚

Multistatic response matrix K maps the currents at the source locations to the scattered fields measured at the detectors.

Subspace is spanned by the singular vectors corresponding to different singular values.

Figure: Damaged structure with uniaxial dielectric (glass-based) or conductive (graphite-based) multilayers

(17)

MUSIC (MUltiple SIgnal Classification) imaging method

Formulations

Standard MUSIC imaging method⁸

φprq “ 1

ř

σ_jăσL

ř₃

v “1

ˇ ˇ

ˇ¯u_j^˚¨ ¯Gvprq ˇ ˇ ˇ

2

Enhanced MUSIC imaging method⁹

φprq “ 1

1 ´ řσ_jąσL

ˇ ˇ

ˇ¯u^˚_j ¨ ¯Gprq ¨¯ ¯atest

ˇ ˇ ˇ

2

with

¯atest“arg max

¯a

ř

σ_jąσL

ˇ ˇ

ˇ¯u_j^˚¨ ¯Gprq ¨¯ ¯aˇ ˇ ˇ

2

ˇ ˇ ˇ¯¯Gprq ¨¯aˇ

ˇ ˇ

2

Choice of σL Exemple for two scatterers T “10³

σi being ordered from the largest to the lowest value

A threshold T arbitrarily chosen σL“ T ˆmax

i pσiq (1)

10⁰ 10¹ 10² 10³

10⁻¹⁵ 10⁻¹⁰ 10⁻⁵ 10⁰ 10⁵

Noiseless 30 dB noise

Figure:

8. H. Ammari et al., SIAM J. Sci. Comput., 29,674-709, 2007 9. X. Chen and Y. Zhong, Inverse Problems, 25, 085003, 2009

(18)

Inverse problem

Configuration of interest

(a)Top view (b)Side view (c)3D view

Figure: Testing configuration

Description of the configuration Measurement configuration

location of sources = location of receivers (x/y/z polarization)

N_receivers“13 ˆ 13 zposition of receivers : 0.5λ0

range of receivers in x-y plane : r´3λ0,3λ0s

Region of interest (ROI) N_cells“21 ˆ 21 ˆ 21

range of ROI along z : r´2.35λ0, ´0.35λ0s range of ROI in x-y plane : r´λ0, λ₀s

(19)

The 1

^st

case - one scatterer

plane cut representation

z = -0.35λ0 z = -0.45λ0 z = -0.55λ0 z = -0.65λ0 z = -0.75λ0

z = -0.85λ0 z = -0.95λ0 z = -1.05λ0 z = -1.15λ0 z = -1.25λ0

z = -1.35λ0 z = -1.45λ0 z = -1.55λ0 z = -1.65λ0 z = -1.75λ0

z = -1.85λ0 z = -1.95λ0 z = -2.05λ0 z = -2.15λ0 z = -2.25λ0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z = -2.35λ0

(20)

Inverse problem

The 1

^st

case - one scatterer

Isosurface = 0.5

Figure: Imaging results by MUSIC for noiseless case (left figure) and 30 dB noisy case (right figure)

Figure: Influence of the choice of the isosurface

(21)

The 2

^nd

case - two scatterers

Isosurface = 0.5

(22)

Inverse problem

The 3

^rd

case - three scatterers

Isosurface = 0.1

(23)

The 4

^th

case - four scatterers

Isosurface = 0.1

(24)

Conclusions

Achievements

Generalized and complete formulation of EM response & Green dyads for undamaged anisotropic multilayers (work at the whole range of frequency for all materials)

Asymptotic formula-based calculation of EM response for 3-D damaged uniaxial multilayers

Challenges

To speed up the interpolation and integration for large source-receiver arrays To handle delaminations at the interfaces as thin planar defects

To extend NdT to detecting larger defects by using MUSIC as the first localization tool for non-linearized procedures

To check the size range of defects possibly detected by the first-order modeling To put more endeavor on experiments for practical applications

(25)

Fast calculation of electromagnetic scattering in anisotropic multilayers and its inverse problem

HAL Id: hal-01101753

https://hal-supelec.archives-ouvertes.fr/hal-01101753

Submitted on 9 Jan 2015

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-

entific research documents, whether they are pub-

lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Fast calculation of electromagnetic scattering in

anisotropic multilayers and its inverse problem

Giacomo Rodeghiero, Ping-Ping Ding, Yu Zhong, Marc Lambert, Dominique

Lesselier

To cite this version:

Giacomo Rodeghiero, Ping-Ping Ding, Yu Zhong, Marc Lambert, Dominique Lesselier. Fast calcu-

lation of electromagnetic scattering in anisotropic multilayers and its inverse problem. ENDE 2014,

Jun 2014, Xi’an, China. �hal-01101753�

Fast Calculation of Electromagnetic Scattering in Anisotropic Multilayers

and its Inverse Problem

Electromagnetic modeling of fiber-based composite structures

ñ

EM response of anisotropic multilayers to distributed sources

EM response of anisotropic multilayers to distributed sources

Two methods for calculating the scattered field

Discretization of the integral equation - 1

method

The techniques for calculating the impedance matrix - 1

method

Padua interpolation-integration technique - calculating fast oscillating

integrals

Padua interpolation-integration technique - calculating fast oscillating

integrals

5th European Congress of Mathematics - Amsterdam, July 14-18 2008

Near-optimal interpolation and quadrature

in two variables: the Padua points

M. C ALIARI a , S. D E M ARCHI a , A. S OMMARIVA b AND M. V IANELLO b

Department of Computer Science, University of Verona, ITALY

Department of Pure and Applied Mathematics, University of Padua, ITALY

Abstract

The Padua points, recently studied during an international

collaboration at the University of Padua, are the first known

near-optimal point set for bivariate polynomial interpolation

of total degree. Also the associate algebraic cubature formulas

are both, in some sense, near-optimal.

Which are the Padua points?

Chebyshev-Lobatto points in [ −1, 1]

C

:= {cos(jπ/n), j = 0, . . . , n}

card(C

) = n + 1 = dim(P

)

Padua points in [ −1, 1] × [−1, 1]

Pad

:= (C

× C

) ∪ (C

× C

) ⊂ C

× C

card(Pad

) = (n + 1)(n + 2)

2 = dim(P

)

Alternative representation as self-intersections and boundary

contacts of the generating curve

g(t) := ( − cos((n + 1)t), − cos(nt)) , t ∈ [0, π]

F

1. The

with their

for n = 12 (left, 91

points) and n = 13 (right, 105 points), also as

(red and blue bullets).

There are 4 families of such points, corresponding to succes-

M. C ÂLIARI â , S. D Ê M ÂRCHI â , A. S ÔMMARIVA ^b ÂND M. V ÎANELLO ^b