Homogenization of the Transmission Eigenvalue
Problem for a Periodic Media
Isaac Harris
Texas A&M University, Department of Mathematics
College Station, Texas 77843-3368
iharris@math.tamu.edu
Joint work with: F. Cakoni and H. Haddar
Research Funded by: NSF Grant DMS-1106972
and University of Delaware Graduate Student Fellowship
April 2016
Homogenization of the TE for Periodic Material
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The Inverse Problem
We consider the time harmonic Acoustic scattering in R3 or
Electromagnetic scattering in R2 (TE-polarization case).
Figure: A periodic domain for three different values of ε.
Inverse Problem
Obtain information about the macro and micro-structure of a
periodic media where the period is characterized by a small
parameter ε 1 from the scattered field.
Homogenization of the TE for Periodic Material
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The Scattering Problem for Periodic Media
The time harmonic scattering by a periodic media with scattered
1 (Rm ) and incident field u i = exp(ikx · d), where the
field uεs ∈ Hloc
total field uε = uεs + u i and scattered field satisfy
∆uεs + k 2 uεs = 0
∇ · A(x/ε)∇uε + k 2 n(x/ε)uε = 0
∂
∂ s
uε = uεs + u i and
uε =
(u + u i )
∂νA
∂ν ε
s
m−1
∂uε
s
2
lim r
− ikuε = 0.
r →∞
∂r
Homogenization of the TE for Periodic Material
in
Rm \ D
in
D
on
∂D
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Convergence of the Coefficients
The matrix A(y) ∈ L∞ (Y , Rm×m ) is Y -periodic symmetric
positive definite and the function n(y) ∈ L∞ (Y ) is a positive
Y -periodic function.
We have that as ε → 0
1
nε := n(x/ε) → nh :=
|Y|
Z
n(y) dy weakly in L∞
Y
Aε := A(x/ε) → Ah in the sense of H-convergence
i.e. for uε * u in H 1 (D) then Aε ∇uε * Ah ∇u in [L2 (D)]m
where Ah is a constant symmetric matrix
Homogenization of the TE for Periodic Material
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Far-Field Operators
It is known that the radiating scattered field which depends on the
incident direction d, has the following asymptotic expansion
1
e ik|x|
∞
u
(x̂,
d;
k)
+
O
usε (x, d; k) =
as |x| → ∞
ε
m−1
|x|
|x| 2
We now define the far field operator as L2 (S) → L2 (S)
Z
(Fg )(x̂; ε) := u∞
ε (x̂, d; k)g (d) ds(d)
S
where S = {x ∈ Rm : |x| = 1} is the unit circle or sphere.
Homogenization of the TE for Periodic Material
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Transmission Eigenvalue Problem
Motivation for the TE-problem
I
Transmission eigenvalues correspond to specific frequencies
where there is an incident field ui that does not scatter
I
They are also connected to the injectivity of the far-field
operator F
I
The Transmission eigenvalues can be used to
determine/estimate the material properties.
Homogenization of the TE for Periodic Material
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The TE-Problem for a Periodic Media
Homogenization is used to study composite periodic media. We
are interested in the limiting case as ε → 0 for the TE-Problem.
Find non-trivial kε , (vε , wε ) ∈ R+ × X (D) such that:
∇ · A(x/ε)∇wε + kε2 n(x/ε)wε = 0
kε2 vε
∆vε +
=0
∂wε
∂vε
wε = vε and
=
∂νA
∂ν
in
D
in
D
on
∂D
Note that the spaces for the solution (wε , vε ) will become precise
later since they depend on whether A = I or A 6= I.
Homogenization of the TE for Periodic Material
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Reconstructing the Real TEs
Question: Can the TEs be determined from the far field data?
Answer: (IH-Cakoni-Sun 2014)
The Far-Field equation in Rm is given by
Z
u∞
ε (x̂, d; k)gz (d) ds(d) = exp(−ikz · x̂)
for a z ∈ D.
S
Let gzα be the regularized solution to the Far Field equation
I
if k ∈ R+ is not a TE then kgzα kL2 (S) is bounded as α → 0
I
if k ∈ R+ is a TE then kgzα kL2 (S) is unbounded as α → 0
Homogenization of the TE for Periodic Material
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Numerical Examples i.e. plot k 7→ ||gzα ||
A = Diag(5, 6) and n = 2 where the domain D is a 2 × 2 Square.
Figure: Plot of the ||gzα ||L2 (S) for 25 points z ∈ D
Homogenization of the TE for Periodic Material
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Numerical Examples
We now compute the TEs using the Far-Field Equation (FFE) and
the FEM where we fix A = Diag(5, 6) and n = 2.
Table: Comparison of FFE Computation v.s. FEM Calculations
Method
Domain
1st TEV
2nd TEV
FFE
FEM
FFE
FEM
square (2 × 2)
square (2 × 2)
circle (R = 1)
circle (R = 1)
1.84
1.84
1.98
1.98
6.60
6.63
7.23
7.13
Homogenization of the TE for Periodic Material
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The case of an Isotropic media (i.e. A = I )
For this case (wε , vε ) ∈ L2 (D) × L2 (D) with uε = wε − vε ∈ H02 (D),
where we have that the difference uε satisfies for nε := n(x/ε)
0 = ∆ + k 2 nε
1
∆ + k 2 uε in D.
nε − 1
The equivalent variational form is given by
Z
1
∆uε + kε2 uε ∆ϕ + kε2 nε ϕ dx = 0 for all ϕ ∈ H02 (D).
nε − 1
D
We can rewriting the variational form to the following form
Aε,kε (uε , ϕ) − kε2 B(uε , ϕ) = 0 for all ϕ ∈ H02 (D).
Homogenization of the TE for Periodic Material
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Tricks of the Trade
Assume that nmin ≤ n(y) ≤ nmax and let uε be the eigenfunction
corresponding to the eigenvalue kε with kuε kH 1 (D) = 1 for all ε
αk∆uε k2L2 (D) ≤ Aε,kε (uε , uε ) = kε2 B(uε , uε ) ≤ kε2 kuε k2H 1 (D)
provided that nmin > 1 or 0 < nmax < 1.
Lemma (Cakoni-Haddar-IH 2015)
Assume that either nmin > 1 or 0 < nmax < 1. Then there exists
an infinite sequence of real transmission eigenvalues kε, j for j ∈ N
such that
I
kj (nmax , D) ≤ kε, j < kj (nmin , D)
if nmin > 1
I
kj (nmin , D) ≤ kε, j < kj (nmax , D)
if 0 < nmax < 1.
Homogenization of the TE for Periodic Material
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Theorem (Cakoni-Haddar-IH 2015)
Assume that nmin > 1 or 0 < nmax < 1, then there is a
subsequence of kε , (vε , wε ) ∈ R+ × L2 (D) × L2 (D) that
converges weakly to (v, w) ∈ L2 (D) × L2 (D) and kε → k that
satisfies
∆w + k 2 nh w = 0 and ∆v + k 2 v = 0
∂v
∂w
=
w = v and
∂ν
∂ν
in
D
on
∂D
provided that kε is bounded, where
Z
1
nh :=
n(y) dy.
|Y|
Y
Homogenization of the TE for Periodic Material
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Reconstructing Material Properties A = I
We can reconstruct the effective material property nh by finding
a n0 such that k1 (n0 ) = k1 (nε ) where
∆w + k 2 n0 w = 0 and ∆v + k 2 v = 0
∂w
∂v
=
w = v and
∂ν
∂ν
in
D
on
∂D.
Now let n(x/ε) = sin2 (2πx1 /ε) + 2
Table: Reconstruction from scattering data
ε
k1 (nε )
nh
n0
0.1
5.046
2.5
2.5188
Homogenization of the TE for Periodic Material
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Theorem (Cakoni-Haddar-IH 2015)
I
Assume that A(y) − I and n(y) − 1 have different sign in Y .
I
Assume that n(y) = 1 and A(y) − I is positive(or negative)
definite.
Then there is a subsequence of
kε , (vε , wε ) ∈ R+ × H 1 (D) × H 1 (D) that converges weakly to
(v, w) ∈ H1 (D) × H1 (D) and kε → k that satisfies
∇ · Ah ∇w + k 2 nh w = 0 and ∆v + k 2 v = 0
∂w
∂v
w = v and
=
∂νAh
∂ν
in
D
on
∂D
provided that kε is bounded, where A(x/ε) → Ah in the sense of
H-convergence as ε → 0.
Homogenization of the TE for Periodic Material
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Reconstructing Material Properties n = 1
We can reconstruct the effective material property Ah = ah I by
finding an a0 such that k1 (a0 ) = k1 (Aε ) where
a0 ∆w + k 2 w = 0 and ∆v + k 2 v = 0 in D
∂w
∂v
w = v and a0
=
on ∂D.
∂ν
∂ν
2
0
1 sin (2πx2 /ε) + 1
Now let A(x/ε) = 3
0
cos2 (2πx1 /ε) + 1
Table: Reconstruction from scattering data
ε
k1 (Aε )
ah
a0
0.1
7.349
0.5
0.4851
Homogenization of the TE for Periodic Material
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Thanks for your Attention:
For more details:
F. Cakoni, H. Haddar and I. Harris
“Homogenization approach for the transmission eigenvalue problem for
periodic media and application to the inverse problem”,
Inverse Problems and Imaging 1025 - 1049, Volume 9, Issue 4,
2015−(arXiv:1410.37297).
In the paper we have also considered:
I The convergence of the interior transmission problem for the cases where
A = I and A 6= I
I Construct a bulk corrector to prove strong convergence for A 6= I
I Numerical test for the order of convergence
Homogenization of the TE for Periodic Material
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Some References
A. Bensoussan, J.L. Lions, G. Papanicolau
Asymptotic Analysis for Periodic Structures
Chelsea Publications, 1978
F. Cakoni and D. Colton,
A Qualitative Approach to Inverse Scattering Theory
Springer, Berlin 2014.
F. Cakoni, B. Guzina and S. Moskows,
“On the homogenization of a transmission problem in scattering theory for
highly oscillating media”,
SIAM J. Math. Analysis (to appear).
I. Harris,
Non-Destructive Testing of Anisotropic Materials
University of Delaware, Ph.D. Thesis (2015).
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Figure: Questions?
Homogenization of the TE for Periodic Material
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