Thin cylindrical conductivity inclusions in a 3-dimensional domain: - - PowerPoint PPT Presentation

Thin cylindrical conductivity inclusions in a 3-dimensional domain: polarization tensor and unique determination from boundary data

Elisa Francini

in collaboration with Elena Beretta, Yves Capdeboscq and Fr´ ed´ eric de Gournay

July 20-24, 2009

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 1 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Low volume fraction inclusions

Ω is a bounded smooth domain, ωǫ ⊂ Ω is an inclusion |ωǫ| → 0 as ǫ → 0 γ0 and γ1 are smooth functions defined in Ω. γǫ = γ0 + (γ1 − γ0)1ωǫ is the conductivity of the body

Ω ωε

Given a current g on ∂Ω, study how the electric potential generated in Ω by this current depends on the presence of the inclusion. Recover information on the inclusion from boundary measurements of the potential.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 2 / 1

Essential references

• A. Friedman and M. Vogelius, Identification of small inhomogeneities
• f extreme conductivity by boundary measurements: a theorem on

continuous dependence,Arch. Rat. Mech. Anal., 1989.

• H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities

from Boundary Measurements, Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004.

• Y. Capdeboscq and M. S. Vogelius, A general representation formula

for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 2003.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references

• A. Friedman and M. Vogelius, Identification of small inhomogeneities
• f extreme conductivity by boundary measurements: a theorem on

continuous dependence,Arch. Rat. Mech. Anal., 1989.

• H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities

from Boundary Measurements, Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004.

• Y. Capdeboscq and M. S. Vogelius, A general representation formula

for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 2003.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references

• A. Friedman and M. Vogelius, Identification of small inhomogeneities
• f extreme conductivity by boundary measurements: a theorem on

continuous dependence,Arch. Rat. Mech. Anal., 1989.

• H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities

from Boundary Measurements, Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004.

• Y. Capdeboscq and M. S. Vogelius, A general representation formula

for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 2003.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Essential references

• A. Friedman and M. Vogelius, Identification of small inhomogeneities
• f extreme conductivity by boundary measurements: a theorem on

continuous dependence,Arch. Rat. Mech. Anal., 1989.

• H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities

from Boundary Measurements, Lecture Notes in Math. 1846, Springer-Verlag, Berlin, 2004.

• Y. Capdeboscq and M. S. Vogelius, A general representation formula

for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal., 2003.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 3 / 1

Compare two potentials

A current g ∈ H−1/2(∂Ω) such that

• ∂Ω g dσ = 0 induces two different

potentials:          div (γ0∇u0) = 0 in Ω, γ0 ∂u0 ∂n = g on ∂Ω,

• ∂Ω

u0 dσ = 0,

Ω

         div (γǫ∇uǫ) = 0 in Ω, γǫ ∂uǫ ∂n = g on ∂Ω,

• ∂Ω

uǫ dσ = 0.

Ω ωε

Can we evaluate uǫ − u0 on ∂Ω?

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 4 / 1

Compare two potentials

A current g ∈ H−1/2(∂Ω) such that

• ∂Ω g dσ = 0 induces two different

potentials:          div (γ0∇u0) = 0 in Ω, γ0 ∂u0 ∂n = g on ∂Ω,

• ∂Ω

u0 dσ = 0,

Ω

         div (γǫ∇uǫ) = 0 in Ω, γǫ ∂uǫ ∂n = g on ∂Ω,

• ∂Ω

uǫ dσ = 0.

Ω ωε

Can we evaluate uǫ − u0 on ∂Ω?

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 4 / 1

Assumptions and notation

Assumption on the behavior of the inclusion : d(ωǫ, ∂Ω) ≥ d0 > 0 for every ǫ and |ωǫ|−1 1ωǫ(·) converges in the sense of measure to µ when ǫ → 0. Assumption on the conductivities γi smooth, c0 < γi(x) < c−1

0 ,

for x ∈ Ω, i = 0, 1. Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N(·, y) be the solution to          divx (γ0(x)∇xN(x, y)) = δy(x) for x ∈ Ω, γ0(x) ∂N ∂nx (x, y) = 1 |∂Ω| for x ∈ ∂Ω,

• ∂Ω

N(x, y) dσx = 0.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1

Assumptions and notation

Assumption on the behavior of the inclusion : d(ωǫ, ∂Ω) ≥ d0 > 0 for every ǫ and |ωǫ|−1 1ωǫ(·) converges in the sense of measure to µ when ǫ → 0. Assumption on the conductivities γi smooth, c0 < γi(x) < c−1

0 ,

for x ∈ Ω, i = 0, 1. Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N(·, y) be the solution to          divx (γ0(x)∇xN(x, y)) = δy(x) for x ∈ Ω, γ0(x) ∂N ∂nx (x, y) = 1 |∂Ω| for x ∈ ∂Ω,

• ∂Ω

N(x, y) dσx = 0.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1

Assumptions and notation

Assumption on the behavior of the inclusion : d(ωǫ, ∂Ω) ≥ d0 > 0 for every ǫ and |ωǫ|−1 1ωǫ(·) converges in the sense of measure to µ when ǫ → 0. Assumption on the conductivities γi smooth, c0 < γi(x) < c−1

0 ,

for x ∈ Ω, i = 0, 1. Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N(·, y) be the solution to          divx (γ0(x)∇xN(x, y)) = δy(x) for x ∈ Ω, γ0(x) ∂N ∂nx (x, y) = 1 |∂Ω| for x ∈ ∂Ω,

• ∂Ω

N(x, y) dσx = 0.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1

Assumptions and notation

Assumption on the behavior of the inclusion : d(ωǫ, ∂Ω) ≥ d0 > 0 for every ǫ and |ωǫ|−1 1ωǫ(·) converges in the sense of measure to µ when ǫ → 0. Assumption on the conductivities γi smooth, c0 < γi(x) < c−1

0 ,

for x ∈ Ω, i = 0, 1. Let N denote the Neumann function of the unperturbed domain: given y ∈ Ω, let N(·, y) be the solution to          divx (γ0(x)∇xN(x, y)) = δy(x) for x ∈ Ω, γ0(x) ∂N ∂nx (x, y) = 1 |∂Ω| for x ∈ ∂Ω,

• ∂Ω

N(x, y) dσx = 0.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 5 / 1

The general result by Capdeboscq and Vogelius

Theorem

There exists a tensor {Mij}n

i,j=1 ∈ L2(Ω, dµ) such that, for y ∈ ∂Ω,

(uǫ − u0)(y) = |ωǫ|

n

• i,j=1
• Ω

(γ1 − γ0)(x)Mij(x)∂u0 ∂xi (x)∂N ∂xj (x, y)dµ(x) +o (|ωǫ|) . Here |ωǫ|−1 o (|ωǫ|) L∞(∂Ω) → 0 as ǫ → 0 uniformly with respect to {g ∈ H−1/2(∂Ω) :

• ∂Ω g dσ = 0 ,

gL2(∂Ω) ≤ 1}.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 6 / 1

Properties of the polarization tensor

One of the possible definition of the polarization tensor is the following:

• Ω

(γ1 − γ0)Mξ · ξ φ dµ = 1 |ωǫ|

• ωǫ

(γ1 − γ0)γ0 γ1 |ξ|2 φ dx + 1 |ωǫ| min

w∈H1

0(Ω)

• Ω

γǫ

• ∇w + 1ωǫ

γ1 − γ0 γ1 ξ

• 2

φ dx + o(1). where ξ is a vector, φ is any smooth positive function and o(1) tends to zero with ǫ. Moreover Mij(x) = Mji(x), min

• 1, γ0(x)

γ1(x)

• |ξ|2 ≤ Mijξiξj ≤ max
• 1, γ0(x)

γ1(x)

• |ξ|2
• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 7 / 1

Properties of the polarization tensor

One of the possible definition of the polarization tensor is the following:

• Ω

(γ1 − γ0)Mξ · ξ φ dµ = 1 |ωǫ|

• ωǫ

(γ1 − γ0)γ0 γ1 |ξ|2 φ dx + 1 |ωǫ| min

w∈H1

0(Ω)

• Ω

γǫ

• ∇w + 1ωǫ

γ1 − γ0 γ1 ξ

• 2

φ dx + o(1). where ξ is a vector, φ is any smooth positive function and o(1) tends to zero with ǫ. Moreover Mij(x) = Mji(x), min

• 1, γ0(x)

γ1(x)

• |ξ|2 ≤ Mijξiξj ≤ max
• 1, γ0(x)

γ1(x)

• |ξ|2
• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 7 / 1

Properties of the polarization tensor

One of the possible definition of the polarization tensor is the following:

• Ω

(γ1 − γ0)Mξ · ξ φ dµ = 1 |ωǫ|

• ωǫ

(γ1 − γ0)γ0 γ1 |ξ|2 φ dx + 1 |ωǫ| min

w∈H1

0(Ω)

• Ω

γǫ

• ∇w + 1ωǫ

γ1 − γ0 γ1 ξ

• 2

φ dx + o(1). where ξ is a vector, φ is any smooth positive function and o(1) tends to zero with ǫ. Moreover Mij(x) = Mji(x), min

• 1, γ0(x)

γ1(x)

• |ξ|2 ≤ Mijξiξj ≤ max
• 1, γ0(x)

γ1(x)

• |ξ|2
• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 7 / 1

Special case: diametrically small inclusions

Let z be a point in Ω and B a bounded domain of volume 1. If ωǫ = z + ǫB then µ = δz and M is the tensor of P´

• lya-Szeg¨
• associated to B. If B is

a ball M = nγ0 γ1 + (n − 1)γ0 Idn

ε

ω

z

In this case (uǫ − u0)(y) = ǫnM∇u0(z) · ∇N(z, y) + 0(ǫn)

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 8 / 1

Special case: thin inclusions

Let σ be a curve in R2 ωǫ = {x : d(x, σ) < ǫ} The limit measure µ is concentrated

• n σ and

M =

• τ ⊗ τ + γ0

γ1 n ⊗ n

• .

σ

ωε

In this case (uǫ − u0)(y) = 2ǫ

• σ

(γ1 − γ0) ∂u0 ∂τ (x)∂N ∂τ (x, y) + γ0 γ1 ∂u0 ∂n (x)∂N ∂n (x, y)

• dσ

+0(ǫ)

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 9 / 1

Cylinders in R3

Ω ⊂ R3 and ωǫ is a cylindrical inclusion with axis σ and basis of the form ǫB2 where B2 is a bidi- mensional domain. The limit measure µ is con- centrated on σ. Let τ be the direction of σ.

ε σ

Theorem

The tensor M has eigenvalue 1 corresponding to the eigendirection τ: Mτ · τ = 1, and, if v · τ = 0, then Mv · v = mv · v, where m is the bidimensional tensor defined by ǫB2.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 10 / 1

Cylinders in R3

Ω ⊂ R3 and ωǫ is a cylindrical inclusion with axis σ and basis of the form ǫB2 where B2 is a bidi- mensional domain. The limit measure µ is con- centrated on σ. Let τ be the direction of σ.

ε σ

Theorem

The tensor M has eigenvalue 1 corresponding to the eigendirection τ: Mτ · τ = 1, and, if v · τ = 0, then Mv · v = mv · v, where m is the bidimensional tensor defined by ǫB2.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 10 / 1

Cylinders in R3

Ω ⊂ R3 and ωǫ is a cylindrical inclusion with axis σ and basis of the form ǫB2 where B2 is a bidi- mensional domain. The limit measure µ is con- centrated on σ. Let τ be the direction of σ.

ε σ

Theorem

The tensor M has eigenvalue 1 corresponding to the eigendirection τ: Mτ · τ = 1, and, if v · τ = 0, then Mv · v = mv · v, where m is the bidimensional tensor defined by ǫB2.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 10 / 1

Idea of proof...

Let us take the cylinder centered at the origin with τ = e3. In order to evaluate Mτ · τ we need to find Φ3

ǫ(x) such that

div

• γǫφ∇Φ3

ǫ

• = div ((γ0 − γ1)1ωǫe3φ) in Ω.

By using regularity estimates it is possible to show that 1 |ωǫ|

• Ω

γǫ

• ∇Φ3

ǫ + 1ωǫ

γ1 − γ0 γ1 ξ

• 2

φ dx = 1 |ωǫ|

• ωǫ

(γ1 − γ0)

• 1 − γ0

γ1

• |ξ|2

+

• (1).
• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 11 / 1

..Idea of proof

In order to evaluate Mei · ei for i = 1, 2 the idea is to approximate the ”correctors” Φi

ǫ for the 3-dimensional problem by correctors of the

corresponding bidimensional section.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 12 / 1

Asymptotic expansion

Let m be the bidimensional tensor associated with an inclusion of the form ǫB2. Then, if we denote be τ the tangent direction to the axis σ, we have, for y ∈ ∂Ω, (uǫ − u0)(y) = ǫ2uσ(y) + o(ǫ2) where uσ(y) =

• σ

(γ1−γ0)(x) ∂u0 ∂τ (x)∂N ∂τ (x, y) + m(x) ∇u0(x) · ∇N(x, y)

• dσx.

where ˜ v = v − (v · τ)τ denotes the non tangential part of v

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 13 / 1

Information from the correction term

The function uσ can be extended from ∂Ω to Ω \ σ and solves div (γ0∇uσ) = 0 in Ω \ σ. The correction term is singular at every point of segment σ. Moreover, γ0 ∂uσ

∂n = 0 on ∂Ω.

Proposition

Let γ0 and γ1 be smooth positive functions. Let Σ be an open subset of ∂Ω and let σ and σ′ be two segments strictly contained in Ω. Let u0 be a smooth solution to div(γ0∇u0) = 0 in Ω such that ∇u0 = 0 in Ω, and let uσ and uσ′ be the correction terms for segments σ and σ′ respectively. If uσ = uσ′

• n

Σ, then σ = σ′.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 14 / 1

Information from the correction term

The function uσ can be extended from ∂Ω to Ω \ σ and solves div (γ0∇uσ) = 0 in Ω \ σ. The correction term is singular at every point of segment σ. Moreover, γ0 ∂uσ

∂n = 0 on ∂Ω.

Proposition

Let γ0 and γ1 be smooth positive functions. Let Σ be an open subset of ∂Ω and let σ and σ′ be two segments strictly contained in Ω. Let u0 be a smooth solution to div(γ0∇u0) = 0 in Ω such that ∇u0 = 0 in Ω, and let uσ and uσ′ be the correction terms for segments σ and σ′ respectively. If uσ = uσ′

• n

Σ, then σ = σ′.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 14 / 1

Reconstruction algorithm

Assumptions: Ω = BR ball of radius R centered at the origin. Linear background potentials: ui

0 = xi for i = 1, 2, 3.

Measurements: ui

ǫ boundary values of the solutions corresponding to

gi = ∂xj

∂n .

Conductivities: γ0 = 1, γ1 constant (say γ1 > 1). First step: identification of the direction τ of the axis of the cylinder. By the asymptotic expansion, the power perturbation can be calculated as, δWij =

• ∂BR
• uǫi − ui

∂xj ∂n ds = |ωǫ|

• Ω

(1 − γ1) M∇xi · ∇xjdµ + o (|ωǫ|) , = |ωǫ| (1 − γ1) Mei · ej + o (|ωǫ|) . The eigenvector of the matrix δWij corresponding to the biggest eigenvalue is the direction τ

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 15 / 1

Reconstruction algorithm: second step

Median plane Consider the background uτ

0 = τ · x and the corresponding solution uτ ǫ

that is completely determined by the previous ones by uτ

ǫ = 3 i=1(τ · ei)ui ǫ. Let A and B be the endpoints of σ.

δWτ =

• ∂BR

(uǫτ − uτ

0) (τ · n)ds = |ωǫ| (1 − γ1) Mτ · τ + o (|ωǫ|) .

D1 =

• ∂BR

(uǫτ − uτ

0) ∂

∂n (x · τ)2 − (x · e1)2 2

• ds

= 1 2|ωǫ| (1 − γ1) Mτ · τ ((A · τ) + (B · τ)) + o (|ωǫ|) . Hence Dψ δWτ = 1 2 ((A · τ) + (B · τ)) + o (1) .

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 16 / 1

Reconstruction algorithm: final steps

By using harmonic test function of the form φa(x) = |x − a|−1 with a on a plane orthogonal to τ and outside BR. Da =

• ∂BR

(uǫτ − x · τ) ∂φa ∂n ds = |ωǫ| (1 − γ1) Mτ · τ

• 1

|A − a| − 1 |B − a|

• + o (|ωǫ|)

By a minimization process with respect to a we identify A and B.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 17 / 1

Numerical simulation

Ω = B1.5, γ0 = 1, γ1 = 3, |σ| = 1, circular cross-section, ǫ = (10√π)−1,

Figure: On the left, the boundary elements of the mesh used for the simulation (Gmsh and Getfem++). On the right, the reconstructed inclusion together with the original one.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 18 / 1

Numerical simulation

Ω = B1.5, γ0 = 1, γ1 = 3, |σ| = 1, circular cross-section, ǫ = (10√π)−1, The midpoint is Xm := (xm, ym, zm) ≈ (0.006, 0.506, 0.150)., τ = (sin(π/3) cos(π/4), sin(π/3) sin(π/4), cos(π/3)). The three dimensional mesh has 12625 nodes. The eigenvalues of δW are λ0 = −1.83777 × 10−2, λ1 = −1.04347 × 10−2, and λ2 = −1.04434 × 10−2. Thus, γ1 > 1 and the first eigenvalue corresponds to the direction of the cylinder. Its eigenvector is ˜ τ = (−0.612494, −0.612253, −0.499998), and |˜ τ + τ| ≈ 2 × 10−4. The computed median plane abscissa is ˜ xτ = −0.387759, and |˜ xτ + Xm · τ| = 1 × 10−3. ˜ L = 0.984. Finally, since λ1 = λ2, it is clear that the equivalent ellipse is a disk: this leads to an estimation of ˜ γ1 ≈ 2.5, and of the radius of the disk of ˜ ǫ ≈

11 100√π.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 19 / 1

Numerical simulation 2

Ω = B1.5, γ0 = 1, γ1 = 0.5, |σ| = 6/5, elliptic cross-section a/b = 2, ǫ = (10√π)−1,

Figure: Simulation for a cylinder with elliptic cross-section. On the left, the mesh. On the right, the reconstructed inclusion together with the original one.

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 20 / 1

Numerical simulation 2

Ω = B1.5, γ0 = 1, γ1 = 0.5, |σ| = 6/5, elliptic cross-section a/b = 2, ǫ = (10√π)−1, Main directions: τ = (0.8324, 0.4009, 0.3827) axis of the cylinder, ν1 = (−0.3448, 0.1660, 0.9239) major axis, ν2 = (−0.4339, 0.9010) minor axis of the ellipse. The midpoint Xm = (−0.001, 0.041, −0.170). The three dimensional mesh has 15306 nodes. The eigenvalues of δW are λ0 = 1.72031 × 10−2, λ1 = 1.22152 × 10−2, and λ2 = 1.39706 × 10−2, Thus, γ1 > 1 and the second eigenvalue corresponds to the direction of the cylinder, the first one to the minor axis, and the third to the major axis. The error in the main direction is 2 × 10−4 and in the cross section, it is 8 × 10−4. Finally, turning to the polarization tensor, from the ratios λ0/λ1 and λ2/λ1, we obtain a ≈

31 100√π and b ≈ 13 100√π and γ1 = 0.58

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 21 / 1

Thank you!

• E. Francini (Universit`

a di Firenze) Thin Cylindrical Inclusions AIP2009,Wien 22 / 1