Direct and Inverse Elastic Scattering Problems for Diffraction - - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings

Johannes Elschner & Guanghui Hu

Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · PICOF’12, April 4, 2012

Content

1

Direct and inverse elastic scattering problems in periodic structures

2

Direct scattering problem: uniqueness and existence

3

Inverse scattering problem: uniqueness for polygonal gratings

4

Inverse scattering problem: a two-step algorithm

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Elastic scattering by diffraction gratings

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Elastic scattering by periodic structures: application in geophysics

Characterize fractures using elastic waves in search for gas and liquids

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Applications

Geophysics

• search for oil, gas and ore bodies

Seismology

• investigate earthquakes

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 5 (27)

Applications

Geophysics

• search for oil, gas and ore bodies

Seismology

• investigate earthquakes

Nondestructive Testings (NDT)

• detect cracks and flaws in concrete structures, such as bridges, buildings,

dams, highways, etc.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 5 (27)

Applications

Geophysics

• search for oil, gas and ore bodies

Seismology

• investigate earthquakes

Nondestructive Testings (NDT)

• detect cracks and flaws in concrete structures, such as bridges, buildings,

dams, highways, etc. Problems:

Understand the reflection and transmission of elastic waves through an interface Design efficient inversion algorithms using elastic waves

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Basic assumptions

The periodic surface Λ is invariant in one direction and 2π-periodic in another

direction, φ = 0.

The elastic medium above Λ is homogeneous, isotropic with the Lamé

constants λ,µ. The mass density ρ = 1.

The elastic displacement-field is time-harmonic, U(t,x) = u(x)e−iωt.

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Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ . Angular frequency: ω > 0 Lamé constants: µ > 0,λ + µ > 0 Compressional wave number: kp := ω/

• 2µ +λ

Shear wave number: ks := ω/√µ

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Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ . Angular frequency: ω > 0 Lamé constants: µ > 0,λ + µ > 0 Compressional wave number: kp := ω/

• 2µ +λ

Shear wave number: ks := ω/√µ Incident angle: θ ∈ (−π/2,π/2) Incident plane pressure wave: uin

p (x) = ˆ

θ exp(ikpx· ˆ θ), ˆ θ := (sinθ,−cosθ), x = (x1,x2)

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Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 8 (27)

Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ .

Quasi-periodicity:

u(x1 +2π,x2) = exp(2iαπ)u(x1,x2), α := kp sinθ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 8 (27)

Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ .

Quasi-periodicity:

u(x1 +2π,x2) = exp(2iαπ)u(x1,x2), α := kp sinθ.

Dirichlet boundary condition on the grating profile:

u = 0

• n

Λ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 8 (27)

Direct problems

Navier equation in 2D (case φ = 0):

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ .

Quasi-periodicity:

u(x1 +2π,x2) = exp(2iαπ)u(x1,x2), α := kp sinθ.

Dirichlet boundary condition on the grating profile:

u = 0

• n

Λ.

An appropriate radiation condition imposed on usc as x2 → ∞.

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Rayleigh Expansion Radiation Condition (RERC) usc = ∇up +− − → curl us, (∆+k2

p)up = 0,

(∆+k2

s )us = 0

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Rayleigh Expansion Radiation Condition (RERC) usc = ∇up +− − → curl us, (∆+k2

p)up = 0,

(∆+k2

s )us = 0 Radiation condition

usc(x) =

∑

n∈Z

Ap,n

• αn

βn

• exp(iαnx1 +iβnx2)

+ ∑

n∈Z

As,n

• γn

−αn

• exp(iαnx1 +iγnx2)

for x2 > Λ+ := max(x1,x2)∈Λ x2. Here, αn := α +n, βn = βn(θ) :=   

• k2

p −α2 n

if |αn| ≤ kp i

• α2

n −k2 p

if |αn| > kp , γn = γn(θ) := k2

s −α2 n

if |αn| ≤ ks i

• α2

n −k2 s

if |αn| > ks . The constants Ap,n, As,n ∈ C are called the Rayleigh coefficients.

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Direct and inverse scattering problems Direct Problem (DP) Given Λ ⊂ R2 and uin, find u = uin +usc ∈ H1

loc(ΩΛ)2 under the boundary,

quasi-periodicity and radiation conditions.

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Direct and inverse scattering problems Direct Problem (DP) Given Λ ⊂ R2 and uin, find u = uin +usc ∈ H1

loc(ΩΛ)2 under the boundary,

quasi-periodicity and radiation conditions.

Uniqueness and existence of solutions Numerical solutions

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 10 (27)

Direct and inverse scattering problems Direct Problem (DP) Given Λ ⊂ R2 and uin, find u = uin +usc ∈ H1

loc(ΩΛ)2 under the boundary,

quasi-periodicity and radiation conditions.

Uniqueness and existence of solutions Numerical solutions

Inverse Problem (IP) Given incident field uin(x;θ) and the near-field data u(x1,b;θ), determine the unknown scattering surface Λ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 10 (27)

Direct and inverse scattering problems Direct Problem (DP) Given Λ ⊂ R2 and uin, find u = uin +usc ∈ H1

loc(ΩΛ)2 under the boundary,

quasi-periodicity and radiation conditions.

Uniqueness and existence of solutions Numerical solutions

Inverse Problem (IP) Given incident field uin(x;θ) and the near-field data u(x1,b;θ), determine the unknown scattering surface Λ.

Can we identify Λ uniquely ? How to recover Λ numerically ?

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Solvability results on the direct scattering problem Theorem

If the grating profile Λ is a Lipschitz curve, then there always exists a solution of

(DP).

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Solvability results on the direct scattering problem Theorem

If the grating profile Λ is a Lipschitz curve, then there always exists a solution of

(DP).

Uniqueness holds for small frequencies, or for all frequencies excluding a

discrete set with the only accumulation point at infinity.

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Solvability results on the direct scattering problem Theorem

If the grating profile Λ is a Lipschitz curve, then there always exists a solution of

(DP).

Uniqueness holds for small frequencies, or for all frequencies excluding a

discrete set with the only accumulation point at infinity.

If Λ is the graph of a Lipschitz function, then for any frequency ω > 0, there

exists a unique solution of (DP).

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Solvability results on the direct scattering problem Theorem

If the grating profile Λ is a Lipschitz curve, then there always exists a solution of

(DP).

Uniqueness holds for small frequencies, or for all frequencies excluding a

discrete set with the only accumulation point at infinity.

If Λ is the graph of a Lipschitz function, then for any frequency ω > 0, there

exists a unique solution of (DP). Elschner J. and Hu G. 2010 Variational approach to scattering of plane elastic waves by diffraction gratings M2AS 33 1924–1941 Elschner J. and Hu G. 2012 Scattering of plane elastic waves by three-dimensional diffraction gratings M3AS 22 1150019

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Reduce the problem to one periodic cell Introduce Γb := {(x1,b) : 0 ≤ x1 ≤ 2π}, Ωb := {x ∈ ΩΛ : 0 < x1 < 2π, x2 < b} Vα = Vα(Ωb) := {u ∈ H1

α(Ωb)2 : u = 0 on Λ}.

D-to-N map

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Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f , then using a periodic Rellich identity: = −2Re

• Ωb

(∆∗ +ω)u·∂2udx =

• Λ +
• Γb
• 2Re(Tu·∂2u)−E (u,u)n2 +ω2|u|2n2 ds

=

• Λ
• µ |∂nu|2 +(λ + µ)|div u|2

n2 ds

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Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f , then using a periodic Rellich identity: = −2Re

• Ωb

(∆∗ +ω)u·∂2udx =

• Λ +
• Γb
• 2Re(Tu·∂2u)−E (u,u)n2 +ω2|u|2n2 ds

=

• Λ
• µ |∂nu|2 +(λ + µ)|div u|2

n2 ds Since n2 = −1/

• 1+|f ′|2 < 0
• n

Λ, we have ∂nu = u = 0

• n

Λ. Applying Holmgren’s theorem leads to uniqueness at arbitrary frequency.

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Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f , then using a periodic Rellich identity: = −2Re

• Ωb

(∆∗ +ω)u·∂2udx =

• Λ +
• Γb
• 2Re(Tu·∂2u)−E (u,u)n2 +ω2|u|2n2 ds

=

• Λ
• µ |∂nu|2 +(λ + µ)|div u|2

n2 ds Since n2 = −1/

• 1+|f ′|2 < 0
• n

Λ, we have ∂nu = u = 0

• n

Λ. Applying Holmgren’s theorem leads to uniqueness at arbitrary frequency.

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Elastic scattering by rough surfaces (∆∗ +ω2)u = g in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u =

• n

Λ, radiation condition as x2 → +∞

O

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Elastic scattering by rough surfaces (∆∗ +ω2)u = g in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u =

• n

Λ, radiation condition as x2 → +∞

O

If g ∈ L2(ΩΛ)2, then there admits a unique solution u ∈ H1(ΩΛ)2.

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Uniqueness of (IP) for polygonal grating profiles A =

• Λf :

f(x1)is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox1

• Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 15 (27)

Uniqueness of (IP) for polygonal grating profiles A =

• Λf :

f(x1)is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox1

• Assume
• 1. Λ1,Λ2 ∈ A , and one of them has a corner point at the origin.
• 2. u1 resp. u2 satisfies the fourth kind boundary conditions on Λ1 resp. Λ2.

(τ ·u = 0, n·Tu = 0)

• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane pressure wave with the

incident angle θ.

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Uniqueness of (IP) for polygonal grating profiles A =

• Λf :

f(x1)is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox1

• Assume
• 1. Λ1,Λ2 ∈ A , and one of them has a corner point at the origin.
• 2. u1 resp. u2 satisfies the fourth kind boundary conditions on Λ1 resp. Λ2.

(τ ·u = 0, n·Tu = 0)

• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane pressure wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 15 (27)

Uniqueness of (IP) for polygonal grating profiles A =

• Λf :

f(x1)is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox1

• Assume
• 1. Λ1,Λ2 ∈ A , and one of them has a corner point at the origin.
• 2. u1 resp. u2 satisfies the fourth kind boundary conditions on Λ1 resp. Λ2.

(τ ·u = 0, n·Tu = 0)

• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane pressure wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)
• 2. If not, can we describe the exceptional classes of grating profiles that generate

the same near field on x2 = b ?

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 15 (27)

Uniqueness of (IP) for polygonal grating profiles A =

• Λf :

f(x1)is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox1

• Assume
• 1. Λ1,Λ2 ∈ A , and one of them has a corner point at the origin.
• 2. u1 resp. u2 satisfies the fourth kind boundary conditions on Λ1 resp. Λ2.

(τ ·u = 0, n·Tu = 0)

• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane pressure wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)
• 2. If not, can we describe the exceptional classes of grating profiles that generate

the same near field on x2 = b ?

• 3. How many incident elastic waves are sufficient to uniquely determine an

arbitrary grating profile Λ ∈ A ?

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Uniqueness under the fourth kind boundary conditions Theorem If u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) then either Λ1 = Λ2 or Λ1,Λ2 ∈ D2(θ,kp). In the latter case, the total field takes the form u = ˆ θ exp(ikpx· ˆ θ)− ˆ θ exp(−ikpx· ˆ θ)−e1 exp(ikpx1)+e1 exp(−ikpx1) in R2, where e1 = (1,0).

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Uniqueness under the fourth kind boundary conditions Theorem If u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) then either Λ1 = Λ2 or Λ1,Λ2 ∈ D2(θ,kp). In the latter case, the total field takes the form u = ˆ θ exp(ikpx· ˆ θ)− ˆ θ exp(−ikpx· ˆ θ)−e1 exp(ikpx1)+e1 exp(−ikpx1) in R2, where e1 = (1,0). Corollary If Λ ∈ A and Λ / ∈ D2(θ,kp), then Λ can be uniquely determined by the near-field data u(x1,b;θ), under the boundary conditions of the fourth kind.

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Uniqueness with minimal number of incident elastic waves Theorem Under the fourth kind boundary conditions, two incident pressure waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional part are excluded, the minimal number is

• ne incident pressure wave.

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Uniqueness with minimal number of incident elastic waves Theorem Under the fourth kind boundary conditions, two incident pressure waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional part are excluded, the minimal number is

• ne incident pressure wave.
• based on the reflection principle for the Navier equation (Elschner & Yamamoto

2010).

• can be extended to the third kind boundary value problem in both 2D and 3D.
• the inverse Dirichlet boundary value problem is challenging.

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Numerical algorithm for (IP): a two-step algorithm Π(x,y) = 1 µ

• Gks(x,y)

Gks(x,y)

• + 1

ω2

• ∂ 2

x1

∂x1∂x2 ∂x2∂x1 ∂ 2

x2

• Gks(x,y)−Gkp(x,y)
• Step 1 Reconstruct the scattered field usc from near-field measurement data.

Making the ansatz for usc in the form usc = 1 2π 2π Π(x1,x2 ;t,0)ϕ(t)dt, x2 ≥ f(x1), we only need to solve the first kind integral equation Jϕ(x1) := 1 2π 2π Π(x1,b ;t,0)ϕ(t)dt = usc(x1,b).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 18 (27)

Numerical algorithm for (IP): a two-step algorithm Π(x,y) = 1 µ

• Gks(x,y)

Gks(x,y)

• + 1

ω2

• ∂ 2

x1

∂x1∂x2 ∂x2∂x1 ∂ 2

x2

• Gks(x,y)−Gkp(x,y)
• Step 1 Reconstruct the scattered field usc from near-field measurement data.

Making the ansatz for usc in the form usc = 1 2π 2π Π(x1,x2 ;t,0)ϕ(t)dt, x2 ≥ f(x1), we only need to solve the first kind integral equation Jϕ(x1) := 1 2π 2π Π(x1,b ;t,0)ϕ(t)dt = usc(x1,b). Step 2 Find f by minimizing the defect ||uin(x1, f(x1))+ 1 2π 2π Π(x1, f(x1) t,0)ϕ(t)dt||2

L2(0,2π) → inf f∈M,

where M = {f(x1) = a0 +

M

∑

m=1

am cos(mx1)+aM+m sin(mx1),a j ∈ R, j = 0,1,··· ,2M}.

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Numerical examples

uin: incident pressure wave θ = 0,

kp = 4.2, ks = 4.5, ω = 5

γ : Tikhonov regularization parameter δ : noise level of the measurement u(x1,b) 2K +1: the number of propagating modes involved in computation K < 4: part of far-field data K = 4: all far-field data K > 4: all far-field data + some evanescent modes

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Example 1: sensitivity w.r.t. parameter K (δ = 0,γ = 10−12)

Figure: K = 1,2,3,4,5,6

1 2 3 4 5 6 1.4 1.6 1.8 2 2.2 2.4 2.6 K=1 computed target initial 1 2 3 4 5 6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 K=2 computed target initial 1 2 3 4 5 6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 K=3 computed target initial 1 2 3 4 5 6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 K=4 computed target initial 1 2 3 4 5 6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 K=5 computed target initial 1 2 3 4 5 6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 K=6 computed target initial

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Example 2: smooth gratings (K = 7,γ = 10−12,δ = 0) Suppose that f(t) = 1.5+0.2exp(sin(3t))+0.3exp(sin(3t)), f ∗(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt), M = 8

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Example 2: smooth gratings (K = 7,γ = 10−12,δ = 0) Suppose that f(t) = 1.5+0.2exp(sin(3t))+0.3exp(sin(3t)), f ∗(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt), M = 8

1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 computed (δ=0) target initial

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Example 2: smooth gratings (K = 7,γ = 10−12,δ = 0) Suppose that f(t) = 1.5+0.2exp(sin(3t))+0.3exp(sin(3t)), f ∗(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt), M = 8

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Example 2: smooth gratings (K = 7,γ = 10−12,δ = 0) Suppose that f(t) = 1.5+0.2exp(sin(3t))+0.3exp(sin(3t)), f ∗(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt), M = 8

1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 computed (δ=0.05) computed (δ=0.1) target initial

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Example 2: different initial guesses (K = 4,γ = 10−12,δ = 0)

1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 K=4,M=8 computed target initial 1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 K=4,M=8 computed target initial Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 23 (27)

Example 3: binary gratings (K = 4,δ = 0,γ = 10−4) A priori information: the unknown surface is a binary grating with a fixed number of corner points

1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 target initial computed 1 2 3 4 5 6 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 target initial computed

Reconstruct a binary grating profile from the far-field data corresponding to three incident angles θ = −π/4,0,π/4 (left) or one incident angle θ = 0 (right)

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Example 4: piecewise linear gratings (δ = 0,γ = 10−4)

1 2 3 4 5 6 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 target initial computed with K=7 or K=4 1 2 3 4 5 6 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 target initial computed with K=3

K = 4: only far-field data (|n| ≤ 4) (left) K = 7: far-field data and part of evanescent modes (4 < |n| ≤ 7) (left) K = 3: part of far field data (|n| ≤ 3) (right)

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Remarks In our reported numerical examples,

the unknown grating profile is given by a finite number of parameters (e.g.

Fourier coefficients or corner points);

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Remarks In our reported numerical examples,

the unknown grating profile is given by a finite number of parameters (e.g.

Fourier coefficients or corner points);

synthetic near-field data are generated by discrete trigonometric Galerkin

method applied to integral equation formulation of direct problem;

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 26 (27)

Remarks In our reported numerical examples,

the unknown grating profile is given by a finite number of parameters (e.g.

Fourier coefficients or corner points);

synthetic near-field data are generated by discrete trigonometric Galerkin

method applied to integral equation formulation of direct problem;

we can readily obtain the singular value decomposition of the first-kind integral

• perator and solve the nonlinear least-squares minimization problem;

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 26 (27)

Remarks In our reported numerical examples,

the unknown grating profile is given by a finite number of parameters (e.g.

Fourier coefficients or corner points);

synthetic near-field data are generated by discrete trigonometric Galerkin

method applied to integral equation formulation of direct problem;

we can readily obtain the singular value decomposition of the first-kind integral

• perator and solve the nonlinear least-squares minimization problem;

we need not solve the direct scattering problem at each iteration.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 26 (27)

Remarks In our reported numerical examples,

the unknown grating profile is given by a finite number of parameters (e.g.

Fourier coefficients or corner points);

synthetic near-field data are generated by discrete trigonometric Galerkin

method applied to integral equation formulation of direct problem;

we can readily obtain the singular value decomposition of the first-kind integral

• perator and solve the nonlinear least-squares minimization problem;

we need not solve the direct scattering problem at each iteration.

Compared to the Kirsch-Kress optimization method based on the combined cost functional, F(ϕ, f) = ||Tϕ −ub||2

L2(0,2π) +ρ||Tϕ +uin||2 L2(Λ f ) +γ||ϕ||2 L2(0,2π),

the two-step algorithm can considerably reduce the computational effort. However, for the combined functional, a convergence result can be proved.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 26 (27)

Thank you very much for your attention !

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 27 (27)