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 ⋅ Workshop 3, Linz, Nov. 24, 2011

Content

1

Mathematical formulations for elastic scattering by diffraction gratings

2

Direct scattering problem: uniqueness and existence

3

Inverse scattering problem: uniqueness for polygonal gratings

4

Inverse scattering problem: a two-step algorithm

5

Conclusions

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 2 (47)

Elastic scattering by diffraction gratings

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 3 (47)

Mathematical formulations

■ Navier equation

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ . Angular frequency: ω > 0 Lame constants: µ > 0,λ + µ > 0 Compressional wave number: kp := ω/ √ 2µ +λ Shear wave number: ks := ω/√µ

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 4 (47)

Mathematical formulations

■ Navier equation

(∆∗ +ω2)u = 0 in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u = uin +usc in ΩΛ . Angular frequency: ω > 0 Lame 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 pressure wave: uin

p (x) = ˆ

θ exp(ikpx⋅ ˆ θ), ˆ θ := (sinθ,−cosθ)⊤, Incident shear wave: uin

s (x) = ˆ

θ⊥ exp(iksx⋅ ˆ θ), ˆ θ⊥ := (cosθ,sinθ)⊤.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 4 (47)

Mathematical formulations

■ Quasi-periodicity

u(x1 +2π,x2) = exp(2iαπ)u(x1,x2) α := ksinθ

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 5 (47)

Mathematical formulations

■ Quasi-periodicity

u(x1 +2π,x2) = exp(2iαπ)u(x1,x2) α := ksinθ

■ Boundary conditions on the grating profile Λ

The first kind (Dirichlet) boundary condition: u = 0 The second kind (Neumann) boundary condition: Tu = 0 The third kind boundary conditions: n ⋅u = 0, τ ⋅Tu = 0 The fourth kind boundary conditions: τ ⋅u = 0, n⋅Tu = 0 Stress operator: Tu = 2µ ∂nu+λ ndiv u+ µτ(∂2u1 −∂1u2) Normal vector: n = (n1,n2)⊤ Tangential vector: τ = (−n2,n1)⊤

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 5 (47)

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

p)up = 0,

(∆+k2

s )us = 0

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 6 (47)

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.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 6 (47)

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

loc(ΩΛ)2.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 7 (47)

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

loc(ΩΛ)2.

Uniqueness and existence results:

■ T. Arens, 1999, integral equation method, smooth periodic profiles, smooth

rough surfaces, the Dirichlet boundary condition, n = 2.

■ J. Elschner, G. Hu, 2010, variational method, Lipschitz grating profiles, including

binary gratings, n = 2,3. A monograph for the boundary value problems of elasticity Kupradze, V. D. et al 1979 Three-dimensional Problems of the Mathematical Theory

• f Elasticity and Thermoelasticity (Amsterdam: North-Holland)

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 7 (47)

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) under the boundary conditions of the first, second, third or fourth kind.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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) under the boundary conditions of the first, second, third or fourth kind. Moreover, uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulating point at infinity.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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) under the boundary conditions of the first, second, third or fourth kind. Moreover, uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulating 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) under the Dirichlet boundary condition.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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) under the boundary conditions of the first, second, third or fourth kind. Moreover, uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulating 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) under the Dirichlet boundary condition.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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) under the boundary conditions of the first, second, third or fourth kind. Moreover, uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulating 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) under the Dirichlet boundary condition. Elschner J. and Hu G. 2010 Variational approach to scattering of plane elastic waves by diffraction gratings Math. Methods Appl. Sci. 33 1924–1941 Elschner J. and Hu G. 2012 Scattering of plane elastic waves by three-dimensional diffraction gratings Mathematical Models and Methods in Applied Sciences (2012)

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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) under the boundary conditions of the first, second, third or fourth kind. Moreover, uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulating 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) under the Dirichlet boundary condition. Elschner J. and Hu G. 2010 Variational approach to scattering of plane elastic waves by diffraction gratings Math. Methods Appl. Sci. 33 1924–1941 Elschner J. and Hu G. 2012 Scattering of plane elastic waves by three-dimensional diffraction gratings Mathematical Models and Methods in Applied Sciences (2012) Remark In general, uniqueness doesn’t hold for the second, third or fourth kind boundary value problems. We can construct non-uniqueness examples for a flat grating.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 8 (47)

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 : usatisfies one of the boundary conditions onΛ}.

D-to-N map

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 9 (47)

The Dirichlet-to-Neumann map T T v = − ∑

n∈ℤ

Wn ˆ vn exp(iαnx1), v = ∑

n∈ℤ

ˆ vn exp(iαnx1) ∈ H1/2

α

(Γb)2 where Wn := 1 i ( ω2βn/dn 2µαn −ω2αn/dn −2µαn +ω2αn/dn ω2γn/dn ) , dn := α2

n +βnγn.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 10 (47)

The Dirichlet-to-Neumann map T T v = − ∑

n∈ℤ

Wn ˆ vn exp(iαnx1), v = ∑

n∈ℤ

ˆ vn exp(iαnx1) ∈ H1/2

α

(Γb)2 where Wn := 1 i ( ω2βn/dn 2µαn −ω2αn/dn −2µαn +ω2αn/dn ω2γn/dn ) , dn := α2

n +βnγn.

Lemma (i) The map T is a bounded linear map from H1/2

α

(Γb)2 to H−1/2

α

(Γb)2.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 10 (47)

The Dirichlet-to-Neumann map T T v = − ∑

n∈ℤ

Wn ˆ vn exp(iαnx1), v = ∑

n∈ℤ

ˆ vn exp(iαnx1) ∈ H1/2

α

(Γb)2 where Wn := 1 i ( ω2βn/dn 2µαn −ω2αn/dn −2µαn +ω2αn/dn ω2γn/dn ) , dn := α2

n +βnγn.

Lemma (i) The map T is a bounded linear map from H1/2

α

(Γb)2 to H−1/2

α

(Γb)2. (ii) ReWn > 0 for all sufficiently large ∣n∣.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 10 (47)

The Dirichlet-to-Neumann map T T v = − ∑

n∈ℤ

Wn ˆ vn exp(iαnx1), v = ∑

n∈ℤ

ˆ vn exp(iαnx1) ∈ H1/2

α

(Γb)2 where Wn := 1 i ( ω2βn/dn 2µαn −ω2αn/dn −2µαn +ω2αn/dn ω2γn/dn ) , dn := α2

n +βnγn.

Lemma (i) The map T is a bounded linear map from H1/2

α

(Γb)2 to H−1/2

α

(Γb)2. (ii) ReWn > 0 for all sufficiently large ∣n∣. (iii) T = T1 +T2, T1v = − ∑

∣n∣≥M

Wn ˆ vn exp(iαnx1), Re{

∫

Γb

−T1u⋅uds} ≥ 0, ∀u ∈ H1

α(Ωb)2.

T2v = − ∑

∣n∣<M

Wn ˆ vn exp(iαnx1),

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 10 (47)

Variational formulation Variational formulation: find u ∈ Vα such that B(u,ϕ) := ∫

Ωb

E (u,ϕ)−ω2u⋅ϕ dx−

∫

Γb

ϕ ⋅T uds =

∫

Γb

(Tuin −T uin)⋅ϕds for all ϕ ∈ Vα, where E (u,ϕ) = (λ +2µ)(∂1u1∂1ϕ1 +∂2u2∂2ϕ2)+ µ(∂2u1∂2ϕ1 +∂1u2∂1ϕ2) +λ(∂1u1∂2ϕ2 +∂2u2∂1ϕ1)+ µ(∂2u1∂1ϕ2 +∂1u2∂2ϕ1).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 11 (47)

Variational formulation Variational formulation: find u ∈ Vα such that B(u,ϕ) := ∫

Ωb

E (u,ϕ)−ω2u⋅ϕ dx−

∫

Γb

ϕ ⋅T uds =

∫

Γb

(Tuin −T uin)⋅ϕds for all ϕ ∈ Vα, where E (u,ϕ) = (λ +2µ)(∂1u1∂1ϕ1 +∂2u2∂2ϕ2)+ µ(∂2u1∂2ϕ1 +∂1u2∂1ϕ2) +λ(∂1u1∂2ϕ2 +∂2u2∂1ϕ1)+ µ(∂2u1∂1ϕ2 +∂1u2∂2ϕ1). Bu = F , F ∈ V ∗

α ,

where (Bu,ϕ)Ωb := B(u,v), (F,ϕ)Ωb := ∫

Γb

(Tuin −T uin)⋅ϕ ds, and (⋅,⋅)Ωb denotes the duality between Vα and V ∗

α.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 11 (47)

Fredholm alternative Theorem Let A be a compact operator from a Hilbert space H1 into a Hilbert space H2, then

■ The equation

(I +A )u = g, g ∈ H2 (1) admits a unique solution, if the homogeneous equation (I +A )u = 0 admits uniquely the trivial solution u = 0.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 12 (47)

Fredholm alternative Theorem Let A be a compact operator from a Hilbert space H1 into a Hilbert space H2, then

■ The equation

(I +A )u = g, g ∈ H2 (1) admits a unique solution, if the homogeneous equation (I +A )u = 0 admits uniquely the trivial solution u = 0.

■ The equation (1) is solvable if and only if

< g,v >= 0 holds for all v ∈ H∗

2 satisfying

(I +A ∗)v = 0.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 12 (47)

Strong ellipticity of the sesquilinear form B B(u,ϕ) := ∫

Ωb

E (u,ϕ)−ω2u⋅ϕ dx−

∫

Γb

ϕ ⋅T uds Korn’s inequality:

∫

Ωb

E (u,u)dx ≥ C∣∣u∣∣2

H1(Ωα)2

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 13 (47)

Strong ellipticity of the sesquilinear form B B(u,ϕ) := ∫

Ωb

E (u,ϕ)−ω2u⋅ϕ dx−

∫

Γb

ϕ ⋅T uds Korn’s inequality:

∫

Ωb

E (u,u)dx ≥ C∣∣u∣∣2

H1(Ωα)2

Properties of the Dirichlet-to-Neumann map T : T = T1 +T2, where Re{ ∫

Γb

−T1u⋅uds} ≥ 0, ∀u ∈ H1

α(Ωb)2,

T2 := T −T1 is a finite dimensional operator.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 13 (47)

Strong ellipticity of the sesquilinear form B B(u,ϕ) := ∫

Ωb

E (u,ϕ)−ω2u⋅ϕ dx−

∫

Γb

ϕ ⋅T uds Korn’s inequality:

∫

Ωb

E (u,u)dx ≥ C∣∣u∣∣2

H1(Ωα)2

Properties of the Dirichlet-to-Neumann map T : T = T1 +T2, where Re{ ∫

Γb

−T1u⋅uds} ≥ 0, ∀u ∈ H1

α(Ωb)2,

T2 := T −T1 is a finite dimensional operator. Conclusion: B is a Fredholm operator with index zero.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 13 (47)

Existence of a solution at arbitrary frequency Consider the operator equation Bu = F, F ∈ V ∗

α,

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 14 (47)

Existence of a solution at arbitrary frequency Consider the operator equation Bu = F, F ∈ V ∗

α,

where (F,ϕ)Ωb = ∫

Γb

(Tuin −T uin)⋅ϕds =

∫

Γb

f0 ⋅ϕds f0 = {

2iβ0kp(λ+2µ) d0

(−α,γ0)⊤eiαx1−iβ0b incident plane pressure wave − 2iγ0ksµ

d0

(β0,α,)⊤eiαx1−iγ0b incident plane shear wave

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 14 (47)

Existence of a solution at arbitrary frequency Consider the operator equation Bu = F, F ∈ V ∗

α,

where (F,ϕ)Ωb = ∫

Γb

(Tuin −T uin)⋅ϕds =

∫

Γb

f0 ⋅ϕds f0 = {

2iβ0kp(λ+2µ) d0

(−α,γ0)⊤eiαx1−iβ0b incident plane pressure wave − 2iγ0ksµ

d0

(β0,α,)⊤eiαx1−iγ0b incident plane shear wave Lemma If ϕ ∈ Vα satisfies B∗ϕ = 0, then Ap,n = 0 for ∣αn∣ < kp and As,n = 0 for ∣αn∣ < ks . (2) Here Ap,n and As,n are Rayleigh coefficients for ϕ ∈ Vα. In particular, Ap,0 = As,0 = 0.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 14 (47)

Existence of a solution at arbitrary frequency Consider the operator equation Bu = F, F ∈ V ∗

α,

where (F,ϕ)Ωb = ∫

Γb

(Tuin −T uin)⋅ϕds =

∫

Γb

f0 ⋅ϕds f0 = {

2iβ0kp(λ+2µ) d0

(−α,γ0)⊤eiαx1−iβ0b incident plane pressure wave − 2iγ0ksµ

d0

(β0,α,)⊤eiαx1−iγ0b incident plane shear wave Lemma If ϕ ∈ Vα satisfies B∗ϕ = 0, then Ap,n = 0 for ∣αn∣ < kp and As,n = 0 for ∣αn∣ < ks . (2) Here Ap,n and As,n are Rayleigh coefficients for ϕ ∈ Vα. In particular, Ap,0 = As,0 = 0. Conclusion: (F,ϕ)Ωb = 0 if B∗ϕ = 0, = ⇒ Existence of a solution !

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Uniqueness Lemma There exists small frequency ω0 > 0 such that Re(Bu,u)Ωb = ReB(u,u) ≥ C∣∣u∣∣2

H1(Ωb),

ω ∈ (0,ω0]. Conclusion: Uniqueness and existence hold for small frequencies !

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 15 (47)

Uniqueness Lemma There exists small frequency ω0 > 0 such that Re(Bu,u)Ωb = ReB(u,u) ≥ C∣∣u∣∣2

H1(Ωb),

ω ∈ (0,ω0]. Conclusion: Uniqueness and existence hold for small frequencies ! According to the Analytic Fredholm theory: either B−1 does not exist for any w ∈ ℝ+,

• r B−1 exists for all w ∈ ℝ+∖D where D is a discrete subset of ℝ+.

Conclusion: Uniqueness and existence hold for all frequencies excluding a discrete set with the only accumulating point at infinity.

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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,

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 16 (47)

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.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 16 (47)

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. Remark Uniqueness holds even if Λ is given by the graph of a Lipschitz function or Λ is a binary grating profile.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 16 (47)

Ongoing work on direct scattering problems Scattering of elastic waves by rough surfaces: (∆∗ +ω2)u = g in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u =

• n

Λ

O

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 17 (47)

Ongoing work on direct scattering problems Scattering of elastic waves by rough surfaces: (∆∗ +ω2)u = g in ΩΛ , ∆∗ := µ∆+(λ + µ)grad div , u =

• n

Λ

O

If the source term g ∈ L2(ΩΛ)2 has a compact support and Λ is the graph of a Lipschitz function, then existence and uniqueness of solutions in H1(ΩΛ)2 can be proved.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 17 (47)

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

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

■ K. Ito and F. Reitich, 1999, conjugate gradient algorithm based on analytic

continuation;

■ F. Hettlich, 2002, iterative regularization; ■ G. Bruckner, J. Elschner, G. C. Hsiao, A. Rathsfeld, 2002-2004, optimization

method;

■ T. Arens, N. Grinberg, A. Kirsch, 2003,2005, factorization method;

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 18 (47)

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

■ K. Ito and F. Reitich, 1999, conjugate gradient algorithm based on analytic

continuation;

■ F. Hettlich, 2002, iterative regularization; ■ G. Bruckner, J. Elschner, G. C. Hsiao, A. Rathsfeld, 2002-2004, optimization

method;

■ T. Arens, N. Grinberg, A. Kirsch, 2003,2005, factorization method;

Uniqueness for the Navier equation (using one incident plane wave):

■ Antonios, C., Drossos, G. and Kiriakie, K. 2001, smooth gratings with a small

height, Dirichlet boundary condition, n=2.

■ J. Elschner and G. Hu, 2011, polygonal or polyhedral gratings, the third and

fourth kind boundary conditions, n=2,3.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 18 (47)

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 ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

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 third (or forth) kind boundary conditions on Λ1 resp. Λ2.
• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane elastic wave with the

incident angle θ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

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 third (or forth) kind boundary conditions on Λ1 resp. Λ2.
• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane elastic wave with the

incident angle θ. Questions:

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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

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 third (or forth) kind boundary conditions on Λ1 resp. Λ2.
• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane elastic wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)
• 2. If not, what kind of geometric characteristics do Λ1 and Λ2 share so as to

generate the same near field on x2 = b ?

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

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 third (or forth) kind boundary conditions on Λ1 resp. Λ2.
• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane elastic wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)
• 2. If not, what kind of geometric characteristics do Λ1 and Λ2 share so as to

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 ?

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

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 third (or forth) kind boundary conditions on Λ1 resp. Λ2.
• 3. u1(x1,b;θ) = u2(x1,b;θ) holds for one incident plane elastic wave with the

incident angle θ. Questions:

• 1. Can we obtain Λ1 = Λ2? (uniqueness)
• 2. If not, what kind of geometric characteristics do Λ1 and Λ2 share so as to

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 ?

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 19 (47)

Uniqueness under the fourth kind boundary conditions Theorem Assume that the fourth kind boundary conditions are imposed on Λ1 and Λ2. if u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) holds for one incident pressure wave, 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)⊤.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 20 (47)

Uniqueness under the fourth kind boundary conditions Theorem Assume that the fourth kind boundary conditions are imposed on Λ1 and Λ2. if u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) holds for one incident pressure wave, 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 the near-field data u(x1,b;θ) corresponding to one incident plane pressure wave can uniquely determine Λ, under the boundary conditions of the fourth kind.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 20 (47)

The unidentifiable class D2(θ,kp).

−4*pi −2*pi 2*pi 4*pi −10 −5 5 10

D2(θ,kp) := ⎧  ⎨  ⎩ Λ ∈ A : Each line segment of Λ lies on the straight line x2 = x1 tanϕ +

2π kp cosθ n for some n ∈ ℤ with

ϕ ∈ { θ

2 + π 4 , θ 2 − π 4 }, and kp(1±sinθ) ∈ ℤ.

⎫  ⎬  ⎭ .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 21 (47)

The unidentifiable class D2(θ,kp), θ = π/6, kp = 2.

−4*pi −2*pi 2*pi 4*pi −10 −5 5 10

Features: (1) Λ ∈ D2(π/6,2) provided Λ lies on the grids. (2) The angle between two neighboring line segments of Λ is fixed (π/2). (3) The set D2(θ,kp) is uniquely determined by the incident pressure wave. D2(θ,kp) := ⎧  ⎨  ⎩ Λ ∈ A : Each line segment of Λ lies on the straight line x2 = x1 tanϕ +

2π kp cosθ n for some n ∈ ℤ with

ϕ ∈ { θ

2 + π 4 , θ 2 − π 4 }, and kp(1±sinθ) ∈ ℤ.

⎫  ⎬  ⎭ .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 22 (47)

Uniqueness under the third kind boundary conditions Theorem If u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) holds for one incident pressure wave with the incident angle θ, then either Λ1 = Λ2 or one of the following cases occurs:

• 1. (a)Λ1,Λ2 ∈ N2(θ,kp), and the total field takes the form

u = ˆ θ exp(ikpx⋅ ˆ θ)− ˆ θ exp(−ikpx⋅ ˆ θ).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 23 (47)

Uniqueness under the third kind boundary conditions Theorem If u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) holds for one incident pressure wave with the incident angle θ, then either Λ1 = Λ2 or one of the following cases occurs:

• 1. (a)Λ1,Λ2 ∈ N2(θ,kp), and the total field takes the form

u = ˆ θ exp(ikpx⋅ ˆ θ)− ˆ θ exp(−ikpx⋅ ˆ θ). (b) Λ1,Λ2 ∈ D2(θ,kp), and 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). 2. 3.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 23 (47)

Uniqueness under the third kind boundary conditions Theorem If u1(x1,b;θ) = u2(x1,b;θ), ∀ x1 ∈ (0,2π) holds for one incident pressure wave with the incident angle θ, then either Λ1 = Λ2 or one of the following cases occurs:

• 1. (a)Λ1,Λ2 ∈ N2(θ,kp), and the total field takes the form

u = ˆ θ exp(ikpx⋅ ˆ θ)− ˆ θ exp(−ikpx⋅ ˆ θ). (b) Λ1,Λ2 ∈ D2(θ,kp), and 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).

• 2. Λ1,Λ2 ∈ N3(θ,kp) with θ ∈ [− π

6 , π 6 ]. The total field takes the form

u = ˆ θ exp(ikpx⋅ ˆ θ)+Rot 2π

3

ˆ θ exp(ikpx⋅Rot 2π

3

ˆ θ)+Rot 4π

3

ˆ θ exp(ikpx⋅Rot 4π

3

ˆ θ).

• 3. Λ1,Λ2 ∈ N4(0,kp), θ = 0, and the total field takes the form

u = −e2 exp(−ikpx2)+e2 exp(ikpx2)+e1 exp(ikpx1)−e1 exp(−ikpx1). Here N2,D2,N3,N4 ∈ A are four classes of unidentifiable grating profiles.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 24 (47)

The unidentifiable classesN3 and N4

Figure: N3(π/6,2)

−2*pi 2*pi −6 −4 −2 2 4 6

Figure: N4(0,4)

−2*pi 2*pi −6 −4 −2 2 4 6 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 25 (47)

Uniqueness with minimal number of incident elastic waves Theorem

■ Under the fourth kind boundary conditions,

two incident pressure waves or four incident shear waves are enough to uniquely determine a grating Λ ∈ A .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 26 (47)

Uniqueness with minimal number of incident elastic waves Theorem

■ Under the fourth kind boundary conditions,

two incident pressure waves or four incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is

• ne incident pressure wave resp. three incident shear waves.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 26 (47)

Uniqueness with minimal number of incident elastic waves Theorem

■ Under the fourth kind boundary conditions,

two incident pressure waves or four incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is

• ne incident pressure wave resp. three incident shear waves.

■ Under the third kind boundary conditions,

four incident pressure waves or two incident shear waves are enough to uniquely determine a grating Λ ∈ A .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 26 (47)

Uniqueness with minimal number of incident elastic waves Theorem

■ Under the fourth kind boundary conditions,

two incident pressure waves or four incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is

• ne incident pressure wave resp. three incident shear waves.

■ Under the third kind boundary conditions,

four incident pressure waves or two incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is three incident pressure waves resp. one incident shear wave.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 26 (47)

Uniqueness with minimal number of incident elastic waves Theorem

■ Under the fourth kind boundary conditions,

two incident pressure waves or four incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is

• ne incident pressure wave resp. three incident shear waves.

■ Under the third kind boundary conditions,

four incident pressure waves or two incident shear waves are enough to uniquely determine a grating Λ ∈ A . If Rayleigh frequencies of the compressional resp. shear part are excluded, the minimal number is three incident pressure waves resp. one incident shear wave.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 26 (47)

Basic tool: reflection principle (Elschner & Yamamoto 2010) Suppose that (∆∗ +ω2)u = 0 in ℝ2 and u satisfies the third (resp. fourth) kind boundary conditions on both lines l0 and l1 (with the angle α). Then

■ u satisfies the same boundary condition on l2,

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 27 (47)

Basic tool: reflection principle (Elschner & Yamamoto 2010) Suppose that (∆∗ +ω2)u = 0 in ℝ2 and u satisfies the third (resp. fourth) kind boundary conditions on both lines l0 and l1 (with the angle α). Then

■ u satisfies the same boundary condition on l2, ■ Rot2αu(x) = u(Rot2αx).

Here Rot is the rotation operator around the origin.

• Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 27 (47)

Other reflection principles Reflection principle for the Helmholtz equation

■ Cheng J., Yamamoto M., Alessandrini G., Rondi L., Liu H., Zou J., Elschner J.

( scattering by bounded obstacles)

■ Elschner J., Schmidt G. , Yamamoto M., Hu G. (scattering by periodic

structures)

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 28 (47)

Other reflection principles Reflection principle for the Helmholtz equation

■ Cheng J., Yamamoto M., Alessandrini G., Rondi L., Liu H., Zou J., Elschner J.

( scattering by bounded obstacles)

■ Elschner J., Schmidt G. , Yamamoto M., Hu G. (scattering by periodic

structures) Reflection principle for the Maxwell equation

■ Liu H., Yamamoto M. and Zou J. 2007 Reflection principle for the Maxwell

equations and its application to inverse electromagnetic scattering

■ Bao G., Zhang H. and Zou J. 2010 Unique determination of periodic polyhedral

gratings to appear in Trans. Amer. Math. Soc.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 28 (47)

Remarks

• 1. Our proofs mainly rely on the reflection principle for the Navier equation applied

to polygonal periodic structures, under the third or fourth kind boundary conditions.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 29 (47)

Remarks

• 1. Our proofs mainly rely on the reflection principle for the Navier equation applied

to polygonal periodic structures, under the third or fourth kind boundary conditions.

• 2. The uniqueness results can be extended to the case of bi-periodic structures. In

ℝ3, there exist seven unidentifiable sets corresponding to one incident elastic wave.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 29 (47)

Remarks

• 1. Our proofs mainly rely on the reflection principle for the Navier equation applied

to polygonal periodic structures, under the third or fourth kind boundary conditions.

• 2. The uniqueness results can be extended to the case of bi-periodic structures. In

ℝ3, there exist seven unidentifiable sets corresponding to one incident elastic wave.

• 3. One incident quasi-periodic point source wave is enough to uniquely determine

a polygonal or polyhedral grating profile under the third or fourth kind boundary conditions.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 29 (47)

Remarks

• 1. Our proofs mainly rely on the reflection principle for the Navier equation applied

to polygonal periodic structures, under the third or fourth kind boundary conditions.

• 2. The uniqueness results can be extended to the case of bi-periodic structures. In

ℝ3, there exist seven unidentifiable sets corresponding to one incident elastic wave.

• 3. One incident quasi-periodic point source wave is enough to uniquely determine

a polygonal or polyhedral grating profile under the third or fourth kind boundary conditions.

• 4. Our method cannot apply to the Dirichlet boundary condition, because we do

not know the corresponding reflection principle.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 29 (47)

Remarks

• 1. Our proofs mainly rely on the reflection principle for the Navier equation applied

to polygonal periodic structures, under the third or fourth kind boundary conditions.

• 2. The uniqueness results can be extended to the case of bi-periodic structures. In

ℝ3, there exist seven unidentifiable sets corresponding to one incident elastic wave.

• 3. One incident quasi-periodic point source wave is enough to uniquely determine

a polygonal or polyhedral grating profile under the third or fourth kind boundary conditions.

• 4. Our method cannot apply to the Dirichlet boundary condition, because we do

not know the corresponding reflection principle.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 29 (47)

Open problem Reflection principle under the Dirichlet boundary condition: Suppose that (∆∗ +ω2)u = 0 in ℝ2 and u = 0 on both l0 and l1 (with the angle α).

■ Can we obtain u = 0 on l2? ■ Does u exhibit any symmetry ?

• Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 30 (47)

Open problem Reflection principle under the Dirichlet boundary condition: Suppose that (∆∗ +ω2)u = 0 in ℝ2 and u = 0 on both l0 and l1 (with the angle α).

■ Can we obtain u = 0 on l2? ■ Does u exhibit any symmetry ?

• If α = π/2, ω = 0, then one solution is u = (−xy2,x2y)⊤.

Our conjecture: u is uniquely determined (up to a constant).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 30 (47)

Numerical algorithm for (IP): Dirichlet boundary condition Inverse problem: Given an incident plane wave uin(x) and the near-field data usc(x1,b) = ∑

n∈ℤ

An exp(iαnx1), An = Ap,n(αn,βn)⊤eiβnb +As,n(−γn,αn)⊤eiγnb, (3) find a grating profile Λ := {(x1, f(x1)) : 0 < x1 < 2π} such that uin +usc = 0

• n

Λ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 31 (47)

Numerical algorithm for (IP): Dirichlet boundary condition Inverse problem: Given an incident plane wave uin(x) and the near-field data usc(x1,b) = ∑

n∈ℤ

An exp(iαnx1), An = Ap,n(αn,βn)⊤eiβnb +As,n(−γn,αn)⊤eiγnb, (3) find a grating profile Λ := {(x1, f(x1)) : 0 < x1 < 2π} such that uin +usc = 0

• n

Λ. Inverse problem: Find f from the knowledge of finite number of Fourier coefficients An, n = 1,2,⋅⋅⋅ ,K.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 31 (47)

Numerical algorithm for (IP): Dirichlet boundary condition Inverse problem: Given an incident plane wave uin(x) and the near-field data usc(x1,b) = ∑

n∈ℤ

An exp(iαnx1), An = Ap,n(αn,βn)⊤eiβnb +As,n(−γn,αn)⊤eiγnb, (3) find a grating profile Λ := {(x1, f(x1)) : 0 < x1 < 2π} such that uin +usc = 0

• n

Λ. Inverse problem: Find f from the knowledge of finite number of Fourier coefficients An, n = 1,2,⋅⋅⋅ ,K. K: the number of propagating modes involved in computation. Far field data: {An : ∣αn∣ < kp or ∣αn∣ < ks}. We always assume that

• 1. βn ∕= 0,

γn ∕= 0 for all n ∈ ℤ. (Rayleigh frequencies are excluded)

• 2. b > f(t) > 0,t ∈ (0,2π).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 31 (47)

Fundamental solutions Fundamental solution to the Helmholtz equation (∆+k2)u = 0: Φk(x,y) = i 4H(1)

0 (k∣x−y∣),

x = (x1,x2),y = (y1,y2) ∈ ℝ2.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 32 (47)

Fundamental solutions Fundamental solution to the Helmholtz equation (∆+k2)u = 0: Φk(x,y) = i 4H(1)

0 (k∣x−y∣),

x = (x1,x2),y = (y1,y2) ∈ ℝ2. Fundamental solution to the Navier equation µ∆u+(λ + µ)grad div u+ω2u = 0: Γ(x,y) = i 4µ H(1)

0 (ks∣x−y∣)I +

i 4ω2 grad xgrad T

x

[ H(1)

0 (ks∣x−y∣)−H(1) 0 (kp∣x−y∣)

] = 1 µ Φks(x,y)I + 1 ω2 grad xgrad T

x

[ Φks(x,y)−Φkp(x,y) ] .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 32 (47)

Quasiperiodic fundamental solutions The α-quasiperiodic fundamental solution to the Helmholtz equation Gk(x,y) =

∑

n∈ℤ

exp(−iα2πn)Φk(x+n(2π,0),y) x−y ∕= n(2π,0) = i 4π ∑

n∈ℤ

1 βn exp(iαn(x1 −y1)+iβn∣x2 −y2∣).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 33 (47)

Quasiperiodic fundamental solutions The α-quasiperiodic fundamental solution to the Helmholtz equation Gk(x,y) =

∑

n∈ℤ

exp(−iα2πn)Φk(x+n(2π,0),y) x−y ∕= n(2π,0) = i 4π ∑

n∈ℤ

1 βn exp(iαn(x1 −y1)+iβn∣x2 −y2∣). The α-quasiperiodic fundamental solution (Green’s tensor) to the Navier equation: Π(x,y) = ∑

n∈ℤ

exp(−iα2πn)Γ(x+n(2π,0),y) = 1 µ Gks(x,y)I + 1 ω2 grad xgrad T

x

[ Gks(x,y)−Gkp(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) ] .

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 33 (47)

A two-step algorithm Step 1 Reconstruct the scattered field usc. 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 Tϕ(x1) := 1 2π ∫ 2π Π(x1,b ;t,0)ϕ(t)dt = usc(x1,b).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 34 (47)

A two-step algorithm Step 1 Reconstruct the scattered field usc. 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 Tϕ(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),b j ∈ ℝ, j = 0,1,⋅⋅⋅2M}.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 34 (47)

Step 1: Solve the first kind integral equation Tϕ(x1) = ub Tikhonov regularization: γϕ +T ∗Tϕ = T ∗ub, (4) γ > 0: regularization parameter, T ∗: the adjoint operator of T.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 35 (47)

Step 1: Solve the first kind integral equation Tϕ(x1) = ub Tikhonov regularization: γϕ +T ∗Tϕ = T ∗ub, (4) γ > 0: regularization parameter, T ∗: the adjoint operator of T. Solution : ϕγ = ∑

∣n∣≤K

ϕ(n)

γ

exp(iαnt), ϕ(n)

γ

:=

2

∑

j=1

σ(n)

j

(σ(n)

j

)2 +γ ( An,U(n)

j

) V (n)

j

,

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 35 (47)

Step 1: Solve the first kind integral equation Tϕ(x1) = ub Tikhonov regularization: γϕ +T ∗Tϕ = T ∗ub, (4) γ > 0: regularization parameter, T ∗: the adjoint operator of T. Solution : ϕγ = ∑

∣n∣≤K

ϕ(n)

γ

exp(iαnt), ϕ(n)

γ

:=

2

∑

j=1

σ(n)

j

(σ(n)

j

)2 +γ ( An,U(n)

j

) V (n)

j

, where U(n) = (U(n)

1 ,U(n) 2 ), V (n) = (V (n) 1

,V (n)

2

), Σ(n) = diag(σ(n)

1 ,σ(n) 2 ).

is the singular value decomposition of M(n) given by M(n) := eiβnb 4πω2βn ( α2

n

αnβn αnβn β 2

n

) + { i 4πµγn I − i 4πω2γn ( α2

n

αnγn αnγn γ2

n

)} eiγnb.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 35 (47)

Step 1: Solve the first kind integral equation Tϕ(x1) = ub Tikhonov regularization: γϕ +T ∗Tϕ = T ∗ub, (4) γ > 0: regularization parameter, T ∗: the adjoint operator of T. Solution : ϕγ = ∑

∣n∣≤K

ϕ(n)

γ

exp(iαnt), ϕ(n)

γ

:=

2

∑

j=1

σ(n)

j

(σ(n)

j

)2 +γ ( An,U(n)

j

) V (n)

j

, where U(n) = (U(n)

1 ,U(n) 2 ), V (n) = (V (n) 1

,V (n)

2

), Σ(n) = diag(σ(n)

1 ,σ(n) 2 ).

is the singular value decomposition of M(n) given by M(n) := eiβnb 4πω2βn ( α2

n

αnβn αnβn β 2

n

) + { i 4πµγn I − i 4πω2γn ( α2

n

αnγn αnγn γ2

n

)} eiγnb.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 35 (47)

Step 2: Find f by minimizing ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 Suppose that a = (a0,a1,⋅⋅⋅ ,aM,⋅⋅⋅ ,a2M) ∈ ℝ2M+1 such that f(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt). for some M ∈ ℕ.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 36 (47)

Step 2: Find f by minimizing ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 Suppose that a = (a0,a1,⋅⋅⋅ ,aM,⋅⋅⋅ ,a2M) ∈ ℝ2M+1 such that f(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt). for some M ∈ ℕ. Then inf

f∈M ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 ⇔

inf

a∈ℝ2M+1 N

∑

j=1

r2

j(a), K ∈ ℕ+

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 36 (47)

Step 2: Find f by minimizing ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 Suppose that a = (a0,a1,⋅⋅⋅ ,aM,⋅⋅⋅ ,a2M) ∈ ℝ2M+1 such that f(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt). for some M ∈ ℕ. Then inf

f∈M ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 ⇔

inf

a∈ℝ2M+1 N

∑

j=1

r2

j(a), K ∈ ℕ+

where, for sj = 2π(j −1)/N, rj(a)2 = 1 N ∣ ˆ θe−iβ f(sj) + ∑

∣n∣≤N

(P(n)ϕ(n)

γ

eiβn f(sj) +S(n)ϕ(n)

γ

eiγn f(sj))∣2

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 36 (47)

Step 2: Find f by minimizing ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 Suppose that a = (a0,a1,⋅⋅⋅ ,aM,⋅⋅⋅ ,a2M) ∈ ℝ2M+1 such that f(t) = a0 +

M

∑

m=1

am cos(mt)+aM+m sin(mt). for some M ∈ ℕ. Then inf

f∈M ∣∣uin(x1, f(x1))+usc(x1, f(x1))∣∣L2 ⇔

inf

a∈ℝ2M+1 N

∑

j=1

r2

j(a), K ∈ ℕ+

where, for sj = 2π(j −1)/N, rj(a)2 = 1 N ∣ ˆ θe−iβ f(sj) + ∑

∣n∣≤N

(P(n)ϕ(n)

γ

eiβn f(sj) +S(n)ϕ(n)

γ

eiγn f(sj))∣2 To solve the least-squares problem F(a) =

N

∑

j=1

r2

j(a) →

inf

a∈ℝ2M+1,

we may use Gauss-Newton Method, Levenberg-Marquardt Method or Trust-Region Reflective Algorithm.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 36 (47)

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). ■ K: the number of propagating modes involved in computation ■ K < 4: partial far-field data ■ K = 4: far-field data ■ K > 4: far-field data + partial evanescent modes

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 37 (47)

Example 1: Fourier gratings (K = 7,γ = 10−12) Example 1 Suppose that f(t) = 2+ζ(cos(t)+cos(2t)+cos(3t)), ζ = 0.05π. f ∗(t) = a0 +a1 cos(t)+a2 cos(2t)+a3 cos(3t)+a4 sin(t)+a5 sin(2t)+a6 sin(3t).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 38 (47)

Example 1: Fourier gratings (K = 7,γ = 10−12) Example 1 Suppose that f(t) = 2+ζ(cos(t)+cos(2t)+cos(3t)), ζ = 0.05π. f ∗(t) = a0 +a1 cos(t)+a2 cos(2t)+a3 cos(3t)+a4 sin(t)+a5 sin(2t)+a6 sin(3t). a0 a1 a2 a3 a4 a5 a6 Target 2 0.157 0.157 0.157 Initial LB

• 5

UB 5 5 5 5 5 5 5 δ = 0 2.0026 0.1590 0.1610 0.1596 δ = 5% 2.0029 0.1585 0.1592 0.1609 0.0004 0.0001 0.0001 δ = 8% 2.0021 0.1594 0.1610 0.1595 0.0003

• 0.0010
• 0.0005

δ = 10% 2.0029 0.1609 0.1601 0.1600 0.0023

• 0.0019

0.0018

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 38 (47)

Sensitivity analysis to the 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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 39 (47)

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.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 40 (47)

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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 40 (47)

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.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 41 (47)

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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 41 (47)

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 ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 42 (47)

Example 3: binary gratings (K = 4,δ = 0,γ = 10−4) A prior information: the unknown surface is a binary grating with a finite 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 single incident angle θ = 0 (right).

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 43 (47)

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 (∣n∣ ≤ 4) (Left) K = 7: far-field data and partial evanescent modes (4 < ∣n∣ ≤ 7) (Left) K = 3: partial far field data (∣n∣ ≤ 3) (Right)

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 44 (47)

Remarks In our reported numerical examples,

■ the unknown grating profile is assumed to have a finite number of parameters

(e.g. Fourier coefficients or corner points);

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 45 (47)

Remarks In our reported numerical examples,

■ the unknown grating profile is assumed to have a finite number of parameters

(e.g. Fourier coefficients or corner points);

■ the near-field data for the direct scattering problem is obtained by a discrete

Galerkin method

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 45 (47)

Remarks In our reported numerical examples,

■ the unknown grating profile is assumed to have a finite number of parameters

(e.g. Fourier coefficients or corner points);

■ the near-field data for the direct scattering problem is obtained by a discrete

Galerkin method Compared to the Kirsch-Kress optimization method for solving a combined cost functional: F(ϕ, f) = ∣∣Tϕ −ub∣∣L2(0,2π) +ρ∣∣Tϕ +uin∣∣L2(Λf ) +γ∣∣ϕ∣∣L2(0,2π) the two-step algorithm can reduce much computational effort, because

■ 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 ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 45 (47)

Remarks In our reported numerical examples,

■ the unknown grating profile is assumed to have a finite number of parameters

(e.g. Fourier coefficients or corner points);

■ the near-field data for the direct scattering problem is obtained by a discrete

Galerkin method Compared to the Kirsch-Kress optimization method for solving a combined cost functional: F(ϕ, f) = ∣∣Tϕ −ub∣∣L2(0,2π) +ρ∣∣Tϕ +uin∣∣L2(Λf ) +γ∣∣ϕ∣∣L2(0,2π) the two-step algorithm can reduce much computational effort, because

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

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

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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 45 (47)

Remarks In our reported numerical examples,

■ the unknown grating profile is assumed to have a finite number of parameters

(e.g. Fourier coefficients or corner points);

■ the near-field data for the direct scattering problem is obtained by a discrete

Galerkin method Compared to the Kirsch-Kress optimization method for solving a combined cost functional: F(ϕ, f) = ∣∣Tϕ −ub∣∣L2(0,2π) +ρ∣∣Tϕ +uin∣∣L2(Λf ) +γ∣∣ϕ∣∣L2(0,2π) the two-step algorithm can reduce much computational effort, because

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

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

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

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 45 (47)

Conclusions

■ Variational approach for the direct scattering problem under the first, second,

third and forth kind boundary conditions. Ongoing work: scattering of elastic waves by non-periodic rough surfaces

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 46 (47)

Conclusions

■ Variational approach for the direct scattering problem under the first, second,

third and forth kind boundary conditions. Ongoing work: scattering of elastic waves by non-periodic rough surfaces

■ Unidentifiable classes within polyhedral or polygonal grating profiles

corresponding one incident plane elastic wave, under the third and forth kind boundary conditions. Ongoing work: reflection principle for the Navier equation under the Dirichlet boundary condition

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 46 (47)

Conclusions

■ Variational approach for the direct scattering problem under the first, second,

third and forth kind boundary conditions. Ongoing work: scattering of elastic waves by non-periodic rough surfaces

■ Unidentifiable classes within polyhedral or polygonal grating profiles

corresponding one incident plane elastic wave, under the third and forth kind boundary conditions. Ongoing work: reflection principle for the Navier equation under the Dirichlet boundary condition

■ A Two-step algorithm applied to the inverse elastic scattering problem, under the

Dirichlet boundary condition. Future work: convergence analysis of the two-step algorithm.

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 46 (47)

Thank you very much for your attention !

Direct and Inverse Elastic Scattering Problems for Diffraction Gratings ⋅ Workshop 3, Linz, Nov. 24, 2011 ⋅ Page 47 (47)