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 Direct and inverse elastic scattering problems in periodic structures 1 Direct scattering problem: uniqueness and existence 2 3 Inverse scattering problem: uniqueness for polygonal gratings 4 Inverse scattering problem: a two-step algorithm Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 2 (27)
Elastic scattering by diffraction gratings Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 3 (27)
Elastic scattering by periodic structures: application in geophysics Characterize fractures using elastic waves in search for gas and liquids Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 4 (27)
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 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 5 (27)
Basic assumptions � The periodic surface Λ is invariant in one direction and 2 π -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 − i ω t . Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 6 (27)
Direct problems � Navier equation in 2D (case φ = 0 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in Angular frequency: ω > 0 Lamé constants: µ > 0 , λ + µ > 0 � Compressional wave number: k p : = ω / 2 µ + λ Shear wave number: k s : = ω / √ µ Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 7 (27)
Direct problems � Navier equation in 2D (case φ = 0 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in Angular frequency: ω > 0 Lamé constants: µ > 0 , λ + µ > 0 � Compressional wave number: k p : = ω / 2 µ + λ Shear wave number: k s : = ω / √ µ Incident angle: θ ∈ ( − π / 2 , π / 2 ) Incident plane pressure wave: u in p ( x ) = ˆ θ exp ( ik p x · ˆ ˆ θ ) , θ : = ( sin θ , − cos θ ) , x = ( x 1 , x 2 ) Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 7 (27)
Direct problems � Navier equation in 2D (case φ = 0 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in 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 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in � Quasi-periodicity: u ( x 1 + 2 π , x 2 ) = exp ( 2 i απ ) u ( x 1 , x 2 ) , α : = k p sin θ . 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 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in � Quasi-periodicity: u ( x 1 + 2 π , x 2 ) = exp ( 2 i απ ) u ( x 1 , x 2 ) , α : = k p sin θ . � Dirichlet boundary condition on the grating profile: u = 0 Λ . on 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 ): ( ∆ ∗ + ω 2 ) u = 0 ∆ ∗ : = µ ∆ +( λ + µ ) grad div , Ω Λ , in u = u in + u sc Ω Λ . in � Quasi-periodicity: u ( x 1 + 2 π , x 2 ) = exp ( 2 i απ ) u ( x 1 , x 2 ) , α : = k p sin θ . � Dirichlet boundary condition on the grating profile: u = 0 Λ . on � An appropriate radiation condition imposed on u sc as x 2 → ∞ . Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 8 (27)
Rayleigh Expansion Radiation Condition (RERC) u sc = ∇ u p + − − → ( ∆ + k 2 ( ∆ + k 2 curl u s , p ) u p = 0 , s ) u s = 0 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 9 (27)
Rayleigh Expansion Radiation Condition (RERC) u sc = ∇ u p + − − → ( ∆ + k 2 ( ∆ + k 2 curl u s , p ) u p = 0 , s ) u s = 0 � Radiation condition � � α n u sc ( x ) ∑ = exp ( i α n x 1 + i β n x 2 ) A p , n β n n ∈ Z � � γ n + ∑ exp ( i α n x 1 + i γ n x 2 ) A s , n − α n n ∈ Z for x 2 > Λ + : = max ( x 1 , x 2 ) ∈ Λ x 2 . Here, � k 2 p − α 2 | α n | ≤ k p if n α n : = α + n , β n = β n ( θ ) : = � α 2 n − k 2 if | α n | > k p , i p � � k 2 s − α 2 if | α n | ≤ k s n γ n = γ n ( θ ) : = � α 2 n − k 2 i if | α n | > k s . s The constants A p , n , A s , n ∈ C are called the Rayleigh coefficients. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 9 (27)
Direct and inverse scattering problems Direct Problem (DP) Given Λ ⊂ R 2 and u in , find u = u in + u sc ∈ H 1 loc ( Ω Λ ) 2 under the boundary, quasi-periodicity and radiation conditions. 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 ⊂ R 2 and u in , find u = u in + u sc ∈ H 1 loc ( Ω Λ ) 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 ⊂ R 2 and u in , find u = u in + u sc ∈ H 1 loc ( Ω Λ ) 2 under the boundary, quasi-periodicity and radiation conditions. � Uniqueness and existence of solutions � Numerical solutions Inverse Problem (IP) Given incident field u in ( x ; θ ) and the near-field data u ( x 1 , 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 ⊂ R 2 and u in , find u = u in + u sc ∈ H 1 loc ( Ω Λ ) 2 under the boundary, quasi-periodicity and radiation conditions. � Uniqueness and existence of solutions � Numerical solutions Inverse Problem (IP) Given incident field u in ( x ; θ ) and the near-field data u ( x 1 , b ; θ ) , determine the unknown scattering surface Λ . � Can we identify Λ uniquely ? � How to recover Λ numerically ? Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 10 (27)
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). Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 11 (27)
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. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 11 (27)
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). Direct and Inverse Elastic Scattering Problems for Diffraction Gratings · PICOF’12, April 4, 2012 · Page 11 (27)
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