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Zentrum f ur Scattering from Perturbed Periodic Structures Technomathematik The Bloch Transform and Scattering from Perturbed Periodic Structures Ruming Zhang, joint with Armin Lechleiter Center for Industrial Mathematics, University of

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch Transform and Scattering from Perturbed Periodic Structures

Ruming Zhang, joint with Armin Lechleiter

Center for Industrial Mathematics, University of Bremen

Linz, October 2016

Ruming Zhang 1 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Non-destructive Testing of Nano-grass

Applications: Optics, sensors, self-cleaning materials, . . .

Ruming Zhang 2 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Non-destructive Testing of Nano-grass

Simple model: Scattering from perturbed periodic structure Maybe possible (but more complicated): Scattering from random structure (?)

Ruming Zhang 3 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Direct Scattering Problems

Given: Periodic structure with local perturbation Incident wave field (non-periodic!) Searched-for: Scattered wave field (analytically/numerically) Setting is less important – possible to look at . . . acoustics and Dirichlet/impedance boundary condition or inhomogeneous medium . . . inhomogeneous electromagnetic medium . . .

Ruming Zhang 4 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Periodic Scattering Problems

Almost classical if everything is (quasi-)periodic but difficult to treat scattering problems in case of modes propagating along structure Fliss, Joly, Li: Scattering in closed periodic waveguides (’05, ’09, ’16) Coatl´ even: Scattering in periodic medium + defect (’12) Hoang, Radosz: Limiting absorption principle for periodic waveguides (’11,’14) Hohage, Soussi: Riesz bases for the translation operator in periodic waveguides (’13)

Ruming Zhang 5 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

1 Dirichlet Scattering Problem 2 Bloch Transform 3 Numerical Analysis

Ruming Zhang 6 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Scattering from Rough Dirichlet Surfaces

Domain Ω = {x ∈ R2 : x2 > ζ(x1)} with ∂Ω = {(x1, ζ(x1)) : x1 ∈ R} for function ζ : R → R Incident field ui solves ∆ui + k2ui = 0 in Ω Seek total field u such that ∆u + k2u = 0 in Ω, u|∂Ω = 0 and us = u − ui radiates: For x2 > H, us(x) =

  • ΓH

e

−i

  • ξx1+√

k2−|ξ|2(x2−H)

  • us(ξ, H) dξ

Ruming Zhang 7 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Existence and Uniqueness

ΩH = {x ∈ R2 : ζ(x1) < x2 < H} ⊂ R2 for ζ : R → R Weighted Sobolev spaces Hs

r(ΩH) with norm

u →

  • (1 + x2

1)r/2u

  • Hs(ΩH),

s, r ∈ R Incident field ui solves ∆ui + k2ui = 0 in Ω and belongs to H1

r (ΩH) for |r| < 1

Seek total field u ∈ H1

r (ΩH) H1 loc(Ω) such that

∆u + k2u = 0 in Ω, u|∂Ω = 0 and us = u − ui radiates Chandler-Wilde & Elschner ’10: Scattering problem solvable in H1

r (ΩH) H1 loc(Ω) if ζ Lipschitz continuous

Ruming Zhang 8 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Existence and Uniqueness

ΩH = {x ∈ R2 : ζ(x1) < x2 < H} ⊂ R2 for ζ : R → R Weighted Sobolev spaces Hs

r(ΩH) with norm

u →

  • (1 + x2

1)r/2u

  • Hs(ΩH),

s, r ∈ R Incident field ui solves ∆ui + k2ui = 0 in Ω and belongs to H1

r (ΩH) for |r| < 1

Seek total field u ∈ H1

r (ΩH) H1 loc(Ω) such that

∆u + k2u = 0 in Ω, u|∂Ω = 0 and us = u − ui radiates Chandler-Wilde & Elschner ’10: Scattering problem solvable in H1

r (ΩH) H1 loc(Ω) if ζ Lipschitz continuous

Ruming Zhang 8 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Existence and Uniqueness

ΩH = {x ∈ R2 : ζ(x1) < x2 < H} ⊂ R2 for ζ : R → R Weighted Sobolev spaces Hs

r(ΩH) with norm

u →

  • (1 + x2

1)r/2u

  • Hs(ΩH),

s, r ∈ R Incident field ui solves ∆ui + k2ui = 0 in Ω and belongs to H1

r (ΩH) for |r| < 1

Seek total field u ∈ H1

r (ΩH) H1 loc(Ω) such that

∆u + k2u = 0 in Ω, u|∂Ω = 0 and us = u − ui radiates Chandler-Wilde & Elschner ’10: Scattering problem solvable in H1

r (ΩH) H1 loc(Ω) if ζ Lipschitz continuous

Ruming Zhang 8 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Main Tool: Floquet–Bloch Transform

Bloch transform “periodic” Fourier transform Period Λ > 0 ⇒ α-quasiperiodicity: f(x + Λ) = e−iα Λf(x) 1D-Bloch transform J defined for f ∈ C∞

0 (R) by

J f(α, x) = Λ 2π 1/2

j∈Z

f(x + Λj)eiα Λj J f(·, x) is periodic with period one ⇒ set W ∗ =

  • − 1

2, 1 2

  • J f(α, ·) is α-qp with period 2π ⇒ set W = (−π, π]

J maps Hs(R) into L2 ; Hs

α()

  • Ruming Zhang

9 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Main Tool: Floquet–Bloch Transform

Bloch transform “periodic” Fourier transform Period 2π ⇒ α-quasiperiodicity: f(x + 2π) = e−2πi αf(x) 1D-Bloch transform J defined for f ∈ C∞

0 (R) by

J f(α, x) =

  • j∈Z

f(x + 2πj)e2πi αj J f(·, x) is periodic with period one ⇒ set W ∗ =

  • − 1

2, 1 2

  • J f(α, ·) is α-qp with period 2π ⇒ set W = (−π, π]

J maps Hs(R) into L2 (−1/2, 1/2); Hs

α( − π, π)

  • Ruming Zhang

9 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Main Tool: Floquet–Bloch Transform

Bloch transform “periodic” Fourier transform Period 2π ⇒ α-quasiperiodicity: f(x + 2π) = e−2πi αf(x) 1D-Bloch transform J defined for f ∈ C∞

0 (R) by

J f(α, x) =

  • j∈Z

f(x + 2πj)e2πi αj J f(·, x) is periodic with period one ⇒ set W ∗ =

  • − 1

2, 1 2

  • J f(α, ·) is α-qp with period 2π ⇒ set W = (−π, π]

J maps Hs(R) into L2 W ∗; Hs

α(W)

  • for all s ∈ R

Ruming Zhang 9 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Bloch Transform on Domains

Ω ΩH Ω2π

H

Periodic domain ΩH = {x ∈ R2 : ζ(x1) < x2 < H} (Partial) Bloch transform in 2D: J u(α, x) =

  • j∈Z

u ( x1+2πj

x2

) e2πi αj J : Hs(ΩH) → L2((−1/2, 1/2); Hs

α(Ω2π H ))

Inverse transform J −1: J −1w ( x1+2πj

x2

) =

  • W ∗ w(α, x)e2πi αj dα,

x ∈ Ω2π

H , j ∈ Z

Ruming Zhang 10 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Bloch Transform on Domains

Ω ΩH Ω2π

H

Periodic domain ΩH = {x ∈ R2 : ζ(x1) < x2 < H} (Partial) Bloch transform in 2D: J u(α, x) =

  • j∈Z

u ( x1+2πj

x2

) e2πi αj J : Hs(ΩH) → L2(W ∗; Hs

α(Ω2π H ))

Inverse transform J −1: J −1w ( x1+2πj

x2

) =

  • W ∗ w(α, x)e2πi αj dα,

x ∈ Ω2π

H , j ∈ Z

Ruming Zhang 10 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Bloch Transform on Domains

Ω ΩH Ω2π

H

Periodic domain ΩH = {x ∈ R2 : ζ(x1) < x2 < H} (Partial) Bloch transform in 2D: J u(α, x) =

  • j∈Z

u ( x1+2πj

x2

) e2πi αj J : Hs

r(ΩH) → Hr p(W ∗; Hs α(Ω2π H )) for s, r ∈ R

Inverse transform J −1: J −1w ( x1+2πj

x2

) =

  • W ∗ w(α, x)e2πi αj dα,

x ∈ Ω2π

H , j ∈ Z

Ruming Zhang 10 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Bloch Transform on Domains

Ω ΩH Ω2π

H

Periodic domain ΩH = {x ∈ R2 : ζ(x1) < x2 < H} (Partial) Bloch transform in 2D: J u(α, x) =

  • j∈Z

u ( x1+2πj

x2

) e2πi αj J : Hs

r(ΩH) → Hr p(W ∗;

Hs

α(Ω2π H )) for s, r ∈ R

Inverse transform J −1: J −1w ( x1+2πj

x2

) =

  • W ∗ w(α, x)e2πi αj dα,

x ∈ Ω2π

H , j ∈ Z

Ruming Zhang 10 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Bloch Transform on Domains

Ω ΩH Ω2π

H

Periodic domain ΩH = {x ∈ R2 : ζ(x1) < x2 < H} (Partial) Bloch transform in 2D: J u(α, x) =

  • j∈Z

u ( x1+2πj

x2

) e2πi αj J : Hs

r(ΩH) → Hr p(W ∗;

Hs

α(Ω2π H )) for s, r ∈ R

Inverse transform J −1 = J ∗: J −1w ( x1+2πj

x2

) =

  • W ∗ w(α, x)e2πi αj dα,

x ∈ Ω2π

H , j ∈ Z

Ruming Zhang 10 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Properties of the Bloch Transform

ΩH Ω2π

H

↓J

J −1↑ Bloch transform J commutes with periodic functions and partial derivatives: ∂xjJ u(α, x) = ∂xj

  • l

u x1+2πl

x2

  • e2πi αl

=

  • l
  • ∂xju

x1+2πl

x2

  • e2πi αl = J
  • ∂xju
  • (α, x)

Hence: w = J u satisfies ∆w(α, ·) + k2w(α, ·) = 0 Transformed function w(α, ·) ∈ H1

α(Ω2π H ) solves

α-quasiperiodic Dirichlet problem Weak formulations for u ∈ H1(ΩH) and w ∈ L2(W ∗; H1

α(Ω2π H )) are equivalent for r ∈ [0, 1)

Ruming Zhang 11 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Properties of the Bloch Transform

ΩH Ω2π

H

↓J

J −1↑ Bloch transform J commutes with periodic functions and partial derivatives: ∂xjJ u(α, x) = ∂xj

  • l

u x1+2πl

x2

  • e2πi αl

=

  • l
  • ∂xju

x1+2πl

x2

  • e2πi αl = J
  • ∂xju
  • (α, x)

Hence: w = J u satisfies ∆w(α, ·) + k2w(α, ·) = 0 Transformed function w(α, ·) ∈ H1

α(Ω2π H ) solves

α-quasiperiodic Dirichlet problem Weak formulations for u ∈ H1

r (ΩH) and

w ∈ Hr

0(W ∗; H1 α(Ω2π H )) are equivalent for r ∈ [0, 1)

Ruming Zhang 11 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

. . . and perturbed periodic structures?

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Perturbed periodic surface Γp defined by ζp : R → R such that ζp = ζ in R \ W Perturbed domains Ωp = {x2 > ζp(x1)} and Ωp

H = {ζp(x1) < x2 < H}

Diffeomorphism Φp : ΩH → Ωp

H such that

Φp = I2 in ΩH \ Ω2π

H′ for some H′ < H

Scattering from Γp ⇔ transformed prb for uT = u ◦ Φp:

  • Ωp

H

  • ∇u · ∇v − k2 uv
  • dx−
  • ΓH

v T +u ds =

  • ΓH

∂ui ∂x2 − T +ui

  • v ds

Ruming Zhang 12 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

. . . and perturbed periodic structures?

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Perturbed periodic surface Γp defined by ζp : R → R such that ζp = ζ in R \ W Perturbed domains Ωp = {x2 > ζp(x1)} and Ωp

H = {ζp(x1) < x2 < H}

Diffeomorphism Φp : ΩH → Ωp

H such that

Φp = I2 in ΩH \ Ω2π

H′ for some H′ < H

Scattering from Γp ⇔ transformed prb for uT = u ◦ Φp:

  • Ωp

H

  • ∇u · ∇v − k2 uv
  • dx −
  • ΓH

v T +u ds =

  • ΓH

fv ds

Ruming Zhang 12 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The “Periodized” Problem

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Recall: uT = u ◦ Φp ∈ H1(ΩH), set vT = v ◦ Φp

  • Ωp

H

  • ∇u · ∇v − k2 uv
  • dx −
  • ΓH

v T +u ds =

  • ΓH

f v ds with coefficients cp(x) = |det∇Φp(x)| and Ap(x) = |det∇Φp(x)|

  • ∇Φp(x)

−1 ∇Φp(x) −T Note: Ap − I2 & cp − 1: compact support in Ω2π

H !

⇒ J (Ap∇uT) = J (∇uT) + J ((Ap − I2)∇uT) = J (∇uT) +

  • j∈Z

[Ap − I2] ( x1+2πj

x2

) ∇uT ( x1+2πj

x2

) eiα2πj

Ruming Zhang 13 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The “Periodized” Problem

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Recall: uT = u ◦ Φp ∈ H1(ΩH), set vT = v ◦ Φp

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds with coefficients cp(x) = |det∇Φp(x)| and Ap(x) = |det∇Φp(x)|

  • ∇Φp(x)

−1 ∇Φp(x) −T Note: Ap − I2 & cp − 1: compact support in Ω2π

H !

⇒ J (Ap∇uT) = J (∇uT) + J ((Ap − I2)∇uT) = J (∇uT) +

  • j∈Z

[Ap − I2] ( x1+2πj

x2

) ∇uT ( x1+2πj

x2

) eiα2πj

Ruming Zhang 13 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The “Periodized” Problem

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Recall: uT = u ◦ Φp ∈ H1(ΩH), set vT = v ◦ Φp

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds with coefficients cp(x) = |det∇Φp(x)| and Ap(x) = |det∇Φp(x)|

  • ∇Φp(x)

−1 ∇Φp(x) −T Note: Ap − I2 & cp − 1: compact support in Ω2π

H !

⇒ J (Ap∇uT) = J (∇uT) + J ((Ap − I2)∇uT) = J (∇uT) + [Ap − I2]∇uT in Ω2π

H

Ruming Zhang 13 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The “Periodized” Problem

Ωp

H

↓

Φp Φ−1

p↑

ΩH Ω2π

H

↓J

J −1↑ Recall: uT = u ◦ Φp ∈ H1(ΩH), set vT = v ◦ Φp

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds with coefficients cp(x) = |det∇Φp(x)| and Ap(x) = |det∇Φp(x)|

  • ∇Φp(x)

−1 ∇Φp(x) −T Note: Ap − I2 & cp − 1: compact support in Ω2π

H !

⇒ J (Ap∇uT) = ∇wB(α, ·) + J ((Ap − I2)∇uT) = ∇wB(α, ·) + [Ap − I2]∇uT in Ω2π

H

Ruming Zhang 13 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds From J (Ap∇uT) = ∇wB + [Ap − I2]∇uT in Ω2π

H ,

  • W ∗
  • ΩH

Ap∇uT · ∇vT dx dα =

  • W ∗
  • ΩH

J −1 ◦ J (Ap∇uT) · ∇vT dx dα Next:

  • W ∗
  • ΓH

vT T +uT ds dα with α-quasiperiodic DtN map

+

ˆ

i(j−α)x

ˆ

i(j−α)x

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds From J (Ap∇uT) = ∇wB + [Ap − I2]∇uT in Ω2π

H ,

  • W ∗
  • ΩH

Ap∇uT · ∇vT dx dα =

  • W ∗
  • ΩH

J ∗ ◦ J (Ap∇uT) · ∇vT dx dα Next:

  • W ∗
  • ΓH

vT T +uT ds dα with α-quasiperiodic DtN map

+

ˆ

i(j−α)x

ˆ

i(j−α)x

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds From J (Ap∇uT) = ∇wB + [Ap − I2]∇uT in Ω2π

H ,

  • W ∗
  • ΩH

Ap∇uT · ∇vT dx dα =

  • W ∗
  • Ω2π

H

J (Ap∇uT) · J ∇vT dx dα Next:

  • W ∗
  • ΓH

vT T +uT ds dα with α-quasiperiodic DtN map ˆ ˆ

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds From J (Ap∇uT) = ∇wB + [Ap − I2]∇uT in Ω2π

H ,

  • W ∗
  • ΩH

Ap∇uT · ∇vT dx dα =

  • W ∗
  • Ω2π

H

[∇xwB + [Ap − I2]∇xuT] · ∇xvB dx dα Next:

  • W ∗
  • ΓH

vT T +uT ds dα with α-quasiperiodic DtN map ˆ ˆ

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds From J (Ap∇uT) = ∇wB + [Ap − I2]∇uT in Ω2π

H ,

  • W ∗
  • ΩH

Ap∇uT · ∇vT dx dα =

  • W ∗
  • Ω2π

H

∇xwB · ∇vB dx dα +

  • Ω2π

H

[Ap − I2]∇xuT · ∇xvT dx Next:

  • W ∗
  • ΓH

vT T +uT ds dα with α-quasiperiodic DtN map ˆ ˆ

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • ΓH

vT J −1 ◦ J

  • T +uT
  • ds dα

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • ΓH

vT J ∗ ◦ J

  • T +uT
  • ds dα

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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SLIDE 34

Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • Γ2π

H

J vT J

  • T +uT
  • ds dα

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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SLIDE 35

Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • Γ2π

H

J

  • T +uT
  • J vT ds dα

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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SLIDE 36

Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • Γ2π

H

T +

α

  • J uT
  • J vT ds dα

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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SLIDE 37

Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • ΩH
  • Ap∇uT · ∇vT − k2 cp uTvT
  • dx−
  • ΓH

vT T +uT ds =

  • ΓH

f vT ds Next:

  • W ∗
  • ΓH

vT T +uT ds dα =

  • W ∗
  • Γ2π

H

T +

α wBvB ds

with α-quasiperiodic DtN map T +

α :

ˆ φjei(j−α)x1 → k2 − (j + α)2 ˆ φjei(j−α)x1

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

ΩH Ω2π

H

↓J

J −1↑ Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • W ∗
  • Ω2π

H

  • ∇xwB(α, ·) · ∇xvB − k2 wBvB
  • dx −
  • Γ2π

H

vB T +

α wB ds

  • dα

+

  • Ω2π

H

  • [Ap − I2]∇uT · ∇vT − k2[1 − cp]uT vT
  • dx

=

  • W ∗
  • Γ2π

H

J f(α, ·) vB(α, ·) ds dα A variational problem in a finite domain: W ∗ × Ω2π

H

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

The Bloch-transformed Problem

ΩH Ω2π

H

↓J

J −1↑ Variational formulation for wB = J uT = J (u ◦ Φp) with test function vB = J (vT) ∈ L2(W ∗; H1

α(Ω2π H )):

  • W ∗
  • Ω2π

H

  • ∇xwB(α, ·) · ∇xvB − k2 wBvB
  • dx −
  • Γ2π

H

vB T +

α wB ds

  • dα

+

  • Ω2π

H

  • [Ap − I2]∇J−1wB · ∇J−1vB − k2[1 − cp]J−1wB J−1vB
  • dx

=

  • W ∗
  • Γ2π

H

J f(α, ·) vB(α, ·) ds dα A variational problem in a finite domain: W ∗ × Ω2π

H

Ruming Zhang 14 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Comments

For r ∈ [0, 1): Existence and uniqueness of solution in Hr

p(W ∗;

H1

α(Ω2π H )) if f ∈ Hr p(W ∗; H−1/2 α

(Γ2π

H )),

i.e., ui ∈ H1

r (ΩH)

If, additionally, ζ, ζp ∈ C2,1(R) & ui ∈ H2

r (ΩH)

⇒ w ∈ Hr

p(W ∗;

H2(Ω2π

H ))

Drawback: Need to know Bloch transform of ∂ui/∂x2 and T +ui (OK for point sources & plane waves)

Ruming Zhang 15 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

1 Dirichlet Scattering Problem 2 Bloch Transform 3 Numerical Analysis

Ruming Zhang 16 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Galerkin Discretization

− 1

2

. . .

1 2

W ∗ Regular, quasi-uniform & periodic mesh of Ω2π

H

⇒ Vh = span{φ(ℓ)

M }M ℓ=1 ⊂

H1(Ω2π

H )

For N ∈ N: ψ(j)

N = 1I(j)

N ∈ L2(W ∗)

Discretization space XN,h ⊂ L2(W ∗; H1

α(Ω2π H )):

  • XN,h =
  • wN,h = e−iαx1

j,ℓ wj,ℓ N,hψ(j) N φ(ℓ) M

  • If w ∈ Hr

0(W ∗; H2(Ω2π H )) for r ∈ [0, 1)

⇒ wN,h − wL2 decays as h(N −r + h)

Ruming Zhang 17 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Galerkin Discretization

− 1

2

. . .

1 2

W ∗ Regular, quasi-uniform & periodic mesh of Ω2π

H

⇒ Vh = span{φ(ℓ)

M }M ℓ=1 ⊂

H1(Ω2π

H )

For N ∈ N: ψ(j)

N = 1I(j)

N ∈ L2(W ∗)

Discretization space XN,h ⊂ L2(W ∗; H1

α(Ω2π H )):

  • XN,h =
  • wN,h = e−iαx1

j,ℓ wj,ℓ N,hψ(j) N φ(ℓ) M

  • If w ∈ Hr

0(W ∗; H2(Ω2π H )) for r ∈ [0, 1)

⇒ wN,h − wL2 decays as h(N −r + h)

Ruming Zhang 17 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Galerkin Discretization

− 1

2

. . .

1 2

W ∗ Regular, quasi-uniform & periodic mesh of Ω2π

H

⇒ Vh = span{φ(ℓ)

M }M ℓ=1 ⊂

H1(Ω2π

H )

For N ∈ N: ψ(j)

N = 1I(j)

N ∈ L2(W ∗)

Discretization space XN,h ⊂ L2(W ∗; H1

α(Ω2π H )):

  • XN,h =
  • wN,h = e−iαx1

j,ℓ wj,ℓ N,hψ(j) N φ(ℓ) M

  • If w ∈ Hr

0(W ∗; H2(Ω2π H )) for r ∈ [0, 1)

⇒ wN,h − wL2 decays as h(N −r + h)

Ruming Zhang 17 / 19 Scattering problem Bloch Transform Numerical Analysis

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Zentrum f¨ ur Technomathematik

Scattering from Perturbed Periodic Structures

Galerkin Discretization

− 1

2

. . .

1 2

W ∗ Regular, quasi-uniform & periodic mesh of Ω2π

H

⇒ Vh = span{φ(ℓ)

M }M ℓ=1 ⊂

H1(Ω2π

H )

For N ∈ N: ψ(j)

N = 1I(j)

N ∈ L2(W ∗)

Discretization space XN,h ⊂ L2(W ∗; H1

α(Ω2π H )):

  • XN,h =
  • wN,h = e−iαx1

j,ℓ wj,ℓ N,hψ(j) N φ(ℓ) M

  • If w ∈ Hr

0(W ∗; H2(Ω2π H )) for r ∈ [0, 1)

⇒ wN,h − wL2 decays as h(N −r + h)

Ruming Zhang 17 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Setting up the Linear System

− 1

2

. . .

1 2

W ∗

       A1 · · · C1 A2 · · · C2 . . . . . . . . . . . . . . . · · · AN CN B1 B2 · · · BN IM       

Regular, quasi-uniform & periodic mesh of Ω2π

H ⇒

  • Vh = span{φ(ℓ)

M }M ℓ=1 ⊂

H1(Ω2π

H )

For N ∈ N: ψ(j)

N = 1I(j)

N ∈ L2(W ∗)

Discretization space XN,h ⊂ L2(W ∗; H1

α(Ω2π H ))

⇒ basis functions (α, x) → e−iαx1ψ(j)

N (α)φ(ℓ) M (x)

Sparse matrix of size M(N + 1) × M(N + 1) contains uh = J−1

Ω wN,h as side condition

One can exactly integrate all α-integrals Precondition by incomplete LU-decom- position of diagonal matrices L = diag{L1, . . . , LN, IM}, U = diag{U1, . . . , UN, IM}

Ruming Zhang 18 / 19 Scattering problem Bloch Transform Numerical Analysis

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Scattering from Perturbed Periodic Structures

Summary

Bloch transform: Scattering from perturbed periodic structures ⇔ Coupled system of quasiperiodic problems System decouples for periodic surfaces Particular choice of discretization space ⇒ simple implementation

  • f inverse Bloch transform

Thanks for your attention!

Ruming Zhang 19 / 19 Scattering problem Bloch Transform Numerical Analysis