Boundary element methods for scattering by fractal screens Andrea - PowerPoint PPT Presentation

W AVES , V IENNA , 26–30 A UGUST 2019 Boundary element methods for scattering by fractal screens Andrea Moiola http://matematica.unipv.it/moiola/ Joint work with S.N. Chandler-Wilde (Reading), D.P . Hewett (UCL) A. Caetano (Aveiro)

Acoustic wave scattering by a planar screen Acoustic waves in free space governed by wave eq. ∂ 2 U ∂ t 2 − ∆ U = 0 . In time-harmonic regime, assume U ( x , t )= ℜ{ u ( x ) e − i kt } and look for u . u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits obstacle Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ { Γ × { 0 }} x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Acoustic wave scattering by a planar screen Acoustic waves in free space governed by wave eq. ∂ 2 U ∂ t 2 − ∆ U = 0 . In time-harmonic regime, assume U ( x , t )= ℜ{ u ( x ) e − i kt } and look for u . u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits obstacle Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ { Γ × { 0 }} x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Acoustic wave scattering by a planar screen Acoustic waves in free space governed by wave eq. ∂ 2 U ∂ t 2 − ∆ U = 0 . In time-harmonic regime, assume U ( x , t )= ℜ{ u ( x ) e − i kt } and look for u . u satisfies Helmholtz equation ∆ u + k 2 u = 0 , with wavenumber k > 0 . Scattering: incoming wave u i hits obstacle Γ and generates field u . Γ bounded subset of Γ ∞ := { x ∈ R n : x n = 0 } ∼ = R n − 1 , n = 2 , 3 ∆ u + k 2 u = 0 D := R n \ { Γ × { 0 }} x 3 x 2 u i ( x ) = e i k d · x Γ u = − u i x 1 u satisfies Sommerfeld radiation condition (SRC) at infinity � r ( 1 − n ) / 2 � (i.e. ∂ r u − iku = o uniformly as r = | x | → ∞ ). 2

Scattering by Lipschitz and rough screens u tot = u + u i Incident field is plane wave u i ( x ) = e i k d · x , | d | = 1 . Classical problem when Γ is open and Lipschitz. What happens for arbitrary (rougher than Lipschitz, e.g. fractal) Γ ? 3

Scattering by Lipschitz and rough screens u tot = u + u i Incident field is plane wave u i ( x ) = e i k d · x , | d | = 1 . Classical problem when Γ is open and Lipschitz. What happens for arbitrary (rougher than Lipschitz, e.g. fractal) Γ ? 3

Waves and fractals: applications Wideband fractal antennas (Figures from http://www.antenna-theory.com/antennas/fractal.php ) 4

Waves and fractals: applications Wideband fractal antennas (Figures from http://www.antenna-theory.com/antennas/fractal.php ) Scattering by ice crystals in atmospheric physics e.g. C. Westbrook Fractal apertures in laser optics e.g. J. Christian 4

Scattering by fractal screens · · · Lots of mathematical challenges: ◮ How to formulate well-posed BVPs? (What is the right function space setting? How to impose BCs?) ◮ How do prefractal solutions converge to fractal solutions? ◮ How can we accurately compute the scattered field? ◮ . . . Note: several tools developed here might be used in the (numerical) analysis of different IEs & BVPs involving complicated domains. 5

Outline ◮ Sobolev spaces on rough sets ◮ BVPs and BIEs ◮ open screens ◮ compact screens ◮ Prefractal to fractal convergence ◮ BEM and convergence ◮ Examples & numerics ◮ Cantor dust: dependence on Hausdorff dimension ◮ Sierpinski triangle: dependence on frequency ◮ Snowflakes: inner and outer approximations ◮ . . . 6

Sobolev spaces on rough subsets of R n − 1 We need fractional (Bessel) Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ H s (Γ) = ( H − s (Γ)) ∗ with equal norms ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 7

Sobolev spaces on rough subsets of R n − 1 We need fractional (Bessel) Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ H s (Γ) = ( H − s (Γ)) ∗ with equal norms ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 7

Sobolev spaces on rough subsets of R n − 1 We need fractional (Bessel) Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ H s (Γ) = ( H − s (Γ)) ∗ with equal norms ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 7

Sobolev spaces on rough subsets of R n − 1 We need fractional (Bessel) Sobolev spaces on Γ ⊂ R n − 1 . For s ∈ R let � � � u ( ξ ) | 2 d ξ < ∞ H s ( R n − 1 )= u ∈ S ∗ ( R n − 1 ) : � u � 2 R n − 1 ( 1 + | ξ | 2 ) s | ˆ H s ( R n − 1 ) := For Γ ⊂ R n − 1 open and F ⊂ R n − 1 closed define [M C L EAN ] H s (Γ) := { u | Γ : u ∈ H s ( R n − 1 ) } restriction H s ( R n − 1 ) � H s (Γ) := C ∞ 0 (Γ) closure H s F := { u ∈ H s ( R n − 1 ) : supp u ⊂ F } support When Γ is Lipschitz it holds that For general open Γ H s (Γ) = ( H − s (Γ)) ∗ with equal norms ◮ � ◮ � � H s (Γ) ∼ � ◮ s ∈ N ⇒ � u � 2 Γ | ∂ α u | 2 L IPSCHITZ ◮ × | α |≤ s ◮ � ( ∼ H s (Γ) = H s = H s 00 (Γ) , s ≥ 0 ) ◮ × IS Γ ◮ H ± 1 / 2 = { 0 } L UXURY ! ◮ × ∂ Γ ◮ { H s (Γ) } s ∈ R and { � H s (Γ) } s ∈ R ◮ × are interpolation scales. 7

BVPs for open and compact screens BVP D op (Γ) for open screens Let Γ ⊂ Γ ∞ be bounded & open. Given g ∈ H 1 / 2 (Γ) (for instance, g = − ( γ ± u i ) | Γ ), find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) satisfying ∆ u + k 2 u = 0 in D , ( γ ± u ) | Γ = g , Sommerfeld RC . γ ± = traces : W 1 ( R n ± ) → H 1 / 2 (Γ ∞ ) 8

BVPs for open and compact screens BVP D co (Γ) for compact scr. BVP D op (Γ) for open screens Let Γ ⊂ Γ ∞ be bounded & open. Let Γ ⊂ Γ ∞ be compact. Given g ∈ � H 1 / 2 (Γ c ) ⊥ Given g ∈ H 1 / 2 (Γ) (for instance, g = − ( γ ± u i ) | Γ ), (e.g., g = − P Γ u i ), find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) satisfying satisfying ∆ u + k 2 u = 0 ∆ u + k 2 u = 0 in D , in D , ( γ ± u ) | Γ = g , P Γ γ ± u = g , Sommerfeld RC . Sommerfeld RC . Orthogonal projection γ ± = traces : W 1 ( R n ± ) → H 1 / 2 (Γ ∞ ) P Γ : H 1 / 2 (Γ ∞ ) → � H 1 / 2 (Γ c ) ⊥ . If Ω bdd open, � H − 1 / 2 (Ω) = H − 1 / 2 , then D op (Ω) &D co (Ω) are equivalent. Ω 8

BVPs for open and compact screens BVP D co (Γ) for compact scr. BVP D op (Γ) for open screens Let Γ ⊂ Γ ∞ be bounded & open. Let Γ ⊂ Γ ∞ be compact. Given g ∈ � H 1 / 2 (Γ c ) ⊥ Given g ∈ H 1 / 2 (Γ) (for instance, g = − ( γ ± u i ) | Γ ), (e.g., g = − P Γ u i ), find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) find u ∈ C 2 ( D ) ∩ W 1 , loc ( D ) satisfying satisfying ∆ u + k 2 u = 0 ∆ u + k 2 u = 0 in D , in D , ( γ ± u ) | Γ = g , P Γ γ ± u = g , Sommerfeld RC . Sommerfeld RC . Orthogonal projection γ ± = traces : W 1 ( R n ± ) → H 1 / 2 (Γ ∞ ) P Γ : H 1 / 2 (Γ ∞ ) → � H 1 / 2 (Γ c ) ⊥ . If Ω bdd open, � H − 1 / 2 (Ω) = H − 1 / 2 , then D op (Ω) &D co (Ω) are equivalent. Ω 8