Hardy space infinite elements for exterior Maxwell problems L. - PowerPoint PPT Presentation

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Hardy space infinite elements for exterior Maxwell problems L. Nannen, T. Hohage, A. Schädle, J. Schöberl Linz, November 2011 1

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics electromagnetic scattering problem � curl u · curl v − εκ 2 u · v dx = g ( v ) Ω κ = 2 . 7684 κ = 2 . 8 2

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics electromagnetic resonance problem Definition (resonance problem) Let ( κ 2 , u ) ∈ C × H loc ( curl , Ω) \ { 0 } with ℜ ( κ ) > 0 be a solution to the eigenvalue problem � � curl u · curl v dx = κ 2 ε u · v dx + BC + RC . Ω Ω Then we call κ a resonance, ℜ ( κ ) > 0 the resonance frequency ℜ ( κ ) and Q := 2 |ℑ ( κ ) | the quality factor. 3

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics outline motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics 4

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics first order ABC Silver-Müller radiation condition: � curl u × x � | x |→∞ | x | lim | x | − i κ u = 0 first order ABC: curl u × ν − i κ u = 0 on Γ := ∂ B a ( 0 ) 5

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics first order ABC Silver-Müller radiation condition: � curl u × x � | x |→∞ | x | lim | x | − i κ u = 0 first order ABC: curl u × ν − i κ u = 0 on Γ := ∂ B a ( 0 ) pros: nothing to implement, no additional dofs cons: poor accuracy, wrong solutions for resonance problems 5

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics first order ABC Silver-Müller radiation condition: � curl u × x � | x |→∞ | x | lim | x | − i κ u = 0 first order ABC: curl u × ν − i κ u = 0 on Γ := ∂ B a ( 0 ) pros: nothing to implement, no additional dofs cons: poor accuracy, wrong solutions for resonance problems reason: factor exp ( ± i κ | x | ) in the asymptotic behaviour of u 5

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics BEM Stratton-Chu formula: � u ( x ) = curl ν ( y ) × u ( y )Φ( x , y ) ds ( y ) Γ − 1 � x ∈ R 3 \ B a ( 0 ) ν ( y ) × curl u ( y )Φ( x , y ) ds ( y ) , κ 2 curl curl Γ with Φ( x , y ) = 1 exp ( i κ | x − y | ) 4 π | x − y | 6

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics BEM Stratton-Chu formula: � u ( x ) = curl ν ( y ) × u ( y )Φ( x , y ) ds ( y ) Γ − 1 � x ∈ R 3 \ B a ( 0 ) ν ( y ) × curl u ( y )Φ( x , y ) ds ( y ) , κ 2 curl curl Γ with Φ( x , y ) = 1 exp ( i κ | x − y | ) 4 π | x − y | pros: boundary integrals, fast convergence, non-convex Γ cons: dependence on κ , Green’s function needed 6

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics classical infinite elements tensor product ansatz: N N � � u ( | x | , ˆ ψ l ( | x | ) e l (ˆ α l ( | x | ) g l (ˆ x )ˆ x ) = x ) + x , | x | ≥ 1 , with l = 0 l = 1 7

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics classical infinite elements tensor product ansatz: N N � � u ( | x | , ˆ ψ l ( | x | ) e l (ˆ α l ( | x | ) g l (ˆ x )ˆ x ) = x ) + x , | x | ≥ 1 , with l = 0 l = 1 ψ 0 ( r ) := 1 r exp ( i κ ( r − 1 )) , � 1 � r l + 1 − 1 ψ l ( r ) := exp ( i κ ( r − 1 )) , r α ( r ) := 1 r l + 1 exp ( i κ ( r − 1 )) . 7

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics classical infinite elements tensor product ansatz: N N � � u ( | x | , ˆ ψ l ( | x | ) e l (ˆ α l ( | x | ) g l (ˆ x )ˆ x ) = x ) + x , | x | ≥ 1 , with l = 0 l = 1 ψ 0 ( r ) := 1 r exp ( i κ ( r − 1 )) , � 1 � r l + 1 − 1 ψ l ( r ) := exp ( i κ ( r − 1 )) , r α ( r ) := 1 r l + 1 exp ( i κ ( r − 1 )) . pros: fast convergence in | x | cons: dependence on κ , complicated theory/ implementation 7

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics classical infinite elements tensor product ansatz: N N � � u ( | x | , ˆ ψ l ( | x | ) e l (ˆ α l ( | x | ) g l (ˆ x )ˆ x ) = x ) + x , | x | ≥ 1 , with l = 0 l = 1 ψ 0 ( r ) := 1 r exp ( i κ ( r − 1 )) , � 1 � r l + 1 − 1 ψ l ( r ) := exp ( i κ ( r − 1 )) , r α ( r ) := 1 r l + 1 exp ( i κ ( r − 1 )) . pros: fast convergence in | x | cons: dependence on κ , complicated theory/ implementation Demkowicz & Pal, An infinite element for Maxwell’s equations , Computer Methods in Applied Mechanics and Engineering, 1998. 7

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics complex scaling unisotropic damping: exp ( i κ | x | ) − → exp ( i κσ | x | ) Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling , Physics Reports, 1998. Berenger, A perfectly matched layer for the absorption of electromagnetic waves , J.Comput.Phy.,1994. 8

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics complex scaling unisotropic damping: exp ( i κ | x | ) − → exp ( i κσ | x | ) Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling , Physics Reports, 1998. Berenger, A perfectly matched layer for the absorption of electromagnetic waves , J.Comput.Phy.,1994. pros: simple to implement, generalized linear eigenvalue problem, well known cons: many parameters to chose, artificial resonances 8

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Helmholtz equation in 1d Helmholtz equation: − u ′′ ( r ) − κ 2 u ( r ) = 0 9

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Helmholtz equation in 1d Helmholtz equation: − u ′′ ( r ) − κ 2 u ( r ) = 0 Laplace transformation: � ∞ e − sr u ( r ) dr , ℜ ( s ) > 0 ˆ u ( s ) := ( L u )( s ) = 0 u ( r ) = C 1 e + i κ r + C 2 e − i κ r � L � � C 1 C 2 ˆ u ( s ) = s − i κ + s + i κ. 9

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Helmholtz equation in 1d Helmholtz equation: − u ′′ ( r ) − κ 2 u ( r ) = 0 Laplace transformation: � ∞ e − sr u ( r ) dr , ℜ ( s ) > 0 ˆ u ( s ) := ( L u )( s ) = 0 ℑ ( s ) u ( r ) = C 1 e + i κ r + C 2 e − i κ r i κ outgoing � L � � ℜ ( s ) C 1 C 2 − i κ incoming ˆ u ( s ) = s − i κ + s + i κ. 9

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Möbius transformation iκ 0 ϕ ( z ) = iκ 0 z +1 z − 1 Definition (Hardy space H + ( S 1 ) ) Let S 1 := { z | | z | = 1 } . Then F ∈ H + ( S 1 ) iff • F ∈ L 2 ( S 1 ) and • L 2 -boundary value of a holomorphic function in D := { z | | z | < 1 } . 10

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics pole condition Definition u is outgoing, iff ML u ∈ H + ( S 1 ) . 11

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics pole condition Definition u is outgoing, iff ML u ∈ H + ( S 1 ) . Schmidt & Deuflhard, Discrete Transparent Boundary Conditions for the Numerical Solution of Fresnel’s Equation , Computers Math. Applic., 1995. Hohage & Schmidt & Zschiedrich, Solving time-harmonic scattering problems based on the pole condition. I. Theory , SIAM J. Math. Anal., 2003. Hohage & Nannen, Hardy space infinite elements for scattering and resonance problems , SIAM J. Numer. Anal., 2009. 11

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Hardy space method in 1d classical formulation: − u ′′ ( r ) − κ 2 u ( r ) = 0 in [ 0 , ∞ ) u ′ ( 0 ) = 1 , ML u ∈ H + ( S 1 ) . 12

motivation transparent boundary conditions Hardy space infinite elements in 1d sequence of infinite elements numerics Hardy space method in 1d classical formulation: − u ′′ ( r ) − κ 2 u ( r ) = 0 in [ 0 , ∞ ) u ′ ( 0 ) = 1 , ML u ∈ H + ( S 1 ) . variational formulation: � ∞ � u ′ v ′ − κ 2 uv � ML u ∈ H + ( S 1 ) . dr = v ( 0 ) , 0 12