Hardy Space Infinite Elements for Scattering and Resonance Problems - PowerPoint PPT Presentation

radiation conditions and TBC pole condition Hardy space method resonances Hardy Space Infinite Elements for Scattering and Resonance Problems Thorsten Hohage Institut für Numerische und Angewandte Mathematik University of Göttingen 19th Chemnitz FEM Symposium

radiation conditions and TBC pole condition Hardy space method resonances collaborators • PD Dr. Frank Schmidt and Lin Zschiedrich (ZIB, Berlin) • Lothar Nannen (Inst. Applied Math., Univ. Göttingen) • Prof. Joachim Schöberl (RWTH, Aachen) • Dr. Maria-Luisa Rapún (Univ. Complutense, Madrid)

radiation conditions and TBC pole condition Hardy space method resonances outline radiation conditions and TBC pole condition Hardy space method resonances

radiation conditions and TBC pole condition Hardy space method resonances scattering vs resonances problems Let K ⊂ R d be smooth, compact and R d ⊂ K connected. scattering problem: For given k > 0 and a given incident field u i find a scattered field u s such that in R d \ K − ∆ u s − k 2 u s = 0 − u s = u i on ∂ K u s satisfies radiation condition resonance problem: Find an eigenpair ( u , k 2 ) such that in R d \ K − ∆ u = k 2 u u = 0 on ∂ K u satisfies radiation condition k is called a resonance. We have Im ( k ) < 0, and u grows exponentially at infinity.

radiation conditions and TBC pole condition Hardy space method resonances transparent boundary conditions (TBC) For finite element computations the infinite domain Ω := R d \ K has to be truncated to a finite computational domain Ω int . At the artificial boundary Γ of Ω int we have to imposed a so-called transparent boundary condition which reflects the radiation condition at infinity.

radiation conditions and TBC pole condition Hardy space method resonances radiation conditions for real wave number k Let K ⊂ R 2 compact, k > 0, and ∆ u + k 2 u = 0 in Ω := R 2 \ K . Then the following conditions are equivalent: • Sommerfeld’s radiation condition: √ r � ∂ u � ∂ r − iku → 0 as r = | x | → ∞ x uniformly for all directions ˆ x = | x | . • In polar coordinates ( r , φ ) ( r > a , 0 ≤ φ < 2 π ) u has a series representation u ( r , φ ) = � ∞ n = −∞ c n e in φ H ( 1 ) | n | ( kr ) . Here H ( 1 ) | n | is the Hankel function of the first kind of order | n | . • u has an integral representation � � � ∂ Φ( x , y , k ) u ( y ) − Φ( x , y , k ) ∂ u u ( x ) = ∂ n ( y ) ds ( y ) | y | = a ∂ n ( y ) in terms of the fundamental solution Φ( x , y , k ) := ( i / 4 ) H ( 1 ) 0 ( k | x − y | ) for a sufficiently large.

radiation conditions and TBC pole condition Hardy space method resonances radiation conditions for Im k < 0 If Im k < 0 and Re k > 0, Sommerfeld’s radiation condition is not a valid characterization of outgoing waves. (In particular, it does not guarantee uniqueness for exterior boundary value problems.) The series representation and the integral representation, however, are still equivalent and lead to well-posed exterior boundary value problems in appropriate norms. (Recall that the solutions grow exponentially at infinity!)

radiation conditions and TBC pole condition Hardy space method resonances classical TBCs • approximation by local boundary conditions, e.g. ∂ u ∂ n = ik u on Γ a Bayliss-Gunzburger-Turkel, Enquist-Majda, Feng, Goldberg, Grote, Keller, ... based on Sommerfeld’s radiation condition or analogous higher order conditions • boundary integral equation method (FEM/BEM coupling) Chandler-Wilde, Costabel, Greengard, Hackbusch, Hsiao, Kress, Nédélec, Rokhlin, Wendland, ... based on integral representation of solution • infinite elements Bettess, Burnett, Demkowicz, Gerdes, Zienkiewicz, ... based on series representation of solution All these TBCs destroy the eigenvalue structure of the problem!

radiation conditions and TBC pole condition Hardy space method resonances complex coordinate stretching/PML We consider the holomor- assumptions: σ ∈ C 1 [ 0 , ∞ ) phic extension of the so- σ ≥ 0, σ ′ ≥ 0, σ ( x ) = 0 for lution u ( r , φ ) in polar co- x ≤ a , ordinates with respect to lim x →∞ σ ( x ) = ∞ . the radial variable r and define u σ ( r , φ ) := u ( r + i σ ( r ) , φ ) Since for d = 1 the holomorphic extension is u ( z ) = c exp (+ ikz ) for an outgoing solution, c exp ( − ikz ) for an incoming solution, we get u σ ( r ) = c exp (+ ikr − k σ ( r )) : exponentially decaying, c exp ( − ikr + k σ ( r )) : exponentially increasing.

radiation conditions and TBC pole condition Hardy space method resonances Perfectly Matched Layer Method (PML) d = 1 : Let γ ( r ) := r + i σ ( r ) . The Imposing a zero Dirich- chain rule applied to u σ = u ◦ γ let condition at some fi- yields nite distance, we obtain a transparent boundary γ ′ ( r ) − 1 u ′ σ ( r ) = u ′ ( γ ( r )) . condition which can be implemented by standard Differentiating again and using fem software. − u ′′ ( z ) = k 2 u ( z ) we obtain the differential equation � � − 1 d 1 γ ′ ( r ) u ′ = k 2 u σ ( r ) . σ ( r ) γ ′ ( r ) dr d > 1 : A similar computation yields a pde for u σ involving an unisotropic damping tensor in radial direction.

radiation conditions and TBC pole condition Hardy space method resonances outline radiation conditions and TBC pole condition Hardy space method resonances

radiation conditions and TBC pole condition Hardy space method resonances pole condition for d = 1 The general solutuion to the 1d Helmholtz eq. u ′′ ( r )+ k 2 u ( r ) = 0 is ∞ e ikr + u − u ( r ) = u + ∞ e − ikr . Its Laplace transform � ∞ 0 e − sr u ( r ) dr is given by ˆ u ( s ) := u + u − ∞ ∞ ˆ u ( s ) = s − ik + s + ik . Note: u is outgoing if and only if the Laplace transform ˆ u of u has no poles in the lower complex half–plane.

radiation conditions and TBC pole condition Hardy space method resonances Laplace transform of the Helmholtz equation in R 2 \ K , K ⊂ { x : | x | < a } compact, k > 0 ∆ u + k 2 u = 0 polar coordinates: x ) := √ ρ u ( ρ ˆ x ∈ S 1 U ( ρ, ˆ ρ > 0 , ˆ x ) , � � ∂ r 2 + k 2 + ∂ 2 1 x + 1 U ( r + a , ˆ ( r + a ) 2 (∆ ˆ 4 I ) x ) = 0 Laplace transform: � ∞ ˆ 0 e − sr U ( r + a , ˆ U ( s , ˆ x ) := x ) dr , Re s > 0 � ∞ ( s 2 + k 2 )ˆ 4 I )ˆ e − a ( s 1 − s ) ( s 1 − s )(∆ ˆ x + 1 U ( s , ˆ U ( s 1 , ˆ x ) + x ) ds 1 s x ) + ∂ = sU ( a , ˆ ∂ρ U ( a , ˆ x ) , Re s > 0

radiation conditions and TBC pole condition Hardy space method resonances pole condition and Sommerfeld radiation condition Definition u satisfies the pole condition if the mapping s → ˆ U ( s , · ) defined on { s ∈ C : Re s > 0 } with values in L 2 ( S d − 1 ) has a holomorphic extension to D := { s ∈ C : Re s > 0 or Im s < 0 } . Theorem A bounded solution to the Helmholtz equation for k > 0 satisfies the pole condition if and only if it satisfies the Sommerfeld radiation condition. T. Hohage, F. Schmidt, L. Zschiedrich: Solving time-harmonic scattering problems based on the pole condition. I: Theory SIAM J. Math. Anal., 35 :183-210 (2003)

radiation conditions and TBC pole condition Hardy space method resonances pole condition for Im k < 0

radiation conditions and TBC pole condition Hardy space method resonances discussion • The pole condition is a unifying radiation condition in particular for • scattering by bounded obstacles • rough surface scattering problems (equivalent to Upward Propagating Radiation Condition proposed by S. Chandler-Wilde as shown by T. Arens & T. Hohage) • scattering problems in wave guides • independent of the differential equation and in particular the wave number • stable representation formula of exterior solution, which is cheap to evaluate (talk by Roland Klose) • leads to several new transparent boundary conditions

radiation conditions and TBC pole condition Hardy space method resonances outline radiation conditions and TBC pole condition Hardy space method resonances

radiation conditions and TBC pole condition Hardy space method resonances Hardy space H 2 − ( R ) Definition A function u, which is holomorphic in the lower complex half-plane C − := { z ∈ C : Im ( z ) < 0 } has L 2 boundary values v = u | R ∈ L 2 ( R ) if � ∞ | u ( x − i ǫ ) − v ( x ) | 2 dx ǫ ց 0 − → 0 . −∞ � � v ∈ L 2 ( R ) : ∃ u : C − → C holomorphic with v = u | R H 2 − ( R ) := . − ( R ) equipped with the L 2 inner product is a Hilbert • H 2 space. • pole condition: ˆ x ) | R ∈ H 2 x ∈ S d − 1 U ( · , ˆ for all ˆ − ( R ) • idea: Galerkin method in H 2 − ( R ) • problem: appropriate basis of H 2 − ( R )

radiation conditions and TBC pole condition Hardy space method resonances Hardy space H 2 − ( S 1 ) Definition Let B 1 := { z ∈ C : | z | < 1 } and S 1 := ∂ B. A holomorphic function u : B 1 → C has L 2 boundary values v = u | S 1 ∈ L 2 ( S 1 ) if � S 1 | u ( rz ) − v ( z ) | 2 | dz | r ր 1 − → 0 . � � v ∈ L 2 ( S 1 ) : ∃ u : B 1 → C holomorphic with v = u | S 1 H 2 − ( S 1 ) := . Lemma − ( S 1 ) equipped with the L 2 inner product is a Hilbert space H 2 with orthonormal basis 1 z j , z �→ √ j = 0 , 1 , 2 , . . . 2 π