Fast iterative BEM for high-frequency scattering

Transcription

1 Fast iterative BEM for high-frequency scattering elastodynamic problems In 3D. Marion Darbas LAMFA UMR CNRS 7352, Amiens, France S. Chaillat (POEMS, CNRS-INRIA-ENSTA) and F. Le Louër (LMAC, UTC) Conference in honor of Abderrahmane BENDALI December 12-14th 2017, Pau M. Darbas Conference A. Bendali, December 13th, 2017

2 Outline

3 Surface scattering problems Considered problems : Scattering of an incident time-harmonic plane wave u inc by an obstacle Ω, a regular bounded domain (C l, l 2) of R 3. Ω + u inc Ω n Γ u + n outwardly unit normal to Ω. The scattered wave u + is solution to an exterior problem Wave propagation equation in Ω + = R 3 \ Ω, Perfectly conducting boundary condition on Γ := Ω, Radiation condition at infinity.

4 Numerical difficulties Geometry Unbounded computational domain Ω + A solution : Integral equation method. Mathematics Integral equations involve the non local Green s kernel dense linear systems. Physics Deal with high-frequency regime very large linear systems. A solution : to use an iterative solver (e.g. GMRES [Saad-Schultz, 86]) cost of O(n iter N 2 ), n iter number of iterations, N number of D.O.F.

5 Goal To accelerate the convergence of GMRES solver : Two desired properties for the matrix of the linear system a "good" eigenvalue clustering in the complex plane, a condition number close to 1. To speed up the computation of matrix-vector products per iteration : Fast Multipole Methods (FMM) [Greengard, Darve, Chew, Sylvand, Darrigrand, Chaillat...] To construct well-conditioned integral operators in the form A = ai + K, a 0, K compact. (e.g. [Levadoux, 01], [Antoine-MD, 05], [Alouges et al, 07], [Pernet, 10], [Bruno et al, 12], [Boubendir-Turc, 14],...)

6 Exterior Navier problem Ω + isotropic homogeneous medium (density : ρ, Lamé parameters : λ, µ) The displacement field u + = u p + u s satisfies the boundary-value problem µ u + + (λ + µ) div u + + ρω 2 u + = 0, in Ω +, u + = u ( inc, on Γ, ) ( lim r up iκ p u p = 0, lim r r r us r r iκ s u s ) = 0, r = x. Angular frequency : ω u p P-waves, u s S-waves Wavenumbers : κ 2 p = ρω 2 (λ + 2µ) 1 and κ 2 s = ρω 2 µ 1 The radiating solution belongs to the space H 1 +( ) := { u H 1 loc(ω + ) : µ u + (λ + µ) div u L 2 loc(ω + ) }.

7 Potential theory Somigliana integral representation [Kupradze, 1965] u + (x) = Du + Γ (x) St Γ(x), x Ω +. Dirichlet trace u + Γ, Neumann trace t Γ := Tu is defined by the traction operator T = 2µ + λn div +µ n curl. n The single- and double-layer potentials Sϕ = Φ(, y)ϕ(y)ds(y), Dψ = [T yφ(, y)] T ψ(y)ds(y), Γ T y = T(n(y), y), [T yφ(x, y)] by applying T y to each column of Φ(x, y). The fundamental solution of Navier equation (3 3 matrix) Φ(x, y) = 1 ρω 2 ( curl curlx { G(κs, x y) I 3 } x div x { G(κp, x y)i 3 }) with G(κ, x y) = e iκ x y /(4π x y ), x, y R 3, x y. Γ

8 Standard CFIE and its drawbacks Combined Fied Integral Equation (CFIE) Find ψ = (t Γ + t Γ inc) H 1/2 (Γ) (where t Γ inc := Tuinc ) solution to ( I 2 + D + iηs) ψ = (t inc ), on Γ. Γ + iηuinc Γ CFIE is well-posed for any frequency and any non-zero real parameter η. Given vector densities ϕ and ψ, the boundary integral operators S and D are defined, for x Γ, by Sϕ(x) = D ψ(x) = Φ(x, y)ϕ(y) ds(y), Γ T x Φ(x, y)ψ(y) ds(y). Γ

9 Standard CFIE and its drawbacks Simulations performed with a FM-BEM code (collocation method) [Chaillat-Bonnet-Semblat, 08]. GMRES solver with no restart and tolerance ɛ = The scatterer is illuminated by incident plane waves of the form u inc (x) = 1 µ eiκs x d (d p) d+ 1 λ + 2µ eiκpx d (d p)d, d S 2, p R 3 p polarization, d propagation vector. Physical parameters : κ s = 1.5κ p, ω = κ s. Coupling CFIE parameter η = 1.

10 Standard CFIE and its drawbacks Diffraction of P-waves (p = d = (0, 0, 1) T ) by the unit sphere. Number of GMRES iterations in function of ω for a fixed density of n λs = 10 points per S-wavelength. #DOFs ω # iter CFIE > 500 The number of GMRES iterations drastically increases with the frequency.

11 Standard CFIE and its drawbacks Spectral analysis - Unit sphere Imaginary part Condition number Condition number Real part Frequency ω Mesh density n λs a) ω = 16π, n λs = 10 b) Vs. ω, n λs = 10 c) Vs. n λs, ω = 6π The condition number of the CFIE deteriorates when the frequency increases.

12 Analytical preconditioning principle Principle A first idea New approximations of the DtN map Dirichlet data is known : u + Γ = uinc Γ 1 Consider the exterior Dirichlet-to-Neumann (DtN) map Λ ex : u + Γ H 1 2 (Γ) t Γ := Λ ex u + Γ H 1 2 (Γ) 2 The Somigliana integral representation is expressed by u + (x) = Du + Γ (x) St Γ(x), x Ω +.

13 Analytical preconditioning principle Principle A first idea New approximations of the DtN map Dirichlet data is known : u + Γ = uinc Γ 1 Consider the exterior Dirichlet-to-Neumann (DtN) map Λ ex : u + Γ H 1 2 (Γ) t Γ := Λ ex u + Γ H 1 2 (Γ) 2 The Somigliana integral representation is expressed by u + (x) = Du + Γ (x) SΛex u + Γ (x), x Ω+.

14 Analytical preconditioning principle Principle A first idea New approximations of the DtN map Dirichlet data is known : u + Γ = uinc Γ 1 Consider the exterior Dirichlet-to-Neumann (DtN) map Λ ex : u + Γ H 1 2 (Γ) t Γ := Λ ex u + Γ H 1 2 (Γ) 2 The Somigliana integral representation is expressed by u + (x) = Du + Γ (x) SΛex u + Γ (x), x Ω+. 3 Taking the Dirichlet trace on Γ, we obtain ( I u + Γ (x) = 2 + D SΛex) u + Γ (x), x Γ.

15 Analytical preconditioning principle Principle A first idea New approximations of the DtN map We get I 2 + D Λ ex S = I, on Γ, and the solution is ϕ = ( t inc Γ ) u inc Λex Γ. Conclusion : The exact adjoint DtN map Λ ex = ( 1 2 I D ) S 1 = ( 1 2 I + D ) 1 N is an ideal "analytical preconditioner" for the CFIE operator. [Steinbach-Wendland, 98], [Christiansen-Nédélec, 00] Method Construct an approximate adjoint DtN Λ and an associated preconditioned CFIE : find ϕ = ( t Γ + t inc Γ ) solution to ( ) I 2 + D Λ S ϕ = ( t inc Γ Λ u Γ inc ), on Γ.

16 Principle A first idea New approximations of the DtN map A first choice of approximate adjoint DtN We consider a low-order approximation which corresponds to Kupradze radiation conditions Λ LO = i((λ + 2µ)κ p I n + µκ s I t ) with I n = n n and I t = I I n. LO-preconditioned CFIE Find ϕ = (t Γ + t Γ inc) H 1/2 (Γ) solution to ( I 2 + D Λ LOS) ϕ = (t inc Γ Λ LOu Γ inc ), on Γ Standard CFIE GG P CFIE Imaginary part Real part

17 Principle A first idea New approximations of the DtN map Diffraction of incident plane P-waves by the unit sphere p = d = (0, 0, 1) t. Number of GMRES iterations for a fixed density of 10 points per wavelength #DOFs ω # iter η = 1 # iter LO CFIE P-CFIE > 500 9

18 Principle A first idea New approximations of the DtN map Diffraction of incident plane S-waves by the unit sphere p = (1, 0, 0) t, d = (0, 0, 1) t. Number of GMRES iterations for a fixed density of n λs = 10 points per S-wavelength. #DOFs ω # iter η = 1 # iter LO CFIE P-CFIE > LO P-CFIE : fast convergence in the spherical case but...

19 Principle A first idea New approximations of the DtN map... this low-order approximation of the adjoint DtN map is less efficient for more complex scatterers. Diffraction of incident plane P-waves by a sphere with cavity #DOFs ω # iter LO P-CFIE >500 C-Shape n λs = 10 Challenge To construct high-order approximations of the adjoint DtN map for a general shape.

20 Relations of Calderón Principle A first idea New approximations of the DtN map Integral relations of Calderón give the following expressions of the adjoint DtN map Λ ex = ( 1 2 I D ) S 1 = ( 1 2 I + D ) 1 N with N the hypersingular boundary integral operator. Approximate DtN in acoustics An efficient approximation is [Antoine-MD-Lu, 06] Λ = 1 ( 2 (P 1 (S)) 1 = 2P 1 (N) = iκ 1 + )1/2 Γ κ 2 with Γ the Laplace-Beltrami operator and P 1 (S), P 1 (N) the principal parts of the "acoustics" boundary integral operators S and N. Particular property in elasticity The adjoint double-layer boundary integral operator D is not compact.

21 New approximations of the DtN map Principle A first idea New approximations of the DtN map Modified potential theory The double-layer boundary integral operator is given by D α ϕ(x) = [(T y αm)φ(x,, y)] T ϕ(y)ds(y), x Γ, Γ where α is a real-valued constant and M = n div + n curl is the n tangential Günter derivative. Modified DtN map The exact exterior Modified Dirichlet-to-Neumann (MDtN) map is defined by Λ ex α : u + Γ H 1 2 (Γ) Λ ex α u + Γ := t Γ αmu + Γ H 1 2 (Γ).

22 Principle A first idea New approximations of the DtN map Representation of the adjoint MDtN map Λ ex α = ( 1 2 I + D α) 1Nα with N α the hypersingular boundary integral operator. D α is compact for the choice α = 2µ2 λ + 3µ [Hähner-Hsiao, 93]. Expression of the adjoint DtN map We have Λ ex = Λ ex α + αm. Expression of the approximate adjoint DtN map Λ = ( I 2 + P 0(D α)) 1P1 (N α ) + αm with α = 2µ2 λ + 3µ.

23 Principle A first idea New approximations of the DtN map P 0 (D α) is decomposed into two terms : P 0 (D α) = I 1 + I 2 with I 1 = i (n ( Γ + κ 2 2 pi ) 1 2 ( div Γ I t Γ Γ + κ 2 s I ) ) 1 2 n I n ( i(2µ α) I 2 = 2ρω 2 n ( Γ + κ 2 s I ) 1 2 ( div Γ I t + n Γ Γ + κ 2 pi ) 1 2 div Γ I t ( + Γ Γ + κ 2 pi ) 1 ( ) ( 2 n I n Γ Γ + κ 2 s I ) 1 2 ( ) ) Γ n In where I n = n n and I t = I I n.

24 Preconditioned CFIEs Principle A first idea New approximations of the DtN map ( I 2 + D Λ S)ϕ = ( t inc Γ Λ u Γ inc ), on Γ. the Low-Order preconditioned CFIE (LO P-CFIE) : Λ = Λ LO = i((λ + 2µ)κ p I n + µκ s I t ) the High-Order preconditioned CFIE with one term (HO(1) P-CFIE) : Λ = 2P 1 (N α ) + αm the High-Order preconditioned CFIE with two terms (HO(2) P-CFIE) : ( I ) Λ = 2 + P 0(D α) 1P1 (N α ) + αm, α = 2µ2 λ + 3µ.

25 Numerical spectral analysis Principle A first idea New approximations of the DtN map Diffraction of P-waves (p = d = (0, 0, 1) T ) by the unit sphere. Distribution of the eigenvalues of the standard and different P-CFIEs (η = 1, κ s = 16π and n λs = 10) Standard CFIE LO P CFIE HO(1) P CFIE HO(2) P CFIE HO(1) P CFIE HO(2) P CFIE Imaginary part Imaginary part Real part Real part

26 Numerical spectral analysis Principle A first idea New approximations of the DtN map LO P CFIE HO(1) P CFIE HO(2) P CFIE 15 LO P CFIE HO(1) P CFIE HO(2) P CFIE Condition number Condition number Frequency ω Mesh density (a) n λs = 10 (b) κ s = 6π To consider the principal part of the modified double-layer boundary integral operator in the preconditioner avoids the dependence of the condition number on the frequency.

27 Some words on square-root operators Regularization of square-root operator Principle A first idea New approximations of the DtN map Damping parameters ε l (l = s, p) are introduced to regularize square-root operators in the zone of grazing rays ( Γ + κ 2 l,ε l I ) ± 1 2 κ l,εl = κ l + iε l, ε l = 0.39κ 1/3 l H 2/3, H the mean curvature of Γ [Antoine-MD-Lu, 06], [Chaillat-MD-Le Louër, 15]. Localization using Padé rational approximants Complex Padé rational approximation of 1 + X of order N p with a rotation angle θ p of the branch-cut {X C I(X ) = 0, R(X ) < 1} [Milinazzo et al, 97] are used N p A j X 1 + X C B j X, X = Γ, C 0, A j, B j C. j=1 k 2 ε

28 Simulations performed with a FM-BEM code [Chaillat-Bonnet-Semblat, 08]. Physical parameters : κ s = 1.5κ p Outer GMRES solver with no restart and tolerance ɛ = 10 3 Inner GMRES solver with no restart and tolerance ɛ = 10 4 Padé approximation : (N p, θ p ) = (35, π/3)

29 Unit sphere Incident plane P-waves, p = d = (0, 0, 1) T. #DOFs ω # iter # iter LO # iter HO(1) # iter HO(2) CFIE P-CFIE P-CFIE P-CFIE (11) (11) (13) (13) > (14) Table Diffraction of P-waves by the unit sphere. Number of GMRES iterations for a fixed density of n λs = 10 points per S-wavelength.

30 Unit sphere Incident plane S-waves, d = (0, 0, 1) T and p = (1, 0, 0) T. #DOFs ω # iter # iter LO # iter HO(1) # iter HO(2) CFIE P-CFIE P-CFIE P-CFIE (10) (11) (14) (15) > (16) Table Diffraction of S-waves by the unit sphere. Number of GMRES iterations for a fixed density of n λs = 10 points per S-wavelength.

31 Cube Incident plane P-waves, p = d = (0, 0, 1) T. #DOFs ω # iter # iter LO # iter HO(1) # iter HO(2) CFIE (η = 1) P-CFIE P-CFIE P-CFIE (13) (13) (12) > (13) Table Diffraction of P-waves by a cube. Number of GMRES iterations for a fixed density of n λs = 10 points per S-wavelength.

32 C-shape Diffraction of incident plane P-waves by a sphere with cavity. Number of GMRES iterations with respect to the frequency ω for the incidence p = d = (0, 0, 1) LO P CFIE HO(1) P CFIE HO(2) P CFIE GMRES iterations Frequency ω

33 Conclusions Derivation of well-conditioned integral equations for the iterative solution of 3D high-frequency elastic scattering problems Highly desirable advantages of the preconditioner Regularizing effect Excellent clustering of the eigenvalues Easy to implement Current works and perspectives To use the new approximations of the DtN map in other numerical methods : ABCs, DDM (with C. Geuzaine and V. Mattesi) MD, F. Le Louër, Well-conditioned boundary integral formulations for high-frequency elastic scattering problems in three dimensions, M2AS, 38 (2015), pp S. Chaillat, MD, F. Le Louër, Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves, CMAME, 297 (2015), pp S. Chaillat, MD, F. Le Louër, Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics, JCP 341 (2017), pp

34 Appendix Λ = ( I 2 + P(D α)) 1P(Nα ) + αm, with P(D α) = I 1 + I 2, D α = D αsm, α = 2µ2 λ + 3µ I 1 = i (n ( Γ + κ 2 2 pi ) 1 2 ( div Γ I t Γ Γ + κ 2 s I ) ) 1 2 n I n ( i(2µ α) I 2 = 2ρω 2 n ( Γ + κ 2 s I ) 1 2 ( div Γ I t + n Γ Γ + κ 2 pi ) 1 2 div Γ I t ( + Γ Γ + κ 2 pi ) 1 ( ) ( 2 n I n Γ Γ + κ 2 s I ) 1 2 ( ) ) Γ n In

35 Appendix P(N α ) = J 1 + J 2 + J 3, N α = N αd M αmd + α 2 MSM J 1 = i 2 ( (λ + 2µ)κ 2 p n( Γ + κ 2 p I) 2 1 n I n + µ ( Γ + κ 2 s I) 2 1 ) (κ ) 2 s It curl Γ curl Γ ( J 2 = i(α 2µ) ( Γ Γ + κ 2 s I) 2 1 ( div Γ I t + n Γ Γ + κ 2 p I) 2 1 ) n I n J 3 = i((α 2µ) 2 2ρω 2 ( n ( Γ + κ 2 s I) 1 2 Γ ( n In ) + n Γ ( Γ + κ 2 p I) 1 2 Γ ( n In ) Γ ( Γ + κ 2 p I) 1 2 div Γ I t + Γ ( Γ + κ 2 s I) 1 2 Γ div Γ I t ).