phase reduction-type preconditioners for acoustic...

Phase Reduction-Type Preconditioners

for Acoustic Scattering Problems

Ibrahim Zangre

Institut Elie Cartan Lorraine (Iecl), Universite de LorraineApplied and Computational Electromagnetics (Ace), University of Liege

ibrahim.zangre@iecn.u-nancy.fr

PhD advisors: Xavier Antoine (Iecl) & Christophe Geuzaine (Ace)

Journee Envol Recherche, Fondation EADS, February 14th 2013, Paris.

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I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 1 / 25

Outline

1 Introduction

2 Shifted Laplace Preconditioners (SLP)

3 Phase-Rduction Models (PRFEM)

4 Conclusion

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 2 / 25

Outline

1 Introduction

2 Shifted Laplace Preconditioners (SLP)

3 Phase-Rduction Models (PRFEM)

4 Conclusion

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 3 / 25

Helmholtz exterior equations

Let us consider the scattering problem ofan incident acoustic wave u

inc

(Ω−)

(Γ)

uinc u

(Ω+) Ω− ⊂ Rd(d = 2, 3)

Γ := ∂Ω−

Ω+ = Rd \ Ω−

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 4 / 25

Helmholtz exterior equations

Then the scattered field u is governed by the system:

−∆u − k2u = 0 in Ω+,

∂nu = −∂nuinc on Γ,

lim|x|→∞

|x| d−12

(

∇u · x

|x| − ıku

)

= 0.

λ: wavelength

k: wavenumber (k = 2π/λ)

The choice of Neumann boundary condition on Γ(sound-hard) is not restrictive

The last equation is the so-called Sommerfeld radiationcondition (imposing that u is outgoing)

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 5 / 25

Approximated Helmholtz problem

For numerical purposes, we bound theinfinite domain by introducing anAbsorbing Boundary Condition (ABC)

(Ω−)

(Γ)

(Ω) (Γ∞)The bounded Helmholtz problem nowreads:

−∆u − k2u = 0 in Ω

∂nu = −∂nuinc on Γ

∂nu = Bu on Γ∞

Weak form: find u ∈ H1(Ω) such that

∫

Ω

(

∇u · ∇v − k2uv

)

dx −∫

Γ∞

Buvdσ = −∫

Γ

∂nuinc

vdσ, ∀v ∈ H1(Ω).

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 6 / 25

Finite Elements Method (FEM) discretization

The FEM discretization of the weak form provides the following linear system

Huh :=(

S − k2M − B

)

uh = bh

h: mesh size

nh : total number of DOF

S: stiffness matrix

M: mass matrix

B: Γ∞-surface mass matrix

nλ = λ/h: density of discretization points per wavelength

uh , bh ∈ Cnh

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 7 / 25

Difficulties

The Helmholtz matrix H is generally highly indefinite

H may be large sized:Strong numerical pollution due to the FEM: consequence is high # of DOFrequired to reach satisfactory accuracy; one should increase nh to limitnumerical pollution related to the FEM)→ use of direct solvers is outcast

Iterative solvers are mostly penalized by the indefiniteness of Hparticularly in the high frequency regimethen divergence or extremly slow convergence are observed→ preconditioners are required to provide a satisfactory convergence

Ideal case: independent (or weakly dependent) convergence with respect tothe wavenumber

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 8 / 25

Few preconditioning technics

Algebraic:

ILU-like factorizations

pARMS (algebraic recursive multilevel)

Linear methods (diagonal,SOR, SSOR)

Analytic:

AILU (analytic ILU) [M.J. Gander, F. Nataf, 2000]:→ Helmholtz operator factorization

H := −∆ − k2 = − (∂x + Λ1) (∂x − Λ2)

Sweeping preconditioner [B. Engquist, L. Ying, 2011]:→ sweeping factorization of the Helmholtz system in a block LDL

t

Shifted Laplace preconditioners [Erlangga & al., 2004]:→ preconditioning by the shifted Laplace operator

A(α) := −∆ − αk2, α = a + ıb ∈ C

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 9 / 25

Outline

1 Introduction

2 Shifted Laplace Preconditioners (SLP)

3 Phase-Rduction Models (PRFEM)

4 Conclusion

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 10 / 25

SLP: spectral properties at continuous level

Let us consider the principal symbol of H and A(α)

σ[H](ξ) = |ξ|2 − k2 and σ[A(α)](ξ) = |ξ|2 − αk

2, ∀ξ ∈ Rd .

Then the symbol of the preconditioned operator A−1(α)H is

σ[A−1

(α)H](z) =

|ξ|2 − k2

|ξ|2 − αk2=

1 + z

α + z, ∀z = −|ξ|2

k2∈ R

−.

→ The “continuous eigenvalues” of A−1(α)H ly on a circular arc with endpoints:

α−1 related to the propagative modes:H :=

(k, ξ) ∈ R × Rd/|ξ| < k

(z → 0)

(1, 0) related to the evanescent modes:E :=

(k, ξ) ∈ R × Rd/|ξ| > k

(z → −∞)

(0, 0) related to the grazing modes:G :=

(k, ξ) ∈ R × Rd/|ξ| ≈ k

(z → −1)

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 11 / 25

SLP: FEM formulation

The discrete Shifted Laplace operator reads

If one considers the classic ABC ∂nu − ıku = 0 on Γ∞:

Aık(α),h := S − αk

2M − ıkB.

or the well-suited ABC ∂nu − ı√

αku = 0 on Γ∞:

Aı√

αk

(α),h:= S − αk

2M − ı

√αkB.

1D Example:

uinc = e

−ıkx1 , Ω = (0, 1), Γ = 0, Γ∞ = 1k = 20π, nλ = 80, α = 1 + 0.5ıSLP preconditioners are solved by an exact LU.

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 12 / 25

SLP: continuous eigenvalues

Real part

Imagin

ary

part

−800 −600 −400 −200 0 200 400 600 800

−600

−400

−200

0

200

400

600

H

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 13 / 25

SLP: eigenvalues clustering

Real part

Imagin

ary

part

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

A−1(α)H (analytic)

zone E

zone G

zone H

[A0(α),h]−1[H]

[Aik

(α),h]−1[H]

[Ai√

αk

(α),h ]−1[H]

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 14 / 25

SLP: limits

+ SLP can be solved efficiently by multigrids methodsIt is likewise well-suited for ILU-like preconditioning(provides factorzation stability)It provides positive definiteness to the Helmholtz problem

− Many eigenvalues remain closed to zero (grazing modes)Convergence remains dependent to the wavenumber (linearly)

→ We propose a FEM formulation based on phase reduction models (PRFEM)to overcome the numerical pollution problemAnd adapt shifted laplace-like preconditioning in the FRFEM framework

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 15 / 25

Outline

1 Introduction

2 Shifted Laplace Preconditioners (SLP)

3 Phase-Rduction Models (PRFEM)

4 Conclusion

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 16 / 25

PRFEM models

Weak formulation of the bounded Sound-Hard Helmholtz problem read: findu ∈ H

1(Ω) such that

A(u, v) = L(v), ∀v ∈ H1(Ω),

(HM)

A(u, v) = (∇u, ∇v)Ω − k2(u, v)Ω − (Bu, v)Γ∞

L(v) = (∂nuinc, v)Γ.

Using the so-called Rytov decomposition of the form

u(x) = aeikφ(x),

where the phase φ is approximated by OSRC technics or by solving the eikonalequation

|∇φ|2 = 1 in Ω,

the wave-envelope a(x) is solution of

A (a, b) = L (b), ∀b ∈ H1(Ω),

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 17 / 25

PRFEM models

where the sesquilinear and linear forms read

(PR)

A (a, b) = (∇a, ∇b)Ω + ik [(a∇φ, ∇b) − (∇φ · ∇a, b)]−k

2((1 − |∇φ|2)a, b)Ω + (. . .)Γ∞ ,L (b) = (e−ikφ∂nu

inc, b)Γ

• First idea of PR-FEM Preconditioning:

Use the damped decomposition

u(x) = aeik(1+iη)φ(x), η ∈ R,

Then the form sesquilineaire above now reads

(DR)

Aη(a, b) = (∇a, ∇b)Ω + ik(1 + iη) [(a∇φ, ∇b) − (∇φ · ∇a, b)]−k

2((1 − (1 + iη)2|∇φ|2)a, b)Ω + (. . .)Γ∞

.

• Second idea of PR-FEM Preconditioning:

Use the damped decomposition

u(x) = aeikηφ(x), kη = k

√1 + iη,

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 18 / 25

PRFEM models

Then the form sesquilinear corresponding to the operator H(α) reads

(SR)

Aα,η(a, b) = (∇a, ∇b)Ω + ikη [(a∇φ, ∇b) − (∇φ · ∇a, b)]−k

2((α − (1 + iη)|∇φ|2)a, b)Ω + (. . .)Γ∞

with α ∈ C; that corresponds to SLP variation of the PRFEM.

2D example: Scattering by the sound-soft unit disc

• k = 10π, nλ = 8, # dof = 35773• FEM: linear P1

• ILUT droptol = 0.05, η = 0.5, α = 1 + 0.5i

• FEM relative error: 8.410−2

• PR-FEM relative error: 3.510−3

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 19 / 25

Numerical Performances

Figure: (a) FEM Solution u

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 20 / 25

Numerical Performances

Figure: (b) PRFEM: Eikonal Equation Phase

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 21 / 25

Numerical Performances

Figure: (c) PR-FEM: Wave-Envelope a

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 22 / 25

Numerical Performances

0 100 200 300 400 500 60010

−12

10−10

10−8

10−6

10−4

10−2

100

102

104

106

Iterations

Bi-

CG

ST

AB

Res

idual

(HM)Fem+Ilut

(HM)Fem+Slp

(PR)PrFem+Ilut

(SR)Prfem+Slp

(DR)Prfem+Damp

Figure: (d) Bi-CGSTAB convergence

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 23 / 25

Outline

1 Introduction

2 Shifted Laplace Preconditioners (SLP)

3 Phase-Rduction Models (PRFEM)

4 Conclusion

I. Zangre (Iecl/Ace) Phase Reduction Preconditioners Feb. 14th 2013 24 / 25

Conclusion

Globlal method based on preconditioning the PRFEM is hopeful

provide satisfactory computation of the numerical solution that would allowto overcome the difficulties (i) and (ii)

Future work will concern generalization to non-convex scatterers

More efficient solvers for the Eikonal equation: Fast-Marching method orhigh-order OSRC technics could improve numerical accuracy andperformance.

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