OSRC-Preconditioning and Fast Multipole Method...

OSRC-Preconditioning and FMM

OSRC-Preconditioning and Fast Multipole Method:Eigenvalues investigation

Eric DARRIGRAND †

joint work with Marion DARBAS ‡ and Yvon LAFRANCHE †

† IRMAR – Universite de Rennes 1‡ LAMFA – Universite de Picardie

Eric Darrigrand Reunion ANR MicroWave – Nancy – Fev. 21-22, 2012


Outline

• Initial motivation: 3-D Helmholtz equation

• Integral equation strategy: CFIE

• Analytical preconditioning strategy: OSRC

• Efficient calculations: FMM

• Numerical resolution results

? Validation on the unit sphere

? A trapping domain

• Eigenvalues numerical investigation

? The unit sphere

? A trapping domain



3-D exterior domain Helmholtz equation

Acoustic scattering in R3:

Ω−

Γu +

n

Ω+ u inc

Find u+ solution of the problem

∆u+ + k2u+ = 0, in Ω+,

u+|Γ = −uinc|Γ or ∂nu+|Γ = −∂nuinc|Γ, on Γ,

lim|x|→+∞

|x|(∇u+ · x

|x|− iku+

)= 0,

with uinc(x) = e−ikθinc·x, θinc the incidence direction, k the wavenumber.



CFIE integral formulation

Find ϕ = −γ+0 (u+ − uinc) ∈ H1/2(Γ) such that(

I

2+M + ηD

)ϕ = −γ+

0 uinc − ηγ+

1 uinc, on Γ,

with η ∈ C∗ the coupling parameter and with the integral operators

Mϕ(x) := −∫

Γ

∂n(y)G(x,y)ϕ(y)dΓ(y), ∀x ∈ Γ

Dϕ(x) := −∂n(x)

∫Γ

∂n(y)G(x,y)ϕ(y)dΓ(y), ∀x ∈ Γ

with G(x, y) = eik|x−y|/(4π|x− y|).

Two essential difficulties:

• implication of dense operators −→ high resolution cost

• bad spectral properties −→ low or non convergence of iterative solvers



OSRC-preconditioned integral formulation

Find ϕ = −γ+0 (u+ − uinc) ∈ H1/2(Γ) such that(

I

2+M − V D

)ϕ = −γ+

0 uinc + V γ+

1 uinc, on Γ.

where V is an approximation of the NtD operator V ex, derived from On-SurfaceRadiation Condition methods:

V =1

ik

(1 +

∆Γ

k2ε

)−1/2

, with kε = k + iε.

Essential remarks:• V evaluated using a Pade approximation.

• V involves only differential operators like ∆Γ.

• spectral properties strongly improved thanks to the relation I/2 +M − V exD = I

• the equation still involves the integral operators M and D.



Use of FMM

Aim: Fast calculation of matrix-vector products with the matrix

[L]i j =

∫Γ

∫Γ

G(x,y)ϕj(y)ϕi(x)dΓ(y)dΓ(x), i, j = 1, · · · , N,

Principle: For i far from j

[L]i j ≈P∑p=1

cp∑

B/B∩suppϕi 6=∅

g(p)i,B

∑B/B∩suppϕj 6=∅

T (p)

B,Bf

(p)

j,B,

with g(p)i,B local moment in box B of target d.o.f. ϕi,

f(p)

j,Bfar moment in box B of source d.o.f. ϕj ,

T (p)

B,Btranslation operator from B to B.

Tools: Gegenbauer series and Funk-Hecke formula.



Precisely:

cp =ik

(4π)2wp , f

(p)

j,B=

∫B∩suppϕj

e−ik<sp,y−CB>ϕj(y)dΓ(y) ,

g(p)i,B =

∫B∩suppϕi

eik<sp,x−CB>ϕi(x)dΓ(x) ,

T (p)

B,B=

L∑`=1

(−i)`(2`+ 1)h(1)` (k|CB − CB |)P`(cos(sp, CB − CB)),

CB center of B ; h(1)` spherical Hankel function ; P` Legendre polynomial;

(wp, sp)p ←→ quadrature rule on the unit sphere;∑Pp=1 ←→ integration on the unit sphere (P = (L+ 1)(2L+ 1));∑L`=1 ←→ Gegenbauer series (L = kd+ C(kd)3).



18 interactions

BT1

BS2

BS3

x11

x12

x13

y21

y22

y23

y31

y32y33 11 interactions

BT1

BS2

BS3

C2

C3

C1

x11

x12

x13

y21

y22

y23

y31

y32y33



Numerical resolution (using Gmsh, MUMPS, and F90-personal codes)

Case of the unit sphere

• For numerical validation (with Mie series)

• Observations:

? Radar Cross Section

? GMRES convergence

Case of a cube with cavity

• A trapping domain

• Observations:

? Radar Cross Section

? GMRES number of iterations

? GMRES residual

? Localization of a resonance frequency



Case of the unit sphereCFIE+OSRC+FMM – fixed discretization density nλ = 10

0 30 60 90 120 150 180−20

−10

0

10

20

30

40

!

Norm

alize

d RC

S

RCS Unit Sphere ; k=23.7

Mie seriesCFIE+FMMCFIE+OSRC+FMM

0 30 60 90 120 150 180−20

−10

0

10

20

30

40

!No

rmal

ized

RCS

RCS Unit Sphere ; k=47.4

Mie seriesCFIE+OSRC+FMM

46080 triangles 184320 triangles



CFIE+OSRC+FMM – at fixed wavenumber

0 30 60 90 120 150 180−10

−5

0

5

10

15

20

25

!

Norm

alize

d RC

S

RCS Sphere ; k=10 ; CFIE+OSRC+FMM

Mie seriesn" = 7

n" = 12

n" = 15

0 30 60 90 120 150 180−15

−10

−5

0

5

10

15

20

25

30

!No

rmal

ized

RCS

RCS Sphere ; k=15 ; CFIE+OSRC+FMM

Mie seriesn" = 8

n" = 16

n" = 32



GMRES convergence

0 10 20 30 40 500

50

100

150

200

k

GM

RES

itera

tions

Unit sphere, n!=10

CFIECFIE+FMMCFIE+OSRC+FMM

5 10 15 20 250

20

40

60

80

100

120

n!

GM

RES

itera

tions

Unit sphere, k=10


Vs. k (nλ = 10) Vs. nλ (k = 10)



CPU costs

k Total CPU time Total CPU time Total CPU time

CFIE CFIE + SLFMM CFIE + SLFMM + OSRC

4.76 7 min 42” 13 min 47” 2 min 42”

11.85 9 h 43 min 4 h 33 min 32 min 40”

23.7 > 15 days 214 h 44 min 6 h 20 min

47.4 – – 48 h 48 min

• Asymptotic behavior of the SLFMM

• Cost related to OSRC operators << cost related to integral operators



Case of a trapping domainCube [−1, 1]3 with the rectangular cavity [0, 1]× [−π/10, π/10]× [−π/10, π/10]

with incident wave (√

3/2, 0, 1/2).



RCS

0 60 120 180 240 300 360−20

−10

0

10

20

30

40

!

Norm

alize

d RC

S

RCS Trapping domain ; k=10.5

CFIE+FMMCFIE+OSRC+FMM

0 60 120 180 240 300 360−20

−10

0

10

20

30

!No

rmal

ized

RCS

RCS Trapping domain ; k=8 ; CFIE+OSRC+FMM

Referencen" = 5

n" = 12.5

n" = 26

41840 triangles, k = 10.5 Various mesh densities, k = 8



GMRES convergence

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

n!

GM

RES

itera

tions

Trapping domain, k=8


0 5 10 15 20 250

100

200

300

400

500

600

k

GM

RES

itera

tions

Trapping domain, n!=10


GMRES iterations vs nλ GMRES iterations vs k



GMRES convergence

0 100 200 300 40010−3

10−2

10−1

100

iteration number

GM

RES

resid

ual

Residuals, k=8

CFIE+FMM, n

!=12.5

CFIE+OSRC+FMM, n!=12.5

CFIE+OSRC+FMM, n!=26



Eigenvalues investigation (using MELINA++ and ARPACK++)

Case of the unit sphere

• For numerical validation – comparisons with:

? the analytical eigenvalues of the operators

? the analytical eigenvalues including Pade approximation

• Behavior vs. wavenumber, mesh density and Pade order

Case of a cube with cavity

• Behavior vs. wavenumber, mesh density and Pade order

• Exhibition of a resonance frequency at k = 5

• Localization of resonance frequencies



Case of the unit sphereDistribution of the eigenvalues, k = 11.85, nλ = 10

−2 0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0

0.5

Real part

Imag

inar

y pa

rt

CFIE ; k=11.85, n!=10

AnalyticalNumerical

0.85 0.9 0.95 1 1.05 1.1−0.15

−0.1

−0.05

0

0.05

0.1

Real partIm

agin

ary

part

CFIE+OSRC ; k=11.85, n!=10, Np=8

AnalyticalNumerical



Numerical eigenvalues vs. Pade order, k = 10, nλ = 11.85

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

−0.2−0.1

00.10.20.30.4

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=2

Padé−analyticalNumerical

0.8 0.9 1 1.1 1.2 1.3

−0.2

−0.1

0

0.1

Real partIm

agin

ary

part

CFIE+OSRC ; k=10, n!=11.85, Np=4


0.8 0.9 1 1.1 1.2 1.3

−0.2

−0.1

0

0.1

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=6


0.8 0.9 1 1.1 1.2 1.3

−0.2

−0.1

0

0.1

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=8


0.8 0.9 1 1.1 1.2 1.3

−0.2

−0.1

0

0.1

Real part

Imag

inar

y pa

rtCFIE+OSRC ; k=10, n

!=11.85, Np=10


0.8 0.9 1 1.1 1.2 1.3

−0.2

−0.1

0

0.1

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=12




Analytical eigenvalues vs. Pade order, k = 10, nλ = 11.85

0.85 0.9 0.95 1 1.05 1.1−0.1

0

0.1

0.2

0.3

0.4

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=2

AnalyticalPadé−analytical

0.85 0.9 0.95 1 1.05 1.1−0.1

−0.05

0

0.05

0.1

0.15

Real partIm

agin

ary

part

CFIE+OSRC ; k=10, n!=11.85, Np=4


0.85 0.9 0.95 1 1.05 1.1−0.1

−0.05

0

0.05

0.1

0.15

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=6


0.85 0.9 0.95 1 1.05 1.1−0.1

−0.05

0

0.05

0.1

0.15

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=8


0.85 0.9 0.95 1 1.05 1.1−0.1

−0.05

0

0.05

0.1

0.15

Real part

Imag

inar

y pa

rtCFIE+OSRC ; k=10, n

!=11.85, Np=10


0.85 0.9 0.95 1 1.05 1.1−0.1

−0.05

0

0.05

0.1

0.15

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=10, n!=11.85, Np=12




Performance of Pade approximation

O

−2θε

θp

2θε

1+

θε = arg(kε)

set of eigenvalues of(

[I] +[∆Γ]

k2ε

)on red line

θp = rotation angle of the usual branch-cut.



Condition number

2 4 6 8 10 12 141.1

1.2

1.3

1.4

1.5

1.6

1.7

Padé order Np

Cond

ition

num

ber

CFIE+OSRC ; k=10, n!=11.85


5 10 15 200

100

200

300

400

k

Cond

ition

num

ber

n!=10

CFIECFIE+OSRC

5 10 15 20 250

50

100

150

200

250

n!

Cond

ition

num

ber

k=10

CFIECFIE+OSRC

a) Vs. Pade order b) Vs. k, nλ = 10 c) Vs. nλ, k = 10



Case of the trapping domainDistribution of the eigenvalues, k = 8, nλ = 9.6, Np = 8

−2 0 2 4 6 8 10 12−2

−1.5

−1

−0.5

0

0.5

Real part

Imag

inar

y pa

rt

CFIE ; k=8, n!=9.6

0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

Real partIm

agin

ary

part

CFIE+OSRC ; k=8, n!=9.6, Np=8



Condition number

5 10 15 20 250

50

100

150

n!

Cond

ition

num

ber

k=4.6

CFIECFIE+OSRC

5 10 15 20 250.1

0.15

0.2

0.25

n!

Smal

lest−m

agni

tude

eig

enva

lue

k=4.6

CFIECFIE+OSRC

5 10 15 20 250

5

10

15

n!

Larg

est−

mag

nitu

de e

igen

valu

e

k=4.6

CFIECFIE+OSRC

2 4 6 8 10 120

50

100

150

200

k

Cond

ition

num

ber

n!=10

CFIECFIE+OSRC

2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

k

Smal

lest−m

agni

tude

eig

enva

lue

n!=10

CFIECFIE+OSRC

2 4 6 8 10 120

5

10

15

k

Larg

est−

mag

nitu

de e

igen

valu

e

n!=10

CFIECFIE+OSRC



Condition number vs. Pade order

2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

Np

CFIE+OSRC ; k=4.2, n!=10

Condition numberLargest mag. eig.Smallest mag. eig.

2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

4

Np

CFIE+OSRC ; k=6.4, n!=10

Condition numberLargest mag. eig.Smallest mag. eig.



Resonance frequency, k = 5

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=5, Np=8

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=5.2, Np=8

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

Real part

Imag

inar

y pa

rt

CFIE+OSRC ; k=5.4, Np=8



Localization of resonances

3 4 5 6 7 8 9 10 110

500

1000

1500

2000

2500

3000

3500

k

Cond

ition

num

ber

Resonance localization



Conclusion and perspectives

• Efficiency of the preconditioning:

? low effect on the cost per iteration of the resolution

? Strong increase of the convergence speed

• Instructive eigenvalues investigation:

? very low condition number for the OSRC-CFIE

? analysis of the effect of the Pade approximation

• Improvements on the strategy:

? Multilevel FMM with FastMMLib

? An alternative to Pade approximation

• Maxwell’s equations

? a short-term task to be done

? join us for the talk given by Marion this afternoon !!!


OSRC-Preconditioning and Fast Multipole Method...

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