The purpose of this article is to present an innovative resolution method for investigating problems of sound scattering by infinite cylinders immersed in a fluid medium. The study is based on the analytical solution of multiple scattering, where incident and scattered waves are expressed in cylindrical harmonics. This modeling leads to dense linear systems, which are made sparse by introducing a cutoff radius around each particle. This cutoff radius is deeply studied and quantified. Numerical resolution is performed using parallel computing methods designed to solve very large sparse linear systems. Comparisons with direct calculations made with another numerical software and homogenization techniques follow and show good agreement with the implemented method. The last part is dedicated to a comparison between the propagation of waves in a circular cluster made of a random distribution of cylinders and the propagation in the corresponding homogenized cluster where the multiple scattering formalism is combined with a statistical analysis to provide an effective medium.

1.
F.
Zàviška
, “
über die Beugung elektromagnetischer Wellen an parallelen, unendlich langen Kreiszylindern
,”
Ann. Phys.
345
,
1023
1056
(
1913
).
2.
S.
Bose
and
A.
Mal
, “
Longitudinal shear waves in a fiber-reinforced composite
,”
Int. J. Solids Struct.
9
,
1075
1085
(
1973
).
3.
M.
Lax
, “
Multiple scattering of waves. II. The effective field in dense systems
,”
Phys. Rev.
85
,
621
629
(
1952
).
4.
V. K.
Varadan
,
V. V.
Varadan
, and
Y.
Pao
, “
Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves
,”
J. Acoust. Soc. Am.
63
,
1310
1319
(
1978
).
5.
P.-Y.
Le Bas
,
F.
Luppé
,
J.-M.
Conoir
, and
H.
Franklin
, “
N-shell cluster in water: Multiple scattering and splitting of resonances
,”
J. Acoust. Soc. Am.
115
,
1460
1467
(
2004
).
6.
C. M.
Linton
and
P. A.
Martin
, “
Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber
,”
J. Acoust. Soc. Am.
117
,
3413
3423
(
2005
).
7.
A. N.
Norris
and
J.-M.
Conoir
, “
Multiple scattering by cylinders immersed in fluid: High order approximations for the effective wavenumbers
,”
J. Acoust. Soc. Am.
129
,
104
113
(
2011
).
8.
G. O.
Olaofe
, “
Scattering by two cylinders
,”
Radio Sci.
5
,
1351
1360
, (
1970
).
9.
J. W.
Young
and
J. C.
Bertrand
, “
Multiple scattering by two cylinders
,”
J. Acoust. Soc. Am.
58
,
1190
1195
(
1975
).
10.
R. P.
Radlinski
and
T. J.
Meyers
, “
Radiation patterns and radiation impedances of a pulsating cylinder surrounded by a circular cage of parallel cylindrical rods
,”
J. Acoust. Soc. Am.
56
,
842
848
(
1974
).
11.
F. A.
Amirkulova
and
A. N.
Norris
, “
Acoustic multiple scattering using recursive algorithms
,”
J. Comput. Phys.
299
,
787
803
(
2015
).
12.
L.
Greengard
and
V.
Rokhlin
, “
A fast algorithm for particle simulations
,”
J. Comput. Phys.
73
(
2
),
325
348
(
1987
).
13.
S.
Koc
and
W. C.
Chew
, “
Calculation of acoustical scattering from a cluster of scatterers
,”
J. Acoust. Soc. Am.
103
,
721
734
(
1998
).
14.
N. A.
Gumerov
and
R.
Duraiswami
, “
Computation of scattering from clusters of spheres using the fast multipole method
,”
J. Acoust. Soc. Am.
117
,
1744
1761
(
2005
).
15.
N. A.
Gumerov
and
R.
Duraiswami
,
Fast Multipole Methods for the Helmholtz Equation in Three Dimensions
(
Elsevier
,
Oxford, UK
,
2004
).
16.
Y. J.
Zhang
and
E. P.
Li
, “
Fast multipole accelerated scattering matrix method for multiple scattering of a large number of cylinders,
Prog. Electromagn. Res.
72
,
105
126
(
2007
).
17.
S.
Balay
,
S.
Abhyankar
,
M. F.
Adams
,
J.
Brown
,
P.
Brune
,
K.
Buschelman
,
L.
Dalcin
,
A.
Dener
,
V.
Eijkhout
,
W. D.
Gropp
,
D.
Kaushik
,
M. G.
Knepley
,
D. A.
May
,
L. C.
McInnes
,
R. T.
Mills
,
T.
Munson
,
K.
Rupp
,
P.
Sanan
,
B. F.
Smith
,
S.
Zampini
,
H.
Zhang
, and
H.
Zhang
, PETSc, available at http://www.mcs.anl.gov/petsc (Last viewed May 1, 2019).
18.
S.
Balay
,
S.
Abhyankar
,
M. F.
Adams
,
J.
Brown
,
P.
Brune
,
K.
Buschelman
,
L.
Dalcin
,
A.
Dener
,
V.
Eijkhout
,
W. D.
Gropp
,
D.
Kaushik
,
M. G.
Knepley
,
D. A.
May
,
L. C.
McInnes
,
R. T.
Mills
,
T.
Munson
,
K.
Rupp
,
P.
Sanan
,
B. F.
Smith
,
S.
Zampini
,
H.
Zhang
, and
H.
Zhang
, “
PETSc users manual
,”
Technical Report No. ANL-95/11—Revision 3.10
, Argonne National Laboratory (2012).
19.
S.
Balay
,
W. D.
Gropp
,
L. C.
McInnes
, and
B. F.
Smith
, “
Efficient management of parallelism in object oriented numerical software libraries
,” in
Modern Software Tools in Scientific Computing
, edited by
E.
Arge
,
A. M.
Bruaset
, and
H. P.
Langtangen
(
Birkhäuser
,
Basel, Switzerland
,
1997
), pp.
163
202
.
20.
P.
Martin
,
Multiple Scattering—Interaction of Time-Harmonic Waves with N Obstacles
(
Cambridge University Press
,
Cambridge, UK
,
2006
).
21.
J. J.
Faran
, “
Sound scattering by solid cylinders and spheres
,”
J. Acoust. Soc. Am.
23
(
4
),
405
418
(
1951
).
22.
N. D.
Veksler
,
Resonance Acoustic Spectroscopy
(
Springer
,
Berlin
,
1993
).
23.
X.
Antoine
,
C.
Chniti
, and
K.
Ramdani
, “
On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders
,”
J. Comput. Phys.
227
,
1754
1771
(
2008
).
24.
T.
Valier-Brasier
and
J.-M.
Conoir
, “
Resonant acoustic scattering by two spherical bubbles
,”
J. Acoust. Soc. Am.
145
,
301
311
(
2019
).
25.
N.
Gould
and
J.
Scott
, “
The state-of-the-art of preconditioners for sparse linear least-squares problems
,”
ACM Trans. Math. Softw.
43
,
1
35
(
2017
).
26.
I. S.
Duff
, “
The impact of high-performance computing in the solution of linear systems: Trends and problems
,”
J. Comput. Appl. Math.
123
,
515
530
(
2000
).
27.
S.
Biwa
,
S.
Yamamoto
,
F.
Kobayashi
, and
N.
Ohno
, “
Computational multiple scattering analysis for shear wave propagation in unidirectional composites
,”
Int. J. Solids Struct.
41
,
435
457
(
2004
).
28.
L. L.
Foldy
, “
The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers
,”
Phys. Rev.
67
,
107
119
(
1945
).
29.
Y.
Pennec
,
B.
Djafari-Rouhani
,
J. O.
Vasseur
,
A.
Khelif
, and
P. A.
Deymier
, “
Tunable filtering and demultiplexing in phononic crystals with hollow cylinders
,”
Phys. Rev. E
69
,
046608
(
2004
).
30.
M.
Chekroun
,
L. L.
Marrec
,
B.
Lombard
, and
J.
Piraux
, “
Multiple scattering of elastic waves: A numerical method for computing the effective wavenumbers
,”
Waves Random Complex Media
22
,
398
422
(
2012
).
31.
B.
Tripathi
,
A.
Luca
,
S.
Baskar
,
F.
Coulouvrat
, and
R.
Marchiano
, “
Element centered smooth artificial viscosity in discontinuous Galerkin method for propagation of acoustic shock waves on unstructured meshes
,”
J. Comput. Phys.
366
,
298
319
(
2018
).
32.
A.
Luca
,
R.
Marchiano
, and
J.-C.
Chassaing
, “
Numerical simulation of transit-time ultrasonic flowmeters by a direct approach
,”
IEEE Trans. Ultrason. Ferroelec. Freq. Control
63
(6),
886
897
(
2016
).
33.
C.
Aristégui
and
Y. C.
Angel
, “
Effective mass density and stiffness derived from p-wave multiple scattering
,”
Wave Motion
44
,
153
164
(
2007
).
34.
M. R.
Haberman
and
A.
Norris
, “
Acoustic metamaterials
,”
Acoust. Today
12
,
31
39
(
2016
).
35.
A.
Aubry
,
A.
Derode
,
P.
Roux
, and
A.
Tourin
, “
Coherent backscattering and far-field beamforming in acoustics
,”
J. Acoust. Soc. Am.
121
,
70
77
(
2007
).
36.
A.
Derode
,
V.
Mamou
, and
A.
Tourin
, “
Influence of correlations between scatterers on the attenuation of the coherent wave in a random medium
,”
Phys. Rev. E
74
,
036606
(
2006
).
37.
D.
Torrent
and
J.
Sánchez-Dehesa
, “
Sound scattering by anisotropic metafluids based on two-dimensional sonic crystals
,”
Phys. Rev. B
79
,
174104
(
2009
).
38.
E.
Reyes-Ayona
,
D.
Torrent
, and
J.
Sánchez-Dehesa
, “
Homogenization theory for periodic distributions of elastic cylinders embedded in a viscous fluid
,”
J. Acoust. Soc. Am.
132
,
2896
2908
(
2012
).
39.
D.
Torrent
and
J.
Sánchez-Dehesa
, “
Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas
,”
Phys. Rev. B
74
,
224305
(
2006
).
You do not currently have access to this content.