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IEEE/OES China Ocean Acoustics Symposium Harbin. China I January
9 11. 2016
Computation of Acoustic Acattering from Objects in Shallow Waters by Conventional and Fast Multipole BEM GONG Jiayuan, AN Junying, CI Guoqing, XU Haiting
Qingdao Branch Institute of Acoustics, Chinese Academy of Sciences Qingdao, P. R. China
[email protected],
[email protected],
[email protected],
[email protected] Abstract-The influences on characteristics
Shallow-water environment
the acoustic are
quite
scattering
different
from
from
has
significant
objects,
those
of
but
the
free-space
problems. A high-performance numerical model of conventional and fast multipole BEM based on Burton-Miller formulation for acoustic scattering from objects in shallow-water waveguides is established for the first time. The numerical model is also combined with the well-developed sound propagation theory for ocean acoustics; the mirror image method is adopted for the near-field problem;
and, the normal mode method (i.e. the
famous open-source code kraken) is chosen for the far-field problem. The time and space complexities of conventional and fast
multi pole
BEM
are
compared
and
analyzed
by
the
computation of different-scale problems. The algorithms are validated by comparison with the scattering results of a rigid sphere situated in a shallow-water waveguide, computed by each method. The horizontal directivity and the frequency responses
objects, which has greatly exacerbated the difficulty of detecting targets. The problem of acoustic scattering from objects in the shallow-water waveguide has attracted many researchers' interests in the past. Ingenito used the integral equation method to study the acoustic scattering from objects in the ideal waveguide [1]. Sarkissian modified the surface integral equation method by method of superposition [2]. Markris et.a!. proposed the spectral method and used sea surface noise for detection and localization [3]. Hong Zhao, Hua Gao, and Junying An used the uni-moment method, T-Matrix method, and the boundary element method to study the acoustic scattering method in shallow waters [4-6]. Wei Fan used the boundary element method to study acoustic scattering from objects in the shallow-water waveguide [7].
of scattering from different objects in shallow-water waveguides
The computational complexities of conventional BEM is
are computed, and the shallow-water target echoes are obtained
O(N\ due to the dense and non-symmetric coefficient matrix
by
created by such a method. In the mid of 80s in the 20 century, the fast multipole method (FMM) was first proposed by Rokhlin and Greengard [8]. Since then, FMM has been combined with the boundary integral method to yield the fast multipole boundary element method (FMM-BEM), which has applied to many areas. The early researches were summarized by Nishimura in a review paper [9]. The first book was published by Yijun Liu, which included the applications of FMM-BEM to elasticity problems, potential problems, electricity and acoustics [10]. Recently, Shande Li applied fast multipole BEM to the large-scale problems [11]. The analytic method for evaluation of moments was applied by Haijun Wu in order to accelerate the efficiency of the diagonal form of FMM-BEM [12].
inverse
FFT.
The
results
conventional BEM is at least is
O(N
show
O(N2),
that,
the
complexity of
and the fast multipole BEM
log N). Combined with these two methods, various-scale
problems can be computed efficiently. The multipath effects of shallow-water waveguides bring in obvious interferences in the space domain; a 'comb filter' effect in the frequency domain; and, also large extension of the target echoes in the time domain. Such characteristics
are
rather
like
that
of
sound
propagation
problems in the ocean waveguide. Further, it can be found that, at least for simple objects, the frequency characteristics of the scattering are controlled by the shallow-water waveguide, and different
objects
only
take
amplitude of the frequency
on
modulation
response,
which
effects are
on
the
completely
different from those of free-space problems.
Keywords-shallow-water waveguide; sound propagation; acoustic scattering; target echo; conventional BEM; fast multipole BEM
I.
INTRODUCTION
In the shallow waters, there exist multiple reflections between sea surface and bottom, and between targets and sea interfaces. The waveguide parameters of depth, bottom, sea state, sound profile and attenuation coefficient have significant influences on the acoustic scattering characteristics of the The work is supported by the National Natural Science Foundation of China (Grant No. 11304344),the National Defense Pre-Research Foundation of China (Grant No. 51303020302-2),the Advanced Area Program for Young Scholars of Institute of Acoustics,Chinese Academy of Sciences,and the Main Direction Program of Knowledge Innovation of Institute of Acoustics, Chinese Academy of Sciences.
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In this paper, the conventional and fast multipole BEM is combined to establish a numerical model for acoustic scattering from objects in the shallow-water waveguides by means of sound propagation theory in the ocean waveguides. Through computation of acoustic scattering from a rigid sphere in the ideal waveguide by the two methods, the algorithms are validated, respectively. The computational complexities of time and memory for conventional and fast multipole BEM are compared and analyzed. Then, the horizontal directivity, frequency responses and target echoes
of scattering from different objects in shallow-water waveguides are computed and analyzed. The results can supply for further study a numerical model for detection, localization and recognition of targets in the oceans. II. BOUNDARY INTEGRAL EQUATION FOR ACOUSTIC SCATTERING FROM OBJECTS IN SHALLOW-WATER WAVEGUIDES The depth of the shallow-water waveguide is H; the water medium is homogeneous; the density is P; and, the sound speedy is c. For the Pekeris waveguide, the sea surface is pressure release, and the medium of the bottom is fluid, whose density is PI. and sound speed Cj. The sound source is a point source located at Xo; the receiving point is at ; x and the target center is at Xt. The boundary condition of the target is rigid, and the boundary surface of the target is denoted as S(y ).
X i x.
Take the derivatives of (1), we can get the hypersingular boundary integral equation (HBIE)
c(x)q(x)= 1J)s[Ksw(x,y,m)q(y) -Hsw(x,y,m)¢(y)JdS(Y)+ q'(x)
(4)
x
8¢(x) 1 x _ c() X =2" ' XES, for S is smooth. () 8n(x) , q 8 ( _8 w x = f � , Ksw - � , Hsw= &G q'() 8n(x) 8n(x) 8n(x)8n(y)'
in which,
'
C
Bottom: PI,
CI
' Z
By Burton-Miller formulation, the fictitious frequency problems can be overcome, and a dual boundary integral equation (CHBIE) can be obtained
z
Fig.l
where Rej and Re2 are the surface and bottom reflection coefficients, respectively. And Rill; ' i = 0,1,2,3 are the ranges between mirror image points and field points.
x
Surface
Water:p,
(3)
CBIE + ,BHBIE
Model of acoustic scattering from a target in the shallow-water
=
0, ,B = i/ k
(5)
waveguide
The conventional boundary integral equation (CBIE) for the shallow-water waveguide is
in
which,
s G w (x,y,w)
¢(x)= L[Gsw(x,y,m)q(y) -¢(y)Fsw(x,y,m)JdS(Y)+ ¢'(x)
(1)
Fsw(x,y,m)= as G w(x,y,m)/8n(y)
And
is
the
shallow-water
waveguide
After discretization, the linear system equation of Burton Miller form of conventional BEM (denoted as CHBEM) for the shallow water waveguide can be derived
Ax=b where A is a NxN coefficient matrix, x
Green's
function. For the far-field problem, the corresponding far-field Green's function can be written as
For the near-field problem, the general near-field Green's function can be obtained by mirror image method
III.
THE FAST MULTIPOLE
(6) is the unknown vector.
BEM FOR ACOUSTIC SCATTERING
FROM OBJECTS IN THE SHALLOW-WATER WAVEGUIDE
The conventional BEM has the benefits that its precision is rather high; the discretization is simple; and, it can be readily applied to the exterior problems. However, the coefficient matrix A is dense and non-symmetric, which makes the computational complexity be O(N2)�O(N3). Because of such a defect, only small-scale problems of several thousands of DOFs can be solved on a single Pc. To overcome the bottleneck, the fast multipole method is combined to accelerate the solution of the boundary element method. Instead of matrix A the matrix-vector product Ax is generated, , and an iterative solver like GMRES is applied to solve the linear equation. Such a scheme can reduce the complexity to O(MogN). The computation procedures of the FMM-BEM include the upward pass (by S2M, M2M translation), downward pass (by
978-1-4673-9978-4/16/$31.00 ©2016
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'k (2n +1 ) f Gsw (x,y,m)q(y)dr(y) _ I 471" n�O
M2L, L2L and L2T translation) and the direct evaluation (by conventional BEM), as shown in Fig. 2. The theory of the FMM-BEM for shallow-water waveguides are introduced simply by taking only the waveguide Green's function
00
=
!!'sf
I
RI�Y [0;' (k, Xi O -Yc) x f M;:J (k'Yc) I( i=O +RP;'(k,x;]-YJ+�0;'(k,x'2 -yJ +R,Rp;'(k,x;3 -Yc) ]
Gsw (x,y,m) as example.
(8)
in which, the defmition of multipole moments is
M;:J (k,yc) L Inm (k,y -yc)q(y)dS(Y) =
(9)
J
B.
M2M (multipole moments to multipole moments) translation When the expansion center moved from y c to y c'
, the
M2M translation formula can be derived
Generate matrix vector product
Ax
n' n+ n' _ m' 2 n +1 ' ( ) I I I ( I ) Wn,n',IU,m',J 1II'=-n'I=I"-II'1 n'=O 00
=
(10)
n+n'-/:even
C.
M2L (multipole moments to local expansion coefficients) translation Denote the local expansion center as
Output Results
x� { XL' YL, zf} , =
j 0,1,2,3 , we can yield the local expansion coefficients by =
M2L translation
Fig. 2 Algorithm flowchart of fast multipole BEM
A.
S2M (sources to multipole moments) translation
r;(k,xO = I(2n'+ I) f I x '�ln-n'l n'�O m'�-n' n+n'-/:even
By multipole expansion method, the waveguide Green's function in (3) can be expanded as
ik Gsw (x, y, m) =I (2n +l) 4
(11)
00
71"
n�O
D. (7)
i=O
x[0;' (k,x;o -yc) + RP;'(k,x;]-Yc) +RP;' (k,X'2 -YJ+RI�O;' (k,xl3 -Yc) ] Then, the integral of the Green's function can be expanded as
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L2L (local expansion coefficients to local expansion coefficients) translation
IEEE
L:(k,x�,) =(-Ir I(2n'+ I) f I '�ln-n'l n'�O m'�-n' n+n'-/:even xWn',n,m',-m,l ]Im-m' ( k,XLij ' - XijL ) Lmn'' ( k'XLij )
(12)
L2T (local expansion coef f icients to targets) translation
E.
90 120 1e-{)7 . '" ' , ,�.' ·
, .. . ,.,.
n
ik =-I (2n+l) I=-n I (RI�r 41T n=O m i=O 00
00
x[Lmn (kxio)lnm(k,xiO_ iO) il E" (k )1n111 (k, _ ) +RIn' +R.J:'1 En" (k,xi2)1n1ll (k 12_ i2) +R R E"n (k Xi3)1-nm(k,xi3 - i3)] X
L
'
ii X L
X
'
X
,X
L
1"2
ii
L
L
X
NUMERICAL COMPUTATION AND ANALYSIS
·
_ " --,,--O ___
-"
�: .,.:. ._
�
- - -NM-SWBEM .-"" ..... ..... NM-FMM-SWBEM
-- Freespace,100 times
enlarged
300 270
L
In the above equations, the notations and symbols can be referenced in [10]. IV.
"{� ,..:i.:.�...;,_�'/_ _·:_-.:.• •...
L
X
L
(13)
(b) Horizontal range between source or receiving point and the sphere center is IOkm
Fig. 3 Validation of horizontal directivity of scattering from rigid objects in shallow waters using different methods
A. Validation of the methods The horizontal directivity of the acoustic scattering field computed by the conventional and fast multipole BEM for the shallow-water waveguides based on the mirror image method and normal mode method are shown in Fig. 3. The depth of the ideal waveguide is 100m, the sound speed of the sea water is lSOOmls, and the density is lOOOkg/m3. The target is a rigid sphere with radius of 1m, and located at the depth of SOm. The incident sound source is a point source, and located at the depth of SOm. The receiving point is at the same place as the source point (monostatic), and the horizontal range between the target center and the source or receiving points is 100m and lOkm, respectively. 90
2.5e-{)5 2e-{)5
Through the horizontal directivities of the scattering field, . It can be found that the results computed by conventional and fast multipole BEM are totally the same, and the results by mirror image method and the normal mode method are also very close to each other. By this means, the algorithms of the conventional and fast multipole BEM for the shallow-water waveguides are validated. B.
10000 --e-CHBEM
rn :0-
60
Analysis of the computational complexities
(l)
E
:l rn C 0
() (l)
-6- FMM-SWBEM,p=5 -*- FMM-SWBEM,p=9 ______ FMM-SWBEM,p= 12
8000 6000 4000
E i=
2000
180
". --NM-SWBEM
,: ,
,-
- - - NM-FMM-SWBEM •
.. ,······· MI -S WBEM
_._--
MI-FMM-SWBEM
0
0
5 DOFs
(a) time complexity 270
(a) Horizontal range between source or receiving point and the sphere center is 100m
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10 4 x10
6
1,400
---e--- CHBEM
co 1,200
--*- FMM-SWBEM,p=9
1f 1,000
� FMM-SWBEM,p= 12
E
:l til c:: 0 () >.
0 E
Q)
2
100Hzl - - -200Hz 1-
5
-&- FMM-SWBEM,p=5
2
-6 x 10
(I) -a :::J
800
4
�3
E « 2
600 400 200 0
0
5 DOFs
10
100
4 x10
(b) space complexity
300
(a) Horizontal range l Okm
Fig. 4 Comparison of computational complexity of conventional and fast multipole BEM
2
The comparison of computational complexities of the conventional and fast multipole BEM is taken in Fig. 4. The time complexity comparison is shown in Fig. 4 (a), from which it can be seen that the time complexity of conventional BEM is O(N2), that is, increases nonlinearly with DOFs. The time complexity of fast multipole BEM increases linearly with DOFs, that is, O(NlogN). As given in Fig. 4 (b), the space complexity of conventional BEM is O(N2) when the DOFs is less than about 10 thousands; and when the DOFs increase more than 10 thousands, matrix A cannot be allocated due to out of memory problem, the matrix-vector product is generated instead, and the iterative solver is applied, which reduces the space complexity to linear relationship of O(N). For the fast mutipole BEM, its space complexity increases almost linearly with the DOFs, or more accurately O(NlogN), and has nothing to do with the expansion order p. C.
200
Scattering Angle(deg)
-6
�
'I I I
1.5
'I ' I
I
(I) -a
,
'I
I
:E
a.. E «
"
I
I I
0.5
, I
I" I
,
,
0
or
\
100
0
200
Scattering Angle(deg)
300
(b) Horizontal range 20km
1
Horizontal Directivities of the Scattering Field
The computation results of the horizontal directivity of acoustic scattering from the benchmark model is shown in Fig. 5. The depth of the Pekeris waveguide is 100m, the sound speed of the sea water is 1500m/s, and the density is 1000kglm3. The sound speed of the bottom is 1800m/s, and density 1800kg/m3. The benchmark model is a standard submarine model with rigid boundary conditions. The incident sound source is a point source located abeam of the benchmark model at the depth of 50m in the Pekeris waveguide. The receiving point is at the same place of the source point, and the horizontal range between the target center and the source or receiving point is lOkm, 20km, and 30km, respectively.
X 10
0.8
x 10
-6
I I
.{g .-2
0.6
a.. E « 0.4
I
I
I
,\
,
, I
, ,,
I
, ,, I I
I I
0.2
100
200
300
Scattering Angle(deg) (c) Horizontal range 30km
Fig. 5 Horizontal directivity of acoustic scattering from the benchmark model object in shallow waters
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From Fig. 5, it can be seen that the horizontal directivity fluctuates very quickly with the scattering angle, and has shown obvious interferences of angles. When the frequency increases, the interferences become stronger. The amplitude of the acoustic scattering pressure in shallow waters attenuates slowly with the increase of horizontal range, which is much different from the characteristic of the free-space problems due to the effect of sound propagation in the waveguide.
Fig. 6 Acoustic scattering from a rigid sphere in the Pekeris waveguide
4
D. Freqency Responses and Target Echoes
150
The computation parameters of the Pekeris waveguide is the same as Fig. 5, and the sampling interval of frequency is O.IHz. One target is a rigid sphere, with its radius 1m. Another target is a rigid spherical-crown cylinder, its length is 10m, and the radius is 3.5m. The transmitting signal is a CW pulse, with its central frequency fo 500Hz, and the filling number Np of cosine waves is equal to 5.
200
250
300 350 Frequency(Hz)
As shown in Fig. 8, the frequency response of the spherical-crown cylinder resembles that of the sphere, but its amplitude is 10 times larger than that of the sphere. From Fig. 6, 7 and 8, it can be found that, the frequency responses take on a 'comb filter' property, and this is the characteristic of the shallow-water sound channel. The results show that, for simple targets at least, the characteristics of acoustic scattering from objects in shallow waters are determined by the characteristics of the shallow-water waveguides, and the targets just take on the effects of amplitude modulation.
450
500
(a) Frequency response
4
x10"
l - fO=500Hz, Np=51
=
The computed results of the frequency responses and the target echoes are shown in Fig. 6 and 7, in which the horizontal range is lIan. From these two figures, we can see that the amplitude of the frequency responses fluctuates very quickly with the frequency, and is completely different from that of the free-space problem. At the horizontal range of lkm, the pulse width of the target echo is already more than 20ms, which indicates that the multipath effect is rather strong.
400
2 o
"5
0
w
1-----�iI1o!j
·2 � L-__�__�__�__�__�__�__�__�__�__� 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700
Time(ms)
(b) Target echo Fig. 7 Acoustic scattering from a spherical-crown cylinder in the Pekeris waveguide
5 '"
-g
.1: is.
4
x10·5 --- Rigid sphere, 10 times enlarged ..····..·..
Spherica l-crown cylinder
3
�2 150
200
250
300
350
400
450
500
Frequency(Hz)
Fig. 8 Comparison of amplitudes of frequency responses between the sphere and spherical-crown cylinder in the Pekeris waveguide
200
300
400
500 600 Frequency(Hz)
700
800
900
1000
x10·7
o
"5 O f-----� W
·1 ·2 ·3 �--�--�--� 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700
Time(ms)
(b) Target echo
978-1-4673-9978-4/16/$31.00 ©2016
CONCLUSION
Combined with the normal mode method and mirror image method, the conventional and fast mutipole BEM model for acoustic scattering from objects in the shallow-water waveguide is established. Through computational results, it can be seen that the time and memory complexities of 2 conventional BEM are O(N ), and the fast multipole BEM is
(a) Frequency response
3
V.
IEEE
O(NlogN). By combining these two method, various-scale problems of acoustic scattering from objects in the ocean can be solved efficiently. The results show that, at least for simple targets, the characteristics of frequency response take on a 'comb-filter' effect, which shows that the characteristics are determined by the ocean waveguide, and the influences of targets are the modulation of amplitudes.
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IEEE
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