INTRODUCTION
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Fast Multipole Burton-Miller Boundary Element Method for
Two and Three-Dimensional Acoustic Scattering
William R. Wolf1,2,*, Sanjiva K. Lele1 6WDQIRUG8QLYHUVLW\6WDQIRUG&$±86$
,QVWLWXWRGH$HURQiXWLFDH(VSDoR6mR-RVpGRV&DPSRV63±%UD]LO
Abstract: A multistage adaptive fast multipole method is used to accelerate the matrix-vector products arising from the Burton-Miller boundary integral equations, which are formed in a boundary element method. The present study considers the scattering of acoustic waves, generated by localized sources from bodies with rigid surfaces. Details on the implementation of a multistage adaptive fast multipole method are described for two and three-dimensional formulations. The code is veri¿ed through the solution of well documented test cases. The fast multipole method is tested for acoustic scattering problems of single and multiple bodies, and a discussion is provided on the performance of the method. Results for engineering problems with complex geometries, such as a multi-element wing, are presented in order to assess the implemented capability.
Keywords: Fast Multipole Method, Boundary Element Method, Burton-Miller Formulation, Acoustic Scattering.
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among all elements inside b and those on L(b) and L(b$OO the interactions from elements on L(b) are calculated using GLUHFW%(0IRUPXODWLRQVVLQFHWKH\FDQQRWEHH[SDQGHG LQPXOWLSROHV+RZHYHUIRUL(b), we compute multipole expansions for all elements inside the boxes in L(b), and, then, compute the effects of these multipoles to each of the elements inside b7KHVHQHDU¿HOGFDOFXODWLRQVFDQEH REVHUYHGLQ³6WHS´RI)LJ
All theorems and analytical tools, which prove the validity of the multipole expansions and translations, and several formal DQDO\VLVRIDFFXUDF\DQGDOJRULWKPFRPSOH[LW\IRUWKH)00DUH DYDLODEOHLQOLWHUDWXUH*UHHQJDUGDQG5RNKOLQ&DUULHUet al (SWRQDQG'HPEDUW/DEUHXFKH'DUYH 2WKHU )00 IRUPXODWLRQV IRU ORZIUHTXHQF\ VFDWWHU -LQJFDQDOVREHIRXQGLQWKHZRUNVIURP'DUYHDQG+DYH 6DNXPDDQG<DVXGDDEDQG*XPHURYDQG 'XUDLVZDPL ,Q WKH IRUPHU PHWKRGV WKH DXWKRUV computed the multipole expansions and translations using plane wave expansions and in the latter, Gumerov DQG'XUDLVZDPLXVHGDIRUPXODWLRQVLPLODUWRWKH RQHXVHGLQWKHSUHVHQWZRUNSOXVUHFXUVLYHUHODWLRQVRI the translation operators and rotation-coaxial translation decomposition of the translation operators to reduce the FRPSXWDWLRQDOFRPSOH[LW\RIWKHDOJRULWKP7KHVHPHWKRGV ZLOOEHVWXGLHGLQDIXWXUHZRUNDQGWKH\DUHEH\RQGWKH VFRSHRIWKHSUHVHQWSDSHU
2D Formulation
,Q WKLV VXEVHFWLRQ IRU WKH VDNH RI FRPSOHWHQHVV ZH SUHVHQWWKHIRUPXODWLRQVXVHGIRUWKH'DOJRULWKP7KHVH HTXDWLRQVDUHZHOOGRFXPHQWHGLQ1LVKLPXUD(TXD -WLRQ FDQ EH ZULWWHQ DV DQ H[SDQVLRQ RI WKH GHJHQHUDWH NHUQHOVVXFKDVLQ(T
( , )
(
)
(
),
G x y
4
i
O ox I
n noy
n
=
3 3-
=-v =-v
/
where the functions O n and I nDUHGH¿QHGDV(TVDQG
( ) ( )
On X i Hn ( ) kr e n
in
1
= i
and
( ) ( ) ( ) .
In X i J kr en n
in
= - i
coordinates of some vector X , which can be ox or oy,
IRULQVWDQFH7KHYHFWRUVox and oy point from some box center, oĺWRDIDU¿HOGVRXUFHORFDWLRQxĺDQGDQHDU¿HOGRQH y
ĺUHVSHFWLYHO\7RKDYHDJRRGDSSUR[LPDWLRQLQ(TRQH
should truncate the series using p terms, where p>k oy 7KXV WKH ERXQGDU\ LQWHJUDOV DSSHDULQJ LQ (T FDQ EH ZULWWHQDV(TVDQG
and
7KHUHIRUHIROORZLQJWKHQRWDWLRQIURP(TRQHFDQ ZULWHPXOWLSROHH[SDQVLRQV0QoĺDV(T
,Q(TLWLVDVVXPHGWKDWWKHRULJLQoĺ is located closely to the boundary element S and, then, |ox| > max|oy_KROGV
7KH00(T0/(TDQG//(TH[SDQ
-VLRQVDQGWUDQVODWLRQVDUHJLYHQE\(TVWR
( ) ( ) ( ),
M on In v o o M o
v v = 3 3 - =-l l
v
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v( ) ( ) ( ),
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v
/
v( ) ( ) ( ) .
L on I o o L o n v v v = 3 3 - =-l l
v
/
v,Q(Tovl is the center of a parent box and oĺ is the center RIRQHRILWVFKLOGUHQDQGLQ(TVDQGoĺ is assumed to be closer to xĺ compared to oĺ)LQDOO\ZHFDQDOVRZULWHWKH boundary and hyper-singular integral equations as functions
RIWKHORFDOH[SDQVLRQV(TVDQG
and
3D Formulation
For the same reason as in the previous subsection, we
pres-HQWWKHIRUPXODWLRQVXVHGIRUWKH'DOJRULWKP7KHVHHTXDWLRQV DUHZHOOGRFXPHQWHGLQ<RVKLGD(TXDWLRQFDQEH ZULWWHQDVDQH[SDQVLRQRIGHJHQHUDWHNHUQHOVVXFKDV(T
( , ) ( ) ( ) ( )
G x y 4ik 2n 1 O ox I oy n nm m n n n m 0 r = + 3 =
=-v =-v
/
/
where the function OmnLVGH¿QHGDV(T
( ) ( ) ( , )
O X h( ) kr Y n
m
n nm
1
i z
=
and Inm
is the complex conjugate of InmGH¿QHGDV(T
( ) ( ) ( , ) .
In X j kr Y m
n n
m
i z
=
,QWKHVHH[SUHVVLRQVjn represents the n-th order spherical
%HVVHOIXQFWLRQRIWKH¿UVWNLQGhn(1) is the n-th order spheri-FDO+DQNHOIXQFWLRQRIWKH¿UVWNLQGDQG<n
m are the spherical KDUPRQLFVGH¿QHGDV(T
( , )
( ) !
( ) !
(cos ) ,
Y n m n m P e n m n m im
i z = i
+
- z
where PnmVWDQGVIRUWKHDVVRFLDWHG/HJHQGUHIXQFWLRQV7KH
terms r, ș, z represent the spherical coordinates of some vector X, which can be ox or oyIRULQVWDQFH7KHYHFWRUV ox and oy point from some box center, oĺWRDIDU¿HOGVRXUFH
location, xĺDQGDQHDU¿HOGVRXUFHORFDWLRQyĺUHVSHFWLYHO\
$JDLQLQ(T_ox| > max|oy_KROGV7KHERXQGDU\LQWHJUDOV
LQ(TFDQEHZULWWHQDV(TXDWLRQVDQG
and
7KHUHIRUHIROORZLQJWKHQRWDWLRQIURP(TRQHFDQ
write multipole expansions Mmn (oĺDVLQ(T
7KH00(T0/(TDQG//(TH[SDQ -VLRQVDQGWUDQVODWLRQVDUHJLYHQE\(TVWR
,Q(Tovl is the center of a parent box and oĺ is the FHQWHU RI RQH RI LWV FKLOGUHQ DQG LQ (TV DQG ovl is
assumed to be closer to xĺ compared to oĺ )RU DOO WKHVH
expressions, the summations in l, l n n n n /=
-+ l l
are performed only for even values of n+n’íl$OVRLQWKHVHIRUPXODVWKHWHUP
( ) ( , )
O h( ) kr Y
nm n nm 1
i z =
I , and the term Wn,n’,m,m’,l are computed XVLQJWKHIRUPXODLQ(T
( )
W l i n n l n
m n m
l t
2 1
0 0 0 , , , ,
n n m m l= + n n l
- + l l
l
l l
l
e oe o
where, t=-m-m’ and a d
b e
c f
e oGHQRWHVWKH:LJQHUMV\PEROZKLFK
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RESULTS
The present section presents results obtained by the %(0 DFFHOHUDWHG E\ WKH DGDSWLYH )00 7KH SUREOHPV DQDO\]HGLQFOXGHD'F\OLQGHUDVSKHUH'DQG'F\OLQ -GHUVZLWKDQGZLWKRXWFDYLWLHVPXOWLSOHVSKHUHVDQGD' PXOWLHOHPHQWZLQJ
Code veri¿cation
7KH ¿UVW SUREOHP RI LQWHUHVW FRQVLGHUV WKH DFRXVWLF VFDWWHULQJDURXQGD'ULJLGF\OLQGHUGXHWRDORFDOL]HG D[LV\PPHWULF F\OLQGULFDO VRXUFH 7KH VROXWLRQ IRU VXFK problem is compared with the analytical one presented by 0RUULV7KHSUREOHPLVGHVFULEHGE\DSRLQWVRXUFH placed at a distance L IURPWKHFHQWHURIWKHF\OLQGHU,Q
the present computations, the distance L PWKHUDGLXV
of the cylinder is set a P DQG WKH ZDYH QXPEHU LV k Pí, which corresponds to a frequency of f Hz
for a speed of sound c PVí
For this test problem, the boundary of the cylinder is GLVFUHWL]HGE\IRXUGLIIHUHQWPHVKHVFRQVLVWLQJRI
DQGHOHPHQWVZLWKFRQVWDQWVKDSHIXQF
-WLRQV7KHFRPSXWDWLRQDOWLPHRIDIDVWDQGRIDGLUHFW%(0 VROYHU IRU WKH ' IRUPXODWLRQ LPSOHPHQWHG LV DVVHVVHG )LJXUH D SURYLGHV D FRPSDULVRQ RI WKH VROXWLRQV LQ WHUPV RI SUHVVXUH GLUHFWLYLW\ REWDLQHG E\ WKH IDVW %(0 method and by the analytical solution for a distance R a
This solution is obtained for a cylinder discretized with FRQVWDQWHOHPHQWVDORQJLWVVXUIDFH)LIW\ERXQGDU\ elements per box is the maximum allowed in this computa-WLRQUHVXOWLQJLQIRXU)00OHYHOV7KHQXPEHURIWHUPVLQ WKH)00VHULHVWKDWSURYLGHVDQDFFXUDWHVROXWLRQIRUWKH
D'F\OLQGHU
(b) Sphere
)LJXUH 3UHVVXUHGLUHFWLYLWLHVIRUWKHDFRXVWLFVFDWWHULQJDURXQGD
wave number and dimension of the scatterer in this prob-lem is n )RUODUJHUka, this number has to be increased LQRUGHUWRKDYHDFFXUDWHVROXWLRQV
The convergence residue of the CGS iterative solver is VHWHTXDOWRIRUDOOWKHWHVWFDVHVLQWKLVSDSHU:LWK
WKHVHFRQGLWLRQVWKHPHWKRGFRQYHUJHGLQ¿YHLWHUDWLRQV 1RSUHFRQGLWLRQHUZDVXVHGWRDFFHOHUDWHWKH&*6VROYHU 2QHFDQREVHUYHWKDWWKHQXPHULFDOVROXWLRQIRUSUHVVXUH directivity matches perfectly with the analytical solution SURYLGHGE\0RUULVD
,Q7DEOHWKHUHDUHWKHFRPSDULVRQVRIFRPSXWDWLRQDO HIIRUWLQFOXGLQJVHWXSWLPHIRUWKHDFFHOHUDWHG%(0DQGWKH GLUHFW%(02QHFDQREVHUYHWKDW5RNKOLQ¶VVLQJOHVWDJH RUWZROHYHO)00LVPRUHH[SHQVLYHWKDQWKHKLJKHUOHYHO )00DQGIRUDVPDOOQXPEHURIHOHPHQWVWKH¿YHOHYHO )00LVPRUHH[SHQVLYHWKDQWKHWKUHHDQGIRXUOHYHO)00 This is a consequence of the several recursive divisions and OLVWFRQVWUXFWLRQVSHUIRUPHGLQWKLVPHWKRG+RZHYHUZKHQ the number of boundary elements is increased, the compu-WDWLRQDOHIIRUWIRUWKHWKUHHOHYHO)00LQFUHDVHVIDVWHUWKDQ IRUWKHIRXUDQG¿YHOHYHOV)00)RUHYHQ¿QHUPHVKHV WKH ¿YHOHYHO )00 EHFRPHV WKH OHDVW H[SHQVLYH PHWKRG VKRZLQJWKDWPXOWLVWDJH)00LVQHFHVVDU\IRUODUJHVFDOH SUREOHPV2QHFDQDOVRREVHUYHWKDWWKHFRPSXWDWLRQDOWLPH XVHGLQWKHGLUHFW%(0VROXWLRQLVPXFKODUJHUWKDQWKHRQH LQ)00VROXWLRQV:KLOHWKHGLUHFW%(0EHKDYHVSURSRU-tional to O(n IRU WKH SUHVHQW VLPXODWLRQV WKH ¿YHOHYHO
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0HWKRG HOHP HOHP HOHP HOHP
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The second problem considers the acoustic scatter-ing around a rigid sphere due to a spherically symmetric VRXUFH7KH VROXWLRQ IRU WKLV SUREOHP LV FRPSDUHG ZLWK WKHDQDO\WLFDOVROXWLRQSUHVHQWHGE\0RUULVE7KH point source is placed at a distance L from the center of the VSKHUH,QWKHSUHVHQWFRPSXWDWLRQVWKHGLVWDQFHL P
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N PíFRUUHVSRQGLQJWRDIUHTXHQF\RII +]IRU
DVSHHGRIVRXQGF PV
For this test problem, the boundary of the sphere is GLVFUHWL]HG E\ IRXU GLIIHUHQW PHVKHV FRQVLVWLQJ RI DQGTXDGULODWHUDOHOHPHQWVZLWKFRQVWDQW VKDSH IXQFWLRQV 7KH FRPSXWDWLRQDO WLPH RI WKH IDVW DQG GLUHFW%(0VROYHUVIRUWKH'IRUPXODWLRQLPSOHPHQWHGLV DVVHVVHG)LJXUHESURYLGHVDFRPSDULVRQRIWKHVROXWLRQV LQWHUPVRISUHVVXUHGLUHFWLYLW\REWDLQHGE\WKHIDVW%(0 method and by the analytical solution for a distance R D
7KLVVROXWLRQLVREWDLQHGIRUDVSKHUHGLVFUHWL]HGZLWK FRQVWDQWHOHPHQWVDORQJLWVVXUIDFH7KHPD[LPXPQXPEHU of boundary elements per box is set equal to one hundred, UHVXOWLQJLQWZR)00OHYHOV7KHQXPEHURIWHUPVLQWKH )00VHULHVLVVHWWRWKUHHLQDOOWKH'FDOFXODWLRQVLQWKLV paper, except for the cavity problem, since we are consider-LQJ ORZ IUHTXHQFLHV 2QH FDQ REVHUYH WKDW WKH QXPHULFDO solution for pressure directivity matches perfectly with the DQDO\WLFDOVROXWLRQSURYLGHGE\0RUULVE
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LQRUPRIWKHHUURUGH¿QHGDV , is SORWWHGDVDIXQFWLRQRIWKHPHVKUH¿QHPHQWLQDORJORJSORW LQ)LJD,QWKHGH¿QLWLRQRI_E|, p is the acoustic pressure computed by the numerical method andpʛLVWKDWFDOFXODWHG
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can be controlled by using more terms in the truncated series
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a plot for the memory requirements for the direct and the fast
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order O(n) in terms of memory requirements, while the fast
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Cylinder with cavity
The third test case presents results for the scattering of pressure waves around a rigid cylinder with a cavity, due to a monopole source located non-symmetrically with
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surface of the cylinder is plotted for a wave number k
mí7KHUDGLXVRIWKHF\OLQGHULVa PLWVOHQJWKLV L PWKHFDYLW\GHSWKLVh PLWVZLGWKLVd P
and its length is l P
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due to a monopole source located non-symmetrically
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accuracy and memory requirements for the acoustic
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one can see the solution for the same problem, but for a cylinder
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WKHRULJLQRIWKH&DUWHVLDQV\VWHP:HFDQREVHUYHWKHVWURQJ LQÀXHQFHRIWKHFDYLW\LQWKHVFDWWHULQJRIWKHSUHVVXUHZDYHV
For the cavity case, the scattering has a preferential direction,
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effects of the cavity on the scattering of pressure waves can be
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observer at a radial distance R PIRUz P7KLV¿JXUH
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the effects of the cylinder with cavity compared to the one with
QRFDYLW\7KHSUHVVXUHGLUHFWLYLW\IRUWKHIRUPHUFOHDUO\VKRZV WKHSUHIHUHQWLDOVFDWWHULQJGLUHFWLRQ7KHVSDWLDOHIIHFWVFDQDOVR EHREVHUYHGDQGRQHVHHVWKDWWKHSUHVVXUHPDJQLWXGHVIRUWKH' FDOFXODWLRQVDUHODUJHUWKDQIRUWKH'DVH[SHFWHG)RUWKH' FDYLW\FRQ¿JXUDWLRQIDU¿HOGSUHVVXUHLVVPDOOHUWKDQWKDWIRUWKH F\OLQGHULQDOOGLUHFWLRQV)RUWKH'FDVHRQHFDQREVHUYHKLJKHU IDU¿HOGSUHVVXUHVIRUDQJOHVEHWZHHQDQGDQGIRU 7KH'GLUHFW%(0DQGWKHWZRWKUHHDQGIRXUOHYHO)00 DUHFRPSDUHGLQWHUPVRIFRPSXWDWLRQDOWLPHLQ7DEOH7KH
time spent by each method for achieving the desired
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cavity due to a monopole source located
non-symmetri-FDOO\ZLWKUHVSHFWWRWKHFDYLW\IRUND
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the required convergence while the two, three and four-level
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box should be always chosen to increase the performance of the
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Multiple bodies
The fourth studied problem assesses the implemented capability in dealing with the scattering of pressure waves around multiple
ERGLHV$PRQRSROHVRXUFHLVSODFHGDWFRRUGLQDWHVx Py P
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These have centers on x PDQGy PDQGUDGLXVa P
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the spheres is plotted for a wave number k Pí)LJVDDQG
(b)VKRZSORWVRISUHVVXUHGLUHFWLYLW\IRU'DQG'FRPSXWDWLRQV
IRUDIDU¿HOGZLWKUDGLDOGLVWDQFHR PIRUz PDQGWKH'
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each method for achieving the desired convergence, obtained in
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distributed inside the computational box if compared with the
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Multi-element wing
This test case consists of acoustic scattering around a
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two monopole sources are placed in the gaps formed between
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the multi-element wing surface for a wave number k
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radial distance R PIRUz P7KHGLUHFWDQGIDVW%(0
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the desired convergence, obtained with ten steps by all the
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-trated in the center of the computational box in the y SODQH
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have a smaller number of operations compared to cases where the scattering body is more uniformly distributed around
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number of empty boxes distant from the center of the
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CONCLUSIONS
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is implemented to deal with the non-uniqueness problems of
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only regular Gaussian quadrature is presented for the solution
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easier, and the equations are discretized along the boundaries
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the matrix-vector products arising from the method become
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provides excellent results matching analytical solutions
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computational time are shown and the multistage algo-rithm has proved to be more efficient than the single-stage
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an increasing number of boundary elements in the
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there is always an optimum number of levels of refinement depending on the number of boundary elements and the
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elements uniformly distributed over the computational box, the improvements in computational cost achieved
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distribution of the boundary elements is concentrated over
a specific region of the computational box, e.g., a wing
located along the center of the box, the results show larger
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problems, a factor of two orders of magnitude reduction
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ACKNOWLEDGEMENTS
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REFERENCES
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