Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics

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1 Enginring Analysis with Boundary Elmnts 8 (00) 8 09 wwwlsvircom/locat/nganabound Applications of th dual intgral formulation in conjunction with fast multipol mthod in larg-scal problms for D xtrior acoustics JT Chn*, KH Chn Dpartmnt of Harbor and Rivr Enginring, National Taiwan Ocan Univrsity, Klung, Taiwan, ROC Rcivd 1 Sptmbr 00; rvisd March 00; accptd 1 April 00 Abstract In this papr, w solv th larg-scal problm for xtrior acoustics by mploying th concpt of fast multipol mthod (FMM) to acclrat th construction of influnc matrix in th dual boundary lmnt mthod (DBEM) By adopting th addition thorm, th four rnls in th dual formulation ar xpandd into dgnrat rnls, which sparat th fild point and sourc point Th sparabl tchniqu can promot th fficincy in dtrmining th cofficints in a similar way of th fast Fourir transform ovr th Fourir transform Th sourc point matrics dcomposd in th four influnc matrics ar similar to ach othr or only som combinations Thr ar many zros or th sam influnc cofficints in th fild point matrics dcomposd in th four influnc matrics, which can avoid calculating rpatdly th sam trms Th sparabl tchniqu rducs th numbr of floating-point oprations from OðN Þ to OðN log a ðnþþ; whr N is numbr of lmnts and a is a small constant indpndnt of N: To spd up th convrgnc in constructing th influnc matrix, th cntr of multipol is dsignd to locat on th cntr of local coordinat for ach boundary lmnt This approach nhancs convrgnc by collocating multipols on ach cntr of th sourc lmnt Th singular and hyprsingular intgrals ar transformd into th summability of divrgnt sris and rgular intgrals Finally, th FMM is shown to rduc CPU tim and mmory rquirmnt thus nabling us apply BEM to solv for larg-scal problms Fiv momnt FMM formulation was found to b sufficint for convrgnc Th rsults ar compard wll with thos of FEM, convntional BEM and analytical solutions and it shows th accuracy and fficincy of th FMM whn compard with th convntional BEM q 00 Elsvir Ltd All rights rsrvd Kywords: Fast multipol mthod; Larg-scal problm; Extrior acoustics; Dual boundary lmnt mthod; Hyprsingular quation; Divrgnt sris 1 Introduction Th boundary lmnt mthod, somtims rfrrd to as th boundary intgral quation mthod, is now stablishing a position as an actual altrnativ to th FEM in many filds of nginring It is ncssary to discrtiz th boundary only instad of th domain, which tas a fwr tim for on-dimnsion rduction in msh gnration Th dual boundary lmnt mthod (DBEM), or so-calld th dual boundary intgral quation mthod dvlopd by Hong and Chn [1,0], is particularly suitabl for th problms with a dgnrat boundary Th dual formulation also plays important rols in som othr problms, g th cornr problm [1], adaptiv BEM [,], th spurious ignvalu of intrior problm [1,1], th fictitious frquncy of * Corrsponding author Tl: þ x10/1; fax: þ addrss: jtchn@mailntoudutw (JT Chn) xtrior problm [,10,11,1], and th dgnrat scal problm [8,9] Thr is considrabl intrst in many applications for th solution of Hlmholtz quation, whn th wav lngth is short or th wav numbr is larg aftr comparing with th siz of boundary, and it is so-calld th larg-scal problm such as th scattring of high frquncy acoustics, Stos flows, molcular dynamics and lctromagntic-wav problms Howvr, th applications of BEM wr limitd in small-scal problms [1,1,,,9,0,,] For larg-scal problms, w nd to modl such problms with a larg numbr ðnþ of boundary lmnts to accuratly rprsnt th gomtry and th solution variation which may not b solvd using a dstop computr Th complxity proportional of th convntional BEM is N and it is xpnsiv in th largscal problm, but th finit lmnt mthod (FEM) is N bcaus of its bandd cofficint matrix [0] Multidomain approach [18] or approximat thoris such as 09-99/0/$ - s front mattr q 00 Elsvir Ltd All rights rsrvd doi:10101/s09-99(0)001-x

2 8 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 th thoris of plats and shlls hav bn mployd to solv th problm using th paralll computrs Whn th siz of th influnc matrix by using BEM is so larg that its storag and solution by Gaussian limination may caus problms for dstop computr Thus, th siz of influnc matrix bcoms th limiting factor that largscal problms can b solvd with a particular computr BEM with itrativ solvrs has bn mployd to dal with th problm [,8] Th major computational cost of th itrativ mthods lis in th matrix vctor multiplication To improv th fficincy in numrical computation of th dual BEM, w will adopt th fast multipol mthod (FMM) to acclrat th spd of calculation This is du to th larg domain and full influnc matrix, and it tas a lot of CPU tim and mmory spac to obtain th influnc matrix To ovrcom th disadvantags, th FMM will b shown to rduc CPU tim and mmory rquirmnt from xponntial ordr to logarithmic ordr thus nabling us apply BEM to rally solv for larg-scal problms Th FMM was initially introducd by Rohlin [] Applications of FMM for BEM analysis hav bn usd by many rsarchrs in various filds of scinc and nginring [,,,,] W will adopt th concpt of FMM to acclrat th calculation of influnc matrix in th dual BEM By adopting th addition thorm, th four rnls in th dual formulation ar xpandd into dgnrat rnls whr th fild point and sourc point ar sparatd Th sparabl tchniqu can promot th fficincy in dtrmining th cofficints as shown in Fig 1, in a similar way of th fast Fourir transform (FFT) ovr th Fourir transform (FT) Th sourc point matrics dcomposd in th four influnc matrics ar similar to ach othr or only som combinations Thr ar many zros or th sam influnc cofficints in th fild point matrics dcomposd in th four influnc matrics Thrfor, w can avoid calculating rpatdly th sam trms Th sparabl tchniqu rducs th numbr of floatingpoint oprations from OððNÞ Þ to OðN log a ðnþþ: To acclrat th convrgnc in constructing th influnc matrix, th cntr of multipol is dsignd to locat on th cntr of ach boundary lmnt Th singular and hyprsingular intgrals ar transformd into th summability of divrgnt sris and rgular intgrals In this papr, th acoustic scattring of gnral structur with th Numann s boundary condition will b considrd Th Burton and Millr formulation by combining th dual boundary intgral quations will b utilizd to solv th xtrior acoustic problms for all wav numbrs in ordr to ovrcom th problm of fictitious frquncy Finally, th CPU tim and mmory rquirmnt will b calculatd using th FMM for th larg-scal problm Th numrical rsults will b compard with thos of convntional DBEM and analytical solutions Mathmatical formulation 1 Hlmholtz quation in xtrior acoustics Lt D, D d b an unboundd rgion, whr d is th numbr of spac dimnsions, d can b 1, or Th boundary of th domain D; dnotd by B; is intrnal and assumd picwis smooth Th outward unit vctor normal to B is dnotd by n: W assum that th boundary, B; admits th partition [,] B ¼ B g < B h ; ð1þ B ¼ B g > B h ; whr B g is th ssntial boundary with spcifid potntial and B h is th natural boundary with spcifid normal drivativ of potntial W intnd to study th ffcts of small disturbanc to a givn bacground flow in such a rgion, undr th usual assumptions that lads to th quations of acoustics Harmonic analysis lads to a boundary-valu problm for th Hlmholtz quation (or rducd wav quation): Find u in th xtrior domain, th spatial componnt for th acoustic prssur or vlocity potntial [,], such that LuðxÞ ¼f ; x in D; ðþ uðxþ ¼g; x on B g ; ðþ uðxþ ¼ ih; x on B n h ; ðþ lim R 1 ðd1þ u R!1 R iu ¼ 0; R at infinity; ðþ whr Lu U u þ u is th Hlmholtz oprator, is th Laplacian oprator and $ 0 is th wav numbr; and in particular u= n U u n is th normal flux and is th gradint oprator; i ¼ 1; R is th distanc from th origin to th fild point In th linarizd quations of motion, vlocity gradints produc a comprssion of th acoustic mdium and prssur gradints ar rlatd to acclration Thus, if th dpndnt variabl is, g th acoustic prssur, thn th Numann boundary condition Eq () rprsnts a prscribd vlocity distribution on that portion of th wt surfac, whr h is proportional to th vlocity and th prsnc of i is a consqunc of diffrntiation with rspct to tim Numann boundary conditions ar thrfor vry common in physical situations that ntail radiation, and in th modl problms and dmonstrativ xampls that ar subsquntly considrd w mphasiz boundary conditions of this typ It is notd that th analysis is valid for any combination of boundary conditions on th wt surfac for th boundary-valu problm Eqs () (), and by no mans is it limitd to Numann problms For scattring problms, a fixd rigid objct is rprsntd by a homognous Numann boundary condition, oftn rfrrd to as a hard scattr Convrsly, th homognous Dirichlt boundary condition, an appropriat rprsntation ðþ

3 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig 1 Comparison of th schm in th calculation of influnc cofficints by using th convntional BEM and th FMM

4 88 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 of a sit of prssur rlas, is trmd a soft scattr An impdanc boundary condition is a linar combination of th two [,] Th govrning quation for an xtrior acoustic problm is th Hlmholtz quation as follows ð þ Þuðx 1 ; x Þ¼0; ðx 1 ; x Þ [ D; ðþ whr f in Eq () is zro (no sourcs), and is th wav numbr, which is th angular frquncy ovr th spd of sound Th boundary conditions can b ithr th Numann or Dirichlt typ Eq () stms from th Sommrfld radiation condition which allows only solutions with outgoing wavs at infinity to b admittd This boundary condition implis an intgral form, th Rllich Sommrfld radiation condition u ðb1 R iu db ¼ 0; ð8þ whr B 1 is th surfac of a sphr with an infinit radius Th radiation condition rquirs nrgy flux at infinity to b positiv, thrby guaranting that th solution to th boundary-valu problm in Eqs () () is uniqu Appropriat rprsntation of this condition is crucial to th rliability of any numrical formulation of th problm In this sction, th statmnt of problm follows th papr by Stwart and Hughs [] sinc thy prsnt a typical formulation Dual boundary intgral formulation Th first quation of th dual boundary intgral quations for th domain point can b drivd from th Grn s third idntity ð puð~xþ ¼ B ð Tð~s; ~xþuð~sþdbð~sþ Uð~s; ~xþ uð~sþ dbð~sþ; B n ~s ~x [ D; ð9þ whr ~x is th fild point ð~x ¼ðx; yþþ; ~s is th sourc point, and Uð~s; ~xþ ¼ pi ðl~s ~xlþ; ð10þ Hð1Þ 0 in which 0 ðl~s ~xlþ is th first ind zroth ordr Hanl function, and Tð~s; ~xþ is dfind by Uð~s; ~xþ Tð~s; ~xþ ; ; ð11þ n ~s in which n ~s dnots th normal vctor at th boundary point ~s; and Uð~s; ~xþ is th fundamntal solution which satisfis Uð~x; ~sþþ Uð~x; ~sþ ¼pdð~x ~sþ; ~x [ D: ð1þ In Eq (1), dð~x ~sþ is th Dirac-dlta function Aftr taing normal drivativ with rspct to Eq (9), th scond quation of th dual boundary intgral quations for th domain point is drivd p uð~xþ ð ð ¼ Mð~s; ~xþuð~sþdbð~sþ Lð~s; ~xþ uð~sþ dbð~sþ; n ~x B B n ~s ~x [ D; ð1þ whr Lð~s; ~xþ ; Uð~s; ~xþ n ~x ; ð1þ Mð~s; ~xþ ; Uð~s; ~xþ ; ð1þ n ~x n ~s in which n ~x rprsnts th normal vctor of ~x: Th xplicit forms for th four rnl functions ar shown in Tabl 1 Th boundary conditions can b ithr th Numann or Dirichlt typ By moving th point x to th boundary, th dual quations for th boundary points ar ð puð~xþ¼cpv Tð~s; ~xþuð~sþdbð~sþ B ð ð1þ RPV Uð~s; ~xþtð~sþdbð~sþ; ~x [ B; B Tabl 1 Th proprtis of th rnl functions for th Hlmhotz quation Krnl Kð~s; ~xþ Uð~s; ~xþ iphð1þ 0 ðlrþ Tð~s; ~xþ ilp Hð1Þ 1 ðlrþ y in i r Lð~s; ~xþ ilp Hð1Þ 1 ðlrþ y i n i r Mð~s; ~xþ ilp l ðlrþ r y i y j n i n j þ Hð1Þ 1 ðlrþ n r i n i Ordr of singularity Oðln rþ wa Oð1=rÞ strong Oð1=rÞ strong Oð1=r Þ hyprsingular Symmtry Uð~s; ~xþ ¼Uð~x; ~sþ Tð~s; ~xþ ¼Lð~x; ~sþ Lð~s; ~xþ ¼Tð~x; ~sþ Mð~s; ~xþ ¼Mð~x; ~sþ Dnsity function yð~sþ u= n u u= n u Potntial typ Singl layr Doubl layr Normal drivativ Normal drivativ of doubl layr Ð of singl layr Kð~s; ~xþyð~sþdbð~sþ Continuous Discontinuous Discontinuous Psudo-continuous continuity across boundary Jump valu No jump pu pð u= nþ No jump Principal valu Rimann Cauchy Cauchy Hadamard Whr Hn ð1þ ðlrþ is first ind of th nth ordr Hanl function, n i dnots th ith componnts of normal vctor on x; rspctivly

5 ð ð ptð~xþ¼hpv Mð~s; ~xþuð~sþdbð~sþ CPV Lð~s; ~xþtð~sþdbð~sþ; B B ~x [ B; ð1þ whr CPV; RPV and HPV dnot th Cauchy principal valu, th Rimann principal valu and th Hadamard principal valu, tð~sþ¼ uð~sþ= n ~s ; B dnots th boundary nclosing D: Th linar algbraic quations discrtizd from th dual boundary intgral quations can b writtn as ½TŠ{u} ¼½UŠ{t}; ð18þ ½MŠ{u} ¼½LŠ{t}; ð19þ whr {u} and {t} ar th boundary potntial and flux, rspctivly Th influnc cofficints of th four squar matrics ½UŠ; ½TŠ; ½LŠ and ½MŠ can b rprsntd as ð U pq ¼ RPV Uðs q ;x p ÞdBðs q Þ; B q ð0þ ð T pq ¼ pd pq þ CPV Tðs q ;x p ÞdBðs q Þ; ð1þ B q ð L pq ¼ pd pq þ CPV Lðs q ;x p ÞdBðs q Þ; B q ðþ ð M pq ¼ HPV Mðs q ;x p ÞdBðs q Þ; B q ðþ whr th subscript p dnots th labl on collocation point, B q dnots th qth lmnt and d pq ¼ 1 if p ¼ q; othrwis it is zro In ordr to ovrcom th problm of fictitious frquncy, th Burton and Millr formulation [] is mployd by combining th dual quations as follows ½TŠþ i ½MŠ {u} ¼ ½UŠþ i ½LŠ {t}: ðþ For all wav numbrs, Eq () can wor wll [] Expanding th four rnls using th multipol xpansion mthod By adopting th addition thorm, th four rnls in th dual formulation ar xpandd into dgnrat rnls which sparat th fild point and sourc point [1] Th rnl function, Uð~s; ~xþ; can b xpandd into Uð~s; ~xþ¼ X >< C m ð~xþr m ð~sþ¼ JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) >: U i ¼ pi U ¼ pi and a ¼ cos 1 ð~s ~pþð~x ~pþ lð~s ~pþllð~x ~pþl : ðþ Th dfinition stch of th coordinat is shown in Fig Th contour plot of potntial for th U rnl can b shown in Fig (a) for th sris form using th dgnrat rnl in Eq () and Fig (b) for th closd-form fundamntal solution of Eq (10) Th rnl function, Tð~s; ~xþ; can b xpandd into Tð~s;~xÞ¼ X1 C m ð~xþ½r m ð~sþ n s Š whr 8 >< ¼ >: ( Hm ð1þ ðl~s ~plþ 1 m J m ðl~x ~plþ cosðmaþ n s þhm ð1þ ðl~s ~plþ cosðmaþ ) ; l~s ~pll~x ~pl; n s T i ¼ pi T ¼ pi 1 m Hm ð1þ ðl~x ~plþ J mðl~s ~plþ n s ; cosðmaþþj m ðl~s ~plþ cosðmaþ n s l~x ~pll~s ~pl; J m ðl~s ~plþ n s ¼ ½J m1ðl~s ~plþj mþ1 ðl~s ~plþš ð8þ ðs ip i Þn i ; ð9þ l~s ~pl Hm ð1þ ðl~s ~plþ ¼ n ~s ½Hð1Þ m1ðl~s ~plþhð1þ mþ1ðl~s ~plþš ðs ip i Þn i ; ð0þ l~s ~pl cosðmaþ ¼msinðmaÞða n i n i Þ; ð1þ s in which n i is th ith componnt of th normal vctor at ~s and a 1 ¼ 1 ðs p Þ ðx 1 p 1 Þðs 1 p 1 Þðs p Þðx p Þ sinðaþ l~s ~pl ; l~x ~pl ðþ 1 m J m ðl~x ~plþ m ðl~s ~plþcosðmaþ; l~s ~pl l~x ~pl; 1 m m ðl~x ~plþj m ðl~s ~plþcosðmaþ; l~x ~pl l~s ~pl; ðþ whr i ¼ 1; J m ðsþ is th first ind mth ordr Bssl function, ~p is th cntr of multipol ( 1 m ¼ 1; m ¼ 0 ) ; ðþ ; m 0 a ¼ 1 ðs 1 p 1 Þ ðx p Þðs 1 p 1 Þðs p Þðx 1 p 1 Þ sinðaþ l~s ~pl ; l~x ~pl ðþ By substituting Eqs (9) (1) into Eq (8), w hav

6 90 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) >< Tð~s; ~xþ >: T i ¼ pi T ¼ pi Fig Th dfinition stch of th coordinat, coordinat transformation and th position for th cntr of multipol 1 m J m ðl~x ~plþ ½Hð1Þ m1ðl~s ~plþ Hð1Þ mþ1 ðl~s ~plþš ðs i p i Þn i cosðmaþþhm ð1þ ðl~s ~plþ l~s ~pl ½m sinðmaþða i n i ÞŠ ; l~s ~pl l~x ~pl; 1 m Hm ð1þ ðl~x ~plþ ½J m1ðl~s ~plþ J mþ1 ðl~s ~plþš ðs i p i Þn i cosðmaþþj l~s m ðl~s ~plþ ~pl ½m sinðmaþða i n i ÞŠ ; l~x ~pl l~s ~pl: ðþ

7 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig Th contour plot of potntial for U rnl (a) Dgnrat rnl of Eq (), (b) Closd-form fundamntal solution of Eq (10) Th rnl function, Lð~s; ~xþ; can b xpandd into Lð~s; ~xþ ¼ X1 8 >< ¼ >: ½C m ð~xþ n x ŠR m ð~sþ L i ¼ pi L ¼ pi 1 m Hm ð1þ J ðl~s ~plþ m ðl~x ~plþ cosðmaþþj n m ðl~x ~plþ cosðmaþ ; x n x l~s ~pl l~x ~pl; ( Hm ð1þ ðl~x ~plþ 1 m J m ðl~s ~plþ cosðmaþþhm ð1þ ðl~x ~plþ cosðmaþ ) ; n x n x l~x ~pl l~s ~pl; ðþ

8 9 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 whr J m ðl~x ~plþ ¼ n x ½J m1ðl~x ~plþj mþ1 ðl~x ~plþš ðx ip i Þn i ; l~x ~pl ðþ Hm ð1þ ðl~x ~plþ ¼ n x ½Hð1Þ m1 ðl~x~plþhð1þ mþ1 ðl~x~plþšðx ip i Þn i ; l~x ~pl ðþ cosðmaþ ¼msinðmaÞðb n i n i Þ; ð8þ x in which n i is th ith componnt of th normal vctor at ~x and b 1 ¼ 1 ðs 1 p 1 Þðx p Þ ðs p Þðx 1 p 1 Þðx p Þ sinðaþ l~x ~pl ; l~s ~pl ð9þ b ¼ 1 ðs p Þðx 1 p 1 Þ ðs 1 p 1 Þðx 1 p 1 Þðx p Þ sinðaþ l~x ~pl : l~s ~pl ð0þ By substituting Eqs () (8) into Eq (), w hav 8 >< Lð~s; ~xþ >: L i ¼ pi whr cosðmaþ n x n s ¼ ½msinðmaÞa in i Š n x ¼a i n i ½msinðmaÞŠ n x þðmsinðmaþþ a in i n x ¼a i n i ½m cosðmaþb i n i Šþ½msinðmaÞŠ ( " ðs n p Þ n 1 1 l~s~pl ðx 1p 1 Þn # 1 x p ðx p Þ " ðs þn 1 p 1 Þðs p Þ n 1 l~s~pl ðx 1p 1 Þn #) 1 x p ðx p Þ : ðþ By substituting Eqs (9) (1), () (8) and () into Eq (), w hav 1 m Hm ð1þ ðl~s ~plþ ½J m1ðl~x ~plþ J mþ1 ðl~x ~plþš ðx i p i Þn i cosðmaþþj l~x m ðl~x ~plþ ~pl ½m sinðmaþðb i n i ÞŠ ; l~s ~pl l~x ~pl; L ¼ pi 1 m J m ðl~s ~plþ ½Hð1Þ m1ðl~x ~plþ Hð1Þ mþ1 ðl~x ~plþš ðx i p i Þn i l~x ~pl ½m sinðmaþðb i n i ÞŠ ; l~x ~pl l~s ~pl: cosðmaþþ m ðl~x ~plþ ð1þ Th rnl function, Mð~s; ~xþ; can b xpandd into 8 M i ¼ pi Mð~s; ~xþ ¼ X1 >< ½C m ð~xþ n x Š½R m ð~sþ n s Š¼ >: m ðl~s ~plþ J 1 m ðl~x ~plþ m cosðmaþ n s n x þj m ðl~x ~plþ cosðmaþ ( þ 1 n m Hm ð1þ J ðl~s ~plþ m ðl~x ~plþ cosðmaþ x n x n s ) þj m ðl~x ~plþ cosðmaþ ; l~s ~pl l~x ~pl; n x n s M ¼ pi cosðmaþ n x ( J 1 m ðl~s ~plþ Hm ð1þ ðl~x ~plþ m cosðmaþþhm ð1þ ðl~x ~plþ n s n x ) þ 1 m J m ðl~s ~plþ ( Hm ð1þ ðl~x ~plþ cosðmaþ n x n s ) þhm ð1þ ðl~x ~plþ cosðmaþ ; l~x ~pl l~s ~pl; n x n s ðþ

9 8 M i ¼ pi 1 m ðhð1þ m1ðl~s ~plþ Hð1Þ mþ1 ðl~s ~plþþ ðs ( i p i Þn i l~s ~pl >< l~s ~pl l~x ~pl; Mð~s; ~xþ M ¼ pi 1 m ðj m1ðl~s ~plþ J mþ1 ðl~s ~plþþ ðs ( " i p i Þn i l~s ~pl >: ðx i p i Þn i l~x ~pl " J m1 ðl~x ~plþ J mþ1 ðl~x ~plþ ( cosðmaþþj m ðl~x ~plþ½m sinðmaþðb i n i ÞŠ þ 1 m Hm ð1þ ðl~s ~plþ ðj m1ðl~x ~plþ J mþ1 ðl~x ~plþþ ðx i p i Þn i ½m sinðmaþða l~x i n i ÞŠ þ J m ðl~x ~plþ½a i n i ðm cosðmaþb i n i Þþðm sinðmaþþ ~pl " " # " # #) n 1 þ n ; ðs p Þ n 1 l~s ~pl ðx 1 p 1 Þn 1 x p ðx p Þ ðx i p i Þn i l~x ~pl ðs 1 p 1 Þðs p Þ n 1 l~s ~pl ðx 1 p 1 Þn 1 x p ðx p Þ Hð1Þ m1 ðl~x ~plþ Hð1Þ mþ1 ðl~x ~plþ ( cosðmaþþhm ð1þ ðl~x ~plþ½m sinðmaþðb i n i ÞŠ þ 1 m J m ðl~s ~plþ ðhð1þ m1ðl~x ~plþ mþ1 ðl~x ~plþþ ðx " i p i Þn i ½m sinðmaþða l~x i n i ÞŠ þ Hm ð1þ ðl~x ~plþ a i n i ðm cosðmaþb i n i Þþðm sinðmaþþ ~pl " " ðs n p Þ n 1 1 l~s ~pl ðx 1 p 1 Þn # " 1 ðs x p ðx p Þ þ n 1 p 1 Þðs p Þ n 1 l~s ~pl ðx!###) 1 p 1 Þn 1 x p ðx p ; l~x ~pl l~s ~pl: JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) ðþ Dual boundary lmnt formulation in conjunction with th FMM By mploying th constant lmnt schm aftr coordinat transformation and moving th cntr of multipol ð~pþ to th cntr of local coordinat on ach boundary lmnt as shown in Fig, ach lmnt of th influnc matrics can b obtaind as follows 1 U rnl For th rgular intgral ði jþ; w hav (a) r i;j 0:l j whr r i;j is th distanc btwn th collocation point on ith plmnt s ffiffiffiffiffiffiffiffi cntr and th jth sourc lmnt s cntr, r i;j ¼ x r þ y r ; x r and y r ar th coordinats of collocation point aftr translation and rotation, l j is th lngth of th jth sourc lmnt Th multipol momnt R m;j is th valu rlatd to th sourc point coordinat and Ci;j;m 1 is th valu rlatd to th fild point coordinat as shown blow: Ci;j;m 1 ¼ pi 1 m m ðr i;jþcosðmaþ; ðþ R m;j ¼ n¼0 J mþnþ1 Þ: ðþ (b) r i;j, 0:l j U i;j ¼ U ds ¼ pi 0:l j J m ðlslþds 0:l j ¼ pi ¼ X1 C 1 i;j;mr m;j ; 1 m m ðr i;jþcosðmaþ 1 m m ðr i;jþcosðmaþ! J mþnþ1 Þ n¼0 ðþ ðr i;j U i;j ¼ U i ðr i;j dsþ U dsþ U i ds 0:l j r i;j r i;j ¼ pi 1 m m ðr i;jþcosðmaþ pi n¼0 J mþnþ1 ðr i;j Þ 1 m J m ðr i;j ÞcosðmaÞ m ðlslþds: r i;j! ð8þ For th waly singular intgral ði ¼ jþ; w rgulariz th intgral by mans of partial intgration and limiting procss ððx r ; y r Þ¼ð0; ÞÞ as follows

10 9 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 ð U i;i ¼ lim U i ð ds þ U ds þ U i ds!0 0:l j ¼ pi ( 1 m n¼1 m ðþcosðpþ X J mþnþ1 ðþþ pi!) 1 m J m ðþcosðpþ Hm ð1þ ðlslþds n¼0 ¼ 0 þ pi! ð0:l J j 0ðÞ 0 ðlslþds ¼ pi ¼ ip 0 l l þ 1 0:l j 0:l j 0 ðlslþds þ ð 0 ðlslþds! ðlslþlsldsþ; ð9þ whr ð lim!0 ð 0 ðlslþds ¼ lim!0 i lnðsþds ¼ 0: p ð0þ T rnl For th rgular intgral ði jþ; w hav (a) r i;j 0:l For th strongly singular intgral ði¼jþ; w rgulariz th intgral by mans of partial intgration, limiting procss and th idntitis from th gnralizd function as shown blow [19] sin m p ¼ p m : ðþ m¼1 W can obtain th intgral as follows T i;j ¼ T ds ¼ pi 0:l j ¼ pi X1 pi 1 m ð0:l m ðr j i;jþcosðmaþ 0:l j ½J m1ðlslþ J mþ1 ðlslþš ð0:l mþ1 ðr j i;jþðm þ 1Þsinððm þ 1ÞaÞ mþ1 ðr i;jþðm þ 1Þsinððm þ 1ÞaÞ m þ J mþ1 ðlslþ 0:l j lsl s 0 þ 0 ð1þ ds lsl 0:l j ½J m ðlslþþj mþ ðlslþšds ¼ C i;j;m½r m;j þ R ðmþ1þ;j Š; ds ð1þ whr C i;j;m¼pi (b) r i;j,0:l mþ1 ðr i;jþðmþ1þsinððmþ1þaþ mþ : ðþ ðr i;j T i;j ¼ T i ðr i;s ds þ T ds þ T i ds ¼ pi X1 ðr mþ1 ðr i;j i;jþðm þ 1Þsinððm þ 1ÞaÞ 0:l j r i;j r i;j r i;j þ pi X1 J mþ1 ðr i;j Þðm þ 1Þsinððm þ 1ÞaÞðÞ r i;j mþ1 ðlslþ lsl ds ¼ pi X1 ðr i;j ½J m þ m ðlslþþj mþ ðlslþšds þ pi X1 J mþ1 ðr i;j Þðm þ 1Þsinððm þ 1ÞaÞðÞ r i;j þ mþ ðlslþšds: J mþ1 ðlslþ ds lsl mþ1 ðr i;jþðm þ 1Þsinððm þ 1ÞaÞ m þ r i;j ½ m ðlslþ ðþ

11 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) ð T i;i ¼ T i ð ds þ T ds þ T i ds ¼ pi X1 0:l j þ pi X1 p ð ffi J mþ1 ðþðm þ 1Þsin ðm þ 1Þ p 1 X1 ds ¼ sm ð1þ m m þ 1 þ X1 mþ1 ðr i;jþðm þ 1Þsin ðm þ 1Þ p ð ð0:lj ð1þ m m þ 1 ¼ X1 mþ1 ðlslþ ds ¼ X1 lsl sin m p þ X1 m m¼1 m¼1 ð1þ m mþ1 ð sin m p m 0 J mþ1 ðlslþ ds lsl s m ds þ X1 ¼ p; ð1þ m mþ1 ðþ whr limj mþ1 ðþ!0 ¼ pffi lim!0 pffi mþ1 ðlslþ lsl ds ðmþ1þðþ mþ1 mþ1 ðmþ1þ! mþ1 ðlslþ lsl L rnl For th rgular intgral ði jþ; w hav (a) r i;j 0:l L i;j ¼ L ds ¼ pi 0:l j whr ds¼0: 1 m ½Hð1Þ m1 ðr i;jþ mþ1 ðr i;jþš x ð0:l i n j i cosðmaþ J r m ðlslþds i;j 0:l j pi 1 m m ðr i;jþðmþsinðmaþ y r n 1 x r n r i;j J m ðlslþds ¼ X1 0:l j C i;j;m ¼ pi 1 m ( C i;j;mr m;j ; ½Hð1Þ m1 ðr i;jþ mþ1 ðr i;jþš x i n i r i;j ðþ ðþ cosðmaþþ m ðr i;jþðmþsinðmaþ y r n 1 x r n ri;j : ð8þ (b) r i;j, 0:l L i;j ¼ ðr i;j ¼ pi 0:l j L i dsþ ðr i;j r i;j L dsþ r i;j L i ds 1 m ½Hð1Þ m1 ðr i;jþ mþ1 ðr i;jþš x i n i r i;j cosðmaþ J m ðlslþds pi 0:l j y r n 1 x ð0:l r n j ri;j J m ðlslþds pi 0:l j ) 1 m m ðr i;jþðmþsinðmaþ 1 m ½J m1ðr i;j Þ J mþ1 ðr i;j ÞŠ x! ð0:l i n j i cosðmaþ m r ðlsldsþ i;j r i;j pi 1 m J m ðr i;j ÞðmÞsinðmaÞ y r n 1 x r n m!: ðlslþds ð9þ r i;j r i;j For th strongly singular intgral ði¼jþ; w rgulariz th intgral by mans of partial intgration and limiting procss as follows ð L i;i ¼ L i ð dsþ L dsþ L i ds 0:l j ¼ pi 1 m ½Hð1Þ m1 ðllþhð1þ mþ1 ðllþšð1þm ð J m ðlslþdsþ pi! J mþ1 ðllþšð1þ m m ðlslþds ¼ X1 ð1þ m ð m¼1 mþ1 0 m¼1 s m dsþ X1 1 m ½J m1ðllþ m¼1 p ffi ð1þ m m1ð m¼1 s m ds ¼ X1 ð1þ m X1 ð1þ m mþ1 1m m¼1 m¼1 ¼ X1 sinðm p Þ 1 X1 sinðm p Þ ¼p; m m whr lim!0 ½J m1ðllþj mþ1 ðllþš pffi Hð1Þ m ðlslþds¼0: M rnl For th rgular intgral ði jþ; w hav (a) r i;j 0:l M i;j ¼ M ds ¼ pi 0:l j whr ðþ ½Hð1Þ m ðr i;jþ mþ ðr i;jþš x i n i r i;j ½ðm þ 1Þsinððm þ 1ÞaÞŠ J m ðlslþ 0:l j lsl ds þ pi ðm þ 1Þcosððm þ 1ÞaÞ y r n 1 x r n r i;j J m ðlslþ 0:l j lsl ds ¼ X1 ðþ mþ1 ðr i;jþ C i;j;m½r m;j þ R ðmþ1þ;j Š; ð0þ ð1þ ðþ

12 9 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Ci;j;m ¼ pi ri;j ðm þ 1Þ y r n 1 x r n r i;j ( ðr i;jþ½ m ðr i;jþ mþ ðr i;jþš x i n i sinððm þ 1ÞaÞþ mþ1 r ðr i;jþðm þ 1Þcosððm þ 1ÞaÞ i;j ) : ðþ (b) r i;j, 0:l ðr i;j M i;j ¼ M i ðr i;j ds þ M ds þ M i ds ¼ pi 0:l j r i;j r i;j ri;j þ mþ1 ðr i;jþðm þ 1Þcosððm þ 1ÞaÞy r n 1 x r n For th hyprsingular intgral ði ¼ jþ; w rgulariz th intgral by mans of partial intgration, limiting procss and using th idntitis from th gnralizd function as shown blow [19] ð1þ m ¼ 1 : ðþ ðm þ 1Þ ðr i;jþ½ m ðr i;jþ mþ ðr i;jþšx i n i sinððm þ 1ÞaÞ m þ ðr i;j r i;j ½J m ðlslþþj mþ ðlslþšds þ pi r i;j ðr i;jþ½j m ðr i;j Þ J mþ ðr i;j ÞŠx i n i sinððm þ 1ÞaÞþJ mþ1 ðr i;j Þðm þ 1Þcosððm þ 1ÞaÞy r n 1 x r n ½ m m þ ðlslþþhð1þ mþ ðlslþšds: r i;j W can obtain th intgral as follows: ( ð M i;i ¼ M i ð ds þ M ds þ M i ds ¼ ip 0:l j ) mþ1 ðlslþ lsl ds þ X1 m¼1 #) þ 1 ðlslþlslds 0:l þ #) þ 1 ðlslþlslds 0:l ðm þ 1Þð1Þ m ¼ ip 1 mþ m¼1 ð 0 ð1þ m1 þ l ip It is intrsting to find that R m;j trm is mbddd in th formula of th four influnc matrics of Eqs (), (1), () and () Construction of th four influnc matrics By using Eqs () and (1), th algbraic systm UT quation of th dual boundary intgral formulation in Eq (18) can b rwrittn as 1 ðlslþ ds þ X1 J lsl m ðþðm þ 1Þð1Þ m m¼1 s m ds ¼ ip ( " l 1 þ l 0 ð1þ m ¼ ip ( " l 1 þ 0 ðm þ 1Þ l ðþ U i;i: ðþ 8 0 C1;;mðR m; þ R ðmþ1þ; Þ C1;N;mðR 98 9 m;n þ R ðmþ1þ;n Þ u 1 C;1;mðR m;1 þ R ðmþ1þ;1 Þ 0 C;N;mðR >< m;n þ R ðmþ1þ;n Þ >= >< u >= þ pi >: >; >: CN;1;mðR m;1 þ R ðmþ1þ;1 Þ CN;;mðR m; þ R ðmþ1þ; Þ 0 u >; N 8 0 C1;;mR 1 m; C1;N;mR m;n t 1 ¼ X1 C;1;mR 1 m;1 0 C;N;mR 1 >< m;n >= >< t >= þ½diagðu i;i ÞŠ : ðþ >: >; >: CN;1;mR 1 m;1 CN;;mR 1 m; 0 t >; N

13 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) By using Eqs () and (), th algbraic systm of th LM quation of th dual boundary intgral formulation of Eq (19) can b rwrittn as 8 0 C1;;mðR m; þ R ðmþ1þ; Þ C1;N;mðR 98 9 m;n þ R ðmþ1þ;n Þ u 1 C;1;mðR m;1 þ R ðmþ1þ;1 Þ 0 C;N;mðR >< m;n þ R ðmþ1þ;n Þ >= >< u >= þ½diagðm i;i ÞŠ >: >; >: CN;1;mðR m;1 þ R ðmþ1þ;1 Þ CN;;mðR m; þ R ðmþ1þ; Þ 0 u >; N 8 0 C1;;mR m; C1;N;mR 98 9 m;n t 1 ¼ X1 C;1;mR m;1 0 C;N;mR >< m;n >= >< t >= þ p½iš : ð8þ >: >; >: CN;1;mR m;1 CN;;mR m; 0 t >; N whr I is th unit matrix By adopting th M þ 1 trms in th sris sum, th four influnc matrics can b rwrittn as R 0;1 0 0 C1;;0 1 C1;;1 1 C 1 1;;M C;1;0 1 C;1;1 1 C;1;M 1 R 1; ½UŠ ¼ þ CN;1;0 1 CN;1;1 1 CN;1;M 1 N ðmþ1þ R M;1 0 0 ðmþ1þ N CN;;0 1 CN;;1 1 CN;;M 1 0 R 0; 0 C1;N;0 1 C1;N;1 1 C 1 1;N;M 0 0 R 0;N 0 R 1; 0 C;N;0 1 C;N;1 1 C 1 ;N;M 0 0 R 1;N þ þ 0 R M; R M;N ðmþ1þ N þ½diagðu i;i ÞŠ N N ; C;1;0 C;1;1 C;1;M ½TŠ¼ CN;1;0 CN;1;1 CN;1;M 0 ðr 0; þr 1; Þ 0 0 ðr 1; þr ; Þ 0 0 ðr M; þr ðmþ1þ; Þ 0 N ðmþ1þ ðmþ1þ N N ðmþ1þ ðr 0;1 þr 1;1 Þ 0 0 C1;;0 C1;;1 C 1;;M ðr 1;1 þr ;1 Þ þ ðr M;1 þr ðmþ1þ;1 Þ 0 0 ðmþ1þ N CN;;0 CN;;1 CN;;M C1;N;0 C1;N;1 C1;N;M 00 ðr 0;N þr 1;N Þ C;N;0 C;N;1 C;N;M 00 ðr 1;N þr ;N Þ þ þ ðr M;N þr ðmþ1þ;n Þ N ðmþ1þ N ðmþ1þ N ðmþ1þ ðmþ1þ N ðmþ1þ N ð9þ þp½iš; ð0þ

14 98 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) C;1;0 C;1;1 C;1;M ½LŠ ¼ CN;1;0 CN;1;1 CN;1;M C1;;0 C1;;1 C1;;M þ CN;;0 CN;;1 CN;;M N ðmþ1þ N ðmþ1þ R 0;1 0 0 R 1;1 0 0 R M; R 0; 0 0 R 1; 0 0 R M; 0 ðmþ1þ N : ðmþ1þ N C1;N;0 C1;N;1 C 1;N;M 0 0 R 0;N C;N;0 C;N;1 C;N;M 0 0 R 1;N þ þ þp½iš; N ðmþ1þ 0 0 R M;N ðmþ1þ N ðr 0;1 þ R 1;1 Þ 0 0 C;1;0 C;1;1 C;1;M ðr 1;1 þ R ;1 Þ 0 0 ½MŠ ¼ : CN;1;0 CN;1;1 CN;1;M N ðmþ1þ ðr M;1 þ R ðmþ1þ;1 Þ 0 0 ðmþ1þ N C1;;0 C1;;1 C1;;M 0 ðr 0; þ R 1; Þ ðr 1; þ R ; Þ 0 þ CN;;0 CN;;1 CN;;M 0 ðr M; þ R ðmþ1þ; Þ 0 N ðmþ1þ C1;N;0 C1;N;1 C 1;N;M C;N;0 C;N;1 C;N;M þ þ N ðmþ1þ : ðmþ1þ N 0 0 ðr 0;N þ R 1;N Þ 0 0 ðr 1;N þ R ;N Þ 0 0 ðr M;N þ R ðmþ1þ;n Þ ðmþ1þ N þ½diagðm i;i ÞŠ N N : ð1þ ðþ It is intrsting that th four influnc matrics in th dual BEM ar all composd of th fild point matrics and th sourc point matrics Th sparabl tchniqu can promot th fficincy in dtrming th influnc cofficints Th sourc point matrics of ½UŠ ar all th sam with ½LŠ; whil th sourc point matrics of ½TŠ ar all th sam with ½MŠ: Bsids, many influnc cofficints in th sourc point matrics of ½TŠ and ½MŠ hav th sam valu with ½UŠ and ½LŠ; or with only som combinations Thr ar many zros or th sam influnc cofficints in th fild point matrics dcomposd in th four influnc matrics Thrfor, w can avoid calculating rpatdly th sam trm Th sparabl tchniqu rducs th numbr of floating-point oprations from OðN Þ to OðN log a ðnþþ: Thus, larg computation tim savings ar achivd and mmory rquirmnts ar rducd, thus nabling us apply BEM to solv larg-scal problms Illustrativ xampls To dmonstrat th validity of th dual intgral formulation in conjunction with th FMM, two xampls for scattring problm by an infinit cylindr with radius ðaþ subjct to th Numann boundary condition ar givn as follows

15 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig Th scattring problm for a cylindr with th Numann boundary condition Exampl 1 Th radius, a; is m and th wav numbr of incidnt wav, ; is p: In th Exampl 1, w solv th problm by applying th LU dcomposition in th dvlopd program and compar with th analytical solution and th convntional dual BEM Th problm was chosn bcaus th analytical solution is nown [,] It is thrfor a good modl problm to tst th accuracy of DBEM by mploying th concpt of FMM Th xampl is shown in Fig and th analytical solution is uðr; uþ ¼ J0 X1 0ðaÞ 0 0 ðaþ Hð1Þ 0 ðrþ i n n¼1 J 0 nðaþ Hn ð1þ0 ðaþ n ðrþcosðnuþ; ðþ whr i ¼ 1: Fig shows th contour plots of th ralpart solutions for th cas of a ¼ 0p: Th unnown boundary solutions of scattring fild using th boundary msh of 100 lmnts, RðuÞ and ImðuÞ; ar plottd in Figs and and 800 lmnts in Figs 8 and 9 Solution using th uniform msh rfinmnt of 800 lmnts convrgs to th xact solution By adopting only four momnt FMM formulation, th rsults ar compard wll with thos of FEM, convntional BEM and analytical solutions Comparison of rror norms for th FMM rsults vrsus diffrnt trms in th sris is shown in Fig 10 Fig 11 shows th rror norms against diffrnt mshs Only a fw trms in th FMM can rach within th rror tolranc Comparison of CPU tim using th FMM with diffrnt trms ar plottd in Fig 1 Fig 1 shows th CPU tim vrsus diffrnt mshs Th trnd of CPU tim in proportional to N and N log : N is found for th convntional BEM and th FMM, rspctivly Exampl Th radius, a; is 0 m and th wav numbr of incidnt wav, ; is p Fig 1 shows th contour plots of th ral-part solutions for th cas of a ¼ 0p: Th unnown boundary solution of scattring fild using th msh of 00 lmnts, RðuÞ and ImðuÞ; ar plottd in Figs 1 and 1 and in Figs 1 and 18 using 1100 lmnts Solution using th uniform msh rfinmnt of 1100 lmnts convrgs to th xact solution By adopting Fig Th contour plot of th ral-part solutions in th cas 1 (a) Exact solution, (b) FMM rsults ðm ¼ Þ: only thr momnt FMM formulation, th rsults ar compard wll with thos of FEM, convntional BEM and analytical solutions Comparison of rror norms for th FMM rsults vrsus diffrnt trms in th sris is shown in Fig 19 Fig 0 shows th rror norms against diffrnt mshs Only a fw trms in th FMM can rach within th rror tolranc Comparison of CPU tim using th FMM with diffrnt trms ar plottd in Fig 1 Fig shows th CPU tim vrsus diffrnt mshs Th FMM can rduc CPU tim thus nabling us apply BEM to solv for

16 00 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Fig Th boundary solution of th scattring fild using 100 boundary lmnts, RðuÞ; for th cas 1 (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ: Fig Th boundary solution of scattring fild using 100 boundary lmnts, ImðuÞ; for th cas 1 (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ:

17 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig 8 Th boundary solution of scattring fild using 800 boundary lmnts, RðuÞ; for th cas 1 (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ: Fig 9 Th boundary solution of scattring fild using 800 boundary lmnts, ImðuÞ; for th cas 1 (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ:

18 0 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Fig 10 Comparison of th rror norms for th FMM rsults vrsus M in th sris for th cas 1 (a) UT quation (b) LM quation (c) Burton and Millr mthod Fig 11 Comparison of th rror norms for th FMM ðm ¼ Þ and th convntional BEM rsults vrsus numbr of lmnts for th cas 1 (a) UT quation, (b) LM quation, (c) Burton and Millr mthod

19 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig 1 CPU tim by using th FMM vrsus M in th sris for th cas 1 Fig 1 CPU tim by using th FMM ðm ¼ Þ and th convntional DBEM vrsus numbr of lmnts for th cas 1

20 0 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Fig 1 Th contour plot of th ral-part solutions in th cas (a) Exact solution, (b) FMM rsults ðm ¼ Þ:

21 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig 1 Th boundary solution of scattring fild using 00 boundary lmnts, RðuÞ; for th cas (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ: Fig 1 Th boundary solution of scattring fild using 00 boundary lmnts, ImðuÞ; for th cas (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ:

22 0 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Fig 1 Th boundary solution of scattring fild using 100 boundary lmnts, RðuÞ; for cas (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ: Fig 18 Th boundary solution of scattring fild using 100 boundary lmnts, ImðuÞ; for th cas (a) UT quation of FMM ðm ¼ Þ; (b) LM quation of FMM ðm ¼ Þ; (c) Burton and Millr mthod of FMM ðm ¼ Þ:

23 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) Fig 19 Comparison of th rror norms for th FMM rsults vrsus M in th sris for th cas (a) UT quation, (b) LM quation, (c) Burton and Millr mthod Fig 0 Comparison of th rror norms for th FMM ðm ¼ Þ and th convntional BEM rsults vrsus numbr of lmnts for th cas (a) UT quation, (b) LM quation, (c) Burton and Millr mthod

24 08 JT Chn, KH Chn / Enginring Analysis with Boundary Elmnts 8 (00) 8 09 Millr formulation by combining th dual boundary intgral quations was utilizd to solv th xtrior acoustic problms for all wav numbrs in ordr to ovrcom th problm of fictitious frquncy Two illustrativ xampls hav bn succssfully dmonstratd by using th FMM for DBEM formulation in th xtrior acoustic problms containing a larg-scal scattr Th numrical rsults wr compard wll with thos of convntional DBEM and analytical solutions Only a fw trms in FMM can rach within th rror tolranc In addition, th CPU tim was rducd in comparison with BEM without mploying FMM concpt Acnowldgmnts Fig 1 CPU tim by using th FMM vrsus M in th sris for th cas Financial support from th National Scinc Council, Grant No NSC E and Wu, Ta-You Award for National Taiwan Ocan Univrsity to th first author, and th Sinotch Foundation for Rsarch and Dvlopmnt to th scond author ar gratfully acnowldgd Rfrncs Fig CPU tim by using th FMM ðm ¼ Þ and th convntional DBEM vrsus numbr of lmnts for th cas a larg-scal problm Th sam trnd of CPU tim in comparison with Fig 1 is obsrvd Conclusions In this papr, th four rnls in th dual formulation wr xpandd into dgnrat rnls whr th fild point and sourc point wr sparatd Th sparabl tchniqu promotd th fficincy in dtrmining th influnc cofficints Th singular and hyprsingular intgrals hav bn transformd into th summability of divrgnt sris and rgular intgrals Th Burton and [1] Abramowitz M, Stgun IA Handboo of mathmatical functions with formulation, graphs and mathmatical tabls Nw Yor: Dovr; 19 [] Amini S, Profit ATJ Analysis of th truncation rrors in th fast multipol mthod for scattring problms J Comput Appl Math 000; 11: 0 [] Amini S, Profit ATJ Analysis of a diagonal form of th fast multipol algorithm for scattring thory BIT Numr Math 1999;9(): 8 0 [] Amini S On th choic of th coupling paramtr in boundary intgral formulation for th xtrior acoustic problm Applic Anal 1990;: 9 [] Burton AJ, Millr GF Th application of intgral quation mthods to numrical solution of som xtrior boundary valu problms Proc R Soc Lond, Sr A 191;:01 10 [] Chn IL, Chn JT, Liang MT Analytical study and numrical xprimnts for radiation and scattring problms using th CHIEF mthod J Sound Vib 001;8():809 8 [] Chn JT, Chn KH, Chn CT Adaptiv boundary lmnt mthod of tim-harmonic xtrior acoustic problms in two dimnsions Comput Mth Appl Mch Engng 00;191:1 [8] Chn JT, Kuo SR, Lin GH Analytical study and numrical xprimnts for dgnrat scal problms in th boundary lmnt mthod for two-dimnsional lasticity Int J Numr Mth Engng 00;:19 81 [9] Chn JT, Lin JH, Kuo SR, Chiu YP Analytical study and numrical xprimnts for dgnrat scal problms in boundary lmnt mthod using dgnrat rnls and circulants Engng Anal Bound Elm 001;(9):819 8 [10] Chn JT, Chn CT, Chn KH, Chn IL On fictitious frquncis using dual BEM for nouniform radiation problms of a cylindr Mch Rs Commun 000;():8 90 [11] Chn JT, Liang MT, Chn IL, Chyuan SW, Chn KH Dual boundary lmnt analysis of wav scattring from singularitis Wav Motion 1999;0: 81

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