DRBEM Applied to the 3D Helmholtz Equation and Its Particular Solutions with Various Radial Basis Functions

Transcription

1 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons, 06, Vol. 4, No., -6 Avalable onlne at Scence and Educaton Publshng DOI:0.69/jpdea-4-- DRBEM Appled to the 3D Helmholtz Equaton and Its Patcula Solutons wth Vaous Radal Bass Functons Hassan Ghassem *, Saeed Fazelfa, Aleza Nade Depatment of Matme Engneeng, Amkab Unvest of Technolog, Tehan, Ian *Coespondng autho: Abstact Ths pape pesents to solve the 3D Helmholtz equaton usng dual ecpoct bounda element method (DRBEM) and ts patcula solutons wth vaous adal bass functons (RBFs). The mpotant functon n ths method s to emplo the RBF. Hee, we fnd the patcula soluton of the Helmholtz equaton ( ± k ) h = f(), whee f() s the RBF. Vaous RBFs ae chosen and the patcula solutons ae obtaned. The dual ecpoct method (DRM) s a method that convets the doman ntegal nto the bounda ntegal. Mathematcal fomulatons and dscetzaton foms ae descbed and dscussed. Numecal esults wth thee RBF wth and wthout polnomal tems ae pesented and dscussed. Algothm of the method s also pesented. Kewods: 3D Helmholtz equaton, adal bass functon, patcula soluton, dual ecpoct method Cte Ths Atcle: Hassan Ghassem, Saeed Fazelfa, and Aleza Nade, DRBEM Appled to the 3D Helmholtz Equaton and Its Patcula Solutons wth Vaous Radal Bass Functons. Intenatonal Jounal of Patal Dffeental Equatons and Applcatons, vol. 4, no. (06): -6. do: 0.69/jpdea Intoducton The bounda element method s a numecal method fo solvng patal dffeental equatons encounteed n mathematcal phscs and engneeng [].The bounda element method can be vewed as some sot of half-wa house between analtcal and numecal methods []. Accodng to wok of Nadn and Bebba (98) [3], thee has been an nceasng nteest n usng the Dual Recpoct Method (DRM) to solve patal dffeental equatons (PDEs) b bounda element methods (Patdge et al. 99 [4] and Golbeg and Chen 997 [5]). The attactveness of the DRM s ts capablt to tansfe doman ntegaton to the bounda ntegaton. In the past, + has been chosen as the ad-hoc bass functon n the DRM [6,7]. In 994, Golbeg and Chen [7,8] povded theoetcal evdence fo the choce of the basc functons b usng the RBFs n the DRM. On the ssue of applcablt, the DRM has been onl appled to the case when the majo dffeental opeato s kept as the Laplace o n hamonc opeato and the est of the tems n the ognal dffeental ae teated as a focng tem. Ths s pmal due to the dffcult n obtanng patcula solutons n a closed fom. As a esult, the DRM s less effectve when the focng tems become too complcated [0,]. In geneal, we would lke to keep focng tem as smple as possble to make bette appoxmaton b RBF. Meanwhle, the smple the focng tem, the moe complcated the dffeental opeato becomes, and also the fundamental soluton become moe nvolved. So fa, the choce of the man dffeental opeato seems to be lmted to Laplace and n hamonc opeatos due to dffcult of poducng patcula solutons n a closed fom. A patcula soluton to the govenng dffeental equaton s then detemned fo each bass functon []. In ths egad, Zhu [3] attempted to usng Helmholtz opeatos as the man dffeental opeato n the DRM. The ke ngedent n dong ths s the ablt to analtcall calculate patcula solutons fo vaous lnea PDEs, Lφ = b. Ths s usuall done b appoxmatng f b N a sees a j j f = j, and then solvng Lhj = f j,< j < N, whee { f j } s an appopate set of lneal ndependent bass functon. Hence, the choce of { f j } s mpotant, and the analss gven n Golbeg et al. (998b). The defned that { f j } needs to povde an accuate appoxmaton to b, and also t should be as a fom that Lhj = f j be solved analtcall. The followng sectons ae oganzed as follows. Secton s descbed the phscal poblem and obtan the govenng equaton. Secton 3 shows that how the DRM method convets the doman ntegal nto the bounda ntegal. Secton 4 fnds analtcal patcula solutons fo vaous RBF lke smple functon f() = +, thn plate Splnes (TPS), f () = log and hghe-ode polhamonc Splnes. Then, applng the patcula solutons s descbed n Secton 5. Secton 6 s dscussed the numecal esults and fnall conclusons ae gven n Secton 7.. Helmholtz Equaton The complete dnamc basc equatons of the flud ae the mass contnut, Nave-stokes and eneg equatons [4]:

2 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons ( φ) ± k φ = b, n Ω () B. C. = known, on Eq. () s the govenng equaton fo the sound feld n the thee-dmensonal flow. The dawbacks of usng the Laplace opeato nstead of the Helmholtz opeato ae: () The nfomaton n the ognal dffeental equaton s patall lost. () The focng tem becomes moe complcated and dffcult to ntepolate b adal bass functons. () The soluton ma not even convege when k becomes lage [5]. 3. Dual Recpoct Method (DRM) In the bounda element method, fo a bod of the bounda ( Γ ) and doman ( Ω ), the ntegal fomulaton of the Eq. () ma be expessed as: whee: φ G e( p) φ( p) = Γ( G φ ) ds Gb( φ) dω n Ω () fo P outsde Suface e( p) = 0.5 fo P on Suface 0 fo P outsde Suface (3) And G s the Geen s functon fom the Helmholtz equaton. Fo the 3D poblems: exp( jk) φ + k φ = b( φ), G = 4π exp( k) φ k φ = b( φ), G = 4 π (4) Eq. () contans the volume ntegal, whch s a dffcult poblem. Theefoe, n ode to ovecome ths dffcult, one of the easest wa s convetng volume ntegal nto the bounda ntegal va DRM [3-6]. Ths method focuses to the tem b( φ ) whch ma be appoxmated b the followng expesson, N+ L bx = = fα (5) whee α, f ae ntepolaton coeffcents and adal bass functon (RBF), espectvel. N s the numbe of collocaton nodes along the bounda, L s the numbe of collocaton ponts nsde the doman, and s defned as the dstance between the node unde consdeaton and the node. Fo each smple souce functon soluton f h needs to be found and satsfed as: h k h f + =, a patcula (6) Heeafte, the patcula soluton of the Eq. (6) was obtaned b some vaous RBFs. Substtutng equatons (6) and (5) nto Eq. () elds: φ G e( p) φ( p) = ( G φ ) ds Γ N+ L G α e( p) h( p) ( Gh h ) ds Γ = (7) Dscetzaton fom of the Eq. (7) can be epesented as follows: N N ˆ lφl = ljφn j ljφj j= j= e L H N+ L N N α ˆ eh l l Lljh j Hljhj = j= j= (8) whee L j and H ˆ j ae nfluence coeffcents, and defned as follows: Llj = Gds Γ ˆ G Hlj = ds Γ (9) These ntegals can be evaluated b numecal and analtcal methods. B vefng t n detal: H 0.5 ˆ lj = δlj + Hlj (0) whee δ j s the Konecke delta, whch s defned as δ j = 0 fo j, and δ j = fo = j. Eq. (8) ma futhe be wtten as: N N N+ L N N H lj j L φ = ljφn j α Hh lj j Lh lj j j= j= = j= j= whee h s obtaned n Secton Patcula Solutons Fst, we ntoduce some RBFs as gven: Lnea classc: TPS: Hghe-ode Splne: 4.. Classcal RBF f = + () () fj = ln (3) [ n] n fj j = (4) B substtutng Eq. () nto Eq. (6), the patcula soluton of the Eq. (6) can be found as follows: ( + k ) h = + (5) Snce ths soluton s axsmmetc wth espect to the souce, t s ndependent of the pola angle θ, and thus Eq. (5) becomes [7]:

3 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons 3 d d ( + + k ) h = + d d A egula soluton of Eq. (6) s obtaned [3,4]: Whee: + cos( k) h = 4 k k = q p 4.. Thn Plate Splne (TPS) (6) (7) (8) We consde an appoach, usuall called the annhlato method as descbed n Ref. [8]. Hee, t s assumed that thee s a lnea patal dffeental opeato M whch satsfes Mf = 0 And commutes wth L ;.e. ML = LM, and then: MLh = Mf = 0 = LMh If the soluton sets, V { v : Lv 0} W = { w : Mw = 0}, ae fnte and dsjont, then s t h = bkβk + ckγk + z k= k= (9) (0) = = and () whee { β k } s a bass fo V, { γ k } s a bass fo W, and Lz = 0. The coeffcents { b k } and { c k } ae detemned b equng Lh = f and addtonal egulat condtons (Golbeg and Chen 997 [5] and Golbeget al. 998a []). B substtutng Eq. (3) nto Eq. (6), the patcula soluton of the Eq. (6) n D can be found as follows: And fo ( + k ) h = ln L _ we have: () L_ h p = log, = P, (3) Now, wth espect to Eqs. (9, 0), and Eq. (): 4 log = 0, > 0, (4) 4 h can be obtaned b solvng L h( p) _ = 0. Fo adall smmetc solutons, t s equvalent to solve: 4 k h = 0 h can be obtaned b solvng: Snce ( k ) ( ) 4 (5) w= 0, k v = 0 (6) s a Bessel opeato (Deck and Gossman 976 [9]): v = AI0 k + BK0 ( k ) (7) whee I 0 and K 0 ae Bessel functons of ode zeo. 4 Snce s a multple of an Eule opeato, p p, 4 p p = p = p p 4, p must satsf the chaactestc equaton p ( p ) = 0, (Deck and Gossman 976 [9]) Thus, = + log + + w a b c d log h = AI0 k + BK0 k + a + b log + c + d log (8) (9) The coeffcents { ABabcd,,,,, } ae found b equng k h = log and the condton that h should be contnuous at = 0. One soluton s gven n Chen and Rashed (998a) b: h ( k ) 4 4log log 4K0, = k k k k 4 4γ 4 k + + log, = k k k (30) whee γ s Eule s constant Hghe-ode Splne Hee, we want to obtan the patcula solutons fo the hghe-ode Splne,.e defned as follows: n log L± h = n (3) n 3 L± h = n To calculate the patcula soluton fo have to solve ± k v = 0 and (3) 3 L n R ±, we n+ 3 w = 0, theefoe n+ 3 n = 0. It s easl shown that the soluton to k v = 0 s (Golbeg et al. [7]): v And fo snh Acosh k B k = + + k v = 0 : v sn Acos k B k = + (33) (34) To obtan solutons, whch ae egula at = 0, we use the Talo sees expansons of cosh( k ) and snh( k ) at = 0, and compang coeffcents gves: h n ( ) n!cosh k n! = n+ 3! n + k = 0 k A smla agument fo L + gves: (35)

4 4 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons n+ n+ n n ( ) ( )! n ( )!( ) = cos( k ) + ( )! h n+ n + k = 0 k (36) The esults fo dffeent odes ae gven n Table and Table. f() Table. Results fo dffeent ode fo L_ 4030 cosh(k) h fo L cosh(k) k 5 k 4 cosh(k) k cosh(k) k 9 70 k 4030 k 0 k k 8 3 k k 6803 Table. Results fo dffeent ode fo L + f() h fo L cos(k) cos(k) + k 4 cos(k) cos(k) k k 0 k 0 k k k 303 k k k 565 k k Fnall, the flowchat algothm of the DRBEM s shown n Fgue. Fst, geomet of the bod s modeled. DRBEM method defnes some ntenal and bounda nodes. Usng RBF, the doman ntegal can be nveted to the bounda ntegal. Ths s the man advantage of DRM to deal wth bounda of the bod. 5. Numecal Results The geomet of poblems s a Squae wth sde of unt length. These examples ae chosen snce the analtcal solutons can be obtaned. Moe complex poblems can be handled n the same DRM fashon wthout an exta dffcult. The nhomogeneous D Helmholtz poblem s govened b equaton: u u + + u = x x (37) ux (, ) = Dx (, ) (38) The analtcal soluton s: u( x, ) = snx + sn + x (39) The Dchlet bounda condtons Dx (, ) can be detemned fom the exact soluton (39). 5.. Wthout Polnomal Tems Fo all dffeent RBF n Table 3, the soluton s obtaned usng 00 bounda elements (50 on each sde) and 5 ntenal nodes. Table 3. Compasons vaous RBF Wthout polnomal tems x RBF Exact DRM % eo log log As t s seen, the esult of + wthout polnomal tem s moe accuate, whle the esults of TPS and polhamonc Splnes of ode wthout polnomal tems ae so bad. 5.. Wth Polnomal Tems To obtan a moe accuate answe, polnomal tems have been added to adal bass functons: f = fj + ζ ( p) (40) whee ζ ( p) s obtaned fom the followng equaton: L± ζ ( p) = pn (4) whee p n can be an polnomal functon. Some of the functons p and ζ ( p) ae gven n Table 4. Table 4. Patcula solutons fo polnomal tems n L + Fgue. Flowchat algothm of the DRBEM p n ζ (p) p n ζ (p) x k x k x x 3 x k 3 k 6 x k x4 k x + 4 x x k x 3 x k 6x x x k x x k x k 4 k k x 3 x 3 k 6x x 3 x 3 k 6x 4 4 k + 4 x x k

5 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons 5 Results fo some ponts, ae compaed n Table 5 b applng polnomal tem, whee p n s a polnomal of total degee n. Table 5. Compasons of the esults fo vaous RBFs wth polnomal tems (x, ) exact acbf TPS+p (n=) TPS+p (n=) (0.5, 0.) (0.5, 0.5) (0.5, 0.75) (0.5, 0.5) (0.7, 0.5) (0.7, 0.65) (0.75, 0.8) Advanced classcal RBF shown n Table 5, s obtaned fom contawse method. In ths method at fst we choce an patcula solutons ( f ), fo example 3 f = +, then wth solvng ths equaton Lf = bf, 4 9 as a esult, the advanced classcal adal bass functon (acbf) s obtaned. 3 acbf = + + k ( + ) 4 9 (4) Accodng to Table 5, t s seen that the esult fom ths method s ve accuate, but t s so dffcult to guessng a patcula soluton that gves an exact answe. Fnall, we compaed the elatve eo (RE) wth vaous RBFs, as shown n Fgue. Although fo all cases the RE s less that 0.03%, but fo acbf s ve small eo. 6. Conclusons In ths pape, a DRBEM fomulaton fo axsmmetc Helmholtz-tpe equaton s befl pesented. We have genealzed pevous wok usng TPS fo fndng patcula solutons to Helmholtz-tpe opeatos b usng hghe-ode Splnes, substantall nceased accuac can be obtaned usng hghe Splnes. Based on ths eseach, followng conclusons can be dawn: Hghe accuac can be obtaned b usng hgheode Splnes wth polnomal tem. Acbf s much bette than othes. The polnomal tems of RBF ae mpotant as shown n Table 3 and Table 5. Wth annhlato appoach, we can obtan patcula solutons of an ode of TPS n D and 3D. Table and Table shows the patcula solutons of 3D Helmholtz-tpe equaton and Table 4 shows the patcula solutons of D polnomals tems fo L +. Numecal examples fo Helmholtz-tpe equaton usng hghe-ode Splnes n 3D wll be examned and compae wth lowe ode Splnes n the next eseach, and that wll be ou futue plan. Fgue. Relatve eo usng vaous RBFs Refeences [] Pozkds, C., A Patal Gude to Bounda Element Method, Boca Raton London New Yok Washngton, D.C. [] Chandle, S., Langdon, W&S. Bounda element methods fo acoustcs, Depatment of mathematcs, 007. [3] Nadn, D. and Bebba, C.A., New Appoach to Vbaton Analss Usng Bounda Elements, n Bounda Element Methods n Engneeng, Computatonal Mechancs Publcatons 98, Southampton and Spnge-Velag, Beln. [4] Patdge, P.W., Bebba, C.A. and Wobel, L., the Dual Recpoct Bounda Element Method, Computatonal Mechancs Publcatons, Southampton and Elseve Appled Scence, London & New Yok, 99. [5] Golbeg M.A. & Chen C.S., Dscete Pojecton Methods fo Integal Equatons Southampton: Computatonal Mechancs Publcatons, 997. [6] Zapletal, J., The Bounda element method fo the Helmholtz Equaton n 3D, Depatment of Appled Mathematcs, 0. [7] Wen, L. & Mng, L. & Chen, C.S. &Xaofeng, Lu., Compactl suppoted adal bass functons fo solvng cetan hgh-ode patal dffeental equatons n 3D, 04.

6 6 Intenatonal Jounal of Patal Dffeental Equatons and Applcatons [8] Golbeg, M.A. & Chen, C.S., The theo of adal bass functons appled to the BEM fo nhomogeneous patal dffeental equatons, Bounda Elements Communcatons, 994, 5(), 576. [9] La, S.J. & Wang, B.Z., Solvng Helmholtz Equaton b Meshless Radal Bass Functons Method, 00. [0] Qu, Z.H., Wobel, L.C. & Powe, H., An evaluaton of bounda Elements XV, Computatonal Mechancs Publcatons, Southampton, 993. [] Popov, V. & Powe, H., A doman decomposton n the dual ecpoct appoach, Bounda Elements Communcatons, 7(), pp. -5, 996. [] Muleshkov, A.S. &Golbeg, M.A. & Chen, C.S., patcula solutons fo axsmmetc Helmholtz-tpe opeatos, 005. [3] Zhu, S., Patcula solutons assocated wth the Helmholtz opeato used n DRBEM, BE Abstacts, 4(6), pp. 3-33, 993. [4] Zhen-Ln J Xue-en Wang, Applcaton of dual ecpoct bounda element method to pedct acoustc attenuaton chaactestcs of mane engne, exhaust slences, Jounal Mane Scence Applcaton, Vol. 7, 008, pp0-0. [5] Chen, C.S. & Bebba, C.A, The dual ecpoct method fo Helmholtz tpe opeatos, Depatment of Mathematcal Scences, Unvest of Nevada, NV 8954, USA, 998. [6] Ramachandan, P.A., Bounda Element Methods n Tanspot Phenomena, Elseve Appled Scence, 994. [7] Chen CS, Rashed YF. Evoluton of thn plate Splne based patcula solutons fo Helmholtz-tpe equaton fo the DRM. Mechancal and Reseach Communcaton 998; 5(), pp95-0. [8] Pee-Debane. Analss of convegence and accuac of the DRBEM fo axsmmetc Helmholtz-tpe equaton, Engneeng Analss wth Bounda Elements, 999, Vol. 3, pp [9] Muleshkov, A.S. &Golbeg, M.A. & Chen, C.S., Patcula Soluton of Helmholtz-tpe Opeatos Usng Hghe-Ode Pol hamonc Splnes, 999. [0] Deck WR, Gossman SI: Elementa Dffeental Equatons wth Applcatons. Addson-Wessle Publshng Co, 976. [] Golbeg, M.A. & Chen CS, Rashed YF., the annhlato method fo computng patcula solutons to patal dffeental equatons, to appea n Eng. Anal. Bound. Elem, 998a.