31 Domain decomposition methods for solving scattering problems by a boundary element method

Transcription

1 Thteenth Intenatnal Cnfeence n Dman Decmpstn Methds Edts: N. Debt, M.Gabey, R. Hppe, J. Péaux, D. Keyes, Y. Kuznetsv c 200 DDM.g 3 Dman decmpstn methds f slvng scatteng pblems by a bunday element methd Y. Bubend, A. Bendal 2 Intductn Integal equatn methds ae wdely used f the numecal slutn f scatteng pblems. Amng the advantages, we mentn dect and smple dealng wth the adatn cndtn, accuacy and eductn f the mesh nly t the bunday. As a cuntepat, ths methd geneates lage dense cmplex matces and n the case f delectc layes may need sme exta auxlay unknwns, namely the equvalent magnetc cuents. Als, the epettn f sme gemetcal pattens can dastcally ncease the sze f the fnal system t be slved n an atfcal way. The am f ths pape s t shw hw these dffcultes can be vecme by a sutable use f a nnvelappng dman decmpstn methd whle hweve keepng the advantages f the bunday ntegal equatns slutns. The man technque used t decmpse the slutn dman nt smalle dmans cnssts n expessng the usual matchng f the Cauchy data f the pblem (the equvalent cuents as they ae geneally efeed t n cmputatnal electmagnetcs) n tems f sme equvalent bunday cndtns f mpedance (als called Rbn) type. The methd als apples t a cnduct cveed by a delectc laye wth nw tw advantages. Fst, at each step, the pblem t be slved has f unknwn the electc cuent nly wheeas the dect slutn als nvlves the magnetc cuent as a supplemental unknwn. Meve, at each step, unknwn nte and exte cuents ae cmpletely uncupled. Anthe nteestng aspect f ths methd s t cuple a fnte element and a bunday element methd. Ths appach has been nvestgated by seveal auths (e.g., [JN80], [Cs87], [dlb95], [Lan94], [HW92]). Hweve, the esultng fnal system s geneally lage and dffcult t slve because t nvlves equatns cmng bth fm the FEM and BEM fmulatns. On the cntay, the methd ppsed n ths pape uncuples cmpletely the tw slutn pcedues. UMR MIP INSA-CNRS-UPS, Cefacs, Fance, bubend@cefacs.f 2 UMR MIP INSA-CNRS-UPS, Cefacs, Fance, bendal@gmm.nsa-tlse.f

2 Y 320 BOUBENDIR, BENDALI Ω Ω 0 n 0 n n Γ Σ Fgue : A typcal gemety Nnvelappng dman decmpstn methd T be specfc, we cnsde the fllwng pblem elated t the scatteng f an TE wave by a cated pefectly cnductng cylnde fnd a suffcently smth such that n! n %$&(' "$ () )* n +,! " $- %(&$' %(&(' " n./ 0243 BA :9<;>=?@= DC )EF HG IKJL? EONP QEO HG IRJSPT* =?M= whee UWV and U@X ae espectvely the unt nmal t. utwadly dected t / and t " (fg. ), s the wave numbe, and, espectvely the ndex and the elatve pemttvty f the delectc medum fllng. Supescpt and 0 ndcate espectve lmts n. fm wthn and ". T uncuple the exte pblem slutn n " and the nte ne n, we use the methds ntated by P.-L. Lns [L90] and late develped f wave ppagatn pblems by B. Despés [Dep9] t wte the tansmssn cndtns n. n the fllwng equvalent fm - % &$Z % W\[], ^_E & ` H"\[], H" n %$&./ ` " a[], " E - %(& Z \[], n./b whee ] s pstve self-adjnt nvesble peat, [cde^ne gf cnphq fj wth and hlkm. Theefe, the cmputatn f the slutn cnssts n slvng the fllwng tw pblems sepaately at each step n 9 p2q s 9 p <4q P * n %(& Z 9 (3a) 4q s n +t %$& Z 9 <4q s 9 \[], <4q s %(& ue ` <4q " \[(]t p2q " n./ (3b) (2)

3 h " DOMAIN DECOMPOSITION METHODS AND BEM 32 Ω 0 Ω n 0 n R n R Γ Σ Fgue 2: A ccula gemety 9 <4q s 9 " p2q s " n " 0423 sa 5 6$5 8:9p; =?@= C 9 4q s " EF G IKJ? 9 EONP 4q s (4a) " EF G IKJ T * =?M= % &7` 9 4q s 9 " a[], 4q P " _E % &(Z 2q \[], " 4q n./b (4b) It s well-knwn (e.g., [CZ92]) that bth pblems (), (3) and (4) ae well-psed n an apppate functnal settng. Obseve that the dect slutn f pblem () eques the detem- natn f the fllwng cupled Cauchy data " n., $- %(& Z %(& Z " n. and m n 9 + (e.g., [BS94]), wheeas, the slutn f pblem (3) eques the detemnatn f 4q P 9 and 4q s 9 and that f pblem (4), the detemnatn f 2q s nly. Cnvegence f the dman decmpstn methd F [,.e. h, the theetcal cnvegence f the algthm (3) and (4) s well knwn [Dep9], [CGJ00]. Hweve, plts f the esdual n fgue 3 clealy ndcates that the dscete vesn f the algthm cnveges f h nly. It seems that nly vaatnal schemes lke fnte element methds can keep 9 the cnvegence ppetes f the algthm at the dscete level (e.g., [Dep9], [dlbfm 98]). Bunday element methd s nt based n such a pncple and thus esults n a nn cnvegent scheme f h. F h, the pf f cnvegence seems t be ut f each f the geneal case. Ths s pbably due t a lack f a sutable way t handle ppagatve and evanescent pats f the slutn sepaately. Hweve, f all cases when a decmpstn f the slutn n ppagatve and evanescent mdes can be dne, we ae able t pve that the algthm wth has a bette behavu than wth h m. The fllwng example athe stkngly llustates ths clam. F a ccula gemety (fg. 2) wth "? =?M= =?@=? we can decmpse the e n mdes fm a Fue-Hankel sees

4 E E " b 322 BOUBENDIR, BENDALI Χ=0.5 Χ= Fgue 3: Behavu f the esduals expansn and analyze sepaately the cnvegence f the ppagatve and evanescent pats f the wave. Settng " t 9<; - ; 9p; - ; (5) pblems (3), (4) ae educed t the fllwng ne-dmensnal pblems % % p < " p " * p; sa % < " EFNP( < " * % < " a[] < " " (6a) (6b) % % p ME < <! % p! - % \[] (7a) (7b) We have assumed that the peat ] s dagnal elatvely t the Fue sees expansn. Slutns t pblems (6), (7) ae espectvely btaned by " s and whee P epesents the Hankel functn f the fst knd and s a slutn f the Bessel equatn f de whch can be expessed by a lnea cmbnatn f the Bessel and Neumann functns f de such that "! m. The teatn peat s chaactezed by the matces %$ "S s & (8)

5 = " k DOMAIN DECOMPOSITION METHODS AND BEM 323 whee "S et P ae defned by "L H " E [] < P H E [] p Fst, we gve a cten chaactezng the cnvegence f the algthm. b (9) Theem The dman decmpstn algthm cnveges f and nly f f all, beng the spectal adus f matx. Pf Let "7 $K. One pssble defntn f the nm n p;. s gven by - - ; - whee 9p; s defned by - ; and 0423 ". The cnvegence f the methd wll be establshed f we can shw that 8 9p; q q - * wth 9p; q - - ; - q b (0) If t exsts " such that `, clealy the methd des nt cnvege. S, we can estct the dscussn t the case whee f all. The matx has tw dstnct egenvalues "S s. S, t can be put n a dagnal fm by -, beng a dagnal matx. Theefe, we btan q - s q b The mst mptant pnt n the pf s that the cndtn numbe - f matx emans unfmly bunded. Elementay aguments then pemt t end the pf. "S s = f each The pevus chaactezatn establshes that the methd cnveges f = t btan the cnvegence f the methd. Slvng pblems (6), (7), we get "S whee ^"/ P! Ë " \N f anphq " an gf P \NPhQ Ë tan gf \NPhQ \N gf anphq!] s " ue $ % ] defned because bth the tw pblems ae well psed. Ppstn - F bth evanescent and ppagatve mdes, = - If cespnds t an evanescent mde, that s, P =. "L =. ", f () whch ae well " lage enugh, then Pf Let " E? \N'&. Clealy, t s enugh t shw that? and & ae bth "S t pve that = =. Sgns f? and & ae espectvely that f ( s! P s and E P! P s. Fm [CK92], t s well-knwn that ( s! P s s equal t the Wnskan ) st +*p. Snce get that &. The ppety?, fm [CK92] we uses a me dffcult agument. Fst, we emak that

6 Y h f 324 BOUBENDIR, BENDALI. Usng Nchlsn s fmula [Wat22], we get P! s L = s = that functn = P = s a stctly deceasng functn, s the quantty s! s s negatve and then? f. We cnclude that f all and h "S, = =. F the pblem n the bunded dman, the pevus sgn detemnatn can be me easly btaned fm cecveness estmates. Let E? N'&. The vaatnal fmulatn f pblem (7) gves and then f! Z =! = E\ s lage enugh, usng cecveness ppety, we get that R = = <! p s b (2) Defntn f < and yelds?. Snce we have cnsdeed that the mateal fllng s wthut lsses (( [ t ) and pefectly eflectng bunday cndtn n +, we ae led the mst sevee case & *. Indeed, n ths case s E? <an h? <an f b F h s (Despé s algthm [Dep9]) = = "L and s = P =. The algthm cnveges as expected fm the study f the geneal case [Dep9]. Obseve hweve that paamete s has n nfluence upn the cnvegence f the algthm and "L gves a less effectve dampng f the evanescent mdes. The nteestng pnt s that takng h als P gves = = f all except a fnte numbe geneally cespndng t ppagatve mdes. But snce f h "S = s =, t s suffcent t tune h f each f these exceptnal mde t btan a maxmal value f h nsung the cnvegence f the algthm. B Numecal esults At each step, pblem (4) n the unbunded dman " has been slved by a bunday element methd [BBC00] and pblem (3) n the bunded dman by an usual ndal fnte element methd. The exte pblem n " s slved by a BEM fllwng the appach ntduced n [Ve99]. The slutn s epesented as a supepstn f a sngle- and a duble-laye ptentals "? t* G IKJM? %& & %(&. & WE? & &. & (3) whee the unknwn denstes and mpedance cndtn ae lnked by the fllwng elatn nduced by the D\[(]* b (4)

7 DOMAIN DECOMPOSITION METHODS AND BEM esdu DDMEI DDMEI sl exacte Fgue 4: Cuplng FEM and BEM The bunday cndtn can then be expessed vaatnnally as %(& ` " EF ".> "./ (5) wth and ae lnked by the same elatn that and. Fmulatng these cnstans thugh a Lagange multple, bth the latte and the magnetc cuents and can be elmnated at the element level when all the unknwns ae appxmated by a -cntnuus BEM, (see [Ve99] f me detals). Plts n fgue 4 gve the esdual and cmpasn between exact and cmputed electc cuent n.. The ncdent wave s a plane wave ppagatng alng the x-axs. The nteestng pnt s that nw, wth h, the dscete algthm cnveges usng ethe a ndal fnte element a bunday element methd. Refeences [BBC00]Y. Bubend, A. Bendal, and F. Clln. Dman decmpstn methds and ntegal equatns f slvng helmhltz dffactn pblem. In Ffth Intenatnal Cnfeence n Mathematcal and Numecal Aspects f Wave Ppagatn, pages , Phladelpha, july SIAM. [BS94]A. Bendal and M. Sulah. Cnsstency estmates f a duble-laye ptental and applcatn t the numecal analyss f the bunday-element appxmatn f acustc scatteng by a penetable bject. Mathematcs f cmputatn, 62(205):65 9, 994. [CGJ00]F. Clln, S. Ghanem, and P. Jly. Dman decmpstn methd f hamnc wave ppagatn: a geneal pesentatn. Cmpute methds n appled mechancs and engneeng, 84:7 2, [CK92]D. Cltn and R. Kess. Invese Acustc and Electmagnetc Scatteng They, vlume 93. Spnge-Velag, 992. [Cs87]M Cstabel. Symmetc methds f the cuplng f fnte elements and bunday elements. In Bunday Elements IV, Vl., Cmput. Mech., pages Bebe, Suthamptn, 987.

8 326 BOUBENDIR, BENDALI [CZ92]G. Chen and J. Zhu. Bunday element methds. Academc Pess Lndn, 992. [Dep9]B. Depés. Méthdes de décmpstn de dmans pu les pblèms de ppagatn d ndes en égme hamnque. PhD thess, Unvesté Pas IX Dauphne, 99. [dlb95]amel de La Budnnaye. Sme fmulatns cuplng fnte element and ntegal equatn methds f helmhltz equatn and electmagnetsm. Numesche Mathematk, 69(3): , [dlbfm 98]Amel de La Budnnaye, Chabel Fahat, Antnn Maced, Fédéc Magulès, and Fançs-Xave Rux. A nn-velappng dman decmpstn methd f exte Helmhltz pblems. In Dman decmpstn methds, 0 (Bulde, CO, 997), pages 42 66, Pvdence, RI, 998. Ame. Math. Sc. [HW92]G. C. Hsa and W. L. Wendland. Dman decmpstn va bunday element methds. Techncal Rept 92-4, Mathematcs Insttut A, Unvestät Stuttgat, 992. [JN80]C. Jhnsn and J. Nedelec. On the cuplng f bunday ntegal and fnte element methds. Math. Cmp., 35: , 980. [Lan94]Ulch Lange. Paallel teatve slutn f symmetc cupled BE/FE equatns va dman decmpstn. Cntemp. Math., 57: , 994. [L90]Pee-Lus Lns. On the Schwaz altenatng methd. III: a vaant f nnvelappng subdmans. In Tny F. Chan, Rland Glwnsk, Jacques Péaux, and Olf Wdlund, edts, Thd Intenatnal Sympsum n Dman Decmpstn Methds f Patal Dffeental Equatns, held n Hustn, Texas, Mach 20-22, 989, Phladelpha, PA, 990. SIAM. [Ve99]Lauent Venhet. Bunday element slutn f a scatteng pblem nvlvng a genealzed mpedance bunday cndtn. Math. Meth. Appl., 22(7): , 999. [Wat22]G. N. Watsn. A teatse n the they f Bessel functns. Cambdge Unvesty Pess, 922.