Numerical Methods for Scattering Phenomena
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1 Nmeal Mehos fo ae heomea Boo Nole Chsa ele eas Wela eeal me oes Uewae oss a Mae eohyss Reseah Ise D-448 el lasofe We -4 emay {booole fele aeaswela}@bwb.o Io häfe elm hm Uesy - Uesy of he eeal me oes amb D-48 olsehofwe 85 emay o.shaefe@hs-hh.e Dffee meal aoahes fo he hysal heomea of sae waes fom he obsale e o a e eao ae esee. They ae base o ffee eal fomlaos. The wae eqao s se feqey oma elmholz-eqao fo he fl. l se eao effes ae meal eaable as well. Ths heomeo s mahemaally base o he Lamé-Nae-Eqao ojo wh he fl. The oeso eal fomlao s he hhoff-eqao feqey oma fo he fl as well as he so alle omlaa-iey fo he elas se. To e oe meal aoah fo ea of sh oblem he e Eleme Meho l so alle fe elemes a afal boaes a be se. Moeoe we se he Boay Eleme Meho ffee aoahes bease he ommefel aao oo maes se ha o efle waes fom boaes a fy o. The aos wae ha s e o a obsale o se s oally eflee. he boay oos ol be a sof oe; ee he se ol be elas. o we oe he sof o elas sae of so. e of he mos saaaes of some meal mehos EM/BEM s he fa ha we oba ffles whe hal he ey hh feqey ae. Ths s bease of lme ome memoy. o hs hh feqey ae he hhoff-ieal eqao s e o sble a sble as es. llmae a o-llmae as. Wh some assmos of efle behao o he se ffee aoahes fo he hhe feqey ae ae obae. Mlle sae aoahes a be omae wh ol a o-ol BEM mehos. Ioo s we ose he wae eqao. oe asfomao leas o he ell elmholzeqao whee s he see of so a s he wae mbe ω ω he la feqey. I hs ae he me hamo fom ˆ ω e s se. Tha ases a oe asfomao of he fom ω f e ω. 4 ollow hs oao we oba he fameal solo e δ X 5 4π whee s he Ela sae bewee he loa o a he fel o X he lae oe s loae o he boay f. δ s he Dasbo. ae fom a abaly shae obsale. BEM-aoah To ee he eal eesso fo he so fel fo a abaly shae obsale whh s e by a ow so soe he homoeeos elmholz eqao q 6 has o be mofe. The homoeeos elmholz oeao s wehe wh a es fo whh lae o wll beome he fameal solo 5 a s eae oe he ee oma f. q. 7 Twe alao of ass eee heoem a we alao of eao by as lea o 7
2 om sm 5 Baes Nole ele Wela häfe q q. 8 The lae em eqao s he e wae. The oeso sysem of eqaos s. 4 e The fao.5 s ow loae o he ma aoal ems of ma. Us eqao 7 eqao 8 yels C q 9 wh whee C.5 smooh s he omlmeay oma wh ese o. I shol be meoe ha eqao 9 s he fameal solo wh ffee ames he oma a o he sfae eal Q a X. ssme ha he so soes ae loae oly a a of he oma q see f. eqao 9 esls C qq. The meal aoah of eqao fo eey loa o o he boay leas o he BEM sysem of eqaos fo he sae oblem. By s he Ele-eqao ωρ oe a sbse he omal ale eloy fo he omal esse eae eqao so ha eqao a be we as ωρ o he boay. q Q qq. q Q Q R X s e : oe q a aee he oma To oba he eal eqao fo sae heomea oe a also ae a loo a he elmholz eal eqao js fo he sae so fel fo a eeo oblem whh esls C ωρ. 5 Now le s mae ha he obsale s o ese. The eo eal fomlao js fo he e wae obae fom he so soe he absee of he obsale a be we as C ωρ 6 whee C s he omlmeay sfae eal o C a he ale of hs eal s.5 o a smooh boay as well. Noe ha eqao 6 he omal eo s sll he same le eqao 5. bao of eqao 6 fom eqao 5 a by ma se of a C C 7 leas o C ωρ. 8 I s lea ha e.. ase 74
3 om sm 5 Baes Nole ele Wela häfe ωρ 9 o moe ese ωρ ωρ ωρ Bease of eqao 7 fs a a omaso wh eqao 5 es s ely ωρ. Noe ha he e wae oes o sasfy he ommefel aao oo le he oal a he saee so fel o. The e wae sasfes he homoeeos elmholz-eqao oma who he saee es. wheeby he sae so fel sasfes he elmholz-eqao he oma : q a hs esls q 4 a wh he fs a of eqao 7 oe oba aa eqao 6.. The BEM-aoah fo flse eao o fl-se eao he omlaa-ey obae fom he Lamé-Nae-eqao jj j j ω 5 wh as he shea a laaoal see of so a as he slaeme eo a be we as C j j j j j ~ ~ 6 wh boay ao eo j a he fameal solo of he Lamé-Nae-eqao j ~ 4 ~ j j j χ δ πµ Ψ. 7 wh shea mols µ. The fos Ψ a χ ee o a a be fo e.. [4]. Le s ow assme ha he saee self s a elas boy. The oeso eqaos fo he ol oee ae [ ] [ ] [ ] [ ] 8 whee aes he o-ol a of he boay a he ol a es. he aal efae. To oba seaess he ol efae we ema ω 9 whee s he omal eo he ol a of he boay. The ole sysem of eqaos a be we as see NLTE [4] [ ] [ ] T ω The mesos of he sbmaes ae [ ] [ ] T ω. The ol ems ae loae o he lef ha se of eqao hey ae efally ow ales. e a oly flee he ol ales by he e eo. 75
4 om sm 5 Baes Nole ele Wela häfe I shol be meoe ha oly he Dhle aa ae allae he efae. To oba he Nema aa he efae eqao 9 a be se aa. fe all hs a fel o allao fo eah oma a be oe. I fe he ee boay s he efae so ha a o-ol boay oes o es hs seal ase ems ae wh a eqaos 8 a. ll o ha eqao 4 esls ll R hhoff-aoah of fs oe I he hhe feqey ae a BEM-aoah fale bease of lme ome memoy. Remembe a fs oe has o sole a sysem of eqao o he boay a afewas a so alle fel o allao a be oe he oma. Moeoe he hhe feqey ae oe a ae some moe assmo o ao he el oess of sol a sysem of eqaos. hs leas o a aoah whh s omaable wh a BEM-fel o allao. The fs assmo s ha he boay s e o a a whh s llmae ll a aohe whh s o-llmae oll by he soe o he bsa ase es. by he eee. The boay eao oee s ow ee o he llmae a oly fe. ohe assmo s a efleo oeffe R o he boay ll so ha he sae so fel a be we as R. The ae omal eo o hs boay s s R. Ths leas o he eal fo he saee fel bease of 7 a 9 ll R R. 4 I has aleay bee show ha he eal oe he e wae fo a o ose he saee - he absee of he saee - ashes e : Moosa ase e : Bsa ase E.. he ase of a lae e wae of esy I Ie 7 hs esls fo << he eal eesso fo he sae so fel 4π ll e R ee o he ala Φ Φ [ os Φ os Φ ] 8. The alas ae ee fe a fe as well. sho ssso of all hese assmo shol be oe. To be osse wh eqaos a he e wae s assme o be zeo o he o-llmae a o he boay. Noe ha wh hs aomao he 76
5 om sm 5 Baes Nole ele Wela häfe eal 5 wll o ash a oseqely he saee fel e o 4 wll also o ash he lm of a ash efleo oeffe. The saee fel fo he moosa ase whee he loao of he loa a he eee o ae oe s obae fom eqao 8 wh os Φ os Φ. I hs ase he ame of he ea eqao 5 ashes a hs yels he ee eal o zeo..4 hhoff-aoah of seo oe The hhoff aoah of fs oe ae - wh some assmos he ma aoal ems of eqao 4 o ao. To moe he hhoff-aoah of he fs oe oble sae o some elemes ae osee see fe 4. Ths s moa ase he se oas oae as o he sfae a he ase of a e oe sfae hs a be elee. I fe 5 aohe es se s ee..5 hhoff-aoah of oe Bease we aleay ee a hhoff-aoah fo he fs a seo oe we oseqely a moe sah fowa o eee a hhoff-aoah of a abaly oe. Coeee ol e los wh a ea oe..6 Tae eh efo o a seay sae sal he moosa f. ae seh s efe as sa T lo 4 wh he saee aos so esse leel a loao a a he ee of he ae. Ths s ealae he fa fel a / λ << whee a eoes he haaes leh of he ae o obsale. I he fa fel T s o ee o. ome ee esls. The BEM aoah fo a sheal saee The BEM-aoah fo a lassal shee s allae. I fe 6 he e esse fel s ee. The saee fel a lae eela o he e wae s show fe 7. e 4: Coe efleo e 5: Tes se e 6: Ie esse fel N/m 77
6 om sm 5 Baes Nole ele Wela häfe 4 mmay I hs ae ffee aoahes fo sae heomea wee eeloe. The ea solo s a BEM-aoah whh aes he fl-se eao o ao. e obas sh a aoah by a BEM-BEM ol meho; BEM-aoah s se fo he fl a as well as fo he se a. The aaae of hs meho os whe wo o moe omas of fe szes ea wh eah ohe l soes he fl oma. ome hhoff aoahes wee ssse as well. e 7: aee esse fel mm λ f f 4 z λ.75 m 5 m/s. The hhoff-aoah The esls fo he oe efleo a fo he es se ae ee fe 8 a 9. e 8: Tae eh fo he oe efleo e 9: Resls fo he es se D-e sfae eeo fo he aae eeo of he llmae a of he sfae Refeees [] C.. Bebba R.D. Csows e. Boay Eleme Mehos oss Elsee le ee Loo New Yo 99 [].. hee R. Be L. loy I. aasalo I. Mallay M. Te Moshze. Vole os ae by a bmae: Resls fom a Behma Tae eh mlao Wosho o.. I. Coess o o a Vbao oholm ly [].. eybe B. oeao.. Rzzo D.. hy. os. o. m. Vol [4] B. Nole Raelemebeehe Nahfelmesse z l--ieao hd-thess Uesy of he eeal me oes amb emay 998 [5].. eybe B. oeao.. Rzzo D.. hy. os. o. m. Vol [6].C. Waema New omlao of os ae. os. o. m. Vol [7] E.. Wllams oe oss aem ess Loo a Deo 999 [8] T.W. W Boay Eleme os WIT ess ohamo Boso [9] Ch. ele ETa Eemeal Tae eh mlao Tehal Reo eeal me oes Uewae oss a Mae eohyss Reseah Ise el emay 4 78
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