A DUAL-RECIPROCITY BOUNDARY ELEMENT METHOD FOR ANISOTROPIC HEAT CONDUCTION IN INHOMOGENEOUS SOLIDS
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1 s Ieo ofeece Fo Scefc ou o ouo Eee s I-SE Ahes 8- Seebe 4 I-SE A DUA-REIPROIY BOUDARY EEE EHOD FOR AISOROPI HEA ODUIO I IHOOGEEOUS SOIDS W.. A K. K. hoo Dvso of Eee echcs Schoo of echc Pouco Eee y echooc Uvesy Reubc of Soe e-: w@u.eu.s web e: h:// Keywos: Du-ecocy bouy eee eho he couco sooy hooeey. Absc. A u-ecocy bouy eee eho s esee fo he uec souo of he obe vov o-sey wo-eso sooc he couco hooeeous sos. e es co he eeue he he fu o he bouy of he so e oe us scouous e eees. he eho foues he obe es of vue obe ovee by syse of fs-oe e oy ffee equos. he uow fucos of e he ffee equos e ve by ehe he uow eeue o he fu bouy o os so he uow eeue seece eo o os. o euce he ffee equos o syse of e ebc equos ech of he uow fucos v he eeue he o os s oe us oyo fuco e ove s e ev. Oce he e ebc equos e sove he eeue c be eee y ese o he eo of he souo o. uec esus fo secfc es obe e ve. IRODUIO Acco o he cssc heoy of he couco he o-sey wo-eso sooc fow of he eey so s ovee by he boc ffee equo ρ c whee s he eeue whch ees o he es co-oes e e he coucvy coeffces such h he syey eo s ssfe he sc equy < hos os he so ρ c e esecvey he esy he secfc he ccy of he so. he Ese coveo of su ove eee e hos fo subscs fo o. Equo s o be sove wo-eso eo R boue by se cose cuve subec o he -bouy coos q f h fo fo fo R > > whee f h e suby ve fucos e o-esec cuves such h q s he he fu efe by q / wh [ ] be he cooes of he u o veco o o wy fo R. he bouy eee eho fo he uec souo of fo he sec cse whch he eeue s e-eee e coss.e. he cse of sey-se he couco hooeeous sos s we esbshe see e.. ees []. I ee eve fo he cse of sey-se he couco f he coeffces vy sy e.. s fucoy e es s hecy ffcu o eve sube fue souo whch c be eoye o ob scy bouy e fouo fo. If he fue souo fo he sey-se wo-eso he couco
2 W.. A K. K. hoo hooeeous sos s use se he esu e fouo cues o oy bouy e bu so o e. o e wh he o e effecve e o o ob eve fouos h o o eque he souo o o be sceze vous oches wee oose fo hooeeous sooc e see e.. ees [] A Kusu ees [3] Kssb Dvo [4] P A [5] suoo Su [6]. I he ese e we cose he s of sov fo he cse whch he coucvy coeffces e he he ee fo 3 whee s ve osve fuco h c be y ffeee es wce wh esec o e o-eve coss ssfy <. he coeso sey-se obe ws ecey sove by A ees Vh [7] us u-ecocy bouy eee eho. Foow cosey he wo [7] we eoy he fue souo fo he sey-se wo-eso he couco hooeeous sooc so whch es he fo of se ohc fuco o eve eo-ffee fouo fo he ffee equo. Wh such fue souo he fouo evby cos o e ove he eo R. he uecocy eho oose by Bebb [8] s e o cove he o e o e e oey. he eho eques us o cooce os R bu he scezo of he eo R o y eees s o eee. hus oy he cuve bouy hs o be sceze. If he bouy s sceze o eees ce es co he bouy eeue he fu oe s sy e fucos ove he eees he eo-ffee equo c be use o efoue oehe wh 3 s vue obe ovee by syse of fs-oe e oy ffee equos. he uow fucos of e he ffee equos e ve by ehe he uow eeue o he fu bouy o os so he uow eeue eo o os. o euce he ffee equos o syse of e ebc equos ech of he uow fucos v he eeue he o os s oe us oyo fuco e ove s e ev. Oce he e ebc equos e sove he eeue c be eee y ese o he eo of he souo o. uec esus fo secfc obe sove us he u-ecocy bouy eee eho escbe hee e esee. IEGRO-DIFFEREIA EQUAIO Pocee s A ees Vh [7] we f h oehe wh 3 ves se o he eoffee equo γ R κ Γ ρc [ ] Φ q Φ s 4 whee γ f R γ f R < γ < f [γ / f es o sooh of ] Φ Re{ [ ]} π Γ π Re
3 W.. A K. K. hoo. κ 5 3 DISOIUOUS IEAR EEES he bouy s sceze o sh e eees eoe by -. he s e os of he e eee e ve by b b esecvey. Fo ccue evuo of he e e 4 e he scezo of o he sh e eees scouous e bouy eees e o be eoye. Soe es o he eeo of such bouy eees y be fou Ps s [9] Fo he scouous e bouy eees wo os o e chose s foows: ve fo b b b. 6 Afe s sceze o e eees f he eeue hs vues esecvey he es o he h fouh es of 4 esecvey e seey oe o vy ey coss ech of he e eees cco o fo ] [ fo ] [ 7 whee 8 wh be he eh of he e eee. Sy fo he he fu q f s vues e ve by q q esecvey he he e q o he fouh e of 4 s oe o vy ey coss ech e eee s. fo ] [ q q q 9 O ehe o q s secfe. I foows h fo ve ehe o q q s ow. hs ves se o uow fucos he bouy oes
4 W.. A K. K. hoo fo. he oos 7 9 fo he oe es o he h he fouh es of 4 e o be subsue o he e e fe s ece by he e eees. he ccuo of he coeffces of q he esu oe eesso fo he e e voves eos ove he e eees. he eos ove he eees c be oe ycy us he ec foue ve A []. 4 DUA-REIPROIY EHOD o y he u-ecocy eho o sfo he o e 4 o e e we fs e he oo [ ] c σ ρ κ whee e coss o be eee [ ] [ ] [ ] [ ] I Re I Re 3/ σ whee e he os o he bouy eees s efe by 6 e seece os he eo of R. We c e be ve by fo o se u syse of e ebc equos. he ebc equos c he be vee o ob [ ] c χ ρ κ whee χ e coss efe by. f f χ σ 3 Us y he u-ecocy eho we f h he oube e 4 c be oey e-we s [ ] [ ] Ψ Φ R c c χ ρ κ ρ κ 4 whee [ ] Γ Φ Ψ s θ β θ γ 5 wh
5 W.. A K. K. hoo [ ] [ ] [ ] [ ] [ ] [ ] 5 / I Re 5 I Re 6 I Re 4 θ 6 θ β. 7 oe h Ψ c be evue uecy by scez o bouy eees. he u-ecocy eho ves se o o uow fucos of e s ve by he eeue he eo os ey fo. As oe ou he ece seco wh he scouous e eees hee e uow fucos os o he bouy. hus f we choose 4 o be ve u by fo he uecocy bouy eee eho hee oey euces he eo-ffee equo o syse of fs oe e oy ffee equos o be sove subec o ow vues of s ve by he eeue. 5 UERIA SOUIO OF ORDIARY DIFFEREIA EQUAIOS o sove he syse of oy ffee equos we oe he o eeue s -h oe oyo fuco of ove he ev < < whee s s osve ube. Secfcy 8 whee fo. Dffee 8 wh esec o we ob: [ ]. 9 If 8-9 s subsue o he syse of oy ffee equos f he syse of ffee equos s chose o be ve u by we ob syse of e ebc equos co fo q fo. If we e he e ebc equos c be sove oehe wh he -bouy coos fo he uow he ques ffee e eves. Fo ee whe we c sove fo he uow ques he e eves use of he ow eeue hece he ow bouy he fu. If we ess o wh we c f he souo he e eves us he us eee eeue. he s c be eee fo 3 oe o eee he eeue hhe hhe e eves.
6 6 A ES PROBE ose he ffee equo wh W.. A K. K. hoo 3 cos 3 3 ρc 3 [ cos ]. 3 Fo secfc obe ovee by he ffee equo wh coeffces ve by we e he souo o o be he sque eo < < < <. We use cu souo of he ffee equo he oe ve by 4 e cos 3 3 o eee he eeue os he souo o he eeue he fu o esecvey he vec he hozo ses of he sque. he u-ecocy bouy eee eho s e o sove wh coeffces ve by se he sque o subec o he bouy coos eee us. If he u-ecocy bouy eee eho ey wos shou be be o ecove oey he eeue ve by. be Ec DRBE o of eo o ob soe uec esus we fsy sceze he sque bouy o bouy eees seec 9 we-sce ou cooco os he eo of he souo o h s 9. I o he ee 6 e e o be.. esecvey. I be he uec vues of he eeue he o.5.6 s obe by us 3 h s by o esecvey s e quc cubc fuco of e e coe wh he ec souo ve by vous e eves. he o of eo of he uec vues ech cou of he be s so coue ve he s ow. I be he esus obe by us fe scezo of he bouy oe eo cooco os wh 4 6 e esee. Fo he o of he eo s obvous h hee s ovee he uec esus be ove hose be. he esus obe us 3 cubc oo of he o eeue s so sfcy oe ccue h hose fo e oo.
7 W.. A K. K. hoo be Ec DRBE o of eo be 3 Ec DRBE o of eo Fo fuhe coso of he uec vues of he eeue.5.6 wh he ec souo o esus e obe by us wh s befoe. Refe o bes 3 4. Fo he bes s obvous h oe ccue uec vues c be obe us hhe vue of. Howeve hee s oy vey sh ovee he ccucy f s cese beyo cu vue. As show bes 3 4 he euco he o of eo s o vey sfc whe he vue of s cese fo 4 o 7.
8 W.. A K. K. hoo be 4 Ec DRBE o of eo REFEREES [] D.. ees 98 Bouy Vue Pobes Govee by Seco Oe Ec Syses P oo [] D.. ees 98 A bouy e equo eho fo he uec souo of seco oe ec ffee equo wh vbe coeffces Jou of he Aus hec Socey Sees B Vo [3] W.. A J. Kusu D.. ees 996 A bouy eee eho fo seco oe ec ffee equo wh vbe coeffces Eee Ayss wh Bouy EeesEee Ayss wh Bouy Eees Vo [4] A. J. Kssb E. Dvo 996 A eeze bouy e equo fo sooc he couco equo wh sy vy he coucvy Eee Ayss wh Bouy Eees Eee Ayss wh Bouy Eees Vo [5] Y. S. P W.. A A coe vbe bouy eee eho fo ec ffee equo wh vbe coeffces oucos uec ehos Eee Vo [6].. suoo Y. Su A u-ecocy bouy eee eho e o he sey-se he couco obe of fucoy e es Eecoc Jou of Bouy Eees BEEQ [7] W.. A D.. ees. Vh 3 A u-ecocy bouy eee eho fo css of ec bouy vue obes fo ohooeeous sooc e Eee Ayss wh Bouy Eees Vo [8]. A. Bebb D. 983 Dyc yss so echcs by eve bouy eee oceue Ieo Jou of So Dycs Ehque Eee Vo [9] F. Ps J. s 997 Bouy Eee eho: Fues Acos Ofo Uvesy Pess Ofo. [] W.. A 4/5 A e-se u-ecocy bouy eee eho fo sooc he ffuso subec o secfco of eey Ae hecs ouo ess. o oo fo he ce oe us s obec efe DOI ee he foow UR o you bowse:- h://.o.o/.6/.c.4..3
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