A multi-domain boundary element analysis technique based on a row elimination-backsubstitution method for solving large-scale engineering problems

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1 Boudry Elemet d Other Meh Reduto Method XXXIII 153 A mult-dom oudry elemet ly tehque ed o row elmto-uttuto method for olvg lrge-le egeerg prolem X.-W. Go & J.-X. Hu Stte Key Lortory of Struturl Aly for Idutrl Equpmet, Dl Uverty of Tehology, P.R. Ch Atrt Th pper preet ovel ly tehque ug the mult-dom oudry elemet method (MDBEM) to olve lrge-le egeerg prolem. Frtly, oudry tegrl equto for olvg geerl het oduto d meh prolem re preeted, whh re etlhed for gle medum d re formulted term of phyl qutte t terl, oudry d terfe pot. The pre ytem of equto formulted term of oly terfe odl qutte emled ed o the three-tep vrle odeg tehque. Flly, rout ler equto oluto method preeted for olvg the pre ytem ed o row elmto--uttuto method (REBSM). Comg REBSM d MDBEM me the oudry elemet method more effet for olvg lrge prtl egeerg prolem. A umerl emple gve to demotrte the effey of the propoed method. Keyword: mult-dom oudry elemet method, Gu elmto method, row elmto--uttuto method, pre ytem of equto. 1 Itroduto The oudry elemet method (BEM) other etevely ued umerl tool olvg egeerg prolem fter the developmet of the fte elemet method (FEM). Aprt from uul dvtge metoed referee (e.g., [1]), few mportt dvtge of BEM over FEM e fgured out : 1) oly WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le) do: /be110141

2 154 Boudry Elemet d Other Meh Reduto Method XXXIII oudry of the prolem eed to e dretzed to elemet d therefore le lor requred for preprg put dt d ey for modelg omplted prolem; 2) t effet olvg th-wlled prolem fter the erly gulr tegrl re urtely evluted [2, 3]; d 3) the grdet of the phyl qutty h the me ury the phyl qutty telf, e t omputtol formulto e lytlly derved from the tegrl equto. However, BEM h heret ddvtge tht the formed oeffet mtre re fully-populted d o-ymmetr, whh lmt the le d peed of olvg egeerg prolem. I order to olve lrge-le prolem, reerher developed the multdom oudry elemet method (MDBEM) [1, 4, 5]. I MDBEM, the omputtol dom of teretg dvded to umer of u-dom; the BEM lger equto re etlhed for eh u-dom; d the glol ytem of equto formed y emlg reult of ll u-dom term of the equlrum d otee odto over ommo terfe ode. The oeffet mtr of the glol ytem of equto ed o MDBEM pre, d therefore the well-developed olver for pre ytem e employed to olve t. The ue of MDBEM ot oly mprove the effey oth prolem le d omputtol peed, ut lo olve frture prolem y dvdg u-dom log r urfe [6] d mult-med prolem y dvdg u-dom log terfe [7]. I MDBEM, the emlg ll of the ytem of equto dretly ffet the omputtol effey. So fr, umer of emlg tehque hve ee propoed [1, 4, 5]. The mple oe to put ll uow t outer oudry ode, d dplemet d trto t terfe ode the uow of the ytem [1]. Suh emlg tehque ey for odg, ut t me the ze of the ytem of equto huge, lmtg the plty of olvg lrge prolem. The effet emlg tehque the vrle odeg method [4, 5], whh ome vrle re elmted frt d oly prt of vrle re erved the fl ytem uow. Amog the vrle odeg method, the three-tep vrle odeg tehque [5] very effet, whh uow t terl d outer oudry ode well trto (or flue) t terfe re elmted tur d oly dplemet (or potetl) t terfe ode re emled uow of the ytem. Th tehque reult mllet ytem of equto d the formed oeffet mtr h hgher prty, utle for olvg lrge-le prolem. Although the ytem oeffet mtr of MDBEM pre, t ot ymmetr. Therefore, the etg powerful equto olver developed for FEM ot e orrowed to olve the MDBEM ytem. New powerful olver for pre o-ymmetr ytem eed to e developed. Uully, there re two type of umerl oluto method for ler ytem of equto: dret d tertve method. I dret method, uh Gu elmto, Gu-Jord elmto, d LU-ftorzto method, the wer e oted predtle umer of operto [8]. I tertve method, uh the Jo method, Cougte grdet, d GMRES, my tep re eery ttemptg to overge to the dered wer [9]. To eep omputtol WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

3 Boudry Elemet d Other Meh Reduto Method XXXIII 155 effey, the etg dret method eed to tore whole oeffet mtr ore d therefore they re ot utle for olvg lrge prolem. A tertve method re operted ed o mtr-vetor produt, lrge ytem e olved. However, tertve method, e the ppromto oluto re modfed t eh tertve tep to pproh the true wer, overget oluto ot gurteed for ll ytem of equto. To overome the defey of the etg dret method the requremet of lrge torge, ovel dret method preeted the pper ed o row elmto--uttuto method (REBSM). I th method, oth elmto d -uttuto proedure re ompleted the me row uder oderto, d therefore, o lter -uttuto proedure requred. Compred to the etg dret method, uh the Gu elmto, the preeted REBSM requre le dt torge, o t e ued to olve lrger ytem of equto. Alo, e REBSM e ppled to ytem of o-ymmetr mtre, t dopted th pper to olve the MDBEM ytem of equto. 2 B oudry tegrl equto het oduto d old meh I th pper, the het oduto d old meh prolem re erved the reerh groud. However, the reult e eteded to other prolem. 2.1 Geerl oudry-dom tegrl equto for het oduto The otrol equto for geerl het oduto prolem e epreed u Q 0 (1) u the temperture, d Q re het odutvte d oure, repetvely. my e the futo of oordte d temperture u o-homogeeou d o-ler prolem. The geerl oudry-dom tegrl equto for eq (1) e derved from the oure olto method [10] u q ud Gqd V ud GQd (2) repreet the oudry of the omputtol dom, G the Gree futo, d q the het flu. 1 1 l for 2D ( 2) 2 r G (3) 1 for 3D ( 3) 4 r WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

4 156 Boudry Elemet d Other Meh Reduto Method XXXIII q q u G / 1 (4) (4) G 2 G G V (4) From eq (4) t e ee tht the oeffet eq (2) the verge vlue of the dgol term of. For otrop prolem wth ott mterl prmeter, redued to the uul odutvty, whle V= Geerl oudry-dom tegrl equto for meh The equlrum equto for old meh e epreed, 0 (5) the ody fore. The reltohp etwee the tre, tr d dplemet u [11] D l l Dlu, l (6) whh D the tre-tr ottutve teor whh ymmetr out l urpt,.e., Dl D l Dl. For o-homogeeou mterl or oler prolem, D l my e the futo of oordte or tree. From the oure olto method [10], the oudry-dom tegrl equto for eq (6) e derved : u T u d U t d V u d U d (7) U the Kelv dplemet fudmetl oluto, t the trto: 1 {-(3 4 )l( r) r, r, } for 2D ( 1) 8(1 ) U (8) 1 {(3 4 ) r, r, } for 3D ( 2) 8(1 ) r U D (9) T, l l WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

5 Boudry Elemet d Other Meh Reduto Method XXXIII 157 V ( U, l Dl ), U, l Dl, U, l D (9) 1 2( 1) D D (10) 2 whh the outwrd orml, ( 2)(1- ) wth eg the Poo rto. From eq (10) t e ee tht the oeffet eq (7) ymmetr, tht. For otrop eltty prolem wth ott mterl prmeter, wth eg the her modulu, d V =0. It oted tht, odg, ll dom tegrl pperg eq (2) d (7) re trformed to oudry tegrl ug the rdl tegrto method (RIM) [12], reultg ell-le BEM ly heme. 3 MDBEM ed o three-tep vrle odeto method The oudry-dom tegrl equto preeted ove re derved for gle dom prolem. However, prtl egeerg prolem uully re ompote truture otg of dfferet mterl. To olve uh prolem, the mult-dom oudry elemet method (MDBEM) uully employed [1, 4]. For th purpoe, the three-tep vrle odeg MDBEM [5] dopted th tudy. Thu, the dom of prolem dvded to umer of u-dom. For eh u-dom, ode re lfed to three type: elfode (ot hred wth other u-dom), ommo terfe ode, d terl ode. Itegrl equto (2) d (7) re ppled to the three type of ode, d followg lger mtr equto e etlhed for eh udom. A H u H u y G t (11) A l H u H u y G t (12) whh urpt, d repreet qutte orrepodg to the elf, ommo d terl ode, repetvely; =+ deote elf plu ommo ode relted to urret u-dom; the uow vetor otg of ll uow dplemet d uow trto t the elf ode of the udom, d y d y re ow vetor formed y multplyg pefed dplemet d trto wth orrepodg elemet of relted oeffet mtre. Frt tep: Elmtg terl dplemet u from eq (11) d (12), t follow A Hu y Gt (13) WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

6 158 Boudry Elemet d Other Meh Reduto Method XXXIII A H G y A y H G H H H H ( H ) ( H ) ( H ) 1 1 ( H ) 1 y A 1 G H (14) Seod tep: Notg tht = + d elmtg uow eq (13) yeld Hˆ u yˆ Gˆ t (15) Hˆ 1 H A ( A ) H Gˆ G 1 yˆ y A ( A ) y Thrd tep: Elmtg ommo odl trto d formg the ytem of equto. Equto (15) hold true for every u-dom. For emlg the glol ytem, t wrtte the followg form for the -th u-dom: ( ) ˆ ( ) 1 ( ) ( ˆ ( ) ( ) ( ) t G H u yˆ ) (17) () The ommo odl dplemet vetor u for the -th u-dom e epreed term of the glolly umered ommo dplemet vetor X y () ug trfer mtr Q ( ) ( ) u Q X (18) A ( A ) () The trfer mtr Q ot of 0 d 1, determed y the otet odto of dplemet t the ommo ode. Coderg otruto of ll u-dom t ommo ode, the equlrum odto of the trto tte tht ( ) 1 G (16) t 0 (19) Suttutg eq (18) to (17), d the reult to (19), the followg ytem of equto e oted A X B (20) 1 ˆ ( ) ˆ ( ) ( ) A ( G ) H Q (21) ( ) 1 ˆ B ( G ) yˆ (22) Solvg eq (20) for ll terfe odl dplemet d uttutg them to prevou epreo, oe ot ll uow. It oted tht the mtr ( ) WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

7 Boudry Elemet d Other Meh Reduto Method XXXIII 159 A eq (20) o-ymmetr pre mtr d, therefore, equto olver for uh ytem eed to e developed. The row elmto--uttuto method( REBSM) dered the followg eto effetve oe for olvg uh type of prolem. 4 Row elmto--uttuto method (REBSM) for olvg o-ymmetr pre ler ytem of equto The ytem of equto (20) e epreed the followg form: 1 the order of the equto et. The ey de of REBSM to fd oluto of the ytem y ompletg oth elmto d -uttuto proedure wth eh row. The m dvtge of th tretmet over the Gu elmto method the le torge requremet of termedte dt. It umed tht, fter the tretmet of the frt -1 row, the followg epreo hve ee oted 1 1 (=1,2,, -1) (24) For the -th equto, we epre t follow (23) (25) Suttutg eq (24) to the frt term of the rght-hd de of the ove equto yeld: 1 1 l l (26) 1 1, (27) l1 Seprtg the -th uow from eq (26) gve 1, ( 1, 2,, ) The, uttutg eq (28) to eq (24), t follow tht l1 l l (28) (29) 1 (=1,2,, -1) (30) WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

8 160 Boudry Elemet d Other Meh Reduto Method XXXIII , ( 1,2,, - 1; 1, 2,, ) (31) Equto (30) d (28) re the ew epreo fter the tretmet of the -th equto, whh the umer of uow o the rght-hd de redued y oe ompred to the epreo of the row -1 how eq (24). Whe the tretmet of the lt row of the equto (23) fhed, the uow of eh rght-hd de dpper d the remg term eome the oluto of the ytem of equto. From the dervto proedure of ove formulto t e ee tht the feture of REBSM e lfed follow: (1) Elmto d -uttuto re performed the me row of the ytem of equto, ey for ue egeerg umerl method uh MDBEM. (2) Dt torge requremet for termedte proe dfferet dfferet row tretmet. For ytem wth full-populted mtr A, the mmum torge our the mddle prt of A d the requred torge ze qurter of A, eg the hlf of wht Gu elmto method requre. (3) From eq (31) t e ee tht oly o-zero elemet eed to e tored for pre ytem, d o ymmetrl d defte properte o A re requred. Therefore, REBSM utle for ue MDBEM. (4) If the oeffet mtr A ot domted y the dgol elemet, the vlue of eq (29) my e zero or very mll. I th e, pvotg eery to eure urte reult. Th ey to fulfll. Wht oly eed to do tht the mmum elemet mog ( 1, 2,, ) determed y eq (27) ped up d ll relted elemet th olum re ehged wth thoe the -th olum. 5 Numerl emple Bed o the method preeted th pper, ode med BERIM h ee wrtte d orrugted dwh truture ueted to dtruted lod (Fg.1) h ee lyzed. The upper d lower over plte of the truture re mde of el lloy wth the Poo rto =0.25 d Yog modulu E=50GMP; the orrugted ret re mde of ttum wth mterl properte of =0.25 d E= 250GMP. The legth, wdth d the of the plte re 4m, 2m d 0.05m, repetvely, d two plte re ped y 1m; the the d p of ret re 0.04m d 0.8m, repetvely. I omputto, the lower over fed d upper over ueted to dtruted preure lod of 0.5MP. The truture dvded to 22 u-dom, how Fg.1. The urfe of the truture dretzed to 7808 eght-oded qudrt oudry elemet (Fg.2) wth oudry ode, mog whh 1782 re ommo ode wth WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

9 Boudry Elemet d Other Meh Reduto Method XXXIII F=0.5MP z y Fgure 1: Corrugted dwh truture uder dtruted lodg. Fgure 2: BEM meh of the orrugted dwh truture. Dplemet uy(mm) ANSYS -3 BERIM X-oordte(m) Fgure 3: Vertl dplemet log md-le of upper plte urfe. WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

10 162 Boudry Elemet d Other Meh Reduto Method XXXIII 4 Dplemet u(10-4m) ANSYS -3 BERIM -4 -oordte(m) Fgure 4: Horzotl dplemet log md-le of upper plte urfe. the degree of freedom eg The mmum d wdth of the ytem of equto 1743 whh how hgher prty. The omputtol tme for olvg uh ytem 107 o PC omputer wth the CPU of 3.0GHz. For ompro, the prolem lo omputed ug the fte elemet oftwre ANSYS wth the model otg of 3360 old 186 r elemet d 3360 ode. Fg.3 d 4 how the urve of omputed vertl d horzotl dplemet log -dreto over the mddle le of the upper urfe. It e ee tht the urret reult (BERIM) re good greemet wth thoe from ANSYS. 6 Coluo A mult-dom oudry elemet method h ee preeted for olvg lrgele egeerg prolem. The ytem of equto emled ug the threetep vrle odeg tehque h the feture of the mllet order d hgher prty; the row elmto--uttuto method (REBSM) effet tehque for olvg o-ymmetr d defte pre ytem of equto, requrg le omputer torge d utle for ug MDBEM. Aowledgemet The uthor grtefully owledge the Ntol Nturl See Foudto of Ch for fl upport to th wor uder Grt NSFC No Referee [1] Bre, C.A. & Domguez, J. Boudry Elemet: Itrodutory Coure, MGrw-Hll Boo Co., Lodo, WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)

11 Boudry Elemet d Other Meh Reduto Method XXXIII 163 [2] Luo J. F., Lu Y.J., Berger E. Aly of two-dmeol th-truture (from mro- to o-le) ug the oudry elemet method. Computtol Meh, 22, pp ,1998. [3] J. Wg d X.W. Go, Struturl mult-le oudry elemet method ed o elemet udvo tehque, Chee Jourl of Computtol Meh, 27(2),pp , [4] Ke J.H., Khv Kumr B.L., Sgl S. A rtrry odeg, oodeg oluto trtegy for lrge le, mult-zoe oudry elemet ly. Comput Meth Appl Meh Eg,79, pp , [5] X.W. Go, L. Guo, Ch. Zhg, Three-tep mult-dom BEM olver for ohomogeeou mterl prolem, Egeerg Aly wth Boudry Elemet, 31, pp ,2007. [6] Ch. Zhg, M. Cu, J. Wg, X.W. Go, J. Slde, V. Slde. 3D r ly futolly grded mterl. Egeerg Frture Meh, 78,pp ,2011. [7] X.W. Go, K. Yg. Therml tre ly of futolly grded mterl truture ug oudry elemet method, Chee Jourl of Theoretl d Appled Meh,43(1), pp , [8] Strg, Glert. Itroduto to Ler Alger (3rd ed.). Welleley, Mhuett: Welleley-Cmrdge Pre, pp.74 76, [9] Sd Y. Itertve Method for Spre Ler Sytem. SIAM, Seod edto, [10] X.W. Go. Soure pot olto oudry elemet method for olvg geerl otrop potetl d elt prolem wth vryg mterl properte, Egeerg Aly wth Boudry Elemet, 34, pp ,2010. [11] X.W. Go d T.G. Dve, Boudry Elemet Progrmmg Meh, Cmrdge Uverty Pre, [12] X.W. Go. A oudry elemet method wthout terl ell for twodmeol d three-dmeol eltoplt prolem. ASME Jourl of Appled Meh, 69, pp , WIT Trto o Modellg d Smulto, Vol 52, 2011 WIT Pre ISSN X (o-le)