Advances in Boundary Element Techniques IX 317

Transcription

1 Advances n Bounday Elemen Technques IX 37 Soluon of Nonlnea Reacon-Dffuson Equaon by Usng Dual Recpocy Bounday Elemen Mehod wh Fne Dffeence o Leas Squaes Mehod G. Meal,a M. Teze-Sezgn,b Depamen of Mahemacs, Mddle Eas Techncal Unvesy, 653,Ankaa, Tukey. a gulnhal@meu.edu., b mun@meu.edu. Keywods: Dual Recpocy Bounday Elemen Mehod, Fne Dffeence Mehod, Leas Squaes Mehod, Nonlnea Reacon-Dffuson Equaon. Absac: In hs sudy, he sysem of me dependen nonlnea eacon-dffuson equaons s solved numecally by usng he dual ecpocy bounday elemen mehod (DRBEM). As he me negaon mehod boh he fne dffeence mehod (FDM) wh a elaxaon paamee and he leas squaes mehod (LSM) ae made use of. The DRBEM s appled fo spaal devaves keepng he nonlnea em and he me devave as nonhomogeny. The esulng me dependen sysem of odnay dffeenal equaons (ODE) s solved by usng he FDM wh a elaxaon paamee as well as he LSM fo obanng accuae and compuaonally effcen esuls. The compuaons ae caed ou fo one nonlnea eacon-dffuson equaon whch has an exac soluon and fo one sysem of eacon-dffuson equaons. The soluon obaned wh boh mehods agee well wh he exac soluon n he fs example and wh he ohe numecal esuls n he second example. The compason of boh me negaon schemes shows ha he FDM wh a elaxaon paamee gves bee accuacy when he opmal value of he paamee s used. Ths makes he soluon pocedue me consumng and compuaonally expensve compang o he LSM whch s a dec applcaon.. Inoducon The nonlnea eacon dffuson equaon as well as he sysem of nonlnea eacon-dffuson equaons ae vey aacve n ecen yeas, snce hey have paccal applcaons n many felds of scence and engneeng. A numbe of combned mehods fo he me dependen paal dffeenal equaons s appled n he leaue. Fo solvng hese poblems classcal mehods dsceze he spaal doman of he poblem wh one of he known mehods such as bounday elemen mehod(bem), fne elemen mehod(fem), dffeenal quadaue mehod(dqm) and fne dffeence mehod(fdm); hen he esulng sysem of me dependen equaons s solved by usng he me negaon schemes such as FDM, RKM(Runge-Kua Mehod),FEM, LSM ec. In he lnea case, fo he one-dmensonal convecon-dffuson equaon and fo he wo-dmensonal dffuson equaon, he FDM s used n boh he space and he me decons n []and [], especvely. Also, he hea and he hea - conducon equaons ae solved by he DRBEM and wh one and wo sep LSM n [3] and [4], especvely. They have also examned he hee and fou sep LS schemes fo he hea equaon n [5] and have found ha one and wo sep mehods ae moe effcen. Recenly, he nonlnea eacon-dffuson equaon s solved by he combned applcaon of DRBEM and FDM n [6] and wh DQM and FDM as well as he one-dmensonal Fshe s equaon n [7]. In he sudy of [8], he wo-dmensonal eacon-dffuson Busellao sysem s solved by usng DRBEM fo space and FDM fo me devaves whch ncludes a lnezaon pocedue. In he pesen pape, he nonlnea eacon-dffuson equaon as well as he sysem of nonlnea eacon-dffuson equaons s solved by usng he DRBEM fo he spaal devaves.then fo he me negaon wo dffeen schemes ae noduced, namely he FDM wh a elaxaon paamee

2 38 Eds: R Abascal and M H Alabad and he LSM. Then he effec on he convegence behavou of he poposed FDM and he compason wh he LSM ae pesened. We also compae he compuaonal effcency (based on he compuaonal me and accuacy) of boh mehods.. Defnon of he Poblem We consde he followng sysem of nonlnea eacon-dffuson equaons u j = ν u j + p j (x, y) Ω, > j = o j = () wh he nal and mxed-ype bounday condons u j (x, y, ) = g j (x, y) () β j (x, y, )u j + γ j (x, y, )q j = (x, y) Γ,> (3) whee Γ s he bounday of he doman Ω and q j = uj, n beng he ouwad nomal on he bounday. n Hee he nonlneay p j depends on he unknowns,.e. fo a sngle equaon he nonlneay s p (u ) and fo he sysem, he nonlneaes ae p (u,u ) and p (u,u ) fo he fs and second equaons n Eq.. 3. DRBEM Fomulaon Eq. can be weghed by he fundamenal soluon u = ln of Laplace opeao n ode o π oban ν(c (u j ) + (q u j u q j )dγ) = ( u j Γ Ω p j)u dω (4) afe he applcaon of he Dvegence Theoem [9]. Hee denoes he souce(fxed) pon, q = u n and c = (x Ω,y,x,y)dΩ. The nonhomogeny,.e. he me devave and he nonlnea em, can be appoxmaed usng adal bass funcons f j (x, y) esulng wh a lnea sysem of equaons o be solved [F ] {α ()} = {b} (5) whee N and L ae he numbe of bounday and seleced neo nodes, especvely, and [F ] conans f j s as coloumns. The adal bass funcons f j ae elaed o ohe dsance funcons û j (x, y) hough he elaon û j = f j n ode o oban he bounday negal only fom afe he applcaon of he Dvegence heoem [].Ths leads us o he followng max-veco fomulaon afe he usage of he elaonshp (5) and subsuon of he nonhomogeny veco b ( ] [ ]) ({ } ) u ν ([H] {u j } [G] {q j })= [H] [Û [G] ˆQ [F ] {p (u)} whee H and G denoe he whole maces of bounday elemens wh kenels q and u, especvely. Û and ˆQ compomse he coodnae funcon column vecos û j and ˆq j. The szes of all he maces n (6) ae (N +L) (N +L) and he vecos ae of sze (N +L). Defnng a new (N +L) (N +L) max C as [C] = ( [H] [Û ] [ ]) [G] ˆQ [F ]. (7) (6)

3 Advances n Bounday Elemen Technques IX 39 Eq. 6 can be eaanged as { } uj [C] + ν [H] {u j } ν [G] {q j } =[C] {p j }. (8) 4. Tme Inegaon 4. Fne Dffeence Mehod The sysem of odnay dffeenal Eq.s 8 fo he unknown u j can be wen as { uj } = {p j } ( βj γ j [Ḡ] + [ H] ) {u j } (9) afe he applcaon of he bounday condon (3) wh he maces [ Ḡ ] = ν [C] [G], [ H] = ν [C] [H]. In hs subsecon we wll employ Eule scheme fo he me devave [] ( ( ) { } { } u m+ j = u m {p } j + m βj [Ḡ] [ {u } ) j + H] m j. () γ j Snce he mehod s explc, he sably poblems can be encouneed and mus be aken caefully. A elaxaon pocedue s employed wh a paamee µ fo he unknown u j n he fom u j =( µ)u m j + µu m+ j posonng he values of u j beween he me levels m and m +.Then he Eq. akes he fom ( I + µ ( d m+ j [ whee d m β m j = j γj m 4. Leas Squaes Mehod [Ḡ] [ { } ( ( [Ḡ] [ { } { } + H])) u m+ j = I ( µ) d m j + H])) u m j + p m j () ] evaluaed a he me level m. When he bounday condons ae appled o he Eq. 8, hen can be ewen as { } uj [C] +[K j ] {u j } =[C] {p j } () whee [K j ]=ν ( [ βj [H]+ γ j ] ) [G]. In a ypcal me elemen of lengh we appoxmae vecos u j and q j as u j Φ () u m j +Φ () u m+ j whee φ () = ξ, φ () =ξ wh ξ = m ae he lnea nepolaon funcons.we consuc he eo funconal Π j as n [4] Π j = m m+ T d (3) whee s he esdual veco, whch s obaned by subsung he appoxmae soluon n Eq.. The LSM soluon s obaned afe mnmzng he unknown u j fom he sysem of equaons as [A] { } u m+ j =[B ] { } u m j +[B ] { } p m j whee (4) ( [A] = [C]T [C]+ ( ) [C] T [K j ]+[K j ] T [C] + ) 3 [K j] T [K j ]

4 3 Eds: R Abascal and M H Alabad µ =.5 =. =. =. =.5 =. =. =.5 = ( 3).( ) 7.7.3(-3).6(-) 4.(-).8.( 3).( 3).3( ).(-3).8(-3).9(-) 4.6(-).9.( 3).3( 3).4( ).(-3) 3.3(-3).3(-) 5.3(-) Table : Maxmum absolue eos a small mes wh FDM fo Poblem µ =8. =. =. =. =.5 =. =. =. =.5 = ( 4) 5.7(-3).( ) ( 6).7(-5) 3.( 5).8 4.5( 4) 8.7( 4) 6.(-3).3( ) 6.( 7) 3.6( 6).8(-5) 3.5( 5).9 4.5( 4) 9.( 4) 6.5(-3).4( ) 6.( 7) 3.8( 6).8(-5) 3.8( 5) ( [B ]= [B ]= Table : Maxmum absolue eos fo seady sae wh FDM fo Poblem [C]T [C] (( ) [C] T [C]+ [K j] T [C] ( ) [C] T [K j ]+[K j ] T [C] ) 6 [K j] T [K j ] ). The soluon of he sysem of nonlnea eacon-dffuson equaons s obaned fom he soluon of he algebac Eq.s o 4. Noe ha n boh me negaon schemes he nonlneay p (u) s appoxmaed only a he me level m n ode o oban a lnea sysem of equaons a he end. 5. Numecal Resuls Poblem.We consde he nonlnea eacon-dffuson equaon [] wh ν =, p (u )= u ( u ) n he squae egon {(x, y) : x, y } fo. The nal condon and he mxed ype bounday condons ae aken appopae wh he exac soluon u (x, y, ) =. +e To measue he qualy of he appoxmae soluons wh boh p(x+y p) mehods we use he maxmum absolue eo whch s defned by max u exac u num whee u exac,u num denoe he exac and numecal soluons fo he poblem. The esuls n ems of hese maxmum absolue eos fo dffeen me ncemens ae pesened n Tables - a he pon x = fo seveal values of elaxaon paamee. Table 3 shows maxmum absolue eos fo he LSM soluon a he same me levels wh he same me ncemens, agan a x =. One can see fom hese ables ha, he me ncemen =. s he gh choce fo boh he mehods bu an opmal value of elaxaon paamee s equed n FDM. Ths s of couse compuaonally expensve pocedue. The LSM s pefeed as a dec mehod alhough hee s a one dg dop fo small mes and wo dgs dop fo seady-sae n he accuacy. Poblem. Nex we solve he sysem of nonlnea eacon-dffuson equaons wh he nonlneaes p (u,u )=+u u 3u, p (u,u )= u u u wh ν = n he same un squae 5 =.5 =. =8. =.. 5.3(-) 6.3(-).7 9.6(-)..8(-) 6.9(-) 6.5(-).6(-4).5.(-).6(-).4(-) 9.6(-4)..3(-).(-).4(-) Table 3: Maxmum absolue eos wh LSM fo Poblem

5 Advances n Bounday Elemen Technques IX 3 n poblem, subjec o he nal and Neumann ype bounday condons ( (u (x, y, ),u (x, y, )) = x 3 x3, y ) 3 y3 (x, y) Ω ( u n, u ) =(, ) (x, y) Γ,>. n Fgue : Soluon of Poblem Fo hs poblem we canno compae he convegence of boh mehods snce we do no have any exac soluon. Bu we see ha boh mehods show he same expeced behavou wh he efeence soluon n [8],.e. u and u / as me nceases. We have found he opmal value of he elaxaon paamee n FDM as.3 fo he soluon. Fg. shows ha he behavou of he soluon agees wh he behavou of he soluon n [8]. The soluon s obaned wh a consdeable small numbe of bounday elemens (N=8) n he use of boh FDM and LSM. 6. Concluson The sysem of nonlnea eacon-dffuson equaons s solved by usng he couplng of he mehod DRBEM n space wh boh he FDM and he LSM n me. The DRBEM esuls n a sysem of ODE s n me and FDM(Eule) wh a elaxaon paamee gves que good accuacy whou he need of vey small me ncemen. The opmal value of elaxaon paamee s me consumng. In he LSM he accuacy dops one o wo dgs n small and lage me values, especvely bu does no eque vey small me ncemen and elaxaon paamee. The DRBEM eques vey small numbe of bounday elemens fo obanng a easonable accuacy when s coupled ehe wh FDM o LSM me negaon schemes. Refeences [] M.M. Chawla, M.A. Al-Zanad, M.G. Al-Aslab, Compues and Mahemacs wh Applcaons, 39,7-84(). [] M.M. Chawla, M.A. Al-Zanad, Compues and Mahemacs wh Applcaons, 4, 57-68(). [3] K. M. Sngh and M. S. Kala, Engneeng Analyss wh Bounday Elemens,8, 73-(993). [4] K. M. Sngh and M.S. Kala, Compue Mehods n Appled Mechancs and Engneeng, 9, -3().

6 3 Eds: R Abascal and M H Alabad [5] K. M. Sngh and M.S. Kala, Communcaons n Numecal Mehods n Engneeng,, 45-43(996). [6] G. Meal, Poceedngs of he Inenaonal Confeence BEM/MRM 7, 33-4(5). [7] G. Meal and M. Teze-Sezgn, Inenaonal Jounal of Compue Mahemacs, (n pn). [8] W.T. Ang, Engneeng Anlayss wh Bounday Elemens, 7, (3). [9] C.A. Bebba and J. Domnguez, Bounday Elemens an Inoducoy Couse, Compuaonal Mechancs Publcaons McGaw-Hll book company(99). [] P.W. Padge, C.A. Bebba and L.C. Wobel, The dual ecpocy bounday elemen mehod, Compuaonal Mechancs Publcaons Elseve Appled Scence(99).

7 Advances n Bounday Elemen Technques IX 33 A Galekn Bounday Elemen Mehod wh he Laplace ansfom fo a hea conducon neface poblem Roman Vodčka Techncal Unvesy of Košce, Faculy of Cvl Engneeng, Vysokoškolská 4, 4 Košce, Slovaka oman.vodcka@uke.sk Keywods: bounday elemens, doman decomposon, non-machng meshes, hea conducon, Laplace ansfom. Absac. The soluon of a hea conducon poblem wh doman decomposon by a Laplace-ansfom mehod based on a bounday elemen echnque s eaed. A sees of he complex valued bounday value poblems fo Helmholz equaon s solved n he fequency doman by he complex Symmec Galekn Bounday Elemen Mehod and subsequenly ansfomed back o he me doman by he nvese Laplace ansfom. The algohm of he doman decomposon soluon ncludes geneally cuved nefaces and ndependen meshng of each sde of an neface poducng non-machng bounday elemen meshes. The examples pesen obaned numecal esuls and hey ae compaed wh analycal soluons. Inoducon Tme-dependen poblems ha ae modeled by nal-bounday value poblems (IBVP) can be eaed by bounday negal equaon (BIE) mehod. Such mehods ae wdely and successfully beng used also fo numecal modelng of poblems n hea conducon. A nce suvey of BIE applcaons fo me dependen poblems s gven n []. The pesen pape noduces an appoach ulzng he Laplace ansfom. Such mehods solve he poblems n fequency doman, usually fo complex fequences. Fo each fxed fequency, he hea conducon poblem educes o a BIE fo a bounday value poblem (BVP) of he modfed Helmholz equaon. The ansfomaon back o he me doman ncludes specal mehods fo nveson of he Laplace ansfom guded by he choce of he me dependence appoxmaon. The spl of he space doman no seveal pas (due o physcal popees of maeals o fo a paallelzaon of he solvng pocess ec.) usually ncludes doman decomposon echnques o be used [6, ]. In he pesen appoach, hese mehods due o nefaces ae appled vey naually o he BIE sysem fo he Helmholz equaon solved n he fequency doman as BIEs use unknowns decly on he bounday so ha echnques deved fo ellpc poblems can be appled. The Bounday Elemen Mehod (BEM) as a numecal ool fo solvng BIE s also used n hs conex [4, 5, 8]. The numecal mplemenaon of a cuved neface s vey mpoan s naually ncluded n he dscussed fomulaon ogehe wh he non-machng meshes along boh sdes of nefaces. As documened n he afoemenoned papes, he cuved boundaes pesen a song ably of each poblem wh an neface appoach. Vaous numecal mplemenaons of he daa ansfe beween wo non-machng meshes va calculaon of negals ove he dscezed cuved sufaces has been gven n []. The pesen appoach uses he mplemenaon of daa ansfe usng an auxlay neface mesh efeed o as common-efnemen mesh. The hea conducon poblem wh an neface Le us consde a body defned by a doman, Ω R n a fxed caesan coodnae sysem x ( =, ), wh a bounded Lpschz bounday Ω =Γ. Noe, ha Γ may nclude cones bu no cacks and cusps. Le Γ S Γ denoe he smooh pa of Γ,.e. excludng cones, edges, pons of cuvaue jumps, ec. Le n denoe he ouwad un nomal veco defned on Γ S. Alhough he developed fomulaon s vald n 3D space as well, fo he sake of smplcy we confne ouselves only o he analyss n D connuum.

8 34 Eds: R Abascal and M H Alabad The pesence of nefaces cause he doman Ω o be spl no seveal pas. Fo he sake of smplcy, le us consde a spl no wo non-ovelappng pas Ω A and Ω B whose especve boundaes we denoe Γ A and Γ B. Thee exss a common pa of boh boundaes Γ A and Γ B, le us denoe hs couplng bounday by Γ c. The nal-bounday value poblem wh an neface (IBVPwI) of a hea conducon poblem fo a empeaue dsbuon u(x,) whou volume hea souces can be saed as u η (x,) a η u η (x,)=, x Ω η, >, η = A, B, (a) u η (x, ) = u η (x), x Ωη, (b) u η (x,)=g η (x,), x Γ η u, q η (x,)= k η uη (x,) = h η (x,), x Γ η n q, (c) u A (x,) u B (x,)=, q A (x,)+q B (x,)=, x Γ c,, (d) whee he maeal paamee a η = kη c η ρ ncludes specfc hea c η, densy ρ η and hemal conducvy k η. The η spl of each bounday Γ η due o he bounday and neface condons can be wen as: Γ η = Γ η u Γ η q Γ c ( =Γ η u Γ η =Γη u Γ c =Γ η Γ c). Theefoe, funcons g η (x,) and h η (x,) noduce gven bounday condons, whle he funcon u η (x) defnes he nal condon. The unlaeal Laplace ansfom s appled o oban he soluon n he fequency doman, le he ansfom empeaue soluon be u(x,p)= u(x,)e p d, Rp σ, () wh σ beng he abscssa of convegence fo he Laplace ansfom. Moeove, le he bounday condons beng defned by funcons wh sepaaed vaables,.e. g η (x,)=gx(x)g η η (), hη (x,)=h η x(x)h η (). Then he IBVPwI () s ansfomed no a doman BVP wh an neface (BVPwI) fo he modfed Helmholz equaon wh he paamee p pu η (x,p) a η u η (x,p)=u η (x), x Ωη, (3a) u η (x,p)=gx(x)g η η (p), x Γη u, q η (x,p)=h η x(x)h η (p), x Γη q, (3b) u A (x,p) u B (x,p)=, q A (x,p)+q B (x,p)=, x Γ c. (3c) Naually, he empeaue soluon u(x,) s fnally found by he nvese Laplace ansfom of u(x,p). The bounday negal equaon sysem The poblem (3) can be solved by a sysem of BIEs fo a fxed paamee p. The fundamenal soluon of he govenng dffeenal equaon and s necessay devaves ae known hey ae gven by he modfed Bessel funcons of he second knd K (z), ( =,, ). All necessay funcons and devaves can by noduced by he followng elaons p U η (x, y,p)= (πk η ) K ( x y ), a Qη (x, y,p)= k η Uη (x, y,p) n x, Q η (x, y,p)= k η Uη (x, y,p) n y, D η (x, y,p)=k η U η (x, y,p) n x n y. (4) The BIE sysem fo BVPwI o be solved ncludes he bounday condons (3b) and also he neface condons (3c) n a weak fom when solved by he Galekn mehod. The fomulaon s based on a vaaon echnque deved n [8]. In ode o we he BIE sysem n a compac and anspaen max fom, we noduce an opeao noaon: ω η Zsw η η s = ω η (y)z η (x, y,p)w η (x,p) dγ(x) dγ(y), (5a) Γ η Γ η s ω η Z η Ω wη = ω η (y)z η (x, y,p)w η (x) dω(x) dγ(y), Ω η (5b) Γ η

9 Advances n Bounday Elemen Technques IX 35 whee ω sands fo a weghng funcon, w sands fo u o q, and s sand fo u, q o c, Z sands fo U, Q, Q o D. Moeove, he nne negal can be egula, sngula o, n he case of (5a), also Hadamad fne-pa negal. Heenafe, ndces u, o c noduce a escon of a funcon o of an opeao o a pa of bounday Γ wh he same ndex. The max fom of he esulng sysem hen eads ϕ A T U u uu A Q A uq Uuc A Q A uc ϑ A q Q A qu D ϕ A qq A Q A qc Dqc A c U ϑ A cu A Q A cq Ucc A IA cc+q A cc Icc AB c Q ϕ B cu Dcq A IA cc+q A cc Dcc A u U ϑ uu B Q B uq Uuc B Q B uc q ϕ Q B qu Dqq B Q B qc Dqc B c Ucu B Q B cq U B cc ϑ B c Icc BA Q B cu Dcq B IB cc+q B cc Dcc B IB cc+q B cc q A u u A q q A c u A c q B u u B q q B c u B c ϕ A T Uuq A u IA uu Q A uu ka U a A uω A ϑ A q IA qq Q A qq D A k uu A a A QA uω ϕ A c Ucq A Tcu A ka = ϑ A U a A cω A c Q A ϕ B cq D A k cu A a A QA cω u ϑ B Uuq B IB uu Q B uu kb U a q B uω B ϕ B c IB qq Q B qq D B k uu B a B QB uω ϑ B Ucq B Q B cu kb U B c a B uω Q B cq Dcu B k B a B QB uω whee I s fomal epesenaon (5) of he deny (I η ) and pojecon (I AB cc h A g A h B g B u A u B, (6), Icc BA ) opeaos. Funcon ϕ η and ϑ η should be chosen o fom he bass of he funcon spaces, whee he funcons q η and u η, especvely, belong o. The found funcons solve (3), whle he soluon of he ognal IBVPwI of hea conducon can be found by nvese Laplace ansfom. Some deals of he numec pocedue appled wll be noed n wha follows. Noes on he numecal soluon Bounday elemen appoxmaon. The numecal soluon of BIE sysem (6) by a bounday elemen echnque ncludes dscezaon of he bounday Γ η of each subdoman Ω η by W η e bounday elemens Γ ηk, k =,,...W η e. Fo makng he expessons smple, le us om he supescp ndex η n hs paagaph. The pesen dscezaon ulzes confomng sopaamec elemens. Le us consde an n-h ode polynomal paameezaon of k-h bounday elemen Γ k gven by he elaon x = N k (ξ), ξ,. Then he appoxmaon of he funcons u(x,p), q(x,p) can be wen n he followng fom W u u(x,p)= ϑ m (x)u m, m= W q q(x,p)= ϕ m (x)q m, (7) whee he nodal shape funcons ϑ m (x), esp. ϕ m (x) ae equal o elemen shape funcons N kj (x) fo some k, j such ha x Γ k, j {,,...n+}. Moeove, he funcons ϑ m should be connuous, whle he funcons ϕ m do no have o be. Neveheless, s clea ha all he funcons defned by he paameezaon ae smooh ogehe wh all he devaves along an elemen. The elaon beween he paameezaon of bounday N k (ξ) and he elemen shape funcons N kj (x) s N kj (x) =N kj (N k (ξ)) = N j (ξ), whee he polynomals N j (ξ) fom he bass of he n-h ode polynomals space. An example of he second ode connuous elemens and he paameezaon s shown on Fgue. I can be seen ha he suppo of a funcon ϑ m (x) can cove up o wo neghbou elemens: compae ϑ m (x) and ϑ m+ (x). Calculaon of negals. The dscezaon of he BIE sysem (6) also ncludes calculaon of he negals (5) n he fom N l (y)z(x, y,p)n kj (x)dγ(x)dγ(y)= N (υ)z(n k (ξ),n l (υ),p)n j (ξ) N k (ξ) N l (υ) dξdυ (8) Γ l Γ k m=

10 36 Eds: R Abascal and M H Alabad ϑ m+ (x) =N (k+) (x) ϑ m+ (x) =N k3 (x) Γ k+ ξ k = Ω Γ k ξ k = Γ q ϑ m (x) =N k (x) N kj (x) =N j (ξ) N (ξ) N (ξ) ξ N 3 (ξ) Fgue : Second ode elemens and he shape funcons fo each pacula negal kenel Z(x, y,p) and fo each pa of bounday elemens Γ k and Γ l. The dea of he calculaon s o spl he negal no a egula pa and no a sngula pa f occus. The egula pas hen can be evaluaed e.g. by a sandad Gauss-Legende quadaue ule, whle he sngula pas ae eaed sepaaely accodng o he ype of he sngulay. The negals wh Z = U and Z = Q can conan only logahmc sngulaes, he ohe case Z = D may esul even n an hypesngula negal, see also [7]. In ode o demonsae he way of calculaon, le us dscuss he mos sngula case n moe deal. The p negal (8), wh espec o (4) and o he noaon =, = x y = N k (ξ) N l (υ) and c = a, eads η = N (υ) K (c) N j (ξ) N n x n k (ξ) N l (υ) dξdυ = N (υ)n j (ξ) N k (ξ) N l (υ) y {[ K (c) ln I (c)+ ] ( T n x )( T n y ) c + [ K (c) ln I (c) ] n T } x n y dξdυ+ c c N (υ)n j (ξ) N k (ξ) N l (υ) [ ln I (c) (T n x )( T n y ) + I ] (c) n T c xn y dξdυ = N (υ)n j (ξ) N k (ξ) N l (υ) [ c (T n x )( T ] n y ) n T xn y dξdυ = D R + D L c D H, (9) whee I, =, ae he modfed Bessel funcons of he fs knd. The ems ae eodeed fo he fs negal D R o conan egula funcons only, he second negal D L o conan a logahmc sngulay fo and he las em D H o be a hypesngula negal f =s a pon of he negaon doman. The logahmc sngulay should be eaed n such a way ha a suable weghed quadaue fomula could be used. Two cases should be dsngushed fo calculaon of he negal D L : fs, when he negals n (8) ae calculaed wh espec o he same elemens,.e. k = l and, second, when he elemens ae neghboung n he mesh, havng one common pon. In he fome case, he dsance funcon endes = N k (ξ) N k (υ) =N k (ξ)(ξ υ) N k (ξ)(ξ υ) + =(ξ υ)g(ξ,υ) () fo a suffcenly smooh non-vanshng veco funcon g(ξ,υ) f ξ = υ. Theefoe D L can be wen as D L = ln G(ξ,υ)dξdυ = ln g(ξ,υ) G(ξ,υ)dξdυ + ln ξ υ G(ξ,υ)dξdυ =D LR +D LL, () wh G(ξ,υ) beng he non-sngula and smooh es of he negand and wh he spl of he esul no he egula pa D LR and he sngula pa D LL. To be able o coecly apply a quadaue ule whch eques he logahmc funcon a he end pon of he neval, followng subsuons ae equed: ξ, υ,ξ : ξ = γ, υ = γ( δ), J= γ, γ, δ, (a) ξ,υ υ, : υ = γ, ξ = γ( δ), J= γ, γ, δ, (b)

11 Advances n Bounday Elemen Technques IX 37 o oban D LL = ln(γδ) G(γ,δ)γdγdδ = γ ln γ G(γ,δ)dγdδ + [ ] ln δ γ G(γ,δ)dγ dδ. (3) In he lae case, he elaon o be sasfed fo bounday paameezaon s e.g. N k () = N l (). The negaon doman s spl no wo angles along he dagonal ξ = υ, wh he followng subsuon o be appled o he angle wh he sngulay: ξ, υ,ξ : ξ = γ( δ)+,υ= γδ, J = γ, γ, δ,. (4) A he vcny of he sngulay, he dsance funcon can be expessed as = N k (γ( δ)+) N l (γδ) =N k () [γ(δ )] N l () (γδ)+ = γh(γ,δ), (5) wh a funcon h(γ,δ) suffcenly smooh and non-vanshng fo γ =, as he Lpschz condon supposed fo he bounday makes he expesson N k () (δ ) N l () δ no o vansh. The negal D L obeys he elaon D L = υ ln G(ξ,υ)dξdυ + D LR + ln(γ h(γ,δ) ) G(γ,δ)γdγdδ = γ ln h(γ,δ) G(γ,δ)dγdδ + γ ln γ G(γ,δ)dγdδ, (6) whee nehe he fs em D LR he negal ove he angle whou sngulay, no he second em conan any sngula funcon and only he las negal ncludes a logahmc funcon wh zeo agumen, hough smoohed by Jacoban γ. The las em n (9), D H, eques exceponal eamen as ncludes hypesngula negal. Ths sngulay has been obaned n a lm pocedue, whee a pon x a he bounday Γ(x) has been appoached by movng a pon y fom he neo of he doman Ω. Theefoe, fo he calculaon of he negal, he bounday Γ(y) can be shfed by a small ε nwads o Ω esulng n an negal, whch does no conan any sngulay. I s also useful o calculae he negal D H noducng a complex funcon z(ξ,υ) = ( N k (ξ) N l (υ)) + ( N k (ξ) N l (υ)) because a smple calculaon va negaon by pas endes followng esul D H = R N (υ)n j z(ξ,υ) z(ξ,υ) (ξ) z dξdυ = (ξ,υ) ξ υ [ ] ξ= [ N j (ξ) ln N (υ)dυ N (υ) ln N j (ξ)dξ + ξ= [ [ ln N (υ)n j (ξ) ] ] ξ= υ= ξ= ] υ= υ= υ= + ln N j (ξ)n (υ)dξdυ. (7) The basc polynomal funcons N j (ξ) and N (υ) can be smoohly dffeenaed along an elemen, moeove hey ae (aleady paameezed) escons o pacula bounday elemens of connuous nodal shape funcons ϑ mj (x) and ϑ m (y), especvely. In he calculaon, he negals ove he suppo of penen ϑ- funcons, whch vansh a he end pons of he suppo, can be gaheed ogehe. Theefoe, he hee fee ems also vansh and only he las negal emans. Fnally, he afoemenoned lm pocedue should be pefomed. The negal, howeve, has emaned only wh logahmc sngulay, whch can be eaed as above. The nvese Laplace ansfom. The algohm of numecal nvese of he Laplace ansfom due o Weeks has been appled, see [9]. I s based on he fac ha he Laplace ansfom of a funcon expessed by a sees wh espec o appopaely scaled ohonomal Laguee funcons can be easly modfed o calculae he appoxmaon of s nvese Laplace ansfom by he quae wave cosne fas Foue ansfom. The numecal pocedue ncludes seveal paamees whch should be caefully chosen o oban a quckly convegen and accuae esuls. The dscusson of he choce of he paamees has been also pesened n [3]. The paamees whch ae necessay fo calculaon ae: abscssa of convegence of Laplace ansfom σ, abscssa of

12 38 Eds: R Abascal and M H Alabad evaluaon of he nvese Laplace ansfom σ, scalng paamees of he Laguee funcons b and of he sees coeffcens evaluaon and also numbe N of funcons used fom he sees, equal o he numbe of pons n fequency doman fo he nvese ansfom evaluaon and he me neval deemned by maxmum equed me nsance max. An example An example has been chosen o documen he numecal behavou of he poposed numecal mehod. I ncludes a smple squae dvded no wo subdomans, see Fgue, wh he neface defned by a cubc splne passng hough he pons E, S and F, whee he coodnaes of S ae as follows: S [.5;.4], S [.;.], S 3 [.5;.6]. Ω B Γ B q S 3 F. Γ B q.4 O E Γ A q Γ c S S x x Γ A u Γ A q Ω A Fgue : A squae doman wh a cuved neface wh he paen of elemen nods n he neface The soluon of he poblem has been chosen so ha s analycally expessed by he sees fomula u(x,)=x x ( e ) [ ] + T m ()sn (m+)πx 4 + T nm ()sn (m+)πx 4 cos nπx, (8a) m= T m () = 3 3π(m+) λm λme +e λ m λ m(λ m ) + 3 T nm () = π (m+) [ n= ( ) m [ 8 e e λ m 3 λ m π(m+) ] +4 λm λme +e λ m λ m(λ m ) ], (8b) [ ] 8( )n π 3 n (m+) λnm λnme +e λnm 4 λ nm(λ nm ) ( ) m e e λnm π 4 n (m+) π(m+) λ nm, (8c) λ nm = n π 4 + (m+) π 6. (8d) I means ha he vanshng nal condons fo smplcy (no need o compue he volume negals n (6) due o (5b)) ae pescbed. The bounday condons, whch ypes ae shown on Fgue, ead u ((x, ),)=, q((,x ),)= 4x ( e ), (9a) q ((,x ),)=, q((x, ),)= 4x ( e ). (9b) The dscezaon by bounday elemens has been made n such a way ha he lenghs of all elemens ae appoxmaely he same. Each sde of he oue squae conou s dvded no lnea elemens. The neface s meshed accodngly: he non-machng mesh aken conans 4 equally spaced elemens along he face of he doman Ω A, he coase mesh, and 7 elemens wh espec o he ohe doman, fne mesh. The paamees of he numecal nvese of he Laplace ansfom has been se o he followng values: N =6pons n he fequency doman, abscssa of he convegence σ se o zeo, maxmum me evaluaon beng uny, paamees =.99, b =6, and abscssa of evaluaon σ beng 8. The esuls obaned n he neface has been focused on: fs, he soluon evoluon n me and s eo along he neface s shown on Fgue 3 fo he empeaue u and on Fgue 4 fo he flux q, second, he

13 Advances n Bounday Elemen Technques IX 39 u A x u An u Aa x Fgue 3: Dsbuon of he empeaue u and of s eo fom he mesh of he doman Ω A along he neface and wh espec o he me q A x q An q Aa x Fgue 4: Dsbuon of he flux q and of s eo fom he mesh of he doman Ω A along he neface and wh espec o he me dsbuon of boh funcons (u and q) and he eos ae evaluaed a a specfc me nsance on Fgue 5. All gaphs use he x -coodnae of neface pons as he abscssa fo he values of penen funcons a a me nsance. Le us commen some obsevaons. The me evoluon gaphs show nce elaon beween he eo magnudes and he hgh gadens of he soluons obaned along he coase mesh of he neface, wh espec o he doman Ω A. The eos ae naually wose fo he fluxes q. The gaphs conan he absolue eos,.e. u n u a, and q n q a, whee he supescp n denoes numecal soluon obaned by he BIE sysem (6) and he nvese Laplace ansfom and he supescp a denoes he analycal soluon obaned by (8a) uncaed whn each sum o ems, howeve, he maxmum elave eos can be esmaed fom hem: s abou. fo he empeaues u and abou.4 fo he fluxes q. The same obsevaon can be also done fom Fgue 5, whch has been made fo he maxmum me value. Moeove, he pcues also nclude he esuls obaned along he fne mesh, so ha boh daa can be compaed muually. As he soluons and he eos ae ploed n he same gaph, he muual elaon beween he magnude of he eo and he descen seepness s even moe obvous. The neface mesh paen shown on Fgue can help o undesand he oscllang chaace n he eo dsbuons: he eos canno concde because he meshes have few common pons. Neveheless, he eos ae no sgnfcan and n he cuen gaphs he esuls of boh funcons u and q, ploed fo he doman Ω B, ncely f wh he analycal soluon.

14 33 Eds: R Abascal and M H Alabad Concluson A es of a bounday elemen appoach fo solvng neface hea ansfe IBVP has been pefomed. The fomulaon ulzng he Laplace ansfom of he soluon solves he poblem n he fequency doman numecally by a sees of BVPs fo Helmholz equaon. The numecal mehod used hee ncludes he complex symmec Galekn BEM. The dea of he appoach has been aken fom a mehod pevously deved by he auhos n [8] fo poblems of elascy and seems o wok well wh he pesened example. The nvesgaons and numecal ess planned fo he poposed mehod wll gve moe goous explanaon fo s applcably n wde ange of he nal-bounday value poblems wh nefaces. u u An u Aa u Bn u Ba u B exac x u n u a q q An q Aa q Bn q Ba q B exac x q n q a Fgue 5: Resuls along he neface fo he me = Acknowledgemen The auho gaefully acknowledge he Scenfc Gan Agency VEGA fo suppong hs wok unde he Gan No. /498/7. Refeences [] Boe, A. de, Zujlen, A.H. van, Bjl, H.: Revew of couplng mehods fo non-machng meshes. Compu. Mehods Appl. Mech. Engg, 96, pp , 7. [] Cosabel, M.: Tme-dependen poblems wh bounday negal equaon mehod. In: Encyclopeda of Compuaonal Mechancs, John Wley & Sons, Eds. Sen, de Bos, Hughes, vol., chap. 5, 4. [3] Gabow, B.S., Guna, G., Lynnes, J.N., Mul, A.: Sofwae fo an mplemenaon of Weeks mehod fo he nvese Laplace ansfom poblem. ACM T. Mah. Sofwae, 4, pp. 63 7, 988. [4] Hsao, G.C., Senbach, O., Wendland, W.L.: Doman decomposon mehods va bounday negal equaons. J. Comp. Appl. Mah., 5, pp ,. [5] Lange, U., Senbach, O.: Bounday elemen eang and neconnecng mehod. Compung, 7, pp. 5 8, 3. [6] Puso, M.A.: A 3D moa mehod fo sold mechancs. In. J. Num. Meh. Engg., 59, pp , 4. [7] Vodčka, R.: On evaluaon of negals asng n SGBEM soluon of modfed Helmholz equaon. In: VIII. vedecká konfeenca Savebnej fakuly v Košcach, TU v Košcach, Savebná fakula, pp. 9 96, 7. [8] Vodčka, R., Manč, V., País, F.: Symmec vaaonal fomulaon of BIE fo doman decomposon poblems n elascy - an SGBEM appoach fo nonconfomng dscezaons of cuved nefaces. CMES Comp. Model. Eng., 7, pp. 73 3, 7. [9] Weeks, W.T.: Numecal nveson of Laplace ansfom usng Laguee funcons. Jonal of he Assocaon fo Compung Machney, 3, pp , 966. [] Wohlmuh, B.I.: Dscezaon Mehods and Ieave Solves Based on Doman Decomposon, Lecue Noes n Compuaonal Scence and Engneeng, vol.7, Spnge, Beln,.

15 Advances n Bounday Elemen Technques IX 33 Assembled Plae Sucues by he Bounday Elemen Mehod D. D. Monnea, a, J. A. F. Sanago, b and J. C. F. Telles, c Exacum Consuloa e Pojeos Lda, Rua See de Seembo, Ro de Janeo, RJ, Bazl Pogama de Engenhaa Cvl - COPPE/UFRJ, Caxa Posal Ro de Janeo, RJ, Bazl a danel.das@exacum.com.b, b sanago@coc.ufj.b, c elles@coc.ufj.bl Keywods: Bounday elemen; Plaes; Ressne s plae heoy. Absac. Ths pape deals wh he analyss of assembled plae sucues, subjeced o abay loadngs, fo whch wo dmensonal plane sess elascy and shea defomable plae bendng heoes ae coupled. To hs end, dec bounday elemen fomulaons, based upon Ressne s plae heoy and -D elascy, ae pesened fo elasosac poblems. The mul-egon echnque s employed o assemble he plaes. Seveal plaes shang common neface boundaes ae accommodaed, ncludng nclned ones. Afe a sandad coodnae ansfomaon, each egon can be combned akng no accoun dsplacemen compably and equlbum equaons n ode o oban he fnal equaon sysem.. Inoducon Sucues composed of assembled plane elemens wh close o open coss secons have been employed n seveal banch of engneeng, such as cvl, mechanc, naval, aeonaucs, ec.; manly due o he advanage of aanng hgh flexual gdy wh low self wegh. Papes by Palemo [] and Palemo e al. [] dscuss plae assembles wh close and open coss secons usng Kchhoff s plae heoy and wo-dmensonal elascy. Dganaa and Alabad [3] and Baz and Alabad [4] also analyzed assembled plae sucues unde abay loadngs usng Ressne s plae heoy and wo egons shang common nefaces. In he pesen wok, he dec bounday elemen fomulaon fo he mul-egon echnque, based upon Ressne s plae and -D elascy heoes ae pesened fo elasosac poblems, consdeng soopc maeals, small defomaons and small dsplacemens. Seveal plaes shang common neface boundaes ae accommodaed, ncludng nclned ones, genealzng pevous analyses [5]. Seveal numecal examples ae pesened and esuls ae compaed wh exac analycal and fne elemen soluons o demonsae he accuacy of he poposed fomulaon.. Bounday Inegal Equaons The equaons wll be pesened hee n ndcal noaon. Hee oman ndces vay fom o 3 and Geek ndces vay fom o. The negal equaons adoped o epesen dsplacemen componens can be wen as ) Ressne s plae bendng: * C w p, x w x d x j j j j w * j *, xp j xdx w3, x * w,, xqxdx ()

16 33 Eds: R Abascal and M H Alabad ) Two dmensonal plane sess elascy: C * * u, xu xdx u, x xdx, () whee he bounday negals on he lef-hand sde ae nepeed n sense of Cauchy s pncpal value. In Equaons and., w ae he oaons abou he x axes, u ae he dsplacemens on he plane xx and w3 s he dsplacemen n he x3 decon. The ems p j ae he bendng momens (j =, ) and shea foce (j = 3), whch ae gven as p M n and p3 Q n, especvely. On he ohe hand epesen n plane acons, gven by N n, q s he dsbued load and / h, h beng he hckness. The coeffcens C j depend on he bounday geomey a he souce pon. The kenels w * j, x and p * x j, epesen he fundamenal soluon fo plae * * bendng, wheeas u, x and, x epesen he fundamenal soluon fo plane elascy. The complee expessons fo hem can be found n efeences [6-8]. Equaons and epesen fve negal equaons pe funconal node of a sucue unde bendng and exensonal effecs n he local coodnae sysem. Thee degees of feedom come fom Resse s plae heoy and wo degees of feedom fom plane sess elascy.. The doman negal of he equaon can be ansfomed n a bounday negal applyng he dvegence heoem. Hee q s consdeed consan (unfomly dsbued load), hence Equaon can be wen as C j * w j pj, xw j xdx w * j *, xp j xdx q,, x * w, xn xdx, (3) * whee s he pacula soluon of he equaon w. The expesson fo can be found n efeence [8]. 3. Assemble of he Equaons Sysem * *, 3 Equaons and 3 epesen he basc expessons fo he soluon of spaal assembled plae poblems usng bounday elemen mehod (BEM). In geneal ems, he boundaes and nefaces of a sucue ae dscezed n elemens, fo whch dsplacemens and acons ae nepolaed by means of funconal node values. The negal equaons fo plane sess elascy and shea defomable plae bendng ae appled o all funconal nodes, n a coespondng local coodnae sysem, fo evey egon, geneang a lnea equaon sysem of he followng max fom: *, H S H u S G S P P P P 5x5 w G 5x 5x5 p 5x 5x s b, (4)

17 whee T S u u u, T P w w w 3 w, T S and T p p p 3 p ae he dsplacemens (u and w) and acons ( and p) fo plane sess elascy and plae bendng, especvely, s he doman load veco and,, and ae he elemen nfluence maces fo plane sess elascy (S) and plae bendng (P), especvely. T q b S G S H P G P H To solve he poblem he equaon sysem of each egon mus be efeed o he same global coodnae sysem. Theefoe, he local equaon sysem (efeed o he ndvdual plae) s ansfomed o he global one by employng he coodnae ansfomaon max as M u u u w w w u u M, u M u M u u T and m m m p p p M, p M p Mp p T (5) whee and T u u u 3 3 u T m m m 3 3 p epesen he dsplacemens and acons vecos, especvely, efeed o he global coodnae sysem. One can obseve n equaons 5 ha a new oaon 3 (o bendng momen ) abou he x 3 m 3-axs s added o he pevous u (o p) veco. Theefoe a new equaon pe funconal node s needed, namely a esan equaon, gven by, m m m (6) n whch j s he cosne of he angle beween x local and x j global axes. Afe a sandad coodnae ansfomaon, he sub-egons can be combned akng no accoun dsplacemen compably and equlbum of acons along he neface boundaes, n ode o oban he fnal global equaon sysem. Noce ha now sx degee of feedom pe funconal node ae consdeed. These equaons (compably and equlbum equaons) can be wen as follows: ) Dsplacemen compably u u u u u u u u u (7) ) Equlbum of acons Advances n Bounday Elemen Technques IX 333

18 334 Eds: R Abascal and M H Alabad m m m m 3 m m 3 m m 3 m, (8) whee he ndex s he sub-egon (ndvdual plae) shang common neface boundaes. Afe he bounday condons ae enfoced, he global equaon sysem can be solved n ode o poduce he global unknowns (dsplacemens and acons) on exenal boundaes and nefaces of he sucue. 4. Numecal Examples Seveal examples of 3-D assembled plae sucues unde flexual and exensonal loads, smulaneously, ae analyzed. The esuls ae compaed wh beam heoy and fne elemen soluons o demonsae he accuacy of he poposed fomulaon. 4.. Canleve I Beam In hs example a canleve beam wh an I coss secon (see fgue ) s suded. Dmensons and popees ae: L = L = 4 cm, L 3 = cm, =.5 cm, = cm, E = kn/cm and =.3. Fgue : Geomecal dmensons of he beam The beam s subjeced o a lnea dsbued load q, vayng fom -kn/cm o kn/cm, on he web of he oppose end of he vecal suppo, as ndcaed n fgue.

19 Advances n Bounday Elemen Technques IX 335 Fgue : Canleve beam subjeced o lnea dsbued load To analyze hs example, 4 elemens and funconal nodes have been employed ove he exenal boundaes and nefaces of he beam. The obaned esuls ae compaed wh beam heoy and he fne elemen mehod. Fo beam heoy he soluon s gven by M ux x, (9) EI whee M s he appled bendng momen and I s he momen of nea wh espec o he neual axs. The obaned esuls fo he vecal dsplacemens of he beam along he neual axs, ae pesened n fgue 3. w (cm), -,5 -,5 -,75 -, -,5 -,5 -,75 -, x (cm) Beam Theoy BEM FEM Fgue 3: Compason of vecal dsplacemens As seen n fgue 3, he beam heoy and boh mehods (BEM and FEM) ae n close ageemen, confmng he valdy of he poposed mehod. 4.. L-shaped Plae Sucue In hs second example hee ecangula plaes wh dffeen szes and he same hckness wee assembled n ode o fom he L-shaped plae sucue, showed n fgue 4

20 336 Eds: R Abascal and M H Alabad The geomec consans ae: L = L = cm, L 3 = L 4 = 5 cm, = 5 cm, = º. The modulus of elascy and Posson s ao ae aken o be 7 kn/cm and.33, especvely. Fgue 4: Plae assembly geomey The L-shaped plae sucue s loaded by he unfomly dsbued load q y = q z =.5 KN/cm, n he y and z decons, along he p edge of he hozonal plae, as depced n fgue 5. Fgue 5: Canleve plae subjeced o he unfomly dsbued loadng n y and z decons The poblem s modelled wh hee sub-egons, each havng 6 elemens and 4 funconal nodes along he nefaces and exenal boundaes. The esuls fo vecal and hozonal dsplacemens along he coss secon a x = 5 cm ae shown n fgue 6. These esuls ae hee compaed wh he fne elemen mehod (FEM). In ode o mpove he compason, fgue 6 pesens he defomed shape n expanded scale wh a faco of. As can be seen, he BEM esul s n excellen ageemen wh he FEM.

21 Advances n Bounday Elemen Technques IX w (cm) v (cm) Undefomed shape BEM (defomed) FEM (defomed) Fgue 6: Defomed sucue 5. Concluson The pesened analyss of sucues composed of 3-D assocaons of plane panels subjeced o co-occung bendng and exenson loads has been consdeed que sasfacoy, wh accuae esuls n compason o exsng alenave pocedues. In addon, he dscussed BEM mplemenaon has been found o lead o accepable soluons even n case of ahe coase dscezaon alenaves, employng a educed numbe of elemens. I can, heefoe, be ecommended fo such panel assembled sucues exsng n cuen engneeng pacce. Refeences [] L. Palemo J., M. Rachd and W.S. Venun: Analyss of Thn Walled Sucues usng he Bounday Elemen Mehod, Engneeng Analyss wh Bounday Elemens. Vol. 9 (99), pp [] L. Palemo J., Analyss of Thn Walled Sucues as Assembled Plaes by Bounday Elemen Mehod (n Pouguese). Thess of Doco of Scence (D.Sc.), Escola de Engenhaa de São Calos / USP, São Calos, SP, Bazl, (989). [3] T. Dganaa and M.H. Alabad, Bounday Elemen Analyss of Assembled Plae Sucues. Commun. Nume. Meh. Engng. Vol.7 (), pp [4] P.M. Baz and M.H. Alabad, Local Bucklng of Thn Walled Sucues by BEM, Advances n Bounday Elemen Technques, pp , (7).

22 338 Eds: R Abascal and M H Alabad [5] D.D. Monnea, Analyss of Assembled Plae Sucues usng he Bounday Elemen Mehod (n Pouguese), Dsseaon of Mase of Scence (M.Sc), COPPE / UFRJ, Ro de Janeo, RJ, Bazl, (8). [6] C.A. Bebba, J.C.F Telles and L.C. Wobel: Bounday Elemen Technques: Theoy and Applcaons n Engneeng, Spnge-Velag, Beln, (984). [7] L.C. Wobel and M.H. Alabad, The Bounday Elemen Mehod, Wley, Chchese, (). [8] F. Vande Weeën, "Applcaon of he Bounday Inegal Equaon Mehod o Ressne's Plae Model", Inenaonal Jounal fo Numecal Mehods n Engneeng. Vol. 8 (98), pp. -., (98).

23 Advances n Bounday Elemen Technques IX 339 Developmen of a me-doman fas mulpole BEM based on he opeaonal quadaue mehod n -D elasodynamcs Takaho SAITOH,a, Sohch HIROSE,b and Takuo FUKUI 3,c Unvesy of Fuku, 3-9-, Bunkyo, Fuku-sh, Fuku, Japan Tokyo Insue of Technology, --,O-okayama, Meguo-ku, Tokyo, Japan 3 Unvesy of Fuku, 3-9-, Bunkyo, Fuku-sh, Fuku, Japan a sao@aku.anc-d.fuku-u.ac.jp, b shose@cv.ech.ac.jp, c ak@aku.anc-d.fuku-u.ac.jp Keywods: Opeaonal Quadaue Mehod (OQM), Fas Mulpole Mehod (FMM), Tme-doman, Elasodynamcs. Absac. Ths pape pesens a new me-doman Fas Mulpole Bounday Elemen Mehod n -D elasodynamcs. In geneal, he use of dec me-doman BEM somemes causes he nsably of me-seppng soluons and needs much compuaonal me and memoy. To ovecome hese dffcules, n hs pape, he Opeaonal Quadaue Mehod (OQM) developed by Lubch s appled o esablsh he sably behavo of he me-seppng scheme. Moeove, he Fas Mulpole Mehod (FMM) s adaped o mpove he compuaonal effcency fo lage sze poblems. The poposed mehod s esed fo lage-sze elasc wave scaeng by many caves. Inoducon Snce he Bounday Elemen Mehod (BEM) s known as a suable numecal appoach fo wave analyss, me-doman ansen poblems have been solved by many eseaches usng BEM by Mansu and Bebba[],andHose[]. In geneal, ansen poblems can usually be solved fo unknown medependen quanes by a dec me-doman BEM wh a me-seppng scheme. Howeve, he use of dec me-doman BEM somemes causes wo poblems. The one s he nsably poblem of meseppng pocedue and he ohe one s compuaonal effcency poblem fo a lage sze poblem. Recenly, o ovecome he fome poblem, he Opeaonal Quadaue Mehod (OQM), poposed by Lubch[3], has been used fo he BEM fomulaon fo some engneeng poblems such as -D scala wave poblem[4], pooelasc poblem[5] and -D ansoopc poblem[6]. In he fomulaon of BEM based on OQM (OQBEM), he convoluon negal s numecally appoxmaed by a quadaue fomula whose weghs ae deemned by he Laplace ansfomed fundamenal soluon and a lnea mulsep mehod. The compuaonal complexy becomes O(LM N) fo he poblem wh M elemens, N me seps, and L expanson ems. On he ohe hand, he lae poblem sll eman because s dffcul o solve a lage scale poblem wh he lage numbe of M by usng OQBEM. In hs pape, we popose a new me-doman fas mulpole BEM based on OQM n -D elasodynamcs. The Fas Mulpole Mehod (FMM), poposed by Geengad and Rokhln[7], s appled o he OQBEM o esolve he compuaonal effcency poblem. Afe he descpon of basc concep and fomulaon of poposed mehod n -D elasodynamcs, numecal examples fo elasc wave scaeng ae demonsaed by usng he poposed mehod. The compuaonal effcency of he poposed mehod s also confmed.

24 34 Eds: R Abascal and M H Alabad Opeaonal Quadaue Mehod Fgue A elasc wave scaeng model. In hs secon, he opeaonal quadaue mehod (OQM) s befly descbed. The Opeaonal Quadaue Mehod (OQM), fs poposed by Lubch, appoxmaes he convoluon f g() by a dscee convoluon usng he Laplace ansfom of he me dependen funcon f( τ). In geneal, he convoluon negal s defned as follows: f g() = f( τ)g(τ)dτ, () whee denoes he convoluon. The convoluon negal defned by Eq. s appoxmaed by OQM as follows: f g(n ) j ω n j ( )g(j ) () whee me was dvded no N equal seps. Moeove, ω j ( ) denoes he quadaue weghs whch ae deemned by he coeffcens of he followng powe sees wh complex vaable z, namely F ( δ(ζ) )= ω n ( )z n. (3) n= In Eq. 3, F s he Laplace ansfom of he me dependen funcon f. The powe sees defned n Eq. 3 can be calculaed by Cauchy s negal fomula. Consdeng a pola coodnae ansfomaon, he Cauchy s negal s appoxmaed by a apezodal ule wh L equal seps π/l as follows: ω n ( ) = F π ζ =ρ ( ) δ(ζ) ζ n dζ ρ n L L l= F ( ) δ(ζl ) e πnl L. (4) whee δ(ζ) s he quoen of he geneang polynomals of a lnea mulsep mehod and ζ l s gven by ζ l = ρe πl/l. In addon, ρ s he adus of a ccle n he doman of analycy of F.

25 Advances n Bounday Elemen Technques IX 34 Tme-Doman BEM Fomulaon n -D Elasodynamcs We consde he -D elasc wave scaeng by a scaee D n an exeo elasc meda D as shown n Fg.. When he ncden wave u n hs he bounday suface S of a scaee D, scaeed waves ae geneaed by he neacon wh he scaee D. Assumng he zeo nal condons,.e., u (x, = ) = and u (x, =)/ =, he govenng equaons and bounday condons ae wen as follows: µu,jj (x,)+(λ + µ)u j,j (x,)=ρ u (x,) n D (5) u =û on S, = ˆ on S, S = S \ S (6) whee u and show he dsplacemen and acon especvely, ρ s he densy of elasc meda D, and λ and µ ndcae Lamé consans. In Eq. 6, û and ˆ ae gven bounday values. The me-doman bounday negal equaon n -D elasodynamcs can be expessed by C j (x)u (x,)=u n (x,)+ S U j (x, y,) j (y,)ds y T j (x, y,) u j (y,)ds y. (7) S In Eq. 7, U j (x, y,) and T j (x, y,) denoe he me-doman fundamenal soluon and s double laye kenel fo -D elasodynamcs and C j s he fee em[8]. Nomally, Eq. 7 s dscezed by usng he appopae nepolaon funcons fo he unknown values and solved by a me-seppng algohm. Howeve, hee ae manly wo dsadvanages of he convenonal me-doman BEM. The fs one s an nsably encouneed n he me-seppng pocedue. The ohe s he dffculy n solvng lage scale poblems. Tme-Doman FMBEM Based on OQM n -D Elasodynamcs To ovecome he dsadvanages of he convenonal me-doman BEM, he Opeaonal Quadaue Mehod (OQM) and he Fas Mulpole Mehod (FMM) ae noduced. BEM Fomulaon Based on OQM In solvng he sysem of he bounday negal equaon (7) numecally, he bounday suface S s dsczed no M elemens due o a pecewse consan appoxmaon of he unknown dsplacemen u and acon. Takng he lm of x D x S and applyng Eq. and Eq. 4 n OQM o he convoluon negals n Eq. 7 yelds he followng dsczed bounday negal equaons fo me ncemen and n sepsasfollows: u (x,n ) =u n (x,n )+ M n α= k= whee A m and B m ae he nfluence funcons whch ae defned by [ A n k j (x, y α ) α j (k ) B n k j (x, y α )u α j (k ) ] (8)

26 34 Eds: R Abascal and M H Alabad A m j (x, y) = ρ m L B m j (x, y) = ρ m L L l= S L l= S Û j (x, y,s l )e πml L dsy (9) ˆT j (x, y,s l )e πml L dsy. () In Eq. 9 and Eq., s l s gven by s l = δ(ζ l )/( ). The paamee ρ has o be ρ< and s aken as ρ L = ɛ whee ɛ shows he assumed eo n he compuaon of Eq. 9 and Eq.. Û j (x, y,s) and ˆT j (x, y,s) ae Laplace doman fundamenal soluons n -D elasodynamcs as follows: Û j (x, y,s)= { K (s T )δ j } [K πµ s (s T ) K (s L )],j () T ˆT j (x, y,s)=n j (y)ρ(c L c T )Ûk,k(x, y,)+ρc T (Ûj,k (x, y,)+ûk,j(x, y,)) n k (y) () whee c L and c T ae he wave velocy of longudnal and ansvesal waves especvely, s gven by = x y, K n s he modfed Bessel funcon of he second knd n Eq. and n (y) s he componen of a ouwad un nomal veco wh espec o y. Noe ha s L and s T ae defned by s L = δ(z)/(c L ) and s T = δ(z)/(c T ) due o he smple expesson. To deemne δ(ζ l ),weuse he backwad dffeenal fomula (BDF) of ode wo as follows: δ(ζ l )=( ζ l )+ ( ζ l ). (3) Noe ha Eq. 9 and Eq. ae dencal o he dscee Foue ansfom. Theefoe, he calculaons of Eq. 9 and Eq. can be evaluaed by means of he FFT algohm. Afe aangng Eq. 8 accodng o he bounday condons, we can oban u (x,n )+ u n (x,n )+ = M [ B j (x, y α )u α j (n ) A j(x, y α ) α j (n ) ] α= M n α= k= [ A n k j (x, y α ) α j (k ) B n k j (x, y α )u α j (k ) ]. (4) Fo he n-h me sep, all he quanes on he gh-hand sde ae known. Theefoe, he unknown values u α and α can be obaned by solvng he above equaon. Unfounaely, we canno solve a lage scale poblem wh he lage numbe of M by he medoman BEM based on OQM because he equed compuaonal complexy and memoy become O(LM N) and O(M L) n Eq. 4, especvely. Theefoe, he me-doman BEM based on OQM s acceleaed by he Fas Mulpole Mehod (FMM) n hs eseach. Tme-Doman Fas Mulpole BEM Fomulaon Based on OQM The FMM poposed by Geengad and Rokhln s a echnque o educe he compuaonal me and memoy fo a lage scale poblem. In ecen yeas, Fas Mulpole BEM, whch s he couplng mehod

27 Advances n Bounday Elemen Technques IX 343 of BEM and FMM, has been developed o mpove he compuaonal effcency fo vaous lage scale poblems n many engneeng felds, e.g., he -D scala wave poblem[9][], and he 3-D sound and envonmenal vbaon poblems[]. Snce FMBEM algohm has been descbed n deal n ohe publshed papes (fo example, see he pape of Nshmua[]), we wll summaze only he essenal fomulas hee. Now, he fundamenal soluon n Eq. s ansfomed no he followng equaon; Û j (x, y,s)= [ Φ U µs, + e 3j Ψ,j] U T (5) whee Φ U and Ψ U ae dsplacemen poenals wh espec o P and S-waves, whch ae defned by Φ U = π K (s L x y ),k (6) Ψ U = e 3kl π K (s T x y ),l. (7) To apply FMM, we consde a pon o nea he souce pon y. Locaons of feld pon x and souce pon y ae expessed as (, θ) and (ρ, φ), especvely n pola coodnae sysem ognaed a he pon o. Usng Gaf s addon heoem, we oban he mulpole expansons of he dsplacemen poenals as follows: Φ U = π Ψ U = π n= n= M U n K n (s L )e nθ (8) N U n K n(s T )e nθ (9) whee he coeffcens M U n and N U n ae called he mulpole momens, whch ae gven by M U n = y k [ In (s L ρ)e nφ] () N U n = e 3kl y l [I n (s T ρ)e nφ ]. () In Eq. and Eq., I n shows he modfed Bessel funcon of he fs knd. Mulpole momens M T n and N T n fo ˆT j (x, y) s smlaly obaned. Once he mulpole momens ae obaned, we can quckly evaluae he max-veco poducs of he dsczed negal equaon (4) usng he fas mulpole algohm[7]. The anslaon fomulas (MM, ML and LL) ae also deved fom Gaf s addon heoem of he fundamenal soluons defned n Eq. and Eq.. The modfed Bessel funcon I n (z), n pacce, ends exponenally o nfny fo lage agumen z. Ths fac somemes causes he nsably of he anslaon fomulas when he cell sze s lage. To esolve he poblem, we noduced he scalng of he mulpole and local expanson coeffcens. Numecal Examples Tme-doman BEM based on OQM (OQBEM) s appled o analyze he ansen behavos of a cavy wh adus a as shown n Fg.. The bounday of he cavy s supposed o be acon fee. The

28 344 Eds: R Abascal and M H Alabad cavy B x Pao and Mow-A OQBEM-A Pao and Mow-B OQBEM-B Pao and Mow-C OQBEM-C ncden wave A a C x u / u c L /a Fgue A scaeng model. Fgue 3 Analyc and numecal soluons u /u a A, B and C. The soluons obaned by OQBEM ae ndcaed by symbols and he analycal numecal esuls of Pao and Mow ae shown by sold lnes. Compung me ( sec ) OQBEM FM-OQBEM 3 4 The numbe of Elemens 5 Fgue 4 The compason of CPU me beween OQBEM and he FM-OQBEM. 3a a ncden wave x cavy Fgue 5 A mulple scaeng model. x numbe of elemens s 64 and me ncemen s c L /a =.65. The paamees N and L ae gven by N = L = 8. In addon, ρ sassumedobeρ = (ɛ = ). The dsplacemen componens of he ncden wave ae gven by u n (x,)=u δ [(c L x a)/a]h(c L x a). () Fg. 3 shows he dsplacemen u /u as a funcon of me a A, B and C on he bounday of he cavy. Ths poblem has been analycally solved n he fequency doman by Pao and Mow[3]. The ansen soluon can be obaned by supeposng he esuls n he fequency doman by means of he fas Foue ansfom. The esuls by OQBEM ae n good ageemen wh he analycal-numecal esuls of Pao and Mow. Fg. 4 shows he CPU me needed n ode o solve scaeng poblems of he ncden waves by he cavy usng me-doman BEM based on OQM (OQBEM) o fas mulpole BEM based on OQM (FM-OQBEM). In hs analyss, he numbe of elemens s adjused by changng he sze of he elemen. We canno solve he case ha he numbe of elemens s 5 o moe wh OQBEM

29 Advances n Bounday Elemen Technques IX 345 because of he escon of he memoy. We can see ha FM-OQBEM s fase han OQBEM when he numbe of elemens s seveal housands o moe as shown n Fg. 4. Fnally, we consde he scaeng poblem of an ncden wave wh wave lengh a/ by 8 8 caves wh he adus a of he ndvdual caves and he cavy spacng 3a beween wo adjacen caves along he x and x axs as shown n Fg. 5. The componens of he ncden wave ae gven by u n (x,)=u ( cos πθ), Θ= x +.5a c L π. (3) The paamees ae aken as N = L = 56, ρ = (ɛ = ) and c L /a =.5. The numbe of DOF n each me sep s 89. Ths poblem canno be solved by OQBEM. Theefoe, he fas mulpole mehod s appled o acceleae he max veco poducs of dsczed bounday negal equaon and o save he memoy. Also, OpenMP wh 8 heads s used o paallelze hs analyss. Fg. 6 (a)-(d) show he me vaaons of he wave felds u /u aound caves. We can see ha scaeed waves ae geneaed by he neacon of he ncden wave and caves. Thus, me-doman fas mulpole BEM based on OQM s vey effecve n boh aspecs of he compuaonal me and equed memoy fo a lage scale poblem. Conclusons In hs pape, he me-doman fas mulpole BEM fomulaon based on OQM was developed fo -D elasodynamcs. The convoluon negals wee dsczed by he opeaonal quadaue mehod and he fundamenal soluons n Laplace doman wee used fo he calculaons of nfluence funcons. The fas mulpole mehod was appled o acceleae he calculaons of max-veco poducs fo he eaded poenal and o educe he memoy equemen. As numecal examples, scaeng poblems of ncden waves by caves wee demonsaed and he compuaonal effcency of he poposed mehod was confmed. In nea fuue, we wll develop he me-doman fas mulpole BEM based on OQM n 3-D elasodynamcs. Acknowledgemen Ths wok s suppoed by he Japan Socey of he Pomoon of Scence. Refeences [] W. J. Mansu and C. A. Bebba Tansen elasodynamcs usng a me-seppng echnque, In; Bounday Elemens, C. A. Bebba, T. Fuagam and M. Tanaka (Eds), (983). [] S. Hose Bounday Inegal equaon mehod fo ansen analyss of 3-D caves and nclusons, Engneeng analyss wh Bounday Elemens, vol.8, No.3, (99). [3] C. Lubch Convoluon quadaue and dscezed opeaonal calculus I, Nume. Mah.,5, 9-45 (988). [4] A. I. Abeu, J. A. M. Cae and W. J. Mansu Scala wave popagaon n D: a BEM fomulaon based on he opeaonal quadaue mehod, Engneeng analyss wh Bounday Elemens, 7, -5 (3).

30 346 Eds: R Abascal and M H Alabad x/ a x/ a x/ a x/ a (a) (b) x/ a x/ a x/ a x/ a ( C ) (d) Fgue 6 Tme vaaons of dsplacemens u /u aound caves. (a) c L /a =6.5 (b) c L /a =.5 (c) c L /a =9.5 (d) c L /a =5.. [5] M. Schanz and V. Suckmee Wave popagaon n a smplfed modelled pooelasc connuum: Fundamenal soluons and a me doman bounday elemen fomulaon, Nume. Mah.,64, (5). [6] Ch. Zhang Tansen elasodynamc anplane cack analyss of ansoopc solds, In. J. Solds and Sucues, vol. 37, (6). [7] L. Geengad and V. Rokhln A fas algohm fo pacle smulaons, Jounal of Compuaonal Physcs, 73, (987). [8] S. Kobayash Wave Analyss and Bounday Elemen Mehods, Kyoo Unvesy Pess (n Japanese), (). [9] T. Fuku and J. Kasumoo Fas mulpole algohm fo wo dmensonal Helmholz equaon and s applcaon o bounday elemen mehod. Poc of he 4h Japan Naonal Symposum on Bounday Elemen Mehods (n Japanese), 8-86 (997). [] T. Saoh, S. Hose, T. Fuku and T. ISHIDA, Developmen of a me-doman fas mulpole BEM based on he opeaonal quadaue mehod n -D wave popagaon poblem, Advances n Bounday Elemen Technques VIII, (6). [] T. Saoh A sudy on effecve 3-D numecal analyss of envonmenal vbaon and nose nduced by a movng an, docal hess n Tokyo Insue of Technology, (6). [] N. Nshmua, Fas mulpole acceleaed bounday negal equaon mehods, Appl. Mech. Rev., 55, (). [3]Y.-H.PaoandC.C.Mow,Dffacon of Elasc Waves and Dynamcs Sess Concenaons, Cane and Russak, New Yok, (973).

31 Advances n Bounday Elemen Technques IX 347 Chaacesc max n he bendng plae analyss by SBEM Panzeca T.,a, Cucco F.,b, Saleno M.,c Dseg, Vale delle Scenze, 98 Palemo, Ialy Va E. Tcom 8, 97 Palemo, Ialy a panzeca@scal., b flppo.cucco@scal., c mgsaleno@scal. Keywods: bendng plae, Symmec Bounday Elemen Mehod, Kchhoff shea foce. Absac. Ths pape deals wh he hn bendng plae analyss by usng he symmec appoach of Bounday Elemen Mehod (SBEM). A fomulaon s used n whch he plae bounday s dscezed no bounday elemens and s subjeced o appopae dsbuons of shea foces and couples, as well as of vecal dsplacemen and oaons. These dsbuons ae he causes and ae modelled hough appopae shape funcons, wheeas he genealzed effecs ae obaned, accodng o he Galekn appoach, as weghng of he dsplacemens and he oaons, as well as of he shea foces and momens. In he equaons sysem he algebac opeao s a symmec max whose coeffcens ae defned as double negals wh hgh ode sngulaes, all compued n closed fom. Inoducon The objec of hs pape s o consde some compuaonal aspecs egadng he hn bendng plae analyss wh he SBEM (Bonne e al. []). Respec o he collocaon BEM, n whch nonsymmec bounday negal fomulaons fo bendng plaes have been suded by vaous auhos (Beskos [], Alabad [3]), he SBEM appoach shows few conbuons abou he plae analyss (Toenham [4], Fang and Bonne [5], Peez-Gavlan and Alabad [6]). The compuaon of he solvng equaon coeffcens pesens consdeable dffcules fo he pesence of hgh ode sngulaes n he kenels of he negals and s cay ou ehe by means of a devave ansfe echnque, employng he negaon by pas o educe he sngulay of he fundamenal soluons (Fang and Bonne [5]), o by means of an appoach based on a lm pocess (Fang and Guggan [7] n he collocaon conex). In he pesen pape a geneal compuaonal mehodology ha makes ease he geneaon and he check of he coeffcens calculus s shown. In hs mehodology, aleady appled o he n-plane loaded plae n Panzeca e al. [8], he bendng plae s suded whou consdeng he acual consan and bounday load condons. Fo he plae an appopae algebac opeao, so-called chaacesc max, whch connecs he knemacal and mechancal quanes along he bounday, s noduced. Ths max s sngula and s coeffcens ae valued by mposng n a sequenal way some dsbuons of causes on he bounday elemens, and by compung he effecs n he same bounday hough a weghng pocess of he esponse. Ths appoach s paculaly useful when he max coeffcens ae deemned; ndeed he gd body echnque allows o vefy he effecveness of he coeffcens compuaon. In he fs secon he peculaes of he chaacesc max ae shown and an appopae eaangemen and employmen of hs max s explaned n ode o oban he algebac opeaos of he mxed value elasc poblems. In he second secon a echnque o compue a coeffcen of he chaacesc max s llusaed. The kenels of he negals ae defned as dsbuons n he Schwaz sense, specfcally he dsbuon defnon s employed as he lm of a successon of funcons. Ths appoach makes

32 348 Eds: R Abascal and M H Alabad possble o naually cancel ou he sngulaes of hghe ode n he cause negaon, wheeas he lowe ode sngulaes ae smoohed by he effec shape funcons and elmnaed by he oue negaon.. Symmec chaacesc opeaos Le us consde he bendng poblem fo a lnealy elasc plae of doman and bounday, dsngushed no consaned and fee. The plae s subjeced o he followng exenal acons (Fg. a): - body foces p nomally appled o he mddle suface n he doman ; T - dsplacemens and oaons u u n n mposed on he consaned bounday ; T - foces and couples f fn cn gven on he fee bounday. The elasc esponse o he known exenal acons may be obaned n ems of bounday quanes, defned along an elemen chaacezed by he ouwad nomal n : T - shea foces and couples f fn cn csn on ; T - dsplacemen and oaons u u n n sn on ; To sudy he plae a geneal saegy s used, based on he noducon of a max, called chaacesc. The plae s embedded n an unlmed doman (Fg. b) havng he same Young s modulus E, Posson s ao and he same hckness h of he plae, and as a consequence s bounday may be consdeed as bounday of o of he complemenay doman \. f n p u n n f n p u n n c n c n a) b) Fg. : a) A polygonal plae, b) he plae embedded n To deve he chaacesc max no dsncon s made beween he consan o fee boundaes. I nvolves ha he ene bounday s subjeced o a dsbuon of layeed mechancal acons and o a dsbuon of double layeed knemacal dsconnues I s known ha he esponse n ems of knemacal u and mechancal quanes a evey pon of he on he bounday s gven by he Somglana Idenes (SI). By mposng he bounday condons u and on, whch eplace he classcal Dcle and Neumann condons u u and u u.. f on, hese SI may be wen n compac fom n he followng way: u u[] f u [ u] uuˆ [p] (.a) + PV PV [] f [ -u] ˆ [p] whee he followng posons ae vald: uf [ ] G fd, u [ u] G ( u)d, uˆ [p] G pd uu PV u uu f (.b) (a-c)

33 Advances n Bounday Elemen Technques IX 349 [] f G fd, [] u G ( u)d, ˆ [p] G pd PV u u (d-f) beng G hk (h,k=u,) he fundamenal soluons maces defned n Panzeca e al. [9]. Le us opeae he dscezaon of he bounday and noduce appopae shape funcons o model he layeed mechancal and he double layeed knemacal quanes: u and f F, u U (3a,b) u whee F and U ae he vecos collecng nodal quanes. Lnea shape funcons ae assumed fo, wheeas quadac ones n he bendng oaon and heman ones n he dsplacemen and osonal oaon dsconnuy ae assumed fo u. Le us pefom he weghng pocess n accodance wh he Galekn appoach n he eqs. (a,b), so obanng he followng bounday negal equaons: + W A ˆ uu FAu ( U) Cu ( U) W (4a) + P A ˆ u FA ( U) Cu FP (4b) whee he followng posons ae se: W u d, Wˆ [ G pd ]d, A [ G d ]d, + T T + T ' uu uu uu + A [ G d ]d, C [ d ], T ' + T + u u u u u + + P d, Pˆ [ G pd ]d, A [ G d ]d, C + T + + T + T ' + u u u u u u [ d ], A T + u u + + [ G d ]d T ' + u u (5a-l) In compac fom one has: BXLˆ (6) wh A A C F ˆ ˆ W B A+C A C X L uu u u ( ),,,, ˆ u A A Cu U P (7a-e) The max A s symmec, wheeas he max C, whch ncludes he fee ems, s emsymmec. The max B s unsymmec and sngula: he sngulay depends on he ccumsance ha he plae may be subjeced o a gd moon. In he pesen pape all he coeffcens of he max B have been compued n closed fom. The max B s used o solve he mxed value elasc poblems. Indeed allows o geneae boh he pseudosffness max and he load vecos due o he mechancal acons appled on he fee

34 35 Eds: R Abascal and M H Alabad bounday and o he knemacal quanes mposed a he consaned bounday, as has been made by Panzeca e al. [8] n he n-plane loaded plae. In ode o ge hs am, a eaangemen of he ows and columns of he chaacesc max s pefomed: ndeed he vecos F and U, W and P ae edefned n he followng way T T T T T T T T T T T,,, F= F F U= U U W = W W P = P P T (8a-d) Le us o noduce he Dchle and Neumann genealzed condons on he boundaes: W on, on (9a-b) P The eaangemen noduced n he eqs. (9a-b) allows o deve he followng equaons: W Buu Buu Bu Bu F Wp uu uu u u p W = B B B B F + W p P Bu Bu B B U Pp u u p P B B B B U P () In compac fom: KX+ Lˆ = () whee he followng posons ae made: K B B X F Lˆ L F L U Buu Bu Wp L ; Lu ; Lp ; Bu B Pp uu u = ; = ; = + u(- )+ Lp p; Bu B U (a-f) In eq. () K s he pseudosffness max, symmec and non sngula, wheeas genealzed load veco. ˆL s he. Coeffcens analycal compuaon n he Kchhoff model. In he Kchhoff model he followng Love-Kchhoff hypoheses have been noduced: - knemacal assumpon: n he plae bounday, havng nomal n and angen s, he osonal oaon s he angenal devave of he vecal dsplacemen; - mechancal assumpon: he dsbued osonal momen along he plae bounday may be eplaced by an appopae dsbuon of ansvesal shea foces, leadng o he so-called Kchhoff shea, whch s he sum of he shea foce and he angenal devave of he osonal momen. To ge a symmec fomulaon, a knemacal quany, assocaed o he Kchhoff shea foce and defned he Kchhoff dsplacemen dsconnuy, has o be noduced. The shape funcons employed fo he modellng pocess ae assumed o be lnea fo he foces and couples, quadac fo he oaon dsconnues, wheeas hey ae assumed o be heman fo he Kchhoff dsplacemens dsconnues. The shape funcons employed fo he weghng pocess

35 Advances n Bounday Elemen Technques IX 35 ae assumed o be lnea fo he genealzed dsplacemens and oaons, quadac fo he genealzed momens, wheeas hey ae assumed o be heman fo he genealzed Kchhoff shea foces. In he evaluaon of he max coeffcens he followng seps ae used: - o mpose a unay value a he node accodng o he local node sysem (Fg. b), - o model he cause along he bounday elemens hough appopae shape funcons accodng o he local sde sysem (Fg. e), - o employ he Somglana Idenes, - o pefom he weghng pocess of he bounday quanes by means of appopae shape funcons accodng o he Galekn saegy. In he case of a ecangula plae, wh sdes paallel o he Caesan axes, le us suppose o deemne he genealzed Kchhoff shea foce T K assocaed wh node as he effec of a Kchhoff dsplacemen dsconnuy U K mposed a he same node (Fg. ) beween he wo + fones and. In he followng schemes he use of he fundamenal soluons s shown, wh he objecve o ake boh he Kchhoff knemacal and mechancal assumpons no accoun: G u knemacal assumpon sn s' u n 3 sn s' 3 n egaon by pas G m n 3 3 G m n m sn 3 3 s' 33 m 33 sn 3 3 s' m mechancal assumpon K sn s u u n n 3 K 3 s' s' 3 3 K K m 33 s s' G 3 K s s' m 33 sn s' m n s' n u n As a consequence, he dsplacemen dsconnuy mposed a node s ansfeed along he bounday sdes nex o he node as he sum of wo dsbuons: one concenng he vecal dsplacemen and he ohe one concenng he angenal devave of he osonal oaon. The genealzed Kchhoff shea foce T K assocaed wh he node can be obaned as he sum of he weghed shea foce along he bounday sdes a and b. On evey sde he weghed shea foce s obaned as he sum of wo conbuons and specfcally he vecal foce and he angenal devave of he osonal momen. A a pon of each bounday sde he Kchhoff fundamenal K soluon, ha has o be modelled by he cause shape funcons and has o be weghed by he effec shape funcon, may be expessed as he sum of fou ems:

36 35 Eds: R Abascal and M H Alabad K s' s s' (3) K whee s he Kchhoff acon a pon x of nomal n caused by: - a dsplacemen dsconnuy -u appled a a pon x ' wh nomal n' ( 3 s ). - a angenal devave of he osonal oaon -sn appled a a pon x ' wh angen s ' ( s' s s' ); 3 33 NODE QUANTITIES a) b a b n s n s s s n x n y z T K =? M x, x My, y y z b) c) T, U x y z U K a x =-U= Dsbuons of shea foces and of angenal devave of he osonal momen BOUNDARY SIDES DISTRIBUTIONS Dsbuons of dsplacemen dsconnues and of angenal devave of he osonal oaon d) b a C, s n e) n F, W C, sn sn n f) b a Fg. : a) Weghed Kchhoff equvalen shea foce, b) node local sysem, c) genealzed Kchhoff dsplacemen dsconnuy, d) dsbuons of he effecs, e) sde local sysem, f) dsbuons of he causes. The causes ae mposed accodng o he node local sysem and he bounday effecs ae evaluaed accodng o he sde local sysem. The genealzed shea foce assocaed wh node as he effec of a Kchhoff dsplacemen dsconnuy U K mposed a he same node deves fom he double negaon of he poduc beween he fundamenal soluon and he shape funcons 'u ( x ') and x. The fundamenal soluon s defned n he bounday elemens a and b, whee s specfed as he followng foms T K K K K K K aa ab bb ba,,,, n whch he double ndces epesen he sdes whee he effec and cause dsbuons ae locaed. Consequenly one has: Inne negals: - The Kchhoff shea foce a a pon of he sde a, caused by he knemacal dsbuons assocaed o he Kchhoff dsplacemen dsconnuy n he sdes a and b: u ( )

37 Advances n Bounday Elemen Technques IX 353 Eh3 4 3(3 ) K K K a aa ua( ')d ' ab ub( ')d ' ( ) 4 ( ) (4) - The Kchhoff shea foce a a pon of he sde b, caused by he knemacal dsbuons assocaed o he Kchhoff dsplacemen dsconnuy n he sdes a and b: Eh3 4 3( 5 ) K K K b bb ub( ')d ' ba ua( ')d ' ( ) (5) 4 ( ) whee ( ) and () ae funcons conanng logahmc sngulay only. eqs. (4) and (5), In he fundamenal soluon K shows a sngulay of ode 4 n he neval (,), nsead he fundamenal soluon shows he same sngulay only fo '. Oue negal: K - Le us compue he pmves T a, heman shape funcons: K K K a ua a b ub b K b K j T of he Kchhoff shea foces K, a K b weghed hough T ( ) d, T ( ) K d, (6) In hese wo pmves he sngulay ode deceases, emanng only he logahmc one. The wo obaned expessons ae funcons of he same naual vaable (,), heefoe he weghed Kchhoff shea foce T K assocaed o he node s obaned as he sum of he wo pmves, defned n (,) neval: E h ( ) T = ( 3 8( ) 3 K K K T a +T b Log[4]) (7) 3. Applcaon. The applcaon concens a squae plae wh wo fee sdes and wo smply suppoed sdes, wh 6 sde and wh he maeal elasc consans E, h. and.3, subjeced o unfom nomal load of un value (Fg. 3). x M, y y M, x x y T, U z Fg. 3: Plae subjeced o a vecal load: consan and load condons. The coeffcens of he doman load veco ae deemned n closed fom by means of a double negaon, he one egadng he cause hough doman negals, and he one egadng he effec hough bounday negals. In Table he esuls obaned ae compaed wh hose obaned usng he classc plae heoy (Tmoshenko []).

38 354 Eds: R Abascal and M H Alabad m y u m x SGBEM 8 nodes Classc heoy (Tmoshenko) u(,) u(,) m x (,) m y (,) m y (,) Table - Dsplacemens and momens n he plae. Refeences [] M. Bonne, G. Mae, C. Polzzoo: Symmec Galekn bounday elemen mehod. Appl. Mech. Rev. 5 (998), p [] D.E. Beskos, (ed.). Bounday Elemen Analyss of plaes and shells, Spnge-Velag, Beln, 99. [3] M.H. Alabad (ed.). Plae bendng analyss wh bounday elemens. In: Advances n Bounday Elemens. Compuaonal Mechancs Publcaons, Souhampon, 998. [4] H. Toenham. The bounday elemen mehod fo plaes and shells. In: P.K. Banejee, R. Buefeld, (Eds.), Developmens n Bounday Elemen Mehods. vol., Elseve, Amsedam, 979. [5] A. Fang, M. Bonne: A Galekn symmec and dec BIE mehod fo Kchhoff elasc plaes: fomulaon and mplemenaon. In. J. Num. Meh. Engng. 4 (998), p [6] J.J. Peez-Gavlan, M.H. Alabad: Symmec Galekn BEM fo shea defomable plaes. In. J. Num. Meh. Engng. 57 (3), p [7] A. Fang, M. Guggan: Bounday elemen analyss of Kchhoff plaes wh dec evaluaon of hypesngula negals. In. J. Num. Meh. Engng. 46 (999), p [8] T. Panzeca, F. Cucco, S. Teaveccha: Symmec bounday elemen mehod vesus Fne Elemen Mehod. Comp. Meh. Appl. Mech. Engng. 9 (), p [9] T. Panzeca, V. Mlana, M. Saleno: A symmec Galekn BEM fo plae bendng analyss, n pess Eu. J. Mech. A/Solds, DOI.6/j.euomechsol.8..4, (8). []S.P. Tmoshenko, S. Wonow-Sky-Kege. Theoy of plaes and shells. McGaw-Hll Book Company, 959.

39 Advances n Bounday Elemen Technques IX 355 Compuaonal aspecs n hemoelascy by he Symmec Bounday Elemen Mehod Panzeca T.,a, Teaveccha S.,b and Zo L.,c Dseg Vale delle Scenze, 98 Palemo, Ialy. a panzeca@scal., b eaveccha@dseg.unpa., c lzo@dseg.unpa. Keywods: hemoelascy, symmec bounday elemen mehod, Galekn appoach, sngula doman negal. Absac. Whn he hemoelascy feld eaed by he Symmec Bounday Elemen Mehod (SBEM) [] some dffcules egad he emoval of he song sngulaes asng n he kenels of he doman negals. I happens because he dffeenal opeao necessay o oban he sess s appled o a sngula dsplacemens feld. When hs mehod s appled o mechancs poblems, seveal compuaonal advanages happen f he doman negals ae eplaced by bounday negals. These advanages ae consdeable especally n he analyss phase because he evaluaon of he doman load ems asng fom a weghed pocess may be easly compued n closed fom. The pesen appoach acs n he dsplacemens feld, a fs subsung he doman negals by bounday ones and successvely applyng he dffeenal opeao o a egula feld n ode o oban he hemal sess feld.. Inoducon In he SBEM he sess sae compuaon n he doman, due o a volumec dsoons dsbuon, epesens one of he opcs moe suded n he plasc analyss. The evaluaon of he sess feld ases fom he Somglana Idenes (S.I.) of he dsplacemens. The dec use of he dffeenal opeao o he sngula feld of he dsplacemens poduces a doman negal whch s nepeed as a Cauchy pncpal value and a jump em (Bu fee em) []. The evaluaon of hs lae negal s made hough a egulazaon pocess. Among he cuen mehodologes, hose based on he ansfomaon of he sngula doman negals egadng san o sess felds no bounday ones, pefomed by Gauss heoem [3] o Radal Inegal Mehod (RIM echnque) [4], ae compuaonally moe advanageous. The advanages of hese mehodologes ae moe useful nsde he SBEM because he load ems due o doman acons, ansfeed on he bounday, pem o pefom he Galekn weghng on he bounday whou doman negals. In he pesen pape, he sess a an nne pon s deal wh a dffeen way n compason wh he Hube [3] and Gao [4] appoaches. In he S.I. of dsplacemens he doman negal s egulazed and he sngula negal s ansfomed no bounday one hough he RIM echnque asfeng so he cause quany on he bounday and he effec pon n he doman. A non-sngula dsplacemen feld s obaned and he dffeenal opeao may be decly appled. By usng he Hooke law, he S.I. of sesses s found; ha lae s fomally he same expesson obaned by Hube [3], bu n he pesen fomulaon he jump em does no appea because he dffeenal opeao s appled o a egula dsplacemens feld. In he pesen fomulaon he nelasc doman acons ae consan, and he poposed appoach pems o deal wh nelasc acons, modelled ehe n he cells o n he body n a smple way. By usng Cauchy fomula and hough a lm appoach he acons ae compued on he bounday; hese lae ae ulzed o oban n a closed fom he weghed values of a pa of he load veco.

40 356 Eds: R Abascal and M H Alabad. Sess feld The sang pon of he elasc poblem s he S.I. of he dsplacemens. When we gnoe he mass foces hs lae expesson akes on he followng fom: uf [, ub,, ] G fd G ( u) d G d () uu u u In he nfne doman he dsplacemens ae caused by mechancal f and knemacal u layeed dsconnues vecos, boh collecng known and unknown funcons, bu also by he volumec dsoon veco, lke empeaue vaaon, whch n he pesen fomulaon ae assumed consan n. In eq.() he G uk ae maces of fundamenal soluons symbolcally noduced by Mae and Polzzoo []. The compably condon s he followng p [, f u, ] p Dx u[, f u, ] p () whee p s he volumec dsoon n P. The dffeenal opeao D x gves se o he followng equaon G f d G ( u) dd G d. (3) u x u p whee he posons G u D x Guu and G D x Gu ae assumed. In he doman negal, whch evaluaes he effecs of he volumec dsoons, he dffeenal opeao nvolves he pesence of hypesngulay n he dsplacemen gaden. G W Ge We e Fg. : Body subjeced o volumec dsoon; ccle of escluson havng adus. Wh efeence o Fg., genecally he lae negal of eq.(3) may be wen as follows D G D G D G x u dlm x u dlm x u d p (4) o n hs way D G D G D G x u dlm ( x u d) lm( x u d p) (5) The expessons (4,5) gve se o fomulaons chaacezed by egulazaon echnques. The soluons poposed by Gao [4] and Hube [3] ae dscussed n he followng subsecons. In he eq.(5) he second lm s null.. Gao appoach The fs negal of eq.(4) s mean as Cauchy Pncpal Value (CPV) and may be wen n a egulazed fom n he followng way lm ( Dx G d) G ( ) dlm G d (6) u p p

41 Advances n Bounday Elemen Technques IX 357 The sngula negal s modfed by he RIM echnque no bounday negal lm G d G Log( ) n d G d (7) p p p no moe sngula. The second negal of eq.(4), by usng Gauss heoem and he elaon Dx(.) Dx' (.), s ansfomed no an negal defned on he bounday of Fg.. Subsequenly, by pefomng a ansfomaon of he Caesan coodnaes no pola ones, hs negal s evaluaed n closed fom so obanng he max J of jump ems (Bu fee em). u ( ) lm d ' d J Dx G N G N' ( ) d (8) u u The expessons of he egolazed san and sess felds ae obaned G f d G ( u) d G ( ) d G d ( JI) (9) u p p p E G f d G ( u) d G ) d G d E( JI) () u p p p. Hube appoach The sngula negal (5), ansfomed no pola coodnaes, s modfed hough he Lebnz heoem so devng he Bu fee em. Subsequenly, afe a new ansfomaon no Caesan coodnaes, a egulazaon of he sngula em s pefomed. One obans lm ( Dx G d) G ( ) dlm Dx G d N' G d () u p u p u p The Gauss heoem and he condon Dx(.) Dx' (.), boh ulzed no he second negal of he pevous equaon, ansfom he expesson as follows lm Dx Gu d G( p) d N' Gu dp N' Gu dp N' Gu dp () so obanng he followng egulazed san and sess felds G f d G ( u) d G ) d N' G d (3) u p u p E G f d G ( u) d G ( ) de( N' G di) (4) u p u p.3 Pesen model In he pesen model he san expesson [ ] gven by eq.(5) s egulazed and he sngula em s ansfeed on he bounday by usng he RIM echnque. Theefoe he dffeenal opeao s appled o a non-sngula dsplacemen feld. The followng expesson of he sans [ ] s obaned Dx G dlm Dx G ( ) dlm Dx G d (5) u u p u p The fs negal of he pevous equaon s egula and he dffeenal opeao may be appled decly o he fundamenal soluon. The RIM echnque s used n he second negal befoe he applcaon of he dffeenal opeao, so obanng:

42 358 Eds: R Abascal and M H Alabad Dx G d G ) ddx G n d (6) u p u p If now we apply he dffeenal opeao o he second negal of eq.(6) and consde he poson Dx(.) Dx' (.), one obans: Dx G n d G n d N' G d (7) u p p u p Because he followng poson s vald fo he ccula negal G n d (8) he san and sess felds ake on he followng egulazed fom: G f d G ( u) d G ( ) d( N' G di) (9) u p u p E G f d G ( u) d G ( ) de( N' G di) () u p u p Ths lae expesson s equal o hose obaned by Hube appoach (4). Though a compason of he expessons povded by Gao () and he Hube (4) appoaches and by he pesen model () he followng equaly s easly demonsable: ( G dej) p EN' Gu d p () Gao Hube, Pesen model The expesson () pems o asse ha he Bu fee em, pesen n Gao fomulaon, does no appea n explc fom n he pesen fomulaon and n he Hube one. When we apply he Cauchy fomula n eq.(), he S.I. of he acon on an elemen havng he slope defned by n s obaned,.e. T G f d G ( u) d G ( ) dn E( N' G di) () u p u p In ode o evaluae he acon on he dscezed bounday elemen, s necessay o make a lm opeaon fom he nne of he body. Indeed he fs negal s mean as CPV o whch he fee em f / mus be added, wheeas he fouh negal, whee he Bu fee em s pesen n mplc fom, gves se o a em equvalen o he CPV and o a fee em as shown n he followng secon.. S.I. of he acon on he bounday The fouh negal n eq.(), below anscbed, s evaluaed hough a lm opeaon T NE NG '. (3) d u p In Fg. P epesens he pon, dsan d fom he bounday, whee he acon has o be compued on a slope chaacezed by a nomal un veco assumed paallel o he nomal veco o he genec bounday elemen. The lm opeaon gves se o a coeffcen obanable n closed fom. Two jump ems ae added o hs coeffcen: he fs of whch coesponds o an half of Bu fee T T em,.e. (/ ) NEJ, he second one NEJ b conans he acangen funcon. Fs jump em: when he pon P appoaches o he bounday ( d ) fom he nne of he body, T he conbuon (/ ) NEJ s obaned pung /, /. Some smple geomecal

43 Advances n Bounday Elemen Technques IX 359 consdeaons pem o obseve ha, whaeve he body geomey may be, he same conbuon ases fo each sde, wheeas he ohe half depends on he emanng bounday. T Second jump em: he addonal em NEJ b depends on he slope of he bounday elemen owads whch he nfnesmal elemen appoaches. y n' a d P n a G b x Fg. : Tacon conbuon on each bounday elemen In deal: T T T T T lm NENG ' ud NE NG ' ud NE( JJb) NE ' ud NG NEH d. (3) 3. Solvng sysem In ode o fomulae he solvng sysem, n he feld of he Galekn symmec fomulaon, he S.I. of he dsplacemens and of he acons, evaluaed on he bounday, ae necessay: T u Guu f dgu ( u) u p u ( ) d G n (4a,b) T T Gu f d f ( ) d ' u d p ( ) p G u N E N G N E H I wen n he hypohess of consan These lae wen n compac fom on and ake on he followng fom u G f d G ( u ) d u [ f, u,] u [ ] uu u G f d G ( u ) d [ f, u,] [ ] u Le us noduce he shape funcons dsoons: and u u (5a,b) u n ode o model he bounday layeed foces and f = F, f = F, u = U, u = U (6a-d) whee ( F, U ) and ( F, U ) ae vecos collecng unknown and known nodal quanes, especvely. In ode o oban he soluon fo he examnng body, he Dchle and Neumann condons have o be mposed,.e. u u compably condon on f equlbum condon on. (7a) (7b) In accodng wh he Galekn saegy, he pevous bounday condons may be wen n he followng genealzed (weghed) fom, so obanng:

44 36 Eds: R Abascal and M H Alabad ( u u )d global compably condon on (8a) u( f)d global equlbum condon on Inoducng he elaons (5a,b) and he modellng (6a-d) n hese lae equaons, one obans: T T T [, ] [ ] d f uu f f u u f u. G d d F G d d ( U ) u F U u U T T T u [, ] [ ] f d G d d F G d d ( U ) u u f u u F U F (8b) (9a) (9b) whose, eaanged and ewen n compac fom as made by Panzeca e al. [5,6], povde he followng solvng sysem: Auu Au F Lu[ F, U] Lu[ ] A A U L [ F, U ] L [ ] u o also AX +L =. (3) The veco X collecs he unknown quanes defned a he bounday nodes, wheeas he veco L collecs also he genealzed dsplacemens and acons due o he volumec dsoons, consan n. (3) 4. Examples The poposed appoach s appled o wo plane sucues whee he hemal loads, vaables n he doman, ae consdeed zone-wse consan by usng he subsucung appoach developed by some of he pesen auhos [5,6] and mplemened n he calculus code Kanak.sGbem [7]. The analyss s made by usng he dsplacemen mehod. A he am o oban some esuls compaable wh he classcal heoes, he body foce s no conddeed. 4. Example A plae havng dmenson (L =, H = ) and un hckness of Fg. 3, subjeced o a quadac empeaue vaaon n y decon, s analyzed. The vaaon low s he followng: T c y c yc x (3) whee he consans ae defned by c (T T T 3), c (T T ), c T3 H H, beng T ; T 4; T. (33a-f) 3 The analycal soluon of hs poblem, hough fo a san plane sae, s he followng: E y xx = T ; T ; ( ) yy u y y y d y ( ) ( ) x = x = (34a-c) -H/

45 Advances n Bounday Elemen Technques IX 36 y H x L Fg. 3: Plae subjeced o quadac hemal load: subdvson n subsucues. The plae s subdvded no subsucues, each of hose s subjeced o a consan hemal vaaon obaned by eq.(3); fo evey subsucue he followng mechancal and physcal chaacescs have been adoped: E dan / cmq,.3; =.. A dscezaon havng consan sep p. cm has been noduced. In he baycene of evey subsucue he sesses x and he vecal dsplacemens u y have been evaluaed by usng he saegy developed n he pesen pape. The esuls obaned ae compaed wh he analycal soluon (see Tab.). y sx analycal sx uy analycal uy e e e e e e e e e e e e e e e e e e e e -5 Table : Analycal soluon x and u y compaed wh he pesen appoach. 4. Example As second example a beam havng dmenson (L = 3 cm, H = 5 cm) and hckness s = 3 cm of Fg. 4 s analyzed. In he mddle secon of he beam a concenaed nelasc dsoon s appled, conssng n a elave oaon k, whee k s he cuvaue due o a lnea hemal vaaon and s he hemal gaden. In he example he concenaed dsoon s smulaed by noducng n he mddle secon of he beam a sp long cm, subdvded no subsucues, each of whose s subjeced o he consan hemal vaaon T x. s The daa of he examnng poblem ae: E 3 dan / cmq,., =., T x =+5, T=-5 x,, k. A dscezaon havng consan sep p.5 cm has been adoped. The analycal soluon, n he case of mono-dmensonal sold, foesees a bul-n momen M (EI / L) 375daN cm and a lnea sess dsbuon along he hegh x (M / I) y n evey secon.

46 36 Eds: R Abascal and M H Alabad y y +5 H x a) L x b) c) Fg. 4: Beam subjeced o concenaed dsoon ; a) subdvson no subsucues; b) empeaue dsbuon. y sx analycal sx M analycal M compued cm Table : Analycal soluon compaed wh he pesen appoach. x -5 Refeences [] Mae G., Polzzoo C., (987), A Galekn appoach o bounday elemen elasoplasc analyss, Comp. Meh. Appl. Mech. Engng., 6, [] Bu H. D., (978), Some emaks abou he fomulaon of hee-dmensonal hemoelasoplasc poblems by negal equaon. In. J. of Solds and Sucues., 4, [3] Hube O., Dallne P., Paeymulle P., Kuhn G., (996), Evaluaon of he sess enso n 3D elasoplascy by dec solvng of hypesngula negals. In. J. Num. Meh. Engg., 39, [4] Gao X. W., (3), Bounday elemen analyss n hemoelascy wh and whou nenal cells. In. J. Num. Meh. Engg., 57, [5] Panzeca T., Saleno M., Teaveccha S., (), Doman decomposon n he symmec bounday elemen mehod analyss. Comp. Mech., 8, 9-. [6] Panzeca T., Cucco F., Teaveccha S., (), Symmec bounday elemen mehod vesus Fne elemen mehod. Comp. Meh. Appl. Mech. Engng., 9, [7] Cucco F., Panzeca T., Teaveccha S., (), The pogam Kanak.sGbem Release..

47 Advances n Bounday Elemen Technques IX 363 Naual convecon flow of mcopola fluds n a squae cavy by DRBEM Sevn Gümgüm,a, Münevve Teze-Sezgn,b Depamen of Mahemacs, İzm Unvesy of Economcs, İzm, Tukey Depamen of Mahemacs, Mddle Eas Techncal Unvesy, Ankaa, Tukey a e-mal:sevn.gumgum@eu.edu., b e-mal:mun@meu.edu.. Keywods: DRBEM, naual convecon, mcopola fluds Absac. The man pupose of hs sudy s o pesen he use of he Dual Recpocy Bounday Elemen Mehod (DRBEM) n he analyss of he unseady naual convecon flow of mcopola fluds n a dffeenally heaed squae cavy. Fo he esulng sysem of odnay dffeenal equaons n me, he fne dffeence mehod (FDM) s made use of. The esuls ae epoed fo he confguaon n whch he cavy s heaed fom he vecal walls whle he hozonal walls ae nsulaed. Soluons ae obaned fo seveal values of mcosucue paamee and Raylegh numbe (Ra). The hea ansfe ae (aveage Nussel numbe) of mcopola fluds s found o be smalle han ha of he Newonan flud. Numecal esuls ae gven n ems of seamlnes, sohems, vocy conous as well as a able conanng Nussel numbe values fo seveal Ra. Inoducon The mcopola flud model, noduced by Engen [], s a genealzaon of he wellesablshed Nave-Sokes model n he sense ha akes no accoun he mcosucue of he flud. Ths model noduces a new knemac vaable called mcooaon whch descbes oaon of pacles. The heoy s expeced o descbe successfully he non-newonan behavo of cean eal fluds, such as lqud cysals, collodal fluds, and lquds wh polyme addves. Physcally, mcopola fluds may be epesened by fluds conssng of dumb-bell molecules o sho gd cylndcal elemens. Naual convecon hea ansfe n enclosues has been of consdeable eseach nees n ecen yeas due o he couplng of flud flow and enegy anspo. Mos of he pevous sudes on naual convecon n enclosues have been elaed o Newonan fluds. The sudy of Lo e al [] epesened a numecal algohm whch has been mplemened o analyze naual convecon n a dffeenally heaed cavy. In he sudy, hghe-ode polynomal appoxmaon used fo dscezng he paal devaves n he DQ mehod has been used o oban accuae numecal esuls whle solvng he velocy-vocy fom of he Nave-Sokes equaons. The penaly fne elemen mehod was appled o solve naual convecon flows n a squae cavy wh non-unfomly heaed walls by Roy and Basak [3] Hsu and Chen [4] numecally nvesgaed he naual convecon of a mcopola flud n an enclosue heaed fom below usng he cubc splne collocaon mehod. They suded he effecs of mcosucue on he convecve hea ansfe and found ha hea ansfe ae of mcopola fluds was smalle han ha of he Newonan flud. In anohe wok, Hsu e al [5] suded naual convecon of mcopola fluds n an enclosue wh solaed hea souces. They obseved ha he hea ansfe ae s sensve o he mcooaon bounday coondons and he aveage Nussel numbe s lowe fo a mcopola flud, as compaed o a Newonan flud. In a ecen sudy, Aydn and Pop [6], numecally nvesgaed he seady naual convecve hea ansfe of mcopola fluds n a squae cavy wh dffeenally heaed walls usng he fne dffeence mehod. They found ha he aveage Nussel numbe nceases wh nceasng Raylegh numbe and an ncease n he maeal paamee educes he hea ansfe. In hs sudy, he dual ecpocy bounday elemen mehod s employed o dsceze he spaal devaves n he seam funcon-vocy fom of he Nave-Sokes equaons, enegy

48 364 Eds: R Abascal and M H Alabad and he mcooaon equaons. DRBEM dea s appled o he Laplace opeao n each equaon by usng he fundamenal soluon of Laplace equaon and keepng all he ohe ems as nonhomogeney. The esulng maces conan negals of logahmc funcon o s nomal devave. The DRBEM educes all calculaons o he evaluaon of he bounday negals only. Ths fac mgh be advanageous n geomecally nvolved suaons ha ae fequenly encouneed n flud flow poblems. DRBEM applcaon of unseady naual convecon flow of mcopola fluds gves se o a sysem of nal value poblems n me. The fne dffeence scheme s made use of solvng hs sysem. Govenng Equaons The non-dmensonal unseady equaons of moon, enegy and mcooaon can be wen as follows [6] ψ = w ( + K) w K N + Ra T P x = w + u w x + v w y P T = T + u T x + v T y () ( + K ) N KN + Kw = N + u N x + v N y whee (x, y) Ω R, >. Ra, P and K ae he Raylegh numbe, Pandl numbe, and maeal paamee especvely. ψ and w ae he seam funcon and he vocy wh u = ψ, u = ψ and w =( v u). The nal and he bounday condons ae aken as y x x y w = T = N = when = x =: y, u = v =, T =.5, N = x =: y, u = v =, T =.5, N = () y =, : x, u = v =, T/ y =, N =. The vocy bounday condons ae deved fom he Taylo sees expanson of he seam funcon equaon. Fo K =, seam funcon, vocy anspo and enegy equaons descbe he classcal poblem of naual convecon of a Newonan flud n a dffeenally heaed squae cavy, fs consdeed by Vahl Davs [8]. The hea ansfe coeffcen n ems of he local Nussel numbe, Nu, and he aveage Nussel numbe, Nu av a he vecal walls ae defned by ( T ) Nu = x=,, Nu av = Nudy. x (3) Applcaon of DRBEM The equaons n () ae weghed hough he doman Ω as n [7], by he fundamenal soluon u = π ln

49 Advances n Bounday Elemen Technques IX 365 of Laplace equaon n wo dmensons n whch s he dsance beween he souce and he fxed pons. Applyng Geen s second deny, we have he followng negal equaons fo each souce pon : c ψ + (ψ q ψ ψ ψ q )dγ = ( w)ψ dω Γ ( + K)c w +(+K) (w Γ q w w w q )dγ = ( w + u w + v w + Ω x y K N Ra P P c T + Γ P (T q T T T q )dγ = Ω Ω ( T + u T x + v T y )T dω T x )w dω ( + K )c N +(+ K ) (N Γ q N N N q )dγ = ( N + u N + v N +KN Kw)N dω Ω x y (4) whee he subscp q ndcaes he nomal devave of he elaed funcon and c = θ /π wh he nenal angle θ a he souce pon. Expandng he nonhomogenees n each equaon n ems of he adal bass funcons f j s w = N+L α j f j (x, y) j= w + u w x + v w y + K N Ra N+L T P x = ᾱ j ()f j (x, y) T + u T x + v T y = j= N+L j= α j ()f j (x, y) (5) N + u N x + v N N+L y +KN Kw = ᾰ j ()f j (x, y) whee α j, ᾱ j, ᾰ j and α j ae undeemned coeffcens. The numbes of bounday and seleced nenal nodes ae denoed by N and L, especvely. The adal bass (coodnae) funcons f j ae lnked o he pacula soluons of each equaon wh he Laplace opeao. Subsung hese expansons n Eq. (4) and he applcaon of Geen s second deny o he gh hand sdes wll esul n max veco equaons fo each unknown ψ, w, T and N. Hψ Gψ q = (H ˆψ G ˆψ q )α ( + K)(Hw Gw q ) = (Hŵ Gŵ q )ᾱ j= P (HT GT q) = (H ˆT G ˆT q ) α ( + K ) (HN GN q ) = (H ˆN G ˆN q )ᾰ (6) whee G and H ae (N + L) (N + L) maces defned by H j = c δ j + (ln( ) π n ) dγ j, G j = ln( π Γ j ) dγ j. Γ j

50 366 Eds: R Abascal and M H Alabad The maces ˆψ, ŵ, ˆT and ˆN ae consuced by akng he coespondng pacula soluons as columns. Evaluaon of he gh hand sdes of each equaon n (5) a all bounday and neo (N +L) pons gves Hψ Gψ q = (H ˆψ G ˆψ q )F { w} { w ( + K)(Hw Gw q ) = (Hŵ Gŵ q )F + u w x + v w y + K N Ra } T P x { P (HT GT q) = (H ˆT G ˆT T q )F + u T x + v T } y ( + K ) { (HN GN q ) = (H ˆN G ˆN N q )F + u N x + v N } +KN Kw y (7) whee F s he (N + L) (N + L) max conanng coodnae funcons f j s as columns. Devaves of w, T and N ae appoxmaed by he DRBEM dea w x = F x F w, w y = F y F w T x = F x F T, N x = F x F N, T y = F y F T N y = F y F N (8) N x = F x F N, N y = F y F N. Subsung convecon ems back no Eq. (7), and fnally eaangng, we end up wh he followng sysem of odnay dffeenal equaons fo w, T and N especvely ẇ Hw + Gw q SF = T H T + G T q = (9) Ṅ H n N + G n N q SF = and a lnea sysem of equaons fo ψ Hψ Gψ q = Sw

51 Advances n Bounday Elemen Technques IX 367 wh S =(H ˆψ G ˆψ q )F and he maces SF, H, G, Hn, Gn, H and G ae SF = Ra T P x K N, SF =Kw ( H = S ( + K) H u F x F + v F ) y F H n = S ( + K ) H ( u F x F + v F y F ) K H = S P H ( u F x F + v F y F ) () G = S ( + K) G, Gn = S ( + K ) G, G = S P G. Fo he devaves of w, T and N n Eq. (9) mplc cenal dffeences ae used assumng he pevous wo me level soluons ae known. Resuls and Dscusson A numecal model was developed o valdae he accuacy fo he soluons of D unseady naual convecon flow of mcopola fluds n a squae cavy gven n Eq. () and (). The no-slp bounday condons of he veloces ae assumed. The hozonal walls ae adabac, whle he vecal walls ae sohemally heaed. Soluons ae obaned by usng N = bounday elemens and L =4neo nodes. Compuaons ae caed ou fo Ra = 3, 4 and 5 wh he me ncemens =.5,. and.3 especvely. The maeal paamee K s aken as,.5, and. An ncease n Raylegh numbe esuls n nensfed cculaon nsde he cavy, and hnne hemal bounday layes fo all he vaables, seam funcon, vocy and sohems nea he heaed and cooled walls. Fo Ra = 3 he voex a he cene was n ccula paen. Wh he ncease n Raylegh numbe he voex changes s shape o ellpcal fom. Snce he vscous foces ae domnang when Ra = 3, hee s no enough convecve moon of he flud whn he cavy. The sohems ae almos vecal n hs case. As he Raylegh numbe nceases, he sohems undego an nveson a he cenal egon of he cavy. These behavos can be seen fom Fg. and Fg.. The effec of vayng Ra on he aveage Nussel numbe a he heaed wall s shown n Table fo some values of K and a fxed value of Pandl numbe, P =.7. I shows ha fo a fxed value of Ra, an ncease n K educes he hea ansfe. In addon, he Newonan flud (K =) s found o have hghe aveage hea ansfe aes han a mcopola flud (K ). Ths s because an ncease n he voex vscosy would esul n an ncease n he oal vscosy of he flud flow, hus deceasng he hea ansfe. The esuls ae n good ageemen wh he esuls gven n [6].

52 368 Eds: R Abascal and M H Alabad Fgue : Seamlnes, vocy conous, sohems fo Ra = 3, K =.5, K =and K =

53 Advances n Bounday Elemen Technques IX 369 Fgue : Seamlnes, vocy conous, sohems fo Ra = 5, K =.5, K =and K =

54 37 Eds: R Abascal and M H Alabad K Ra = 3 Ra = 3 Ra = 4 Ra = 4 Ra = 5 Ra = 5 Pesen [6] Pesen [6] Pesen [6] Table : The effec of K on he aveage Nussel numbe Nu av fo dffeen values of Ra Concluson The unseady naual convecve hea ansfe of mcopola fluds n a dffeenally heaed squae cavy s compuaonally suded usng he DRBEM. Tme devave s dscezed usng mplc cenal dffeence scheme. The esuls ae obaned fo all vaables, seam funcon, vocy and empeaue also fo a Newonan flud fo compason. Smulaons ae pefomed o nvesgae he effecs of he Raylegh numbe, Ra, and he maeal paamee, K, on he momenum and hea ansfe. As he Raylegh numbe nceases bounday laye fomaon sas and he aveage Nussel numbe nceases. Howeve, an ncease n he maeal paamee educes he aveage Nussel numbe. Refeences [ ] AC. Engen, Theoy of mcopola fluds, J. Mah. Mech., 6 (966); -8. [ ] D.C Lo, D.L Young and C.C Tsa, Hgh esoluon od D naual convecon n a cavy by he DQ mehod, JCAM, 3 (7); [3 ] S. Roy and T. Basak, Fne elemen analyss of naual convecon flows n a squae cavy wh non-unfomly heaed walls, In. J. Engg. Sc., 43 (5); [4 ] T.H Hsu and C.K. Chen, Naual convecon of mcopola fluds n a ecangula enclosue, In. J. Engg. Sc., 34(4) (996); [5 ] T.H. Hsu, P.T. Hsu and S.Y. Tsa, Naual convecon flow of mcopola fluds n an enclosue wh hea souces, In. J. Hea Mass Tansfe, 4, No.7 (997); [6 ] O. Aydn and I. Pop, Naual convecon n a dffeenally heaed enclosue flled wh a mcopola flud, In. J. of Themal Scences, 46 (7); [7 ] P.W. Padge, C.A. Bebba and L.C. Wobel The Dual Recpocy Bounday Elemen Mehod, Comp. Mech. Pub. Souhampon and Elseve Sc., London, (99). [8 ] G. Vahl Davs, Naual convecon n a squae cavy: A benchmak soluon, In. J. Nume. Meh. Fluds, 3 (983);

55 Advances n Bounday Elemen Technques IX 37 Two-dmensonal Themo-Poo- mechanc fundamenal soluon fo unsauaed sols Pooneh Maghoul a, Behouz Gam c,b, Dens Duhamel a a Unvesé Pas-Es, Insu Nave, LAMI, Ecole des Pons, Pas, Fance Emal: Maghoulp@lam.enpc.f b Unvesé Pas-Es, Insu Nave, CERMES, Ecole des Pons, Pas, Fance Emal: Gam@enpc.f c Depamen of Cvl Engneeng, Unvesy of Tehan, Tehan, Ian Keywods: Bounday elemen mehod; fundamenal soluon; me doman; unsauaed sol; poous meda; hemo-poo-elasc behavou Absac. In hs acle, afe a bef dscusson on he unsauaed sols govenng dffeenal equaons ncludng he equlbum, a, mosue and hea ansfe equaons, he closed fom ansen hemal fundamenal soluon of he govenng dffeenal equaons fo an unsauaed wo-dmensonal defomable poous medum wh lnea elasc behavou fo a symmec pola doman have been noduced. The deved fundamenal soluon has been vefed mahemacally by compason wh he pevously noduced coespondng fundamenal soluon. Inoducon Thee ae numeous meda encouneed n engneeng pacce whose behavou s no conssen wh he pncples and conceps of classcal sauaed sol mechancs. An unsauaed poous medum can be epesened as a hee-phase (gas, lqud, and sold) o hee-componen (wae, dy a, and sold) sysem. The lqud phase s consdeed o be pue wae conanng dssolved a and he gas phase s assumed o be a bnay mxue of wae vapou and dy a [, ]. The a n an unsauaed sol may be n an occluded fom when he degee of sauaon s elavely hgh. A a lowe degee of sauaon, he gas phase s connuous. Commonly s he pesence of moe han wo phases ha esuls n a medum ha s dffcul o deal wh n engneeng applcaons. The phenomenon of coupled hea and mosue ansfe n a defomable paly sauaed poous medum s mpoan n elaon o seveal poblems, ncludng undegound soage of ho and cold maeals, dsposal of hgh-level nuclea wase and so on. Themallynduced mosue movemens can lead o changes n boh he hemal and sohemal popees of he sol whch can subsequenly affec he funconng of he sol fo s nended pupose. In ode o model unsauaed sol behavou, fsly he govenng paal dffeenal equaons should be deved and solved. Because of he complcaed foms of govenng paal dffeenal equaons he dffeen numecal mehods ae pesened fo solvng hem. Among hem, he bounday elemen mehod as he mos effcen s gong o be employed fo moe complcaed and coupled ones egadng he behavou and consequenly he govenng dffeenal equaons. As n hs mehod, dung fomulaon bounday negal equaons, he appled mahemacs concep of he Geen funcons has been employed. Ths ype of fundamenal soluons fo he govenng paal dffeenal equaons should be fs deved. Indeed, aempng o solve numecally he bounday value poblems fo unsauaed sols usng bounday elemen mehod leads one o seach fo he assocaed Geen funcons [3]. The compehensve sae-of-he-a evew by [4, 5, and 6] povdes clealy pesened nfomaon on he fundamenal soluon appled n he sauaed sol. Fo unsauaed sols, he fs Geen funcons fo he nonlnea govenng dffeenal equaons fo sac and quas-sac pooelasc meda fo boh wo and hee-dmensonal poblems have been deved by [6, 7]. The hemo-poo-elasc Geen funcons fo he nonlnea govenng dffeenal equaons fo sac and fo boh wo and hee-dmensonal poblems have been deved by [8]. The pesen eseach s an aemp o deve hese Geen funcons fo wo-dmensonal defomable quas-sac unsauaed sol. Followng some easonable and necessay smplfcaons, he fundamenal soluons wll be noduced n boh fequency and me domans.

56 37 Eds: R Abascal and M H Alabad Govenng Equaons Fo an unsauaed maeal nfluenced by hea effecs, he govenng paal dffeenal equaons consdeed ae of fou man goups: equlbum equaons, mosue ansfe equaons, a ansfe equaons and hea dffuson equaons [9]. Sold Skeleon. The equlbum equaon and he consuve law fo he sol s sold skeleon ncludng he effecs of sucon and empeaue []: j jpa P,, j a b () dj j Pa Dd sjdpa Pw TjdT () Whee s DD. s and T DD. T. Wh D s sm D T Tm e s epa Pw and e T e T and m The lnea elasc max D s wen as: jkl j kl k jl l jk D (3) Consdeng he san-defomaon elaons: j u, j uj, (4) One can conclude: u u P P T b (5) j, j, s a, j s w, j T, j j Connuy and ansfe equaons fo mosue. Ths pa wll be dvded no vapou ansfe and lqud ansfe fomulaon as follows: Lqud phase ansfe. Accodng o [] he unsauaed flow equaon can be wen as U qw / w Kw. z (6) The capllay poenal vaes wh mosue conen and empeaue. The capllay poenal n a efeence empeaue n ems of sucon wll ake he followng fom: ( ) / Pg Pw (7) w The vaaon of capllay poenal accodng o he empeaue s consdeed by he noducon of suface enson (T ). (, T) ( ). ( T)/ (8) Whee ( T ) s suface enson of he wae and and ( ) ae especvely, he suface enson of he wae and he capllay poenal n a efeence empeaue. Subsung fom eqs (7, 8) and afe manpulang some mahemacal opeaons a new sucon-based fomulaon of wae movemen equaon can be found. U qw / w DTwT DPw Pg Pw Dw z (9) ( ) d ( T) ( T ) Whee DTw s hemal lqud dffusvy Kw, D Pw s sohemal lqud dffusvy K w and dt. w Dw s gavaonal dffusvy, K w. Vapou ansfe. The equaon of vapou dffuson n poous meda accodng o [] heoy s gven as qvap D... v n vap () In ode o fnd vap as a funcon of empeaue and mosue conen, local hemodynamc equlbum should be assumed. Unde hs assumpon he followng hemodynamc elaonshps can be noduced:

57 Advances n Bounday Elemen Technques IX 373 vap. h. g h exp( ) () RT. Whee s he densy of sauaed wae vapou and h s s he elave humdy. In hs equaon s a funcon of T only and h s a funcon of only. Then: d dh (3) vap h T dt d Subsung Eq (3) n Eq () and consdeng he hypoheses pesened n [3]: q / D T D P P (4) vap w Tv Pv g w Whee D ( T) a d... vn h., n D Tv, w T dt, n and D Pv D g ( T)... vnvap, w RT. w, n n () Toal mosue ansfe. The oal mosue movemen n unsauaed sol due o empeaue gaden and s esulng mosue conen gaden s equal o he sum of he flows whch ake place n boh phases, vapou and lqud. Thus q / V U D T D P P D z (5) w T P g w w Whee DT s he hemal wae dffusvy, and s equal o D Tvap D Tw, and D P s sohemal wae dffusvy and s equal o D Pvap D Pw. Mosue mass consevaon. The consevaon law fo mosue mass s wen: w. ns. vap. n.( S) dv( w. VU) (6) I seems easonable o dspense wh he vaaons of Ka and Kw due o he vaaons of S and consequenly of Pg Pw fo smplcy, snce devng he consdeed Geen funcons wll become oo dffcul, a leas wh usual mehods, due o he nonlneay of o exsence of non-consan coeffcens n he govenng dffeenal equaons. In hs manne, he effecs of S have been consdeed n a and wae coeffcens of pemeably by assumng K a and K w as a mul-lnea funcon of Pg Pw fo each fne doman [6]. Howeve, Eq (6) may be wen as:.( ) n. ws vap S n w vap S wdt T wdp Pg wdp Pw (7) Consdeng ha, he poosy equals he volumec san: n kk uk, k (8) And he defnon of he degee of sauaon n he lnea fom: S bs( Pg Pw) ds( T T ), (9) The govenng equaon fo he mosue becomes: u ( P, g P ) kk w T ws vap( S) n. w vap. g n. w vap. g wdt T wdp Pg wdp Pw () Whee g S / Pg Pwand g S / T. Connuy and ansfe equaons fo a. Consdeng he genealzed Dacy s law, he a flow equaon can be gven as: Vg qg / g Kg. Pg / g z () Consdeng ha P g s a funcon of empeaue, hs equaon can be wen as

58 374 Eds: R Abascal and M H Alabad Kg Pg Vg.. T Kg. Pg / gz T g Usng he hemodynamc sae equaons fo gases, he Pg. can be eplaced by T Pg Pg Pam T T 73. g g pg Ths yelds P g Vg Kg. pg. T Kg. z (4) g Wh. g K. g..( ) d g c e S (5) g c and d ae consans. Wh he same appoach pesened fo he mosue equaons, he mass consevaon of a can be wen as: dvv (6) uk, k ( P P ) a w T. H S. nh g. n H a a a g (7) k a. HD. P HD. P k. HD. T a Pw w Pw a a Pg Tw a Hea flow. Toal flow of laen and sensve hea n an unsauaed poous medum s gven based on Phlp and De Ves heoy as: mw w mv w v mg g g w fg v v fg g q. T [ C.. U C.. V C.. V ] T T. h. V. h. V (8) Enegy Consevaon Equaon. The dffeenal equaon fo hea flow s a descpon, n mahemacal ems, of he law of consevaon of enegy. The enegy consevaon equaon n a poous medum can be expessed by (9) dv( q) In whch q s hea flux and s he volumec bulk hea conen of medum whch can be defned by ct. T T n. v. h (3) fg Whee c T s he volumec hea capacy of unsauaed mxue and can be wen as: ct ( n). s. Cms. w. Cmw ( n). vcmv ( n ). gcmg (3) Combnng he above equaons yeld he geneal dffeenal equaon of enegy consevaon as: u P kk, g Pw T (3) g g. g 3 4 T 5 Pg 6 Pw Whee C. C. S. C.( S ) C.( S ). T T. h.( S ) Cmw. n. w Cmv. n. v Cmg. n. g T T v. hfg. n Cms. s.( n) Cmw. S. w Cmv.( S ) v Cmg. n.( S ). g ms s mw w mv v mg g v fg 3 C.. D C.. D C. K.. T T. h. D. h. K. 4 m mw w Tw mv w Tv mg g Pg g w fg Tv v fg g Pg g () (3)

59 Advances n Bounday Elemen Technques IX 375 K g 5 Cmw. w. DPw Cmv. w. DPv Cmg. Kg. g T T w. hfg. DPv v. hfg. C.. D C.. D T T. h. D 6 mw w Pw mv w Pv w fg Pv Se of govenng equaons. The govenng paal dffeenal equaons based on he lneazaon assumpons consdeed may be summazed and smplfed as:. ( U)( x, ). U( x, ) ( x, ), x S, (33) T T Whee ( x, ) b b, U( x, ) u u Pw Pg T, x ( xy, ), D and he componens of ae: j kl a 3 6. a 33 7 a 34 8 a 35 9 a 4 3. a 43 4 a 44 5 a 45 6 a 5. a 53 a 54 a 55 3 And he componens of ae: a.. j. j j a a 3 3. a 4 4. a 5 5. lk a 33 a 34 a 35 a 43 7 a 44 8 a 45 9 a a a Whee, j, k,, l 35,, and s Laplace opeao. Laplace ansfom doman fundamenal soluon The objecve of hs secon s o deve he fundamenal soluon assocaed wh equaon (33) whch s he esponse of he medum o un pon excaon (connuous un lne excaon n D). The geneal soluon pocedue developed by Kupadze [4] s used n hs sudy fo he devaon of he fundamenal soluon [5]. Fo a connuous un lne foce n he h decon suddenly appled a he ogn,.e. ( x, ) ( x). H( ) whee H () s he Heavsde sep funcon, he Laplace ansfom of whch s / p ( x). Then, one can ewe equaon (33) n he followng fom: p.. Dxp (, ). Dxp (, ) I ( x), x R, (34) p ( x, p) D ( x, p) I ( x), p (35) Whee I denoes he un max of ode 5, D= D j s he ansfomed fundamenal soluon max and 5 5 ( x, p) p. ( x) ( x) s he dffeenal opeao max wh he componens as follows: (, ).. j x p a j j a 3( x, p) a3 4( x, p) a4 5( x, p) a5 3j ( x, p) p. a6 j 4 j ( x, p) p. a3 ( x, p) p. a 5j ( x, p) p. a a. j ( x, p) p. a a ( x, p) p. a a ( x, p) p. a4 a7. 44 ( x, p) p. a5 a8. 45 ( x, p) p. a6 a9. 53 ( x, p) p. a a4. 54 ( x, p) p. a a5. 55 ( x, p) p. a3 a6. / x,,. s he Laplacan opeao. and The fs sage s o deemne ( x, p), adjon dffeenal opeaos of ( x, p ), whch s defned by ( x, p) ( x, p) de ( x, p) (37) k kj j In whch he deemnan of ( x, p) s gven by x p D p D p D3 p D4 (38) de (, ) g j (36)

60 376 Eds: R Abascal and M H Alabad Whee D, D, D3and D 4 ae consans ncludng above a j coeffcens. Bul fom he cofacos of ( x, p), he elemen of dffeenal opeao ( x, p) can be expessed as: * D ( x, p) B. p. B. p. B. p. B. B. p.. B. p... B. p... B... j j j 6 j 7 j 8 j 4 6 D * 4 6 3( x, p) B9. p.. B. p.. B.. D * 4( x, p) B. p. B3. p.. B 4.. D * ( x, p) B. p.. B. p.. B.. D * ( x, p) B. p.. B. p.. B. p D * (, ) x p B p B p B3 D * ( x, p ) B. p. B. p. B D * ( x, p ) B. p. B. p. B. D * ( x, p) B. p.. B. p.. B. p D * (, ) x p B33 p B34 p B3 5 D * ( x, p ) B. p. B. p. B D * ( x, p ) B. p. B. p. B. D * ( x, p) B. p.. B. p.. B. p D * (, ) x p B45 p B46 p B4 7 D * ( x, p ) B. p. B. p. B D * ( 55 x, p ) B p B5 p B5 3 (39) Fo he second sage, we assume ha, ps a scala soluon o he equaon de ( x, p), p x (4) p Whch gves * ( x, p) ( x, p) I ( x) (4) p Consequenly, we ge * D x, p (4) Equaon (4) enables us o deemne he weny fve funcons D j by applyng he dffeenal opeao * x, po he sngle unknown funcon, p. Now, equaons (38) and (4) may be combned o yeld 6 D3 4 D D 4. p.. p.. p p. 4. D4 D4 D 4 3 D x (43) 4 Ths elaon, wh he noducon of as p. D4., leads o he followng dffeenal equaon: 3 x Afe manpulang some mahemacal opeaons, one may oban he, pfuncon as follows: (44) K. K. K 3. (45) p. D In whch: m p, m p, m p (46) 3 3 And he m coeffcens n eq (46) ae: D 3 D3 3. D3 3. m h, m h. h, m3 h. h 3D 4 3D4 3D4 m4, m5, m6 m m m m m m m m m m m m Whee D D 3 D3 D D D 3 3 h q, q, q 3 ( ) / 9, 9. 7 ( ) / 54 D D D D D D * Fnally, by applyng he dffeenal opeao x, po, p and by defnon of he nem funcons, we ge he D fundamenal soluon of (33) as: C. C. C. C C. C. C. C (47) (48) (49)

61 Advances n Bounday Elemen Technques IX 377 C. C. C. C x.... x x xj j j D x ( x, p )... D ( x, p). C. C. C. j j x x D ( x, p). C. 7 C. 5 C. x D ( x, p). C. 7 C. 5 C. D ( x, p) C. C. C x D ( x, p). C 5. C. 3 C 7. x D ( x, p). C. 7 C. 5 C. D ( x, p). C. C. C D33 ( x, p) C. C. C (5) D 35 ( x, p) C7. C 8. C D 43 ( x, p) C33. C34. 3 C35. 4 D 44 ( x, p) C36. C 37. C D 45 ( x, p) C39. C4. 3 C4. 4 D 53 ( x, p) C45. C 46. C D 54 ( x, p) C48. C49. 3 C5. 4 D 55 ( x, p) C5. C 5. C In whch he C j coeffcens ae consans and he kl nem funcons ae: 3 3 p K. K. K. K K. K. 3. p K.. K.. K K.. K. 3. K 3.. p p. K. K. K K. K. K 3. p K.. K. 3. K K.. K. 3. K 3.. p K.. K. 3. K 3. p K.. K. 3. K K. K. K 3. p (5)

62 378 Eds: R Abascal and M H Alabad Tansen fundamenal soluon To oban he me doman fundamenal soluon, one needs o evaluae he analycal nveson of D j. Fs, s necessay o fnd ou he nvese ansfom kl, of he funcons conanng he modfed Bessel funcons kl,. By he use of he followng fomulas [5]: K m j.. p m j. mj., L, p 4 K m j.. p mj. mj. mj. mj., L exp., (5) p. p K m.. j p mj. mj., L exp p m. 4. j Expessons of he nem funcons ae obaned: m4 m5 m 6, L m., m., m3., m m m 3, L m4 m., m 5 m., m 6 m3., 3, L 3m4. m. m., m5. m. m., m6. m3. m3., 4, L 4m4. m. m., m5. m. m., m6. m3. m3., m4 5 6,., m m L m m., m3., m m m m m3 m 3 m m m m m m , L m., m., m., (53) 3, L m4 m m m5 m m m6 m 3 m., L 4m4 m m m m5 m m m m6 m3 m 3 m, L m4 m m m5 m m m6 m 3 m., L 6m4 m m m m5 m m m m6 m3 m 3 m ,...,.. 3, 4...,...,.. 3., ,...,.. 3, 6...,...,.. 3., m4 m5 m 6 7, L 7 m., m., m3., m m m 3 Fnally, he fundamenal soluons ae obaned: x.... x x xj j j D x ( x, p )... D ( x, p). C. C. C. j j x x D3 ( x, p). C8. 7 C9. 5 C. 6 x D5 ( x, p). C4. 7 C43. 5 C44. 6 D ( x, p) C. C. C x D 5 ( x, p). C5. C6. 3 C7. 4 x D 4 ( x, p). C3. 7 C3. 5 C3. 6 D ( x, p). C. C. C. D 33 ( x, p) C. C. C D 35 ( x, p) C7. C 8. C D 43 ( x, p) C33. C34. 3 C35. 4 D 44 ( x, p) C36. C 37. C D 45 ( x, p) C39. C4. 3 C4. 4 D 53 ( x, p) C45. C 46. C

63 Advances n Bounday Elemen Technques IX 379 D 54 ( x, p) C48. C49. 3 C5. 4 D 55 ( x, p) C5. C 5. C (54) 4 Whee B. B. B3. 3 B4. 4 B5. B6. B7. 3 B8. 4 (55) 3 B5. B6. B7. 3 B8. 4 Fo nsance, he deved Geen funcons ae shown hough Fgs. o :..5 z y -5-6 x z Fg : Geen funcon D Sold skeleon dsplacmen n decon one due o a un pon load n decon one. Fg : Geen funcon D34 Vaaons of wae pessue due o a un ncemen n a pessue. Vefcaon Fsly, he new ansen fundamenal soluon s compaed o he seady THHM fundamenal soluon [8] by subsung he coeffcens of ems n whch he me vaaon s pesen, by zeo: 8 6 F xx j Dj ( x, p) j F. F... j j LnF F F,, j,, 4. p. D 4 6 F3 D 3( x, p) F3.. x Ln 4. p. D 6 F 4. p. D 4 4 D 4( x, p) F4.. x Ln 5 D 5( x, p) F5.. x Ln D ( x, p), 3, 5, j,, j 4 6 F 4. p. D 8 D ( x, p) F. Ln F. p. D 33 F. p. D 8 D 34( x, p) F. Ln 8 3 D 35( x, p) F 3. Ln 4 F. p. D F. p. D (56) 8 3 D 43( x, p) F3. Ln 8 3 D 44( x, p) F3. Ln F. p. D 4 4 F. p. D 8 33 D 45( x, p) F3 3. Ln 8 4 D 53( x, p) F4. Ln 4 F. p. D 4 4 F. p. D 8 4 D 54( x, p) F4. Ln 8 43 D 55( x, p) F4 3. Ln F. p. D 8 Whee 5 Lnand Fj ae he consan coeffcens. 3, 538, p. D4 Secondly, f he coeffcens epesenng he hemal behavou of he phenomenon and Heny s coeffcen appoach o zeo he seady fundamenal soluon (eq. 56) wll appoach he coespondng sohemal soluons [7]: 4

64 38 Eds: R Abascal and M H Alabad ( ) ( 3).ln.. j ( ). x. xj D j ( x, p) 8..( ). w. s. x. ln D 3 ( x, p) 8. p.( ). K g.( s ). x D 4 ( x, p). ln D 8. p..( ). K 3 D 4 (57) w.ln D 33 ( x, p). p. K 34 D43 w g g.ln D 44 ( x, p). p. K D D 5 D 5,, j, 4, Also, s evden ha whle s appoaches zeo, he fundamenal soluon n eq (57) appoaches elasosac fundamenal soluon [6, 7]: ( ) ( 3 ).ln.. j ( ). x. xj D j ( x, p) g. x D 4 ( x, p). ln 8..( ). 8..( ). K g. g w.ln D g.ln 33 ( x, p) D 44 ( x, p) D 3 ( x, p) D 3 D 4 D 34 D 43. K.. p. K. w w g g D 5 D 5j,, j, 4, (58) Refeences [] D.W Pollock Wae esouces eseach, (5), (986). [] S.Olvella, J.Caea, A.Gens, E.E.Alonso Tanspo Poous Meda, 5, 7 93(994). [3] E.Jabba, B.Gam, 7 h Inenaonal Confeence on Bounday Elemen Technques (B.Gam, A.Selle, M.H.Alabad), EC:, Pas, (6). [4] B.Gam, M.Kamalan Inenaonal Jounal of Geomechancs (4), (). [5] B.Gam, K.V.Nguyen Communcaons n Numecal Mehods n Engneeng (3), 9 3 (5). [6] B.Gam, E.JabbaInenaonal Jounal of Solds and Sucues4, (5). [7] B.Gam, E.Jabba 5 h Inenaonal Confeence on Bounday Elemen Technques (M.H.Alabad, V.M.A.Leão), EC:, Lsbon, 7- (4). [8] E.Jabba,B.Gam Inenaonal Jounal of Compue Modellng n Engneeng and Scences8(), 3-43 (7). [9] B.Gam, P.Delage, M.Ceolaza Advances n Engneeng Sofwae 9(), 9-43 (998). [] B.Gam, P.Delage, h Inenaonal Confeence on Unsauaed Sols (E.E.Alonso, P.Delage), EC:, Pas, (995). [] L.A.Rchads J. Physcs, (93). [] J.R.Phlp, D.A.de Ves Tans. Am. Geophys 38, -3 (957). [3] J.Ewen, H.R.Thomas Géoechnque, 39(3), (989). [4] V.D.Kupadze e al. Thee-dmensonal Poblems of he Mahemacal Theoy of Elascy and Themoelascy, Noh-Holland, Nehelands (979). [5] M.Abamowz, I.A.Segun Handbook of Mahemacal Funcons, Naonal Bueau of Sandads, Washngon, D.C. (965). [6] P.K.Banejee The Bounday Elemen Mehods n Engneeng, McGaw-Hll Book Company, England (994). [7] G.Bee Pogammng he Bounday Elemen Mehod, John Wley and Sons, England (). j g w

65 Advances n Bounday Elemen Technques IX 38 A Thee-Sep MDBEM fo Nonhomogeneous Elasc Solds X.W. Gao, a, J. Hong,b and Ch. Zhang,c Depamen of Engneeng Mechancs, Souheas Unvesy, Nanjng 96, PR Chna Depamen of Cvl Engneeng, Unvesy of Segen, D-5768 Segen, Gemany a xwgao@seu.edu.cn, b hongjun34567@6.com, c c.zhang@un-segen.de Keywods: Bounday elemen mehod; Nonhomogeneous elasc solds; Mul-doman echnque; Funconally gaded maeals (FGMs). Absac. In hs pape, a hee-sep bounday elemen mehod (BEM) s pesened fo solvng bounday value poblems n wo-dmensonal (D) and hee-dmensonal (3D) nonhomogeneous and lnea elasc solds by usng he mul-doman bounday elemen mehod (MDBEM). Fundamenal soluons fo homogeneous and lnea elasc solds ae adoped n he MDBEM. Bounday-doman negal equaons expessed n ems of nomalzed dsplacemens and acons ae fomulaed fo each sub-doman. The fs sep s he elmnaon of nenal vaables, and he second one s he elmnaon of bounday unknowns used only by ndvdual sub-domans, and he las sep s he esablshmen of a sysem of lnea algebac equaons accodng o he connuy/dsconnuy condons of he dsplacemens and he acons a common nodes on he nefaces. Dsconnuous elemens ae ulzed o model he acon dsconnuy a cone nodes. Numecal examples ae pesened o demonsae he accuacy and he effcency of he pesen hee-sep MDBEM. Inoducon The maeal popees of connuously nonhomogeneous solds such as funconally gaded maeals (FGMs) ae dependen on spaal posons []. Alhough he bounday elemen mehod (BEM) has been successfully developed and appled o homogeneous and lnea elasc solds snce many yeas, s exenson and applcaons o connuously nonhomogeneous and lnea elasc solds ae no sagh-fowad. The man eason s he fac ha he equed fundamenal soluons o Geen's funcons fo geneal nonhomogeneous and lnea elasc solds ae ehe mahemacally oo complcaed o no avalable, whch makes an easy and effcen numecal mplemenaon dffcul. One emedy fo hs dffculy s he use of fundamenal soluons fo homogeneous and lnea elasc solds n he BEM fomulaon, whch nvolves doman-negals n he bounday-doman negal equaons. Alhough he doman negals can be conveed o global bounday negals [] o local ones [3], he esulng sysem of algebac equaons nvolves unknown quanes conssng of boh bounday unknowns and nenal dsplacemens. The numecal soluon of such a sysem wh mxed bounday and nenal vaables s n geneal vey me consumng o even unfeasble fo lage-scale poblems usng he sngle-doman BEM. An effcen numecal way o solve bounday value poblems n nonhomogeneous and lnea elasc solds s he mul-doman bounday elemen mehod (MDBEM) [4,5], whee he consdeed doman s dvded no seveal sub-domans. Thee exs wo dffeen effcen soluon echnques n he MDBEM. One s he vaable condensaon echnque and he ohe s he eave echnque. The fs echnque esuls n a small sysem of equaons by elmnang some vaables n an nemedae sep, whle he lae solves he lage sysem of algebac equaons by usng a fas eave scheme such as he Kylov s eaon mehod [6]. In hs pape, a hee-sep BEM s pesened fo solvng bounday value poblems n wo-dmensonal (D) and hee-dmensonal (3D) nonhomogeneous and lnea elasc solds by usng he MDBEM [5]. Fundamenal soluons fo homogeneous and lnea elasc solds ae adoped n he MDBEM. Bounday-doman negal equaons wh nomalzed dsplacemens and acons ae fomulaed fo sub-domans. The fs sep of he mehod s he elmnaon of nenal vaables fo each sub-doman. The second sep s he elmnaon of bounday unknowns defned ove nodes used only by he sub-doman self. And he hd sep s he esablshmen of a sysem of lnea

66 38 Eds: R Abascal and M H Alabad algebac equaons accodng o he connuy/dsconnuy condons of he dsplacemens and he acons a common nodes on he nefaces. Dsconnuous elemens ae mplemened o popely model he acon dsconnuy a cone-nodes. The pesen hee-sep MDBEM has wo mpoan feaues, namely, only nefacal dsplacemens ae unknowns n he fnal sysem of algebac equaons, and he coeffcen max s blocked and spase. Consequenly, lage-scale D and 3D bounday value poblems n nonhomogeneous and lnea elasc solds can be deal wh effcenly. Inegal equaons fo funconally gaded maeals (FGMs) In soopc, connuously nonhomogeneous, and lnea elasc solds, such as funconally gaded maeals, he shea modulus s a funcon of spaal coodnaes, whle, fo mos cases, he Posson s ao can be egaded as a consan. Unde hs assumpon, negal equaons fo bounday and nenal nodes can be deved by usng Gauss dvegence heoem as follows [5,7] cu ( x p ) U ( x, x p ) ( x) d T ( x, x p ) u ( x) d V ( x, x p ) u ( x) d, () j j j j j j whee c= fo nenal pons and c=/ fo smooh bounday pons, U j and T j ae he Kelvn dsplacemen and acon fundamenal soluons [8] fo an soopc, homogeneous and lnea elasc sold wh, and Vj, k, k[( ) j,, j ] ( )(,, j, j, ), () 4 ( ) whee = fo D and =3 fo 3D poblems and =-. In Eqs. () and (), u j ( x) and ae he nomalzed dsplacemens and shea modulus defned by u ( x) ( x) u ( x), ( x) log ( x). (3) Fom Eq. (), can be seen ha no dsplacemen gadens ae nvolved n he bounday-doman negal equaons. Ths s abued o he use of he nomalzed quanes defned n Eq. (3). Compason of Eq. () o he convenonal bounday negal equaons fo soopc, homogeneous and lnea elasc solds [8] shows ha hee s a doman negal appeang n he negal equaons. Ths doman negal s conveed no an equvalen bounday negal usng he adal negal mehod (RIM) []. Snce he unknown vaables u j ae ncluded n he doman negal, some nenal pons ae equed o be placed nsde he doman o mpove he compuaonal accuacy n he appoxmaon of u j n ems of he adal bass funcons (RBFs) n he applcaon of RIM [,7]. Consequenly, he esulng sysem of algebac equaons ncludes boh bounday unknowns and nenal nomalzed dsplacemens as he sysem unknowns. To solve such a sysem usng a sngle doman echnque, he compuaonal me and he equed memoy soage would be huge fo complcaed 3D poblems. Theefoe, a obus MDBEM soluon echnque s desed fo solvng such lage-scale poblems. Thee-sep MDBEM fo soopc, nonhomogeneous and lnea elasc solds As shown n Fg., he doman of concen s dvded no a numbe of sub-domans. The nodes used fo each sub-doman ae classfed no hee ypes: self nodes, nenal nodes, and common nodes. To effcenly explo he MDBEM echnque, he ode of he hee ypes of nodes s aanged n such a way ha he self nodes ae numbeed fs, followed by he common nodes and fnally he nenal nodes. To model he acon dsconnuy a a cone o an edge, dsconnuous elemen [5] s used and moe han one nodes ae defned a an nenal cone a whch dffeen sub-domans mee

67 Advances n Bounday Elemen Technques IX 383 and a a bounday cone a whch a leas one of he componens s specfed wh he dsplacemen bounday condon (see Fg. ). j j u u k u u self node nenal node common node j Fg.. Defnon of hee ypes of nodes Afe usng he node aangemen saegy descbed above and applyng RIM [] o ansfom he doman negal no a bounday negal, he bounday-doman negal equaons () can be conveed no a sysem of algebac equaons fo each sub-doman, whch can be wen n he max fom as fo bounday nodes, and H uh u H u G G (4) bs s bc c b bs s bc c HuHu HuG G (5) s s c c s s c c fo nenal nodes. In Eqs. (4) and (5), he subscp b denoes quanes fo bounday nodes conssng of self nodes and common nodes, and he subscps s, and c epesen quanes fo self, nenal and common nodes, especvely. Also, u s, u c, u, s and c ae dsplacemen and acon vecos coespondng o he hee ypes of nodes. I s noed ha fo pecewse homogeneous solds, he max H s an deny max and H b s a zeo max. Afe nvokng all specfed dsplacemen and acon bounday condons n Eqs. (4) and (5), he followng equaon se can be obaned fo each sub-doman A xh u H u y G, (6) bs s bc c b b bc c Ax s shu c c Hu y G c c, (7) whee x s s he veco conssng of unknown dsplacemens and unknown acons ove he self nodes, and y b and y ae he known vecos fomed by mulplyng all gven bounday dsplacemens and acons wh he coespondng max elemens. To solve Eqs. (6) and (7) fo he unknown vecos x s, u c, c and u, a hee-sep soluon echnque s appled, whch s descbed n he followng.

68 384 Eds: R Abascal and M H Alabad Sep : Elmnang nenal unknowns fo each sub-doman The max H n Eq. (7) s a squae max and well-posed, so elmnang he nenal dsplacemens u fom Eqs. (6) and (7) follows fo each sub-doman whee A xh u y G, (8) bs s bc c b bc c A A H H A bs bs b ( ) s, H H H H H bc bc b ( ) c, G G H H G bc bc b ( ) c, y y H H y b b b ( ). I s noed agan ha fo pecewse homogeneous solds, he max H s an deny max and s a zeo max, and heefoe s unnecessay o fom Eq. (8) snce he maces o be fomed n Eq. (9) educe o he ognal maces. (9) Hb Sep : Elmnang bounday unknowns fo each sub-doman Snce fo each sub-doman he bounday nodes ae composed of he self nodes and he common nodes, Eq. (8) can be dvded no wo ses of equaons fo self nodes and common nodes,.e., AxHuy G, () ss s sc c s sc c AcsxsHccuc yc Gccc. () All maces n Eqs. () and () ae sub-maces of he coespondng maces n Eq. (8). Now, elmnaon of x s fom Eqs. () and () yelds Hˆ u yˆ Gˆ, () whee cc c c cc c ˆ Hcc Hcc Acs ( Ass ) Hsc, ˆ Gcc Gcc Acs ( Ass ) Gsc, yˆ c yc Acs ( Ass ) ys. (3) Sep 3: Assemblng he sysem of equaons fom all sub-doman s conbuons Equaons () and () can be appled o evey sub-doman. Fo he n-h sub-doman, he acon veco c fo he common nodes can be expessed as ( Gˆ ) ( Hˆ u yˆ ). (4) ( n) ( n) ( n) ( n) ( n) c cc cc c c Assemblng all sub-doman s conbuons fo he global common nodes and applyng he acon ( n) equlbum condon c esuls n he fnal sysem of algebac equaons as whee n K U cc c c Y, (5)

69 Advances n Bounday Elemen Technques IX 385 whee K ( ˆ G ) Hˆ Q, (6) ( n) ( n) ( n) cc cc cc n Y ( ˆ G ) yˆ, (7) ( n) ( n) c cc c n ( n) Q s he locaon max conssng of and, whch elaes he local dsplacemen veco ( n ) u c o he global one c U. Solvng Eq. (5) fo U c, we can oban he dsplacemens a all common neface nodes, and hen subsung back no Eq. (4) we can compue he acons a each doman s common nodes. Usng hese esuls, he bounday unknowns a self nodes of each sub-doman can be calculaed by applyng Eq. (). Equaon (5) shows ha he numbe of degees of feedom of he sysem s only he numbe of degees of feedom of he common neface nodes, whch s much smalle han hose of all bounday and nenal nodes. I s noed ha fo pecewse homogeneous solds, he pesen hee-sep soluon echnque educes o a wo-sep soluon echnque conssng of he las wo seps as descbed above. Numecal example The numecal example consdeed hee s a mul-plana ubula DX-Jon as depced n Fg.. Ths example has been analyzed n efeence [8]. The DX-Jon consss of a lage damee ube (chod) neseced ohogonally by wo smalle damee ubes (baces). The oue ad of he chod and baces ae 8.6mm and 8.58mm, especvely, whle he nne ad ae one-half of hese values. Fo smplcy, only a sho secon of he ubes s analyzed and he egh-fold symmey s exploed (Fg. 3). The half-lenghs of he ubes ae gven by Lx 9mm and Ly Lz 66mm. A consan Posson s ao =.3 s used and a ensle load F=.34GPa s appled o each end of he DX-Jon. F F F F F Fg.. Mul-plana ubula DX-Jon subjeced o axal loads F

70 386 Eds: R Abascal and M H Alabad The compuaonal doman s dvded no hee sub-domans as shown n Fg. 3. The BEM mesh consss of 66 lnea quadlaeal bounday elemens wh 66 bounday nodes (ncludng 56 neface elemens and 6 neface nodes) and 39 nenal nodes. z 3 x y Fg. 3. BEM mesh of he DX-Jon Case : Homogeneous DX-Jon Fo he pupose of he valdaon of he pesen MDBEM, numecal calculaons ae fs caed ou fo he homogeneous case by seng he Young s modulus E = GPa fo all he hee sub-domans. Fgue 4 shows he dsbuon of he dsplacemen u x along he mddle lne of he oue suface of he chod. Fo compason, he esuls usng he sngle-doman code BEMECH lsed n [8] ae also gven. I can be seen ha he wo ses of numecal esuls ae n vey good ageemen. Dsplacemen (mm) BEMECH Pesen x (mm) Fg. 4. Dsplacemen fo homogenous DX-Jon wh E = GPa

71 Advances n Bounday Elemen Technques IX 387 Case : Funconally gaded DX-Jon x y y z z The second compuaon s pefomed by assumng E EL x, E E L e and E3 E L e fo he sub-domans, and 3, especvely. The paamees, and ae deemned by x y z EL E log( EL / E) log( EL / E),,, L L L x x y z whee E=GPa, E E, E 5E and E E. Fgue 5 plos he dsbuon of he L L L dsplacemen u x along he mddle lne of he oue suface of, and Fg. 6 shows he dsplacemens u y and u z along he mddle lnes of he oue sufaces of and 3, especvely. Fgue 7 llusaes he axal sesses yy and zz along he mddle lnes of he domans and 3, especvely. Fom Fgs. 6 and 7 can be seen ha he axal dsplacemens along domans and 3 ae que dffeen, snce he vaaon of he Young s modulus s dffeen fo he wo domans. On he ohe hand, Fg. 7 shows he same paen fo he axal sesses of he wo domans and 3. Ths ndcaes ha he dsbuon of he axal sesses manly depends on he appled loads n he consdeed case. Dsplacemen (mm) y ux (ove suface of doman ) x (mm) Fg. 5. Vaaon of u x ove he oue suface of z Dspacemens (mm) uy (ove suface of doman ) uz (ove suface of doman 3) y o z ove suface o 3 (mm) Fg. 6. Vaaons of u y and u z ove he oue sufaces of and 3

72 388 Eds: R Abascal and M H Alabad Sesses Sgma-yy (ove doman ) Sgma-zz (ove doman 3) y o z ove suface o 3 (mm) Fg. 7. Vaaons of axal sesses along mddle lnes of and 3 Summay A hee-sep MDBEM s pesened n hs pape fo he numecal soluon of bounday value poblems n D and 3D soopc, connuously nonhomogeneous and lnea elasc solds. Fundamenal soluons fo soopc, homogeneous and lnea elasc solds ae mplemened n he pesen MDBEM. Bounday-doman negal equaons expessed n ems of nomalzed dsplacemens and acons ae fomulaed fo each sub-doman. The doman-negals ae ansfomed o bounday negals by usng he adal negaon mehod (RIM). Though a wo-sep elmnaon pocedue, a sysem of lnea algebac equaons fo he dsplacemens a common nodes s esablshed. Dsconnuous elemens ae adoped o model he acon dsconnuy a cone nodes. The numbe of unknowns n he pesen hee-sep MDBEM s much smalle han ha of he classcal sngle-doman BEM. Numecal examples show ha he pesen hee-sep MDBEM s effcen and suable fo solvng lage-scale poblems n soopc, connuously nonhomogenous and lnea elasc solds. Acknowledgemen Suppo by he Geman Reseach Foundaon (DFG) unde he pojec numbe ZH 5/- s gaefully acknowledged. Refeences [] X.W. Gao, Ch. Zhang, J. Sladek and V. Sladek: Compos. Sc. Tech. Vol. 68 (8), p. 9. [] X.W. Gao: Eng. Anal. Bound. Elem. Vol. 6 (), p. 95. [3] J. Sladek, V. Sladek and Ch. Zhang: Buldng Reseach Jounal, Vol. 53 (5), p. 7. [4] S. Ahmad and P.K. Banejee: In. J. Nume. Meh. Engng., Vol. 6 (988), p. 89. [5] X.W. Gao, L. Guo, and Ch. Zhang: Eng. Anal. Bound. Elem. Vol. 3 (7), p [6] K. Davey, S. Bounds, I. Rosndale and M.T.A. Rasgado: Comp. & Suc. Vol. 8 (), p [7] X.W. Gao, Ch. Zhang and L. Guo: Eng. Anal. Bound. Elem. Vol. 3 (7), p [8] X.W. Gao and T.G. Daves: Bounday Elemen Pogammng n Mechancs. Cambdge Unvesy Pess,.

73 Advances n Bounday Elemen Technques IX 389 DBEM fo Facue Analyss of Sened Cuved Panels (Plaes and Shallow Shells Assembles) P.M. Baz and M.H. Alabad Depamen of Aeonaucal Engneeng, Impeal College London Souh Kensngon campus, London SW7 AZ Keywods: Facue Mechancs, Shea Defomable, Plae and Shallow Shell Assembles. Absac. Ths pape pesens applcaons whee he DBEM fomulaon pesened by Dganaa and Alabad [3] s combned wh he mul egon BEM pesened ecenly by Baz and Alabad [], fo he analyss of cacked shea defomable plaes and shallow shell assembles. Sess nensy facos ae obaned usng he CTOD echnque. Seveal examples ae solved o demonsae he capables of he poposed echnque. Compang DBEM wh FEM models, was clea ha good accuacy and ecency can be acheved wh he pesen mul egon DBEM appoach. Inoducon. Cacks ae pesen n mos sucual membes ehe as a esul of he manufacung pocess o due o a localzed damage dung sevce lfe. These cacks may gow by fague, cooson o ceep, deceasng he sengh and leadng o he falue of he sucue. The dual bounday elemen mehod (DBEM) s based on he use of deen equaons on each cack suface (dsplacemen and acon negal equaons). Dung he pas yeas, he dual bounday elemen mehod has emeged as a obus numecal mehod fo facue mechancs poblems []. Applcaons of he dual bounday elemen mehod o facue mechancs of shea defomable plaes have been epoed ndependenly by Rashed, Alabad and Bebba [6] and Ahmad-Booghan and Weang [8] whle DBEM fo shea defomable shallow shells have been deved by Dganaa and Alabad [3]. Mul-egon BEM (Plae and Shallow Shells). Les consde M assembled cylndcal shallow shells o plaes joned a J as shown n Fgue a. The global coodnae sysem s gven by -x -x -x 3, and he local coodnae sysems fo each egon by -x -x -x 3 (m =;M). The plaes o shallows shells have an unfom hckness h, Young s modulus E, Posson s ao º. As shown n Fgue c, w epesen oaons of he mddle suface, w 3 denoes he ou-of-plane dsplacemen, and u epesen n-plane dsplacemens. And genealzed acons ae denoed as: p due o he sess couples, p 3 due o shea sess esulan and due o membane sess esulans. Shallow shells ae defned usng a cuvlnea coodnae sysem. Ths means ha conay o fla plaes whch have a fx nomal (local coodnae sysem) hough he whole plae, he local coodnae sysem of a shallow shell changes wh he cuvaue (see Fgue b). In he smple case of wo shallow shells o plaes wh he same axs oenaon a he juncon lne (see Fgue c), he connuy and equlbum equaons along he jon can be wen as follows: u = u + ; w = w + ; X = () = X p = Because wo o moe angled plaes o shallow shells joned ogehe ae consdeed, an appoach smla o ha poposed n [] s developed. To smplfy hs appoach, he local coodnae sysems of each egon s assumed o be defned such ha he x decons ae all algned wh he global decon x, followng he mplemenaon fo plae assembles pesened by Wen e. al. [9] o D =

74 39 Eds: R Abascal and M H Alabad x 3 m m J n Shallow Shell Base Plane m x c J n w 3 m x m x m x 3 c u x 3 x m x 3 x R c x a) m m x m+ x 3 b) x m+ m x m+ 3 x m+ x m m x x 3 m+ x x x 3 x x Dsplacemens c) m x 3 m+ x m m+ w 3 w m m+ 3 w w m m+ w w m u m+ u m m+ u u m m+ p m p p m+ 3 p 3 Tacons m m m+ m+ p p m m+ x D Fgue : a) Plae and Shallow Shell Assembly, b) Local Coodnae Sysem of Shallow Shell, c) Smple Assembles. Psa [4]. Based on he above smplfcaon, w 3 and u dsplacemens fo any gven shallow shell a a juncon lne (J ) can be pesened as shown n Fgue b. Theefoe, compably equaons fo each pa of adjacen shallow shells (e.g. m =and m =) could be wen as follows: u (n n + n 3n 3)+w3(n 3n + n 33n 3) = u (n n + n 3n 3)+w3(n 3n + n 33n 3) u (n n 3 + n 3n 33)+w3(n 3n 3 + n 33n 33) = u (n n 3 + n 3n 33)+w3(n 3n 3 + n 33n 33) u = u w = w w = w = () whee n ae he componens of he oaon max of he shallow shell base plane m fom local o global coodnaes [], and n ae he componens of he oaon max fom he cuvlnea coodnae sysem o he shallow shell base plane m. The componens of n ae gven by: n = cos( ); n =; n 3 =cos(9+ ) n = ; n =; n 3 = n 3 = cos(9 ); n 3 =; n 33 =cos( ) (3) whee s measued wh espec o he shallow shell base plane, as shown n Fgue b. Equaons n () esul n a sysem of 5M 4 compably condons, and have o be supplemened

75 Advances n Bounday Elemen Technques IX 39 wh 4 equlbum condons as follows: X [ (n n + n 3 n 3)+p 3 (n 3 n + n 33 n 3)] = = X [ (n n 3 + n 3 n 33)+p 3 (n 3 n 3 + n 33 n 33)] = = X = = X p = (4) o poduce he equed 5M equaons. Ths appoach eles on he assumpon ha he plae o shallow shell flexual gdy n s own plane s so lage ha s possble o gnoe s assocaed defomaon, n anohe wods, hee s no dllng oaon. Bounday Inegal Fomulaon. The dual bounday elemen mehod s based on he use of wo ndependen equaons, he dsplacemen and acon bounday negal equaons, a each pa of concden souce pons on he sufaces ha defne a cack. The dsplacemen negal equaons fo collocaon pons on one cack suface (x + + ), can be wen as follows [3]: (x + )+ Z Z (x )+ (x + x) (x) (x) = (x + x) (x) (x) Z 3 x + X (X)+ (X)+ (X) (X) Z 3 x + X (( ) + ) 3 (X) (X) Z + 3(x + X) 3 (X)(X) (5) and, Z + = (x + )+ Z (x )+ (x + x) (x)(x) (x + x) [ ( )+ ] 3 (x) (x)(x) Z (x + X) [ ( )+ ] 3 (X)(X) Z Z = (x + x) (x)(x)+ (x + X) (X)(X) (6) In ode o avod an ll-condoned sysem, he acon negal equaons ae used fo collocaons on he ohe cack suface (x )[3]: (x ) Z Z (x + )+ (x )= (x x) (x)(x)+ (x ) 3(x x) 3 (x)(x) Z Z = (x ) (x x) (x)(x)+ (x ) 3(x x) 3 (x)(x) Z (x ) µ (X)+ (X)+ (X) 3(x X)(X)

76 39 Eds: R Abascal and M H Alabad Z (x ) (( ) + ) 3 (X) 3(x X)(X) Z + (x ) 3(x X) 3 (X) (7) 3(x ) Z Z 3(x + )+ (x ) 3(x x) (x)(x)+ (x )= 33(x x) 3 (x)(x) Z Z = (x ) 3(x x) (x)(x)+ (x ) 33(x x) 3 (x)(x) and Z (x ) Z (x ) µ (X)+ (X)+ (X) 33(x X)(X) (( ) + ) 3 (X) 33(x X)(X) Z + (x ) 33(x X) 3 (X) (8) (x ) Z (x + )+ (x )= (x x) (x)(x) Z + (x ) (x x) [ ( )+ ] 3 (x) (x)(x) Z (x ) (x X) [ ( )+ ] 3 (X)(X) Z Z = (x ) (x x) (x)(x)+ (x ) (x X) (X) + (x ) [( ) + ] 3 (x ) (9) Equaons (5-6) and (7-9) epesen dsplacemen and acon negal equaons on he cack sufaces, especvely; and ogehe wh he dsplacemen negal equaons (see equaons 3. and 3. n [3]) fo collocaon on he es of he bounday, fom he dual bounday negal fomulaon n shallow shell poblems. I s woh noce ha as he souce pons x + and x ae concden, exa feeemsappeanequaons(5-9)focollocaononbohcacksufaces. Soluon Saegy. The mplemenaon of he dual bounday elemen fomulaon eques ha bounday o be dscezed. In he case of shallow shell egons seveal unfomly dsbued doman pons ae equed fo he applcaon of he dual ecpocy mehod (DRM). In he case of he bounday: connuous, sem dsconnuous and dsconnuous quadac sopaamec bounday elemens ae used o descbe he geomey of each egon (plae o shallow shell). The dealed modellng saegy s smla o he one descbed by Poela Alabad and Rooke [5] and can be summazed as follows [3]: ² The cack boundaes ae dscezed wh dsconnuous quadac elemens (each node of one cack suface s concden wh anohe node on he oppose cack suface). ² Connuous quadac elemens ae appled along he emanng bounday of he sucue, excep a he nesecon beween a cack and an edge o a cones, whee sem-dsconnuous elemens ae equed n ode o avod a common node a nesecons. ² The acon negal equaons ae used fo collocaon on one cack suface (x ).

77 Advances n Bounday Elemen Technques IX 393.E-4 ABAQUS DBEM x /h.e+.e+.5e+.e+ u /h -.E-4-4.E-4-6.E-4 E=7.6 GPa v=.33 q=.6 MPa b=.75 m h =.5 m h =.5 m h 3=.75 m h x 3 x h 3.75 m.35 m x h.4 m Fgue : Cack Openng Dsplacemen fo Cuved Sened Panel. ² The dsplacemen negal equaons ae used fo collocaon on he oppose cack suface (x + + ). ² In he non-cack boundaes (x ) he common dsplacemen negal equaons ae employed. ² Fo shallow shell, seveal unfomly dsbued DRM pons ae used n he doman. Ths smple saegy s vey obus, makng he DBEM an eecve ool fo he modelng of geneal edge o embedded cack poblems. Numecal Example. Acuvedsened panel wh a cene cack as shown n Fgue s analyzed. The maeal popees and dmensons ae also pesened n Fgue. The cuved panel s smply suppoed along all he bounday and s subjeced o an unfom nenal pessue q. In ode o valdae he DBEM fomulaon n shallow shell and plae assembles, esuls ae compaed wh FEM soluons. The FEM half model has a oal of 59 elemens and 833 nodes. The DBEM model conans 7 shallow shells and 6 fla plaes, wh a oal of 48 elemens and 4 DRM pons (4 elemens pe cack sde). Fgue pesens he n-plane dsplacemen u along he symmey lne of he cenal cacked shallow shell (cack openng dsplacemen). Fom Fgue s evden he good ageemen beween boh numecal soluons (DBEM and FEM). Concluson. In hs wok, applcaons of he DBEM fo he facue mechancs analyss of shea defomable plae and shallow shells assembles was pesened. A mul-egon echnque was used o model plae and shallow shell assembled sucues subjeced o abay loadng. Addonal equaons wee obaned by mposng compably and equlbum equaons along he neface boundaes. The DBEM shallow shell fomulaon was developed by couplng bounday elemen fomulaons of shea defomable plae bendng and wo dmensonal plane sess elascy; as a esul, doman negals appea n he fomulaon and ae eaed wh he Dual Recpocy Technque.

78 394 Eds: R Abascal and M H Alabad Tacon negal equaons wee appled on one cack suface and he usual dsplacemen negal equaons on he ohe cack suface and non-cack boundaes. Specal cack p elemens ae used o model accuaely he dsplacemen feld. These dsplacemens wee used fo he evaluaon of SIF usng he CTOD echnque. Compang DBEM wh FEM models, was clea ha good accuacy and ecency can be acheved wh he pesen mul egon DBEM appoach. Refeences [] Alabad, M.H., The Bounday Elemen Mehod, vol II: applcaon o solds and sucues,chchese, Wley (). [] P.M. Baz, M.H. Alabad, Local Bucklng of Thn-Walled Sucues (Plae and Shallow Shell Assembles) by he Bounday Elemen Mehod. Submed o Inenaonal Jounal of Solds and Sucues. [3] Dganaa, T., Alabad, M.H., Dual bounday elemen fomulaon fo facue mechanc analyss of shea defomable shells, Inenaonal Jounal of Solds and Sucues, 8, (). [4] D Psa, C., Bounday Elemen Analyss of Mul-layeed Panels and Sucues, PhD Thess, Depamen of Engneeng, Queen May Unvesy of London (5). [5] Poela, A., Alabad, M.H. and Rooke, D.P., The dual bounday elemen mehod: eecve mplemenaon fo cack poblems, Inenaonal Jounal fo Numecal Mehods n Engneeng, 33, (99). [6] Rashed, Y. F., Alabad, M. H. and Bebba, C. A., Hype-sngula bounday elemen fomulaon fo Ressne plaes, Inenaonal Jounal of Solds and Sucues, 35, 9-49 (998). [7] Ressne, E., On a Vaaonal Theoem n Elascy, Jounal of Mahemacs and Physcs, 9, 9-95 (95). [8] Weang, J.L., Ahmad-Booghan, S.Y., Facue analyss of plae bendng poblems usng bounday elemen mehod, n Plae Bendng Analyss wh Bounday Elemens, Advanced n Bounday Elemen Sees, M. H. Alabad (Ed.), Compuaonal Mechancs Publcaons, Souhampon (998). [9] Wen, P.H., Alabad, M.H., Young, A., Cack gowh analyss fo mul-layeed afame sucues by bounday elemen mehod, Engneeng Facue Mechancs, 7, (4). [] Zenkewcz, O.C., Taylo, R., The Fne Elemen Mehod, Vol : Sold Mechancs, B-H, Oxfod, ().

79 Advances n Bounday Elemen Technques IX 395 An Incemenal Technque o Evaluae he Sess Inensy Facos by he Elemen-Fee Mehod P.H. Wen and M.H. Alabad Depamen of Engneeng, Queen May, Unvesy of London, London, UK, E 4NS Depamen of Aeonaucs, Impeal College, London, UK, SW7 BY Absac In hs pape an ncemenal echnque was developed o evaluae sess nensy facos accuaely by he use of he elemen-fee Galekn mehod based on he vaaon of poenal enegy. The sffness max s evaluaed wh a doman negal by he use of adal bass funcon nepolaon whou elemens n he doman. The Laplace ansfomaon echnque and he Dubn nveson mehod ae used o oban he me doman physcal values. The applcaons of he poposed ncemenal echnque o wo-dmensonal facue mechancs have been pesened. Compasons have been made wh benchmak analycal soluons and bounday elemen mehod. Key wods: Elemen-fee Galekn mehod, sess nensy faco, Laplace ansfomaon mehod, Facue mechancs, movng leas squae nepolaon.. Inoducon Alhough he FEM and BEM have been vey successfully esablshed and appled n engneeng as numecal ools, he developmen of new advanced mehods nowadays s sll aacve n compuaonal mechancs. Meshless appoxmaons have eceved much nees snce Nayoles e al [] poposed he dffuse elemen mehod. Lae, Belyschko e al [] and Lu e al [3] poposed elemen-fee Galekn mehod and epoducng kenel pacle mehods, especvely. One key feaue of hese mehods s ha hey do no eque a sucued gd and ae hence meshless. Recenly, Alu and hs colleagues pesened a famly of Meshless mehods, based on he Local weak Peov-Galekn fomulaon (MLPGs) fo abay paal dffeenal equaons [4] wh movng leas-squae (MLS) appoxmaon. MLPG s epoed o povde a aonal bass fo consucng meshless mehods wh a geae degee of flexbly. Howeve, Galekn-base meshless mehods, excep MLGP pesened by Alu [5] sll nclude seveal awkwad mplemenaon feaues such as numecal negaons n he local doman. A compehensve evew of meshless mehods (MLPG) can be found n he book [6] by Alu. A vaey of local nepolaon schemes ha nepolae he andomly scaeed pons s cuenly avalable. The movng leas squae and adal bass funcon nepolaons ae wo popula appoxmaon echnques ecenly. Wh compasons of hese wo echnques, he movng leas-squae appoxmaon s geneally consdeed o be one of he bes schemes wh a easonable accuacy, paculaly fo sac elascy demonsaed by Wen e al [7]. In hs pape, he mesh fee Galekn mehod s pesened wh he adal bass funcon nepolaon and an ncemenal echnque has been developed o calculae he sess nensy facos wh hgh accuacy. In addon, he enched adal bass funcon and hgh densy of node dsbuon

80 396 Eds: R Abascal and M H Alabad nea he cack fon ae no needed n hs appoach. The accuacy of poposed mehod has been demonsaed hough benchmak examples.. Vaaon of poenal enegy and MLS Based on he vaaon of poenal enegy, he elemen fee Galekn mehod s developed on he bass of fne elemen mehod by he use of adal base funcon nepolaon n hs pape o evaluae sac and dynamc sess nensy faco wh an ncemenal echnque. Fo a lnea wo dmensonal elascy, he equlbum equaons can be wen as j j f u, () whee j denoes he sess enso, f he body foce, s he mass densy, u u / he acceleaon. Consdeng he vaaon of he oal poenal enegy, wh espec o each nodal dsplacemen, and he elaons u uˆ, Buˆ and D yelds a lnea algebac equaon sysem n a max fom as K ˆ ˆ NN u N M NN u f N () whee N s he oal numbe of node n he doman. The sffness and mass maces ae: T T K B ( x, y) D( y) B( x, y) d( y), M ( x, y) ( x, y) d( y) (3) n whch x x,,... N, and nodal foce veco s defned by T T f ( x, y) b( y) d ( y) ( x, y) ( y) d( y) (4) n( y) whee y, x) ( y, x ), ( y, x ),..., ( y, x ) and uˆ u ˆ, uˆ,..., uˆ T ( n ( y) n( y) ae veco of ( k ) ( k ) shape funcons and nodal values of dsplacemen. The collocaon pons x k x, x, k,,..., n( y), ae he shape funcons and n(y) he oal numbe of nodes n he local k doman named as suppoed doman as shown n Fgue. Fo a wo dmensonal plane sess case, we can eaange he above equaon n a max fom as T n( y) n( y) u( y) u ˆ ˆ ˆ ˆ T, u ˆ ˆ ˆ ( y, x)ˆ; u u u, u, u, u..., u, u (5) Fo convenence of analyss, he lde (^) s emoved n he followng dscusson. Applyng he Laplace ansfom o he equaon () yelds ~ K s M u~ f (6) whee s s he Laplace paamee. We assume ha he dsplacemens u(y) a he pon y can be appoxmaed n ems of he nodal values n a local doman (see Fgue ) as n( y) k u ( y) k ( y, x k ) uˆ ( y, x)ˆ u (7) k n( y) T whee ( y, x) ( y, x), ( y, x ),..., n ( y) ( y, x n( y) ), ˆ ˆ, ˆ,..., ˆ u u u,, ( k ) ( k ) u ˆ ( x) s he nodal values a pon x x, x, k,,..., n( ) u and k y. Fo he wo dmensonal plane sess case, we can eaange he above elaon as follows

81 Fgue. Sub-doman y fo MLS/RBF nepolaon of he feld pon y and suppo domans.. ˆ, ˆ..., ˆ, ˆ, ˆ, ˆ ˆ, ), ( )ˆ,, (, ) ( T ) ( ) ( ) ( ) ( T y y y y u x y u x y y u n n n n u u u u u u u u (8) 3. Radal bases funcon The dsbuon of funcon u n he sub-doman y ove a numbe of andomly dsbued noes ) (,,...,, y x n can be nepolaed, a he pon y, by ) ( ), ( ), ( ) ( T ) ( y a x y R x y y y n R a u (9) whee ), ( ),...,, ( ),, ( ), ( ) ( T x y x y x y x y R y R n R R s he se of adal bass funcons cened a he pon y, ) ( y n k k a ae he unknown coeffcens o be deemned. The adal bass funcon has been seleced o be he followng mul-quadcs ), ( k k c R x y x y () wh a fee paamee c and n hs pape, we selec c=h (h s specfed lengh n each example). Fom he nepolaon saegy n Eq. (9) fo RBF, a lnea sysem fo he unknowns coeffcens a s obaned by u a R () I s appaen ha he nepolaon of feld vaable s sasfed exacly a each node. As he RBFs ae posve defne, he max R s assued o be nveble. Theefoe, we can oban he veco of unknowns fom Eq. () ) ( ) ( x u x R a () So ha he appoxmaon u(y) can be epesened, a doman pon y, as feld pon y node x sub-doman y Advances n Bounday Elemen Technques IX 397

82 398 Eds: R Abascal and M H Alabad n( y) u T ( y) R ( y, x) R ( x) u( x) ( y, x) u u k k k (3) whee he nodal shape funcon ae defned by T ( y, x) R ( y, x) R ( x) (4) I s woh nocng ha he shape funcon depends unquely on he dsbuon of scaeed nodes whn he suppo doman and has he Konecke Dela popey. As he nvese max of coeffcen R ( x) s a funcon only of dsbued node x n he suppo doman, s much smple o evaluae he paal devaves of shape funcon. Fom Eq. (3), we have n( y) u ( y) ( y, x u u (5). k,k ), k 4. Incemenal echnque To deve he negal fo sess nensy faco fo dynamc poblem, one sac efeence poblem has o be deemned fs. Le and u be he acon and dsplacemen boundaes especvely. If hee s an ncemen of cack suface a a cack p, an ncemen of dsplacemen uk n he doman and on he acon bounday and k on he dsplacemen bounday wll occu. The sess nensy faco n he sac case can be wen as / ( ) uk k K I k d uk d (6) a a u The numecal esuls of sac case can be used decly o he dynamc poblem. The elaonshp beween he sess nensy facos fo he efeence poblem and he eal dynamc poblem can be wen as ~ u k u K ~ k d u~ k I k k d s u~ k d (7) ( ) K I a a a u In ode o evaluae he sess nensy faco n he me doman, he Dubn s nvese mehod s employed n hs pape K e ~ ~ k k f ( ) f ( ) Re f exp (8) T k T T whee ~ f ( ) s he ansfomed vaables n he Laplace ansfom doman when he paamee s k s k k / T. The selecon of paamees and T affecs he accuacy slghly. In he compuaons, we have chosen 5/ and T / n he followng examples, whee h / c, h s he hegh of cacked shee and c E( ) / ( )( ). Two numecal examples ae gven o demonsae he accuacy and effcency of he poposed echnque.

83 Advances n Bounday Elemen Technques IX [8] h y y a cack p KI/a.3... Incemen echnque handbook n b (a) (b) Fgue. Squae plae wh a cenal cack (h=b) unde enson : (a) a quae of he plae; (b) nomalzed sess nensy faco, whee a/a= -n KI/a Ths mehod BEM KI/a Ths pape BEM..5. c /h c /h (a) (b) Fgue 3. Squae plae wh a cenal cack subjeced o dynamc enson H ( ) : (a) h=b; (b) h=b. 5. Examples 5. A cenal cack n ecangula shee unde unfom sac load A squae plae of wdh b and hegh h conanng a cened cack of a subjeced o a unfom shea load on he op and he boom s analysed. Due o he symmey, a quae of plae s consdeed as shown n Fgue (a). Hee Posson s ao =.3. A se of ( N oal ) unfomly dsbued nodes s used and he negaon s pefomed by dvdng he squae no cells wh 4 4 Gauss pons. The suppo doman s seleced as a ccle of adus d y ceneed a feld pon y, whch s deemned such ha he mnmum numbe of nodes n he sub doman n( y ) N, hee he numbe N s seleced o be fo all followng examples. Fgue (b) shows he convegence of he nomalzed sess nensy faco -.5

84 4 Eds: R Abascal and M H Alabad K I / a agans he lengh of he ncemen of cack suface a. Excellen ageemen wh Refeence [8] 3 can be acheved when a / a. 5. A Sngle cenal cack n ecangula plae unde enson Consde a ecangula plae of wdh b and lengh h wh a cenally locaed cack of lengh a. I s loaded dynamcally n he decon pependcula o he cack by a unfom enson H ( ) on he op and he boom. Due o he symmey, a quae of plae s consdeed as shown n Fgue (a). Posson s ao =.3 and Young s modulus s un. Two geomees of ecangula plae ae consdeed n hs example,.e. h=b and h=b. To demonsae he accuacy of mesh fee mehod, he esuls gven by Wen [9] usng he ndec bounday elemen mehod (fcous load mehod) ae ploed fo compason. Nomalze dynamc sess nensy facos K I / a by hese wo echnques ae shown n he Fgues 3(a) and 3(b). Appaenly befoe he aval me of dlaaon wave avelng fom he op of plae, he sess nensy faco should eman o be zeo. The ageemen beween he soluons s consdeed o be good. 6. Refeences [] B. Nayoles, G. Touzo & P. Vllon, Genealzng he fne elemen mehod: dffuse appoxmaon and dffuse elemens, Compuaonal Mechancs,, 37-38, 99. [] T. Belyschko, Y.Y. Lu & L. Gu, Elemen-fee Galekn mehod, In. J. Numecal Mehods n Engneeng, 37, 9-56, 994. [3] W.K. Lu, S. Jun & Y. Zhang, Repoducng kenel pacle mehods, In. J. Numecal Mehods n Engneeng,, 8-6, 995. [4] S.N. Alu & T. Zhu, A new meshless local Peyov-Galekn (MLPG) appoach o nonlnea poblems n compuaonal modellng and smulaon, Compu Model Smul Engng, 3, 87-96, 998. [5] S.N. Alu & T. Zhu, The meshlesss local Peyov-Galekn (MLPG) appoach fo sovlng poblems n elaso-sacs, Compu Mech, 5, 69-79, 999. [6] S.N. Alu, The Meshless Mehod (MLPG) fo Doman and BIE Dscezaons, Fosyh, GA, USA, Tech Scence Pess, 4. [7] P.H. Wen and M.H. Alabad, An Impoved Meshless Collocaon Mehod fo Elasosac and Elasodynamc Poblems, Communcaons n Numecal Mehods n Engneeng, 7 (o appea). [8] D.P. Rooke and D.J. Cawgh, A Compendum of Sess Inensy Facos, HMSO, London, 976. [9] P.H. Wen, Dynamc Facue Mechancs: Dsplacemen Dsconnuy Mehod, Compuaonal Mechancs Publcaons, Souhampon, 996.

85 Advances n Bounday Elemen Technques IX 4 Sess analyss of compose lamnaed plaes by he bounday elemen mehod F. L. Tosan, A. R. Gouvea, E. L. Albuqueque, and P. Solleo Faculy of Mechancal Engneeng, Sae Unvesy of Campnas , Campnas, Bazl, [osan,adana,edelma,solleo]@fem.uncamp.b Keywods: Plaes, bounday elemen mehod, lamnaed composes, and sess analyss. Absac. Ths pape pesens a bounday elemen analyss of sesses n lamnae compose plaes followng Kchhoff hypohess. Sess negal equaons ae deved fom he ansvesal dsplacemen negal equaon. All devaves of ansoopc hn plae fundamenal soluons ae compued analycally. Sesses ae compued n each lamna a nenal pons of he plae. A numecal example s pesened n ode o assess he poposed mehod. Resuls ae compaed wh soluons found n leaue, showng good ageemen. Inoducon The maeal ansoopy pesens wo dffeen hands. On one hand, uns he maeal analyss exemely had due he lage numbe of vaables necessay o epesen s mechancal popees. On he ohe hand, he use of ansoopc maeals allows he desgne o conol he mechancal popees along each decon, nceasng he maeal sengh whou nceasng he wegh. Wh he demand by opmzaon of naual esouces and he lage offe of compuaonal esouces, he desgne age ae changng fom smple analyss o opmzed pefomance. In hs way, he use of hgh pefomance compose maeals s an neesng opon because hey allow he conol of he mechancal popees ehe by he choce of he componens, max and enfocemen, o by he componen ode nsde he maeal. Togehe wh he demand by hgh pefomance compose maeals, has also nceased he demand by elable and accuae numecal pocedues fo hs maeals analyss. The complexy of he ansoopc maeal analyss s evden n leaue. I can be noed ha he numbe of efeences n whch he bounday elemen mehod s appled fo ansoopc maeals s sgnfcanly smalle han hose eang soopc maeals. Howeve, n he las en yeas, mpoan advances on bounday elemen echnques appled o ansoopc maeals wee publshed n he leaue. Fo example, plane elascy poblems wee analyzed by [, ], [3], and [4, 5, 6, 7], ou of plane elascy poblems by [8], -dmensonal poblems by [9,, ], and Kchhoff plaes by []. Bounday elemen fomulaons have been appled o plae bendng ansoopc poblems consdeng Kchhoff as well as shea defomable plae heoes. [3] pesened a bounday elemen analyss of plae bendng poblems usng fundamenal soluons poposed by [4] based on Kchhoff plae bendng assumpons. [5] poposed a fomulaon n whch he sngulaes wee avoded by placng souce pons ousde he doman. [6] pesened an analycal eamen fo sngula and hypesngula negals of he fomulaon poposed by [3]. Shea defomable plaes have been analyzed usng he bounday elemen mehod by [7, 8] wh he fundamenal soluon poposed by [9]. In hs wok he calculaon of nenal pon sesses of ansoopc plaes usng he bounday elemens mehod s pesened. Sess negal equaons ae deved fom he ansvesal dsplacemen negal equaon. Sesses ae compued n evey lamna a nenal pons of he plae. Resuls ae compaed wh soluons found n leaue, showng good ageemen.

86 4 Eds: R Abascal and M H Alabad Bounday negal equaon As shown by [], he bounday negal equaon fo ansvesal dsplacemens w n a bounday pon of an ansoopc plae can be wen as: [ ] Γ w(q)+ Vn (Q, P )w(p ) ) N c m n (Q, P ) w(p dγ(p )+ Rc n (Q, P )w c (P ) = ] [V n (P )w (Q, P ) m n (P ) w Γ n (Q, P ) N c dγ(p )+ R c (P )wc (Q, P )+ = g(p )w (Q, P )dω. () Ω whee n s he devave n he decon of he ouwad veco n ha s nomal o he bounday Γ; m n and V n ae, especvely, he nomal bendng momen and he Kchhoff equvalen shea foce on he bounday Γ; R c s he hn-plae eacon of cones; w c s he ansvese dsplacemen of cones; P s he feld pon; Q s he souce pon; and an asesk denoes a fundamenal soluon. Fundamenal soluons equed a equaon () ae gven by []. San and dsplacemen n lamnae compose plaes Lamnaes ae fabcaed such ha hey ac as an negal sucual elemen. To assue hs condon, he bond beween wo lamnae n a lamnae should be nfnesmally hn and no shea defomable o avod he lamnae slp ove each ohe, and o allow dsplacemen connuy along he bond []. Thus, we could consde ha sans ae connuous along s hckness. Howeve, as each lamnae s compounded by dffeen maeals, sesses pesen dsconnues along lamnae nefaces. In Kchhoff plaes, sans ae gven by: ε x = z w x, ε y = z w x, γ xy = z w x y. () So, n ode o oban sans, second ode devaves of negal equaons () need o be compued. Fo nenal pons, hese devaves ae gven by: [ ] w(q) Vn x = Γ x (Q, P )w(p ) m n ) N c (Q, P ) w(p dγ(p )+ x n = [ V n (P ) w x (Q, P ) m n(p ) 3 w ] n x (Q, P ) N c dγ(p )+ Γ = Rc x (Q, P )w c (P ) R c (P ) wc (Q, P )+ x Ω g(p ) w (Q, P )dω, (3) x

87 Advances n Bounday Elemen Technques IX 43 [ ] w(q) Vn y = Γ y (Q, P )w(p ) m n ) N c (Q, P ) w(p dγ(p )+ y n = [ ] V n (P ) w y (Q, P ) m n(p ) 3 w n y (Q, P ) N c dγ(p )+ and Γ Γ = [ ] w(q) x y Γ = Vn x y (Q, P )w(p ) m n ) N c (Q, P ) w(p dγ(p )+ x y n = [ V n (P ) w x y (Q, P ) m n(p ) 3 w ] n x y (Q, P ) N c dγ(p )+ = Rc y (Q, P )w c (P ) R c (P ) wc (Q, P )+ y Ω g(p ) w (Q, P )dω, (4) y R c x y (Q, P )w c (P ) R c (P ) wc (Q, P )+ x y Ω g(p ) w (Q, P )dω. (5) x y As pesened by [], sesses a each lamnae can be evaluaed fom san gven by equaon () as followng: σ x Q Q Q 6 ɛ x σ y τ = Q Q Q 6 ɛ y xy Q 6 Q 6 Q 66 γ, (6) xy [ ] whee max Q s gven by: The ansfomaon max [T ]sgvenby: [ ] Q =[T ] [Q][T]. (7) cos θ sn θ snθcos θ [T ]= sn θ cos θ snθcos θ, (8) sn θ cos θ sn θ cos θ cos θ sn θ whee θ s he angle beween he fbe oenaon and he decon of axs x. The sffness max [Q] s gven, n ems of engneeng consans, by: [Q] = E L ν LT ν TL ν LT E T ν LT ν TL ν LT E T ν LT ν TL E T ν LT ν TL G LT, (9) whee E L s he elasc modulus n he paallel o he fbe decon, E T s elasc modulus n he ansvesal o he fbe decon, G LT s he shea modulus n he plane of he lamnae, and ν LT s he pncpal Posson ao n he plane of he lamnae. Numecal esuls To valdae he pocedues mplemened, a nne-layeed, symmecal angle-ply lamnae wh sackng sequence [+θ/ θ/+ θ/ θ/+ θ/ θ/ + θ/ θ/ + θ]wh θ 45 o was chosen. The plae s squae wh edge lengh a = m. All edges ae smply-suppoed and all layes have he same hckness. The oal hckness s equal o h =. m and maeal popees ae: E L = 7 GPa, E T =5.

88 44 Eds: R Abascal and M H Alabad GPa, G LT = 3. GPa, and ν LT =.5. Fgues and show he effec of he vaaon of θ on he dsplacemen and bendng sess esulans, especvely, a he cene of he plae. They ae compaed wh fne elemen esuls obaned by []. I s woh o say ha he fne elemen fomulaon consdes he effec of he shea defomaon. As can be seen, n boh cases he ageemen beween he bounday elemen hn plae and he fne elemen shea defomable plae s vey good. (wet h 3 /qa 4 ) w, hs wok w, Refeence [] θ (degees) Fgue : Effec of he oenaon θ n he ansvesal dsplacemen esponse a he cene of he plae. (Mx/qa ), (My/qa ) Mx, hs wok My, hs wok Mx, Refeence [] My, Refeence [] θ (degees) Fgue : Effec of he oenaon θ n he momen esponse a he cene of he plae. Fgue 3 shows he sess dsbuon (σ x ) along he hckness of he plae. I can be seen ha sess ae dsconnuous a he neface and vay lnealy along each lamnae.

89 Advances n Bounday Elemen Technques IX 45.5 ( σx h )/q z/h.4.5 Fgue 3: Sess dsbuon (σ x ) along he hckness fo θ =45 o a he cene pon of he plae. Conclusons Ths pape pesened a bounday negal fomulaon fo he compuaon of sess n nenal pons of ansoopc hn plaes. An negal equaon fo he second dsplacemen devave s developed and all devaves of he fundamenal soluon ae compued analycally. The obaned esuls ae n good ageemen when compaed wh fne elemen hck plae esuls. Acknowledgmen The auhos would lke o hank he Sae of São Paulo Reseach Foundaon (FAPESP) fo fnancal suppo fo hs wok (gan numbe: 3/9498-). Refeences [] P. Solleo and M. H. Alabad, Facue mechancs analyss of ansoopc plaes by he bounday elemen mehod. Inenaonal Jounal of Facue, 64: 69-84, 993. [] P. Solleo and M. H. Alabad, Ansoopc analyss of compose lamnaes usng he dual bounday elemens mehods. Compose Sucues, 3:9-34, 995. [3] A. Deb, Bounday elemens analyss of ansoopc bodes unde hemo mechancal body foce loadngs. Compues and Sucues, 58:75-76, 996. [4] E. L. Albuqueque, P. Solleo and M. H. Alabad, The bounday elemen mehod appled o me dependen poblems n ansoopc maeals. Inenaonal Jounal of Solds and Sucues, 39:45-4,. [5] E. L. Albuqueque, P. Solleo and P. Fedelnsk, Dual ecpocy bounday elemen mehod n Laplace doman appled o ansoopc dynamc cack poblems. Compues and Sucues, 8:73-73, 3. [6] E. L. Albuqueque, P. Solleo and P. Fedelnsk, Fee vbaon analyss of ansoopc maeal sucues usng he bounday elemen mehod. Engneeng Analyss wh Bounday Elemens, 7: , 3.

90 46 Eds: R Abascal and M H Alabad [7] E. L. Albuqueque, P. Solleo and M. H. Alabad, Dual bounday elemen mehod fo ansoopc dynamc facue mechancs. Inenaonal Jounal fo Numecal Mehod n Engneeng, 59:87-5, 4. [8] Ch. Zhang, Tansen elasodynamcs anplane cack analyss of ansoopc solds. Inenaonal Jounal of Solds and Sucues, 37:67-63,. [9] M. Kogl and L. Gaul, A bounday elemen mehod fo ansen pezoelecc analyss. Engneeng Analyss wh Bounday Elemens, 4:59-598,. [] M. Kogl and L. Gaul, A 3-d bounday elemen mehod fo dynamc analyss of ansoopc elasc solds. CMES-Compue Modelng n Engneeng and Scence, :7-43,. [] M. Kogl and L. Gaul, Fee vbaon analyss of ansoopc solds wh he bounday elemen mehod. Engneeng Analyss wh Bounday Elemens, 7:7-4, 3. [] E. L. Albuqueque, P. Solleo, W. S. Venun and M. H. Alabad, Bounday elemen analyss of ansoopc kchhoff plaes. Inenaonal Jounal of Solds and Sucues, 43:49-446, 6. [3] G. Sh and G. Bezne, A geneal bounday negal fomulaon fo he ansoopc plae bendng poblems. Jounal of Compose Maeals, :694-76, 988. [4] B. C. Wu and N. J. Aleo, A new numecal mehod fo he analyss of ansoopc hn plae bendng poblems. Compue Mehods n Appled Mechancs and Engneeng, 5: , 98. [5] C. Rajamohan and J. Raamachandan, Bendng of ansoopc plaes by chage smulaon mehod. Advances n Engneeng Sofwae. 3: , 999. [6] W. P. Pava, P. Solleo and E. L. Albuqueque, Teamen of hypesngulaes n bounday elemen ansoopc plae bendng poblems. Lan Amecan Jounal of Solds and Sucues, :49-73, 3. [7] J. Wang and K. Schwezehof, Sudy on fee vbaon of modeaely hck ohoopc lamnaed shallow shells by bounday-doman elemens. Appled Mahemacal Modellng, : , 996. [8] J. Wang and K. Schwezehof, Fee vbaon of lamnaed ansoopc shallow shells ncludng ansvese shea defomaon by he bounday-doman elemen mehod. Compues and Sucues, 6:5-56, 997. [9] J. Wang and K. Schwezehof, The fundamenal soluon of modeaely hck lamnaed ansoopc shallow shells. Inenaonal Jounal of Engneeng and Scence, 33:995-4, 995. [] B. D. Agawal and L. J. Bouman, Analyss and pefomance of fbe composes. nd Edon, John Wley & Sons Inc, New Yok, 99. [] H. V. Lakshmnaayana and S. S. Muhy, 984. A shea-flexble angula fne elemen model fo lamnaed compose plaes. Inenaonal Jounal fo Numecal Mehods n Engneeng, : 59 63, 984.

91 Advances n Bounday Elemen Technques IX 47 Bounday Elemen Fomulaon fo Dynamc Analyss of Cacked Shees Repaed wh Ansoopc Paches M. Maule,P.Solleo, E. L. Albuqueque 3 Faculy of Mechancal Engneeng, Sae Unvesy of Campnas , Campnas, Bazl, [mammn,solleo,edelma 3 ]@fem.uncamp.b Keywods: Bounday Elemen Mehod, Ansoopy, Bonded Repa, Dynamc Facue Mechancs. Absac. The am of hs pape s o pesen he bounday fomulaon and a soluon pocedue o pefom he dynamc analyss of cacked shees epaed wh an ansoopc pach. The numecal mehod ha s used o pefom he modelng of he cack s he dual bounday elemens mehod (DBEM). The neacon eec beween he shee and he ansoopc epa s modeled usng he dual ecpocy bounday elemens mehod (DRBEM). The neal eecs ae also modeled usng he DRBEM. A ansen soluon pocedue s pesened. Inoducon The pesence of cacks n mechancal o sucual componens unde dynamc loads deceases s mechancal and fague essance due o he hgh sess concenaon a he cack p. Facue mechancs poblems ae a majo concen n he aeonauc ndusy, snce hee s he need of pojecs wh hgh elably, hgh essance, and low coss. Aeonaucal sucues ae usually composed by meallc shees and snes. Cacked shees n aeonaucal sucues ae usually epaed by bondng, bolng, o scewng a meallc epa ove he cack egon. The use of boled o scewed epas ceaes holes n he sucue, whch ae majo sess concenaos and whee cacks ae lkely o nae a. The bonded epas have beng used successfully and ae egaded by he ndusy as an ecen soluon fo hs knd of poblem. In hs pape, he bounday elemen mehod (BEM) s appled fo he analyss of he dynamc esponse of cacked shees epaed wh an ansoopc pach. The elasosac esponse of hs sysem has been pevously pesened by Useche, Solleo and Albuqueque []. Due o song geomecal sngulaes, s no suable o oban good esuls fo dsplacemens and acons n he ousks of he cack usng he convenonal BEM fomulaon. Theefoe, anohe echnque mus be used n ode o oban good esuls fo acons and dsplacemens a hs egon. The dual bounday elemen mehod (DBEM) s a bounday modellng echnque amng facue mechancs poblems, allowng o dsceze he cack n a sngle egon. Ths echnque has been successfully descbed and mplemened by Dganaa [], and egaded as an ecen echnque o smulae facue mechancs poblems. The consdeaon of neacon and neal eecs of he componens noduces doman negals n he bounday equlbum equaons, whch mus be ansfomed no bounday negals. The dual ecpocy mehod (DRBEM) has been successfully used o ovecome hs poblem fo soopc and ansoopc shees, as shown by Kögl and Gaul [3] and Albuqueque, Solleo and Alabad [4]. In ode o have a boade ange of esuls, a ansen soluon pocedue s adoped, allowng o oban esuls fo deen load ccles. Ths pocedue has been descbed and mplemened by Houbol [5] and Loee and Mansu [6]. Howeve, a ansen soluon s usually moe me consumng han a soluon on he fequency doman.

92 48 Eds: R Abascal and M H Alabad Bounday Elemen Fomulaon Fgue : Cacked plae adhesvely bonded wh ansoopc pach Consdeng ha he shee shown n Fg. s unde a dynamc load, he negal equaon fo he shee (S) n a souce pon (x ) s gven by: c S ( j x ) u S ( j x ) ˆ + Tj S ( x,x ) u S ( j x ) ˆ dγ= Uj S ( x,x ) S ( j x ) dγ+ h S ˆ Ω R Γ S U S j ( x,x ) b S ( j x ) ˆ dω+ Ω S Γ S U S j ( x,x ) ρü S ( j x ) dω. () Smlaly, he negal equaon fo he epa (R) s obaned subsung he (S) ndexes n eq.() by (R). The coecen c j (x ) depends on he poson of he souce pon (x ) n elaon o he bounday whch s beng negaed, j (x ) and u j (x ) ae he acons and dsplacemens of he sysem, T j (x,x) and U j (x,x) ae he fundamenal soluons fo acons and dsplacemens, h s he hckness of he componen and ρ s he mass densy of he componen maeal. The s hee ems fom eq.() efe o he classcal elasosacs fomulaon, he fouh em efes o he assemblng of he plae and he pach, and he las em efes o he eec of body foces due o he masses of he shee and he pach unde dynamc load. The shea eacons n he adhesve b j (x ) wll be calculaed by he deence beween he dsplacemens of he shee and he pach[]: b j (x) = S A { ( u S h j x ) u R ( j x )}, () A whee S A s he shea module of he adhesve maeal and h A s he hckness of he adhesve laye. I s mpoan o emembe ha he ndexes (S) and (R) n eq.(), eq.(), and fuhe equaons do no mply on any summaon, and ae used only o addess whch componen s beng consdeed. Dual Bounday Elemen Mehod (DBEM) The DBEM echnque consss on applyng a dsplacemen equaon n one of he sdes of he cack, and a acon equaon on he emanng sde of he cack. Snce he only cacked componen n hs analyss s he soopc shee, hs pocedue s no appled o he ansoopc pach. The dsplacemen equaon s gven by eq.(). The acon equaon, obaned by he deenaon of eq.() [], s gven by:

93 Advances n Bounday Elemen Technques IX 49 ( S j x ) ( + n () x ) ˆ h S n ( () x ) ˆ Ω R Γ S D S jk Sjk S ( x,x ) u S ( k x ) ( dγ=n () x ) ˆ ( x,x ) b S ( k x ) ( dω+n () x ) ˆ Ω S Γ S D S jk Djk S ( x,x ) S ( k x ) dγ+ ( x,x ) ρü S ( k x ) dω, (3) whee S jk (x,x) and D (x,x) ae lnea combnaons of devaves of T j (x,x) and U j (x,x). When x x, S jk (x,x) exhbs hypesgulay O ( ), and D (x,x) exhbs song sngulay O ( ), whee (x,x) s he dsance beween he souce node and he negaon pon and n (x ) s a unay veco, nomal o he souce pon bounday. The eq.(3) s known as hypesngula equaon fo plane elascy, and, alongsde wh eq.(), consues he bass of he DBEM echnque. Dual Recpocy Bounday Elemen Mehod (DRBEM) The eq.() conans boh doman and bounday negals. The DRBEM allows appoxmang a doman negal by a sum of bounday negal funcons. Applyng he DRBEM n hs elasodynamc poblem consss on appoxmae he neal and neacon eecs of eq.() by: b j ( x ) = D d= αk d f jk d ( x,x ) ( and ρü j x ) = E e= βk e ( qe jk x,x ), (4) whee D s he numbe of nodes shaed by he pach and he shee and E s he oal numbe of nodes of he componen whch s beng consdeaed.the coecens αk d and βe k ae nepolaon coecens and fjk d (x,x) and qjk e (x,x) ae nepolaon funcons, whch, fo he soopc shee, ae gven by: ( ) x d,x =( ) δ jk and qjk e (xe,x)=( ) δ jk. (5) f d jk Fo he ansoopc pach[3, 4], he nepolaon funcons ae gven by: ( ) fjk d x d,x = C jlm [c(,m, δ lk + δ m δ lk )] and qjk e (xe,x)=c jlm [c(,m, δ lk + δ m δ lk )], (6) whee C jlm s he elasc consan enso fom he equlbum equaon of he pach. The consan c s chosen andomly when pacula soluons fo dsplacemens û kj (x ) and acons ˆ kj (x ) ae adoped[3]: û kj = c 3 δ kj and ˆ kj = σ kjm n m, (7) whee n m s a unay veco, nomal o he souce pon bounday and σ kjm s gven by: [ ] σ kjm = C kms c 3 (,s δ j +, δ js ). (8) In ode o oban useful soluons fo fjk d and qe jk, s mpoan o emembe ha he pacula soluons fo û kj mus espec he equlbum equaons gven by[3]: f d jk = C jlmû d lk,m and q e jk = C jlmû e lk,m. (9)

94 4 Eds: R Abascal and M H Alabad Couplng he eq.(4) wh he eq.(), he equlbum equaon fo he shee s gven by: c S ( j x ) u S ( j x ) ˆ + Tj S ( x,x ) u S ( j x ) ˆ dγ= Uj S ( x,x ) S ( j x ) dγ+ h S D αk d c S j d= E e= β e k ( x d) û ds kj c S j (x e )û es Γ S ( x d) ˆ + Γ R ˆ kj (xe )+ Γ S U S j Γ S ( ) ( x d,x ds kj x d) ˆ dγ ˆ Uj S (x e,x) es kj (xe ) dγ Γ S Γ R ( ) Tj S x d,x ( û ds kj Tj S (x e,x)û es kj (xe ) dγ x d) dγ + (). Smlaly, he equlbum equaon fo he pach s obaned subsung he (S) ndexes n eq.() by (R). Dscezed Bounday Elemen Fomulaon In ode o oban he elasodynamc esponse of he sysem, he bounday s dvded no bounday elemens. The elemens aken n consdeaon ae quadac connuous elemens o model he pach and quadac dsconnuous elemens o model he shee. Fo maes of convenence, fou vecos wh dmenson ( E) ae ceaed: u = ϕu () ; û = ϕû () ; = ϕ () ; ˆ = ϕˆ (), () whee ϕ s he veco of quadac shape funcons, u ( E) and ( E) ae he vecos of nodal dsplacemens and acons of he sysem, and û ( E) and ˆ ( E) ae he vecos of paculaly soluons fo nodal dsplacemens and acons of he sysem. Couplng he eq.() wh eq.(), and callng ˆ UϕdΓ =G Γ j and ˆ T ϕdγ =H, Γ j () s possble o ewe he equlbum equaon fo he shee as: H S lj us j = G S lj S j + h S D [ d= H S lj uds j G S lj ds j ] α S d + E e= [ H S lj u es j G S ] lj es j β S e. (3) In a smla way, he equlbum equaon fo he pach s obaned subsung he (S) ndexes n eq.(3) by (R). H and G ae smla o H and G, bu obaned by negaon on he epa bounday. The D vecos α d ( ) and he E vecos β e ( ) can be assembled n wo vecos α (D ) and β (E ). Theefoe, he eq.(4) can be ewen as: b = Fα and p = Qβ, (4) whee b conans he shea eacons of he adhesve fo he nodes shaed by he pach and he shee and p conans he body foces of he componen unde consdeaon fo each one of s nodes. The max F conans he values of he funcon fj d (x,x) fo he nodes shaed by he pach and he shee. Smlaly, he max Q conans he values of he funcon qj e (x,x) fo he nodes of he componen whch s beng consdeaed. The maxfom fo he equlbum equaons of he componens ae obaned couplng he eq.(4) and wh eq.(3), and can be wen as: H S Γ us Γ G S Γ S Γ = A S Γα S Γ B S Γ ps Γ H R Γ ur Γ G R Γ R Γ = A R Γ α R Γ B R Γ pr Γ

95 Advances n Bounday Elemen Technques IX 4 u S Ω H S Ω us Ω = A S Ωα S Ω B S Ω ps Ω u R Ω H R Ω ur Ω = A R Ωα R Ω B R Ω pr Ω, (5) whee A and B ae gven by: ] ] A = [HÛD G ˆT D F and B = [HÛE G ˆT E Q. (6) ˆT and Û ae maxes of acon and dsplacemen fundamenal soluons. Makng use of he DRBEM and he elaon gven by eq.() s possble o ewe he em fo he neacon sheeepa as: u S Ω u R = h A S A F S α S and u R u S Ω = h A S A F R α R. (7) Fnally, couplng he equaons fo he shee and he epa usng he DRBEM, he equaon sysem, whch ules he poblem, s gven by: [ ] { (H A) S A S } { u S G S (H A) R A R u R = S + B S p S } B R p R (8) Tansen Soluon Pocedue To solve he lnea sysem gven by eq.(8), a ansen soluon pocedue s used. Ths pocedue was poposed by Houbold[5], whch s he mos ndcaed o solve poblems of me negaon alongsde wh he DRBEM [6]. Consdeng ha he neal eecs of he componens ae due an acceleaon eld gven by: p = ρü = Qβ, (9) he eq.(8) can be ewen fo an nsan of me τ + τ as: [ ] { (H A) S A S u S } { τ+ τ G S S (H A) R A R u R = τ+ τ + B S ρ S ü S } τ+ τ τ+ τ B R ρ R ü R τ+ τ. () In ode o poceed wh he me negaon, he τ peod s dvded n N me seps, whee: τ = N τ. () Assumng ha he soluon fo he eq.() s known a τ =, τ, τ..., he acceleaon a τ + τ s appoxmaed by he expesson[5]: ü τ+ τ = τ ( uτ+ τ 5u τ +4u τ τ u τ τ ). () Inseng eq.() no eq.(), he followng sysem of equaon s obaned: [ (H A) S ρ S τ B S] A S { (H A) [A R R ρ R τ B R] u S τ+ τ u R τ+ τ } = ) G S S τ+ τ + BS ρ ( 5u S S τ τ +4u S τ τ us τ τ ) B R ρ ( 5u R R τ τ +4u R τ τ ur. (3) τ τ

96 4 Eds: R Abascal and M H Alabad Conclusons In hs pape, a fomulaon fo dynamc analyss of cacked shee epaed wh adhesvely bonded paches was pesened, and a pocedue o solve he poblem was poposed as well. The pocedue descbed n hs pape s moe me consumng han a pocedue on a fequency doman. Howeve, allows a boade ange of soluons wh applcaon of deen load ccles. Ths pocedue also allows a fuhe mplemenaon of dynamc sess nensy faco deemnaon, whch s an mpoan vaable n he desgn and pojec of aeonaucal sucues. Acknowledgmens The auhos would lke o hanks FAPESP (The Sae of São Paulo Reseach Foudaon) and CNPq (Naonal Reseach Councl) fo he nancal suppo of hs wok. Refeences [] J. Useche, P. Solleo and E. L. Albuqueque, Bounday elemen analyss of cacked shees epaed wh adhesvely bonded ansoopc paches, BeTeq: Inenaonal Conf. On Bounday Elemen Technques, Pas, (6). [] T. Dganaa, Bounday elemen analyss of cacks n shea defomable plaes and shells, Topcs n Engneeng, V.43, Souhampon, WTI Pess.. [3] M. Kögl and L. Gaul, Abounday elemen mehod fo ansen pezoelecc analyss. Engn. Anal. Wh Bounday Elemens, 4: () [4] E. L. Albuqueque, P. Solleo and M. H. Alabad The bounday elemen mehod appled o me dependen poblems n ansoopc maeals, In. Jounal of Solds and Sucues, 39; 45-4,. [5] J. C. Houbol, Aecuence max soluon fo he dynamc esponse of elasc acaf, Jounal of Aeonaucal and Scence, 7:54-55 (95) [6] C. Loee and W. J. Mansu, Analyss of me negaon schemes fo bounday elemen applcaons o ansen wave popagaon poblems. In Bebba and W. S. Venun, edos, Bounday Elemen Technques: Applcaons n sess analyss and hea ansfe, 5-, Compuaonal Mechancs Publcaons, Souhampon. (987)

97 Advances n Bounday Elemen Technques IX 43 Homogenzaon of nonlnea composes usng Hashn-Shkman pncples and BEM P.Pochazka,a and Z. Shaf Khodae,b Socey of Scence, Reseach and Advsoy, Czech Assocaon of Cvl Engnees Czech Techncal Unvesy n Pague, Faculy of Cvl Engneeng, Depamen of Mechancs, Pague, Czech Republc a pe.poch@gmal.com b Zaha.shaf@gmal.com Keywods: Homogenzaon of composes, nonlnea poblem, Hashn-Skman pncple, bounday elemens, egenpaamees, Lppmann-Swnge equaon Absac. Bounday elemen mehods suffe fom one dsadvanage, whch s descpon of nhomogeneous and plasc sucues. I appeas ha one useful ck can bdge hs ssue. Ths s geneaed by genealzed Hashn-Skman vaaonal pncples, n whch he egenpaamees ae nvolved. Such an appoach leads o sepaaon of phases n he composes. In vey many cases one phase behaves lnealy ( s mosly he fbe) and he ohe (max n pevalng cases) nonlnealy. Usng an equvalen negal fomulaon of he H-S pncples yelds excluson of fbe nfluence n he poblem. Ths s an mpac of denfcaon of maeal popees of he fbe wh ha of compaave medum. New unknown sans o sesses n he max occu nsead of lnea mechancal popees defned n he fbes. When consdeng nonlnea maeal behavo of he max, hese sans o sesses play a ole of new quanes appeang n eaon seps needed fo denfcaon of plasc behavo.. Inoducon The appoach fo calculang plasc defomaons (o alenavely elaxaon sesses) seleced n hs pape sas wh he dea of Hashn-Skman vaaonal pncples, [], whch lead o vaaonal bounds of lnea composes. Exendng he pncples by noducng egenpaamees no he fomulaon, [], new fee paamees occu n he posulaon of he poblem and hey wll seve plasc defomaon o elaxaon sesses. They wee successfully used n opmzaon of pesess n lamnaed composes n [3], fo example. Suvey of access o he homogenzaon echnques of peodc composes usng numecal pocedues can be found n [4]. In hs pape basc elaonshps ae deved and he appoach s befly descbed. An example s pesened n he end of hs pape. The plasc behavo s descbed n [5] fo he Mses hypohess.. Basc consdeaons Befoe we ackle he fomulaon of he poblem of fndng oveall maeal popees of a compose nonlnea sucue usng bounday elemen mehod a useful appoach deved n a smla way as Hashn-Shkman vaaonal pncples wll be menoned. Fo hs eason he H-S pncples wll be befly menoned n he sequel. Only pmay pncple s consdeed fo applcaon o lowe bounds (lowe esmae of he oveall popees defomaon mehod) whle he dual pncple can be appled o foce mehod of fndng uppe esmae of he effecve maeal popees.

98 44 Eds: R Abascal and M H Alabad Le a doman R3 wh s bounday u p, u p, be gven, descbng a shape of he body unde consdeaon. Moeove, le us assume ha along he bounday u of he body dsplacemens u,,, 3, be pescbed and along he bounday p acons be gven. Influence of volume wegh foces s negleced. The appoach s spl no wo seps: s sep: dsplacemens u { u }, acons p { p }, sans { j } and sesses { j } (, j,, 3 ) ae calculaed fo homogenzed, soopc compaave medum and hey ae consdeed o be known n he nex. Sacal equaons fo he sess feld and geomec bounday condons ae vald: x j j, u u on u, p p on p () and also homogeneous and soopc Hooke s law holds vald wh maeal sffness max L : jkl kl j L n () nd sep: quanes fo nhomogeneous ansoopc medum ae o be deemned,.e. dsplacemens u, acons p, sans and sesses ae unknown. In hs sep geomecally same body s assumed wh pescbed bounday dsplacemens u and gven acons p fom he fs sep. Hooke s law s now vald fo maeal sffness max of he whole body (no moe soopc homogeneous) nvolvng also egensess feld : j Ljklkl j n, u u on u, p p on p (3) Peodc un cell s consdeed, fo whch holds: u ( x ) E x u( x), ( x) E ( x) E ( u( x)), (4) j j j whee E j ae componens of he oveall (macoscopc) san enso, u ae componens of flucuang dsplacemen veco. In peodc sucue holds fo he un cell: u s same a each bounday pon n he decon of nvaance of he laye, whle acons ae hee ansymmec. Smlaly o he classcal H-S pncple symmec polazaon enso s noduced, defned as: j j j j j j jkl kl L j (5) Inasmuch as s sacally admssble, also holds: ( L jklkl j ) x Nex, subacng (5) and (3), consdeng (4), povdes: j (6) jkl kl L L [ L ] L E n, L (7) j jkl kl j

99 Advances n Bounday Elemen Technques IX Lppmann-Swnge equaon fo plascy In hs secon negal fomulaon o he H-S vaaonal poblem s ceaed. Mulplyng (7) successvely by es funcons,,, 3, negang he esul ove he doman, applyng wo mes Geen s heoem and seng fo he kenel dsplacemen componens yeld: whee * * * u ( ) u ( x, ) p ( x)d ( x) p ( x, ) u ( x)d ( x) ( x, ) ( x)d( x), (8) k k k k kl kl * kl * k u ( x, ) ( x, ), x j * k p ( x, ) L jlm * lk u ( x, ) n j L x m * jlmlmk ( x, ) n j Hee and n wha follows geomecally and maeally symmec un cell s consdeed, so ha he bounday dsplacemens u vansh. Posonng on he bounday and akng no accoun he bounday condons (4), he negal equaon equvalen o (6) s hen: Dffeenang (8) wh espec o * * u ( x, ) d ( x) ( x, ) ( x)d( x), (9) k p k kl j povdes he expesson * * ( ) h ( x, ) p d ( x) ( x, ) ( x)d( x), () j jk k and he volume negal s aken n Hadamad s sense. Equaons (9) and () can be expessed n a compehensve fom: hee s an opeao G() whch elaes he flucuang san and polazaon enso as: ( x) G( ( x)) () Le us subsue fo kl (x) fom (7) no he lae equaon and spl he negaon ove fbe and max, whch povdes: jkl kl kl ( ) j * jk h ( x, ) pd ( x) k f * jkl m * jkl ( x, ){[ L ( x, ){[ L jkl ] kl jkl ( ) L ]( ( ) µ ( )) L kl jklekl j jklekl }d( x), }d( x) () as hee s no egensan (plasc san) nsde of a fbe. If we se L smplfes as: f jkl L jkl, he lae equaon

100 46 Eds: R Abascal and M H Alabad ( ) j * jk h ( x, ) pd ( x) k L m * jkl ( x, ){[ L f * jklekl jkl m kl L f kl ( x, )d( x), ]( ( ) µ kl ( )) L f jkl E }d( x) Nex, equaon (3) can be wen n ncemen fom (noe ha no dffeenaon s caed ou, s no pemed hee fo he sngula negals nvolved n such a pocess do no adm ): kl (3) ( ) h ( x, )pd ( x) ( x, ){[ L L ]( ( ) µ ( ))}d( x), j * jk k m * jkl m jkl f jkl kl j m (4) and he pons of obseve belong now only o he max. Fo compleeness le us we negal equaons whch ae equvalen o (9): u ( x, )pd ( x) ( x, )[ L L ]( ( ) µ ( ))d( x), * k k * kl m m kl f jkl kl kl m (5) Equaons (9) and (3) esablsh a smulaneous sysem fo compuaon of elasc sae n nhomogeneous maeal composed fom fbe and max due o un san mpulses. The lae wo equaons (4) and (5) ceae a smulaneous sysem fo mpovemen of bounday acons and flucuang sans due o plascy. In concluson, he san componens a any pon of he doman ae dependen on bounday acons and sans and egensans n max only, and he bounday acons ae dependen on he sans and egensans n max. The lae equaons lead us o an appoach, whch s descbed n moe deals n he followng secon. 4. Calculaon of plascy effec Fs, le us consde geomey of a compose un cell. The soluon of such a cell n peodc sucue s concsely descbed n [4], fo example. Applyng appopae un dsplacemens o he bounday of he cell, s elasc maeal popees ae denfed wh ha of fbe, null supescp quanes ae saghfowadly obaned, and even on much moe complcaed geomey han ous. The values of dsplacemens, sans and sesses ae also speculaed a sang levels. Fgue : Un cell used n he sudy Dvdng he exenal bounday of he un cell no subsufaces n 3D o subnevals n D as N M p ( ) p ( ) ( I ), ( ) ( ) ( Q ), µ ( ) µ ( ) ( Q ) (6) j j j j jk k k j M jk k k

101 Advances n Bounday Elemen Technques IX 47 whee ae he chaacesc funcons beng equal o one fo nsde he subegon I o ohewse, I s a bounday subegon, s a subegon n he doman, doman subegons and bounday subegons. If we pu j and zeo M, N ae especvely numbes of j p ( j ) p j s ceneed n j p ( ), k k jk ( ) j( k ) j and µ jk ( ) µ j ( k ) µ j, k k, all quanes ae unfomly dsbued nsde he subegons (elemens). Obvously, I, s dencal wh he cene of gavy of he subegon I s a bounday elemen whle s an nenal cell. Applyng hese appoxmaons, pung fs µ oveall, fom () k and (3) sans, acons and sesses follow. Then sesses ae obaned fom (5). Tesng hem fo plasc ules, dsbuon of µ j s specfed and (4) wh (5) delves he ncemens of sans usng only he doman of he max. Ths s he geaes advanage of he above descbed appoach, whch consss n concenaon of he poblem of mpovemen of bounday acons and flucuang san o he doman of max only, whle fbe s no longe nvolved no compuaon. 5. Example A plane squae un cell s consdeed wh fbe volume ao equal o.6 accodng o Fg., so ha only fs quae s assessed. The followng elasc maeal popees of phases ae assumed: Young's modulus of fbe E f f = GPa, Posson's ao =.6; on he max E m = 7 GPa, and m =.3. The Mses plascy s consdeed wh plasc coeffcen k MPa. Dsplacemens n he plana pncpal decons ae dawn n Fg. fo un dsplacemen excaon n x-decon. j Fgue : Dsbuon of elasc and plasc dsplacemens n he fs quae of he un cell

102 48 Eds: R Abascal and M H Alabad 6. Conclusons In hs pape useful eamen of plasc behavo on a un cell s descbed when usng bounday elemen mehod. Ths splendd numecal mehod suffes fom one mpoan poblem: nhomogeneous maeal popees (so s plascy, fo example) ae hadly nvolved n he compuaon. I appeas ha fo pacula poblems hee s a way on how o bdge hs poblem. Exended Hashn-Shkman vaaonal pncples ae used fo equvalen fomulaon, whch coves polazaon enso. A specal choce of elasc maeal popees (denfcaon of compaave medum wh fbe) leads us o elmnaon of fbes fom he nonlnea compuaons and basc smplfcaon of eave pocedue. Fom he pcues descbng dsbuon of dsplacemens n he fs quae of he cell a elasc and plasc saes s seen ha pncpal edsbuon of exeme dsplacemens s aaned. Ths esul s obaned n vey sho compuaonal me, as only wo eaons ae necessay o ge sgnfcan values of he dsplacemens. The connuaon of eaons s no moe necessay, s nsgnfcan. Moeove, n each eaon sep only mulplcaon of vecos and maces s equed; hee s no soluon of equaons. Only n he elasc sae sandad lnea algebac equaons ae solved. Acknowledgmens: The fnancal suppo of hs wok povded fo he fs auho by gans No. GACR 3/8/97 and CZE MSM of he Gan Agency of he Czech Republc ae gaefully acknowledged. Refeences [] Z. Hashn, S. Shkman: On some vaaonal pncples n ansoopc and nonhomgeneous elascy, J Mech Phys Solds (3), (96), p [] P.P. Pocházka, J. Šejnoha: Exended Hashn-Shkman vaaonal pncples, Applcaons of Mahemacs 49 (4), (4), p [3] G.J. Dvoak, P.P. Pocházka: Thck-walled compose cylndes wh opmal fbe pesess, Composes Pa B 7B, (996), p [4] P.M. Suque: Homogenzaon echnques fo compose meda, Lecue Noes n Physcs 7, Beln, Spnge (985), p [5] M. Duvan, J.-P. Lons: Vaaonal nequales n mechancs, Dunod, Pas (978).

103 Advances n Bounday Elemen Technques IX 49 Tansen Dynamc Analyss of Ineface Cacks n -D Ansoopc Elasc Solds by a Tme-Doman BEM Sefane Beye, a, Chuanzeng Zhang, b, Sohch Hose 3, c, Jan Sladek 4, d and Vladm Sladek 4, e Enegy Seco, Semens AG, D-86 Gölz, Gemany Depamen of Cvl Engneeng, Unvesy of Segen, D-5768 Segen, Gemany 3 Depamen of Mechancal and Envonmenal Infomacs, Tokyo Insue of Technology, Tokyo 5-855, Japan 4 Insue of Consucon and Achecue, Slovak Academy of Scences, 8453 Baslava, Slovaka a sefane.beye@semens.com, b c.zhang@un-segen.de, c shose@cv.ech.ac.jp, d jan.sladek@savba.sk, e Vladm.sladek@savba.sk Keywods: Ineface cacks; Layeed ansoopc elasc solds; Dynamc sess nensy facos; Tme-doman BEM; Mul-doman echnque. Absac. In hs pape, ansen elasodynamc analyss of an neface cack n wo-dmensonal (-D), layeed, pecewse homogenous, ansoopc and lnea elasc solds subjeced o an mpac loadng s pesened. A me-doman bounday elemen mehod (BEM) s developed fo hs pupose. Dsplacemen bounday negal equaons (BIEs) n conjuncon wh a mul-doman echnque ae appled n he pesen me-doman BEM. Collocaon mehod s used fo boh he spaal and he empoal dscezaons. Numecal examples fo compung he complex dynamc sess nensy faco ae pesened and dscussed. Inoducon Ineface cacks and neface debondng ae he mos dsnc falue mechansms n compose maeals because of he msmach n he maeal popees. To chaaceze he cack-p sess and defomaon felds, sess nensy facos (SIFs) and enegy elease aes ae ofen used n lnea elasc facue mechancs. Because of he mahemacal complexy of he neface cack poblems, only vey few nvesgaons on neface cacks n geneally ansoopc and lnea elasc solds unde mpac loadng condons can be found n leaue. Alhough he me-doman bounday elemen mehod (BEM) can be ulzed n pncple fo hs pupose, s numecal mplemenaon and applcaons o dynamc analyss of neface cacks n homogeneous, geneally ansoopc and lnea elasc solds have been epoed n leaue only vey ecenly. The man eason fo hs defcency s due o he fac ha he equed elasodynamc fundamenal soluons n he me-doman BEM do no have explc closed-fom analycal expessons and hey have a vey complex mahemacal sucue. Ths pape pesens a ansen dynamc analyss of neface cacks n wo-dmensonal (-D), layeed, ansoopc and lnea elasc solds. A me-doman BEM s developed fo hs pupose. A mul-doman BEM s appled, whch dvdes he nhomogeneous layeed ansoopc sold no homogeneous and ansoopc layes. Tme-doman elasodynamc fundamenal soluons fo homogeneous, ansoopc and lnea elasc solds and dsplacemen bounday negal equaons ae used fo each laye. Fo boh empoal and spaal dscezaons of he bounday negal equaons, collocaon mehods ae adoped. By usng he connuy/dsconnuy condons of he dsplacemens and he sesses on he nefaces and he cack-faces and nal condons, an

104 4 Eds: R Abascal and M H Alabad explc me-seppng scheme s obaned fo compung he dscee unknown bounday daa ncludng he cack-openng-dsplacemens (CODs). An effcen echnque fo compung he complex dynamc SIFs fom he numecally calculaed CODs s pesened and dscussed. To demonsae he accuacy and he effcency of he pesen me-doman BEM, numecal examples ae pesened and dscussed. Inal bounday value poblem and me-doman BIEs We consde a layeed, ansoopc and lnea elasc sold wh an neface cack as shown n Fg.. In he absence of body foces, he layeed sold sasfes he equaons of moon j, j u, () Hooke s law j Ejkluk, l, () he nal condons u( x, ) u ( x, ) fo, (3) he bounday condons () () f ( x, ), x c c c, (4) * f( x, ) f ( x, ), x f, (5) * u( x, ) u ( x, ), x u, (6) (,) an he connuy/dsconnuy condons on he neface n and he cack-faces c In Eqs. ()-(8), u, componens, () () () () u ( x, ) u ( x, ), j ( x, ) j ( x, ), x n, (7) () () () () (8) u( x, ) u ( xc, ) u ( x c, ), x c. j and f j n j epesen he dsplacemen, he sess and he acon n j s he ouwad nomal veco, s he mass densy, E jkl s he elascy enso, () () c c c denoes he cack-faces, ex f u sands fo he exenal bounday wh f and neface, and * u beng he bounday pas wh pescbed acons f and dsplacemens u *, n s he s he cack-openng-dsplacemens (CODs), especvely. A comma afe a u quany epesens spaal devaves whle a do ove a quany denoes me dffeenaon. Geek ndces ake he values and, whle Lan ndces ake he values, and 3. Unless ohewse saed, he convenonal summaon ule ove epeaed ndces s mpled. Maeal Doman n () c () c n ex Maeal Doman Fg. : An neface cack n a layeed ansoopc and lnea elasc sold

105 Advances n Bounday Elemen Technques IX 4 To each sub-doman of he cacked sold, he followng me-doman dsplacemen BIEs ae appled whee G j j G G cu( x, ) u f - u ds, j j j j j j y (,) x ex n c, (9) G c s he fee-em dependng on he smoohness of he bounday, u ( x, y ;, ) and ( x, y ;, ) ae he elasodynamc dsplacemen and acon fundamenal soluons fo homogeneous, ansoopc and lnea elasc solds, x and y epesen he obsevaon and he souce pons, and denoes Remann convoluon f g f( ) g( ) d. () Fo smooh boundaes cj.5 j, whee j s he Konecke-dela funcon. The elasodynamc fundamenal soluons fo homogenous, ansoopc and lnea elasc solds deved by Wang and Achenbach [] ae mplemened n he pesen me-doman BEM. Noe hee ha he elasodynamc fundamenal soluons fo homogenous, ansoopc and lnea elasc solds canno be gven n closed foms n conas o homogeneous, soopc and lnea elasc solds. In -D case, hey can be epesened by lne-negals ove a un ccle. I should be also emaked hee ha he dsplacemen BIEs (9) have a song sngulay n he sense of Cauchy-pncpal value negals. Numecal soluon pocedue To solve he songly sngula dsplacemen BIEs (7), a numecal soluon pocedue s developed. Collocaon mehod s appled fo boh he empoal and he spaal dscezaons. The dsplacemens and he acons ae appoxmaed by u M P p p (, ) m( u) ( ) ( u) ( ) u m m p y y, () M P p p (, ) m( f )( ) ( f )( ) f m m p f y y, () whee ( ) p m() y s he spaal shape funcons, () ( ) s he empoal shape funcon, ( u ) p m and ( f ) p mae dscee values a he m-h collocaon pon and p-h me-sep. Also, M s he oal elemen numbe and he me s dvded no P equal me-seps,.e., P. In hs analyss, consan spaal and lnea empoal shape funcons ae adoped fo smplcy,.e., j, ( y) ( y), mu m f y y m m, (3) p p ( u) ( ) ( f )( ) ( p ) H ( p) p H p ( p ) H ( p ), whee H[] s he Heavsde sep funcon. In Fg., he used spaal and empoal shape funcons ae depced. (4)

106 4 Eds: R Abascal and M H Alabad ( ) m( ) y p () () m m m 3 l l l3 y ( p ) p ( p ) Fg. : Spaal and empoal shape funcons By subsung Eqs. () and () no he me-doman BIEs (9), applyng he dscezed BIEs o each sub-doman and nvokng he nal condons (3), a sysem of lnea algebac equaons can be obaned as N - N N N-n n N-n n A u B f B f A u, (5) n n whee A and B n ae he sysem maces, u N s he veco conanng he bounday N dsplacemens, and f s he acon veco fo he exenal bounday and he cack-faces. By consdeng he bounday condons (4)-(6) and he connuy condons (7), equaon () can be eaanged as - N N N N n n N n n x C D y B f A u, (6) n N N n whch x epesens he veco wh unknown bounday daa, whle y denoes he veco wh known bounday daa. The explc me-seppng scheme (6) s appled fo compung he unknown bounday daa me-sep by me-sep. By usng consan spaal and lnea empoal shape-funcons (3) and (4), me and spaal negaons can be pefomed analycally. Songly sngula negals ae compued analycally by a specal egulazaon echnque. Only he lne-negals ove he un ccle n he elasodynamc fundamenal soluons have o be compued numecally by usng sandad Gaussan quadaue fomula. Moe deals on he numecal mplemenaon of he me-doman BEM can be found n he ecen wok of Beye e al. [] and Beye [3]. Compuaon of he complex dynamc sess nensy faco Fo an neface cack, he dsplacemen and he sess felds nea he cack-p can be chaacezed by a complex sess nensy faco whch s defned as K K K, (7) whee K and K ae he eal and he magnay pas of he complex sess nensy faco. The amplude and he phase angle of he complex dynamc sess nensy faco can be obaned by usng he followng equaons

107 Advances n Bounday Elemen Technques IX 43 K () K() K () K () 4cosh 4 u (, ) u (, ) d u (, ) d u (, ) d d d d u u K (, )/ (, ) () an ( ). K ( ) u (, )/ u (, ) In Eqs. (8) and (9), s a small dsance o he cack-p, s he b-maeal consan, and he consans d, d, (, ) can be found n [4]. Snce he sess feld nea he p of an neface cack shows a vey complcaed oscllang sngulay, no specal cack-p elemens ae mplemened n he pesen me-doman BEM fo smplcy. Fo hs eason, he local behavo of he cack-openng-dsplacemens (CODs) canno be descbed popely by usng he pesen me-doman BEM. To mnmze he numecal eo n he compuaon of he complex dynamc sess nensy faco fom he CODs, a leas-squaes echnque based on he mnmzaon of he quadac devaons of he dsplacemens on he cackfaces s appled. Numecal examples As fs numecal example, we consde an neface cack of lengh a n a ecangula plae conssng of wo homogeneous, ansoopc and lnea elasc maeals as depced n Fg. 3. The plae s subjeced o an mpac ensle loadng of he fom H (), whee s he loadng amplude and H () s he Heavsde sep funcon.the geomey of he cacked plae s defned by w =,mm, h = 4,mm and a = 4,8mm. The oue bounday of he plae s dscezed by 8 consan elemens wh elemens fo each sde, and he cack s dscezed by 5 consan elemens. A me-sep.3s s appled. Plane san condon s assumed n he numecal calculaons. Two dffeen Gaphe-epoxy composes ae consdeed n he fs example, whch have he followng maeal consans C () j C () j GPa, () = 6kg/m sym GPa, () = 6kg/m sym Numecal esuls fo he nomalzed amplude of he complex dynamc sess nensy faco s K ()/ K ( s I KI a ) ae pesened n Fg. 4. A compason wh he numecal esuls of, (8) (9)

108 44 Eds: R Abascal and M H Alabad Wünsche [5] shows a vey good ageemen. Wünsche used a me-doman BEM based on a collocaon mehod fo he empoal dscezaon and a Galekn-mehod fo he spaal dscezaon. Fgue 4 shows ha he nomalzed amplude of he complex dynamc sess () () nensy faco s zeo befoe he wave aval me a he cack-p h/ cl 6.5s, whee c L s s he velocy of he quas-longudnal wave of he doman. Afe he wave aval, K ()/ K I nceases apdly and eaches s maxmum value K ()/ K s I 3.8 a 9.9s. Then deceases o a local mnmum and heeafe nceases agan. A second peak s obaned a 8.3s h a w () K ~ ()/ K I s WÜNSCHE [5] TDBEM [s] Fg. 3: An nne neface cack Fg. 4: Nomalzed amplude of he complex dynamc sess nensy faco K ~ ()/ K I s a = 3 mm a = 4 mm a = 4 mm a = 6 mm a = 7 mm a = 8 mm [s] K ~ ()/ K I s a=9mm a=mm a=mm a=mm a=3mm a=4mm [s] Fg. 5: Effecs of he cack-lengh on he nomalzed amplude of he complex dynamc sess nensy faco

109 Advances n Bounday Elemen Technques IX 45 By usng he same empoal and spaal dscezaons, he effecs of he cack-lengh on he nomalzed amplude of he complex dynamc sess nensy faco ae nvesgaed. The coespondng numecal esuls ae pesened n Fg. 5. Fgue 5 eveals ha he cack-lengh has s sgnfcan nfluences on K ()/ K I. Boh he maxmum value and he coespondng me nsan s ae dependen on he cack-lengh. Fo small cack-lengh he fs peak of K ()/ K I s also s maxmum value, whle fo lage cack-lengh he second peak becomes s maxmum. In he second example, we consde an edge neface cack of lengh a n a ecangula plae conssng of wo dffeen ansoopc and lnea elasc maeals as depced n Fg. 6. The geomey of he cacked plae s defned by w =,mm, h = 4,mm and a = 4,8mm. The plae s subjeced o an mpac ensle loadng of he fom H (). The oue bounday of he plae s dscezed by 8 consan elemens wh elemens fo each sde, and he cack s dscezed by 5 consan elemens. A me-sep.4s s chosen. Plane san condon s assumed n he numecal calculaons. The same maeal combnaon as n he fs example s nvesgaed. Fgue 7 shows he nomalzed amplude of he complex dynamc sess nensy faco vesus he me. Hee agan, a compason of he pesen numecal esuls wh ha of Wünsche [5] shows s agan a good ageemen. The K ()/ K I -cuve shows a moe smooh ncease wh me afe he () wave aval me h/ cl 6.5s han n he fs example fo a cenal neface cack. The s maxmum value of K ()/ K I s aaned a abou 3.6s and a second peak s obseved a abou 9.s () 3. h a K ~ ()/ K I s.5. w.5..5 WÜNSCHE [5] TDBEM [s] Fg. 6: An nne neface cack Fg. 7: Nomalzed amplude of he complex dynamc sess nensy faco Fnally, he effecs of he lengh of he edge neface cack on he nomalzed amplude of he complex dynamc sess nensy faco ae nvesgaed by usng he same empoal and spaal s dscezaons. Fgue 8 pesens he coespondng numecal esuls fo K ()/ K I. Smla o he fs example fo a cenal neface cack, he cack-lengh may affec he behavou of he s K ()/ K I -cuve consdeably. The numecal esuls gven n Fg. 8 confm agan ha boh he maxmum value and he assocaed me depend songly on he cack-lengh. Fom Fg. 8 can be s concluded ha n compason o he K ()/ K I -cuve fo a cenal neface cack as shown n Fg.

110 46 Eds: R Abascal and M H Alabad s 5, he vaaons of he K ()/ K I -cuve wh me ae even moe complex n he case of an edge neface cack. K ~ ()/ K I s a = 3,mm a = 4,mm a = 5,mm [s] K ~ ()/ K I s a = 6,mm a = 7,mm a = 8,mm [s] Fg. 8: Effecs of he cack-lengh on he nomalzed amplude of he complex dynamc sess nensy faco Summay Ths pape pesens a me-doman BEM fo ansen elasodynamc analyss of an neface cack n -D, layeed, pecewse homogeneous, ansoopc and lnea elasc solds. Tme-doman dsplacemen BIEs n conjuncon wh a mul-doman echnque ae appled n he pesen medoman BEM. Fo boh he empoal and he spaal dscezaons, a collocaon mehod s adoped. Consan spaal shape funcons and lnea empoal shape funcons ae used fo smplcy, whch allow us o pefom he empoal and he spaal negaons analycally. Only he lne-negals appeang n he elasodynamc fundamenal soluons have o be compued numecally. An explc me-seppng scheme s obaned fo compung he unknown bounday daa numecally. An effcen leas-squaes echnque s appled fo accuaely compue he complex dynamc sess nensy faco fom he CODs. Numecal esuls fo he complex dynamc sess nensy faco ae pesened and compaed wh avalable efeence soluons. Acknowledgemen Ths wok s suppoed by he Geman Reseach Foundaon (DFG) unde he pojec numbes ZH 5/5- and ZH 5/5-, whch s gaefully acknowledged. Refeences [] C.-Y. Wang and J.D. Achenbach: Geophys. J. In. Vol. 8 (994), p [] S. Beye, Ch. Zhang, S. Hose, J. Sladek and V. Sladek: Sucual Duably & Healh Monong Vol. 3 (7), p. 77. [3] S. Beye: PhD Thess (n Geman), TU Begakademe Febeg, Gemany, 8. [4] S.B. Cho, K.R. Lee, Y.S. Choy and R. Yuuk: Engneeng Facue Mechancs Vol. 43 (99), p. 63. [5] M. Wünsche: PhD Thess (n Geman), TU Begakademe Febeg, Gemany, 8.

111 Advances n Bounday Elemen Technques IX 47 Modellng of opogaphc egulaes fo sesmc se esponse F. J. Caa, B. Beno, I. Del Rey, E. Alacón. ETSI Indusales Unvesdad Polécnca de Madd José Gueez Abascal, 86, Madd, Span fjcaa@es.upm.es ETSI Topogafía, Geodesa y Caogafía Unvesdad Polécnca de Madd Caeea de Valenca, km 7 83, Madd, Span Keywods: BEM, wave popagaon, layeed meda, egula nefaces, wave scaeng, se effecs Absac. Sesmc evaluaon mehodology s appled o an exsng vaduc n he souh of Span, nea Ganada, whch s a medum sesmcy egon. The nfluence of boh geology and opogaphy n he spaal vaably of gound moon ae suded as well as sesmc hazad analyss and gound moon chaacezaon. Afcal hazad-conssen gound moon ecods ae synhessed applyng sesmc hazad analyss and se effecs ae esmaed hough a dffacon sudy. Dec BEM s used o calculae he valley dsplacemen esponse o vecally popagang SV waves and ansfe funcons ae geneaed allowng he ansfomaon of fee feld moon o moon a each suppo. A closed fomulae s used o esmae hese ansfe funcon. Fnally, he esuls obaned ae compaed. Inoducon Pefomance Based Engneeng s he eassuemen of he classcal lne whch es o use he mos advanced and compehensve pocedues o gve confdence o he desgne and quanfy sucual damages n ems ha boh owne and socey can undesand he nvolved sks. Among ohes, hs new paadgm ncludes hazad analyss and se effecs fo mpoan bdges. The hazad analyss wll poduce wo man esuls. The fs s he chaacezaon of he hazad self. The second one s he denfcaon of mos-lkeable scenaos, allowng selecon o geneaon of ecods compable wh hem. Sesmc hazad analyss s beyond he scope of hs pape, alhough a bef summay s pesened below. Then se effecs wll be nvesgaed, assessng he need of consdeng mulple suppo excaons. Dec Bounday Elemen Mehod (DBEM) wll be used o compue ansfe funcons allowng he ansfomaon of fee feld moon o moon a he foundaon of each pe. The nfluence of opogaphc deals n eahquake acceleaons has been ecognzed fo a long me. As he analycal soluon s lmed o vey smple geomec ypes, all eseach on ealsc cases s based on numecal echnques ha can be he so-called Indec Bounday Elemen mehod (IBEM) [], a pue Dec Bounday Elemen Mehod (DBEM) [,3], o a mxue of Fne and Bounday elemens [4]. The bounday elemen mehod s especally well sued fo he analyss of he sesmc esponse of valleys of complcaed opogaphy and sagaphy. Infne egons ae naually epesened, and he adaon of waves owads nfny s auomacally ncluded n he model whch s based on an negal epesenaon vald fo nenal and exenal egons. The man focus of hs wok s elaed o he evaluaon of he dffeen acceleaons a he pe foundaons of bdges, and geneang sem analycally ecommendaons ha can be appled o he geneal esponse speca.

112 48 Eds: R Abascal and M H Alabad Descpon of he vaduc and he valley The bdge s pa of he Spansh Hghway newok, whch means ha he Spansh code consdes as an mpoan nfasucue wh a desgn lfe of yeas. The bdge has wo-decks, suppoed by u-shaped pes n he ops. The oal lengh s 35 m dsbued n sx spans: 45,5 m + 4x53,5 m + 45,5 m. The pe heghs ae (fom lef o gh) 7, 64 m, 74, 79 m and 5 m. The valley geomey can be seen n Fgue. The lef hllsde has a slope of appoxmaely 3º and he gh one s abou 4º. The cenal pa s almos hozonal aound he ve bed. Fgue. Valley geomey. The geology s shown n Fgue. Thee s an eosve conac beween he Plocene and he Teay ocks ha poduces a shallow laye of conglomeaes on a dome of schss. Insde hs one a dolomc ncluson has been deeced nea pe numbe 4. Boh abumens ae founded on conglomeaes. Fgue. Valley geology. S waves velocy popees, he densy and he shea modulus G of he dffeen maeals ae shown n Table. Table. Maeal popees. Cs (m/s) ( kg/m 3 ) G ( GPa) Conglomeaes, Schss 8 4 7,76 Dolomes 4 7 5,55

113 Advances n Bounday Elemen Technques IX 49 Sesmc hazad analyss Sesmc hazad assessmen a he bdge se s compued followng he sandad zonfed pobablsc appoach, whee eahquake occuence s modelled as a possonan pocess and eahquake ecuence s chaacezed by a doubly-uncaed Guenbeg-Rche elaon [5]. Thee s a elavely poo knowledge on poenal faul souces n he sudy aea. Howeve, geomophologc and neoeconc analyses povde ough esmaes on pesen-day acvy and maxmum possble magnudes of some fauls. Accodng o hese sudes, s possble o elae a magnude Mw , epcenal dsance Rep 3-4km even wh he Venas de Zafaaya and Pnos-Puene pncpal fauls. Tha scenao was used o pefomance smulaons of hazad-conssen acceleaon-me hsoes. Se modellng Sucual analyss of he bdge showed ha one of he lm condons was elaed o he longudnal bdge behavou. So was decded o conduc a bdmensonal sudy n he plane conanng he bdge vecal and longudnal decons, and o sudy fo he valley esponse o he ncdence of vecally popagang SV waves. The objecve s o oban, n he fequency doman, he ansfe funcons beween he dsplacemens a evey pe foundaons and a efeence pon n a fcous oucop of schss fa fom he se ha wll be supposed o be subjeced o he sesmc hazad defned n he pevous pon. The use of a DBEM s especally appopae fo hs knd of analyss. In hs case 4 subegons have been dscezed. The bounday elemen mesh has been neuped m fom he cene of he valley whee he behavou s smla o a monodmensonal column of safed sol. The oal numbe of elemens s 64, wh paabolc nepolaon of dsplacemens and acons. A he suface neface, he elemen sze s equal o 5 mees and, a nenal nefaces, he elemen sze has been chosen akng no accoun he maxmum shea wave velocy of he wo maeals sepaaed by he neface. Fgue 3. Tme doman dsplacemen of he valley o ncden vecal SV waves. The ncden me sgnal s a Rcke wavele. The saons ae locaed along he fee suface of he half-space, a a hozonal dmensonless hozonal coodnae x=a.

114 43 Eds: R Abascal and M H Alabad Synhec sesmogams wee compued usng he FFT algohm fo a Rcke wavele. Tme sees wee obaned fom he ansfe funcons esmaed a eceves placed along he fee suface. Fgue 3 shows a sample of he synhec sesmogams, compued a he suface of he model defned n Fgue, fo he vecal ncdence of SV waves (Rcke wavele s cenal fequency s f p =.c s /a =.9 Hz). The esponse speca of he hozonal componen ae ploed n Fgue 4b a evey suppo of he bdge. The esponse speca have been calculaed fom a smulaed acceleaon-me hsoy and ae pesened n ems of pseudo-acceleaon as a funcon of me. Fgue 4b shows he compason beween he speca geneaed a evey suppo combnng he fee feld moon and he ansfe funcons fom Fgue 4a and he speca compued decly fom he fee feld acceleaon me hsoy. In all cases, he dffeences ae noceable, and a some peods ae as hgh as he 5%. These esuls show ha s mandaoy o conduc a dffacon sudy. (a) (b) Fgue 4. (a) Hozonal dsplacemen ansfe funcons obaned fo BEM model. (b) Elasc esponse speca geneaed a evey suppo akng no accoun BEM ansfe funcons and fee feld esponse specum. Closed fom soluon Bdge desgnes no always have he me o he specfc echnques o develop complex dffacon poblems. In addon, he unceanes nvolved n he quanfcaon of sesmc acon sugges he possbly of usng he closed fom soluon assocaed o he ansfe funcon of a layeed meda o esmae he elave dsplacemen beween pes. In addon, man nees of desgnes s cened aound esponse speca mehod, so ha hs secon es o compae he elave dffeences ha can be found when he appoach s appled o hs complcaed laye meda. Consde a sol depos conssng of hozonal layes esng on a sem-nfne meda. Assumng ha an ncden hamonc SV vecal wave s popagang n he sem-nfne meda, sasfacon of he equemens of equlbum and compably a each neface gve se o a sysem of smulaneous equaons, whch allows he ampludes fo he efleced and efaced waves o be expessed n ems of he amplude of he ncden SV wave, so ha, one can defne a ansfe funcon H f(h,,c s,,h,,c s,, 3,c s, 3 3, ) elang he hozonal dsplacemen amplude a he fee bounday o he ncden waves (whee H s laye hegh, s maeal densy, c s s SV-wave popagaon velocy and s he maeal dampng ao, fo supefcal laye (), nemedae laye () and sem-nfne meda (3); s he angula fequency of he ncomng waves). Applyng Haskell popagao mehodology [6]:

115 Advances n Bounday Elemen Technques IX 43 H e e e e kh kh kh kh () Gk Gk (), Gk Gk 3 3 kh kh e e kh kh e e k c n s n n Fgue 5 shows he ansfe funcons obaned wh he closed fom soluon, whee, cs ae he same fom Table, s equal o.5 n all cases, H s equal o m and usng fo H he deph of he saum unde each pe. (3) (4) (a) (b) Fgue 5. (a) Hozonal dsplacemen ansfe funcons obaned fo closed fomulae. (b) Elasc esponse speca geneaed a evey suppo akng no accoun BEM ansfe funcons, closed fomulae ansfe funcons and fee feld esponse specum. Fgue 5b shows he compasons beween he speca geneaed hough hose ansfe funcon a evey suppo and hose fom Fgue 4b. Fo all paccal puposes, hese esponse speca ae smla o hose obaned wh BEM, and heefoe, fo paccal desgn he appoach seems vald. Summay Pncpal sesmc codes defne he sesmc acons by means of elasc esponse specum. Theefoe, bdge desgnes use elasc esponse specum n bdge pojecs and, a leas, fo he geneal popoon of sucual membes, hey ae manly neesed fo smle pocedues. Ths pape pesens how BEM can be used o oban acceleaon-me hsoes a evey bdge pe foundaons akng no accoun se effecs, valleys of egula geomey and complcaed saa wh egula nefaces. Elasc esponse speca obaned wh hese ansfe funcons have been compaed wh fee feld elasc esponse specum. The esuls show ha fee fle specum noduces sgnfcan eos and jusfed he consdeaon of se effecs. Fnally, has been shown ha close fom soluons poduce accepable esuls fom he engneeng paccal vew pon, especally f, as usual, befoe been appled hose speca ae smoohed.

116 43 Eds: R Abascal and M H Alabad Acknowledgemen Ths wok has been poduced as a pa of he eseach founded by Spansh Mnseo de Fomeno, whn he Naonal Plan of Scenfc Reseach, Developmen and Technologcal Innovaon 4-7, wh numbe of pojec 87/A4. The auhos gaefully acknowledge he kndness of Pof. R. Gallego who povded he bounday elemen code LPoSo [5]. Refeences [] Sánchez-Sesma FJ, Campllo M. Dffacon of P, SV, and Raylegh waves by opogaphc feaues: a bounday negal fomulaon. Bull Sesmol Soc Am 99; 8: [] Domnguez, J. Bounday elemens n dynamcs. ElSeve & CMP 993. [3] Alvaez-Rubo S., Beno J.J., Sanchez-Sesma F.J. and Alacon E. The use of dec bounday elemen mehod fo ganng nsgh no complex sesmc se esponse. Compues & Sucues 83; [4] Faccol E, Paolucc R, Vann M. TRISEE 3D se effecs and solfoundaon neacon n eahquake and vbaon sk evaluaon. In: Euopean Commsson, Decoae Geneal XII fo Scence, Reseach and Developmen Envonmens and Clmae Pogamme-Clmae and Naual Hazads Un [5] Gaspa-Escbano J.M., Beno, B. Gound-Moon Chaacezaon of Low-o-Modeae Sesmcy Zones and Implcaons fo Sesmc Desgn: Lessons fom Recen Mw 4.8 Damagng Eahquakes n Souheas Span. Bullen of he Sesmologcal Socey of Ameca, Ap 7; 97: [6] Kame, S.L. Geoechncal Eahquake Engeneeng. Pence Hall. New Jesey [5] Gallego R. BEM pogam LPoSo. Inenal epo. ETSI Camnos, Canales y Pueos. Unvesdad de Ganada.

117 Advances n Bounday Elemen Technques IX 433 A fas 3D BEM fo ansoopc elascy based on heachcal maces I. Benede,a, A. Mlazzo,b, M.H. Alabad,c Dpameno d Tecnologe ed Infasuue Aeonauche, Vale delle Scenze, Edfco 8, 98 Palemo - Ialy Depamen of Aeonaucs, Impeal College London, Souh Kensngon Campus, Rodec Hll Buldng, Exhbon Road, SW7AZ, London, UK a.benede@unpa., b albeo.mlazzo@unpa., c m.h.alabad@mpeal.ac.uk Keywods: Fas BEM solves, heachcal maces, ansoopc elascy. Absac. In hs pape a fas solve fo hee-dmensonal ansoopc elascy BEM poblems s developed. The echnque s based on he use of heachcal maces fo he epesenaon of he collocaon max and uses a pecondoned GMRES fo he soluon of he algebac sysem of equaons. The pecondone s bul explong he heachcal ahmec and akng full advanage of he heachcal foma. The applcaon of heachcal maces o he BEM soluon of ansoopc elascy poblems has been numecally demonsaed hghlghng boh accuacy and effcency leadng o almos lnea compuaonal complexy. Inoducon. The use of compose maeals n many engneeng applcaons enables mpoved desgn fo sucues, equpmen and devces. The pefomance of such nheenly ansoopc maeals mus be caefully evaluaed o mee nceasng equemens n ccal engneeng applcaons. Much effo has been devoed o expemenal sudes fo compose maeals chaacezaon. On he ohe hand numecal modelng and analyss have ecenly gaheed sgnfcan momenum. Compuaonal mehods such as he fne dffeence mehod (FDM), he fne elemen mehod (FEM) and he bounday elemen mehod (BEM) have been wdely exploed o cay ou numecal analyses of sucual poblems nvolvng boh soopc and ansoopc maeals. The bounday elemen mehod s paculaly appealng fo many sucual applcaons, alhough s exensve ndusal usage, especally when lage scale compuaons ae nvolved, s hndeed by some lmaons, manly elaed o he feaues of he soluon max. Such max s geneally fully populaed, hus esulng n nceased memoy soage equemens as well as nceased soluon me wh espec o ohe numecal mehods fo poblems of he same ode. Moeove, he analyss of hee-dmensonal ansoopc elasc solds n he famewok of he BEM eques some addonal consdeaons. The lack of ansoopc Geen s funcons fo he consucon of he bounday negal epesenaon [] esuls n he use of ehe he negal expesson of he fundamenal soluons [-5] o explc expessons wh complex calculaons [6-4]. Due o he fom of he 3D fundamenal soluons, BEM echnques fo ansoopc elascy applcaons esuled n slowe compuaons wh espec o he soopc case, fo whch analycal closed fom fundamenal soluons ae known. Many nvesgaons have been caed ou o ovecome such lmaons. In pacula, fas mulpole mehods (FMMs) have been developed o solve effcenly bounday elemen fomulaons fo elascy poblems [5]. Alhough FMMs ae vey effecve, hey eque he knowledge of he kenel expanson n advance n ode o cay ou he negaon and hs s paculaly complex fo ansoopc elascy poblems, fo whch analyc closed fom expessons of he kenels ae no avalable. In he pesen pape he Fas BEM based on heachcal maces and he algeba poposed n efeence [7] fo hee-dmensonal soopc elascy s exended o ansoopc applcaons. The man sep s he consucon of he appoxmaon of suable blocks of he bounday elemen max based on he compuaon of only few enes of he ognal blocks. Ths appoxmaon, n conjuncon wh he use of Kylov subspace eave solves, leads o elevan numecal advanages, namely educed memoy soage equemens and educed compuaonal me fo he soluon. The effecveness of he echnque fo he analyss of ansoopc solds s numecal demonsaed n he epoed applcaons.

118 434 Eds: R Abascal and M H Alabad The bounday elemen model The bounday negal equaon govenng he behavo of a hee-dmensonal ansoopc body wh bounday, n absence of appled body foces, s gven by,, c x u x T x x u x d U x x x d () j j j j j j whee u j and j ae he bounday dsplacemen and acons. U j and T j ae he ansoopc fundamenal soluon kenels, whose expesson s gven n efeences [] whee dffeen appoaches fo he compuaon ae dscussed. Afe sandad BEM dscezaon [] eq. () leads o a lnea sysem of he fom Hu G () whee u and ae he vecos collecng he componens of he nodal values of dsplacemen and bounday acons, especvely. The soluon of sysem (), afe focng he bounday condons n ems of pescbed nodal values, povdes he values of he unknown dsplacemen and acons on he body bounday. The sysem of algebac equaons pesens a coeffcen max whch s fully populaed and nehe symmec no defne. Ths esuls n nceased memoy equemens as well as nceased assembly and soluon me wh espec o ohe numecal mehods fo poblems of numecal compaable sze. Moeove, n he case of ansoopc elascy, all he max enes need o be compued negang kenels wh complex expesson, due o he lack of closed fom Geen s funcons fo hee-dmensonal ansoopy. The use of heachcal maces fo he appoxmaon and soluon of BEM sysems of equaons asng n soopc elascy applcaons has been poposed n [6,7]. Such echnque seems vey appealng fo ansoopc BEM poblems whee he compuaon and negaon of he negal equaon kenels s vey nvolved. In he followng he basc pncples and he paccal seps needed o geneae he BEM max appoxmaon usng heachcal maces ae summazed. The eade s efeed o he efeences [6,7] fo moe deals. The consucon of he fas BEM solve s based on a heachcal epesenaon of he collocaon max. Such epesenaon s bul by epesenng he max as a collecon of sub blocks, some of whch adm a specal appoxmaed and compessed foma. Such blocks, efeed o as low ank blocks, can be appoxmaed by compung only some of he enes of he ognal blocks (8) hough adapve algohms known as Adapve Coss Appoxmaon (ACA) (9,). Low ank blocks epesen he numecal neacon, hough asympoc smooh kenels, beween ses of collocaon pons and cluses of negaon elemens whch ae suffcenly fa apa fom each ohe. The dsance beween cluses of elemens enes a cean admssbly condon, based on some seleced geomecal ceon, fo he exsence of a low ank appoxman. The blocks ha do no sasfy such condon ae called full ank blocks and hey need o be compued and soed enely. The low ank epesenaon of he collocaon max allows o educe memoy soage equemens as well as o speed up opeaons nvolvng he max. The pocess leadng o he subdvson of he max no low and full ank blocks s based on geomecal consdeaons on he bounday mesh of he analyzed body, as schemacally llusaed n fg. (). Each block n he collocaon max s elaed o wo ses of bounday elemens, he one conanng he collocaon pons coespondng o he max ow ndces and he one goupng he elemens ove whch he negaon s caed ou, ha conans he nodes coespondng o he max columns. If hese wo ses of bounday elemens ae sepaaed, hen he block wll be epesened and soed n low ank foma and s called admssble, whle wll be enely geneaed and soed n full ank foma ohewse. The admssbly of a canddae block s based on he nequaly mn dam, dam ds(, ) (3) xo x xo x whee s a paamee nfluencng he numbe of admssble blocks on one hand and he convegence speed of he adapve appoxmaon of low ank blocks on he ohe hand []. The max block-wse subdvson and classfcaon s based on a pevous heachcal paon of he max ndex se amed a goupng subses of ndces coespondng o conguous nodes and elemens on he bass of some compuaonally effcen geomecal ceon. The paon s soed n a bnay ee of ndex subses, o cluse ee, ha consues he bass fo he followng consucon of he heachcal block subdvson ha wll be soed n a quaenay block ee. A possble algohm, leadng o geomecally balanced ees, s gven n efeence [] and consdes fo smplcy boxes famng he consdeed cluses.

119 Advances n Bounday Elemen Technques IX 435 Fgue. Schemac of he bounday subdvson. As he admssble blocks have been locaed, he appoxmaon s geneaed hough ACA algohms whch allow o each adapvely he a po seleced accuacy c. Addonally, o opmze memoy soage equemens and educe he oveall compuaonal complexy, he low ank blocks ae ecompessed whou accuacy penales, akng advanage of he educed Sngula Value Decomposon (SVD) [3]. Moeove, snce he nal max paon s geneally no opmal, once he blocks have been geneaed and ecompessed, he ene sucue of he heachcal block ee can be modfed hough a coasenng pocedue, whch educes he soage equemens and speeds up he soluon mananng he pese accuacy [4]. As an almos opmal epesenaon s obaned, he soluon of he sysem can be ackled ehe decly, hough heachcal max nveson [5], o ndecly, hough eave mehods [6]. In boh cases, he effcency of he soluon eles on he use of a specal ahmec,.e. a se of algohms ha mplemen he opeaons on maces epesened n heachcal foma, such as addon, max-veco mulplcaon, max-max mulplcaon, nveson and heachcal LU decomposon. A collecon of algohms ha mplemen many of such opeaons s gven n [] whle he heachcal LU decomposon s dscussed n [6]. The use of eave mehods akes full advanages of he heachcal epesenaon explong he effcency of he low-ank max-veco mulplcaon. The convegence of eave solves can be mpoved, o somemes obaned fom a non convegen scheme, by usng suable pecondones. In he pesen appoach an LU pecondone max s bul n heachcal foma sang fom a coase appoxmaon wh accuacy p of he collocaon max [6]. An eave GMRES algohm s fnally used n conjuncon wh such pecodone fo solvng he sysem. Numecal expemens and dscusson The heachcal compuaonal scheme descbed n [7] has been modfed o pefom analyses on ansoopc bodes. In pacula he ansoopc fundamenal soluons ae compued by usng he echnque poposed by Wlson and Cuse n efeence [5]. In hs echnque he dependence of he fundamenal soluon kenels on he souce pon o feld pon decon s obaned by nepolaon fom a daabase conanng he so called modulaon funcons fo he fundamenal soluon dsplacemen and dsplacemen devaves. The daabase s acually consued by ables conanng he modulaon funcons fo dffeen souce pon o feld pon decons, descbed n ems of wo angles and n sphecal coodnaes. All he compuaons have been pefomed usng an Inel Coe TM Duo pocesso T93 (.5 GHz) wh GB of RAM. In ode o compae he obaned esuls wh hose obaned fo soopc bodes, he same confguaon poposed n efeence [7] wh ansoopc maeal popees has been analyzed. The analyzed mechancal elemen loaded an-symmecally a he holes and clamped a he cene cylndes s shown n fg. wh he feaues of he consdeed meshes.

120 436 Eds: R Abascal and M H Alabad Fgue. Geomey and mesh daa. N of Nodes N of Elemens Mesh A 4 38 Mesh B Mesh C A fs se of analyses has been caed ou on he mesh C o nvesgae he effec of he pecondone accuacy p on he convegence of he heachcal GMRES eave soluon. Fo hs pupose, he accuacy of he collocaon max has been se o 5, he admssbly paamee has been chosen as, c 8 he mnmal block sze has been se o n mn 36 [7] and he GMRES elave accuacy has been se o. The esuls obaned ae shown n Table whee he pecondone pecenage of soage, he soluon mes and he soluon speed up aos wh espec o sandad ansoopc BEM ae gven. A second se of analyses has been pefomed o sudy he nfluence of he admssbly paamee on he soluon. The esuls obaned ae pesened n Table n ems of pecenage soage befoe and afe coasenng, speed up aos and accuacy wh espec o he sandad BEM soluon. The soage memoy equesed by he fas heachcal BEM s ndependen fom he admssbly paamees. The same s no ue fo he soluon me and consequenly fo he speed up of he soluon whch ae affeced by he chosen value of. Theefoe, as expeced, hee s an opmum value of whch se he bes block paon of he max and mnmze he me equesed fo he soluon. Fo a dealed dscusson abu he nfluence of see efeence [7]. Table 3 epos memoy equemens befoe and afe coasenng, assembly me and speed up ao, soluon me and speed up ao and he accuacy of he fnal soluon a dffeen values of he collocaon max equesed accuacy wh he ohe paamee se a he values shown n he able. The memoy equemens, assembly mes and soluon mes decease when he pese accuacy deceases, as he aveage ank of he appoxmaon s educed. Table. Effec of he pecondone max accuacy (mesh C, 5 c,, n mn 36 ) Pecondone Pecondone Pecond. GMRES Toal Soluon Soluon Toal p Soage % Tme (s) LU me me me (s) Speed up Speed up Table. Soage, mes and speed up aos fo dffeen (mesh C, 5 c, p, n mn 36 ) Soage % befoe coas. Soage % afe coas. Assembly me (s) Soluon me (s) Assembly Speed up Soluon Speed up Toal Speed up L nom Table 3. Effec of heachcal max accuacy (mesh C, 4,, n 36 ) Soage % c befoe coas. Soage % afe coas. Assembly me (s) Soluon me (s) p Assembly Speed up mn Soluon Speed up Toal Speed up L nom 5.7

121 Advances n Bounday Elemen Technques IX Table 4. Effec of fundamenal soluon descpon accuacy (mesh C, 5, 4,, n 36 ) Soage % befoe coas. Soage % afe coas. Assembl y me (s) Soluon me (s) c Assembl y Speed up Soluon Speed up p Toal Speed up mn L Nom Regadng he collocaon max heachcal appoxmaon, an neesng ssue whch need o be nvesgaed s he effec of he degee of accuacy n he nepolaon of he modulaon funcons of he fundamenal soluons. In pacula, wh efeence o he appoach employed n he pesen pape, hs effec has been suded by pefomng analyses wh hee dffeen accuacy of he ables descbng he modulaon funcons. Each ables s chaacezed by a dffeen numbe of enes fo he souce pon o feld pon decon, obaned by seng he angula sepaaon ough he dffeen decons o a fxed value. The esuls ae shown n Table 4, whee s evdenced ha he degee of accuacy n he descpon of he kenels affecs he pefomance of he ACA block appoxmaon. Hgh accuacy n he kenel descpon deemnes a lowe aveage ank n he ACA appoxmaon of admssble blocks, wh bee esulng pefomances of he mehod. Fnally, he pefomances of he mehod ae hghlghed n Fg. 3, whee he memoy usage and he soluon me obaned by analyzng hee dffeen meshes ae shown. The vaaon of memoy usage and soluon me wh espec o he degees of feedom clealy shows ha, also fo ansoopc bodes, he pesened appoach eques almos lnea compuaonal complexy. Is effcency mpoves wh he poblem dmenson and appeas vey appealng fo lage scale sysems..8 6 Memoy Usage [GB].6.4. Sandad BEM Fas Heachcal BEM Soluon Tme [sec] 8 4 Sandad BEM Fas Heachcal BEM DoF DoF Fgue 3. Memoy usage and soluon me wh espec o poblem DoF. Conclusons A Fas solve fo 3D ansoopc elascy BEM poblems based on heachcal maces has been developed. The pefomed ess demonsaed he applcably of he echnque o he analyss of ansoopc bodes. The pefomances of he mehod, namely elevan memoy soage and soluon me savngs, pevously demonsaed fo soopc BEM sysems, make he echnque appealng also fo ansoopc poblems. Almos lnea compuaonal complexy a nceasng degees of feedom has been evdenced. Such feaues make he echnque vey appealng fo lage scale applcaons.

122 438 Eds: R Abascal and M H Alabad Refeences [] M.H. Alabad, The Bounday Elemen Mehod: Applcaons n Solds and Sucues, vol.. John Wley & Sons Ld (). [] D.M. Bane, Phys. Sa. Sol. (b), 49, (97). [3] L.J. Gay, A. Gffh,L. Johnson, P.A. Wawzynek, Elecon. J. Bound. Elem,, (3). [4] S.M. Vogel, F.J. Rzzo, J. Elas., 3, 3-6 (973). [5] R.B. Wlson, T.A. Cuse, In. J. Nume. Meh. Eng.,, (978). [6] T. Chen, F.Z. Ln, Compu. Mech.,5, (995). [7] T.C.T. Tng, V.G. Lee, Q. J. Mech. Appl. Mah., 5, (997). [8] G. Nakamua, K. Tanuma, Q. J. Mech. Appl. Mah., 5, 79-94, (997). [9] E. Pan, F.G. Yuan, In. J. Nume. Meh. Eng., 48, -37 (). [] N.A. Schcla, Ansoopc analyss usng bounday elemens. Comp. Mech. Publ. (994). [] M.A. Sales, L.J. Gay, Compu. Suc., 69, (998). [] F. Tonon, E. Pan, B. Amade, Compu. Suc.,79, (). [3] V.G. Lee, Mech. Res. Commun., 3, 4-49 (3). [4] C.Y. Wang, M. Denda, In. J. Solds Sucues., (7). [5] V. Popov, H. Powe, Eng. An. Bound. Elem., 5, 7 8 (). [6] M. Bebendof, R. Gzhbovksks, Mah. Meh. Appl. Scences, 9, (6). [7] I. Benede, M.H. Alabad, G.Davì, In. J. Solds Sucues, 45, (8). [8] E. E. Tyyshnkov, Calcolo, 33, 47-57, (996). [9] M. Bebendof, Numesche Mahemak, 86, , (). [] M. Bebendof, S. Rjasanow, Compung, 7, -4, (3). [] S. Böm, L. Gasedyck and W. Hackbusch, Eng. An. Bound. Elem., 7, 45 4, (3). [] K. Gebemann, Compung, 67, 83-7, (). [3] M. Bebendof, Effzene numesche Lösung von Randnegalglechungen une Vewendung von Nedgang-Mazen, Ph.D. Thess, Unvesä Saabücken,. dsseaon.de, Velag m Inene, ISBN , () [4] L. Gasedyck, Compung, 74, 5-3, (5). [5] L. Gasedyck, W. Hackbush, Compung, 7, , (3). [6] M. Bebendof, Compung, 74, 5-47, (5).

123 Advances n Bounday Elemen Technques IX 439 A BEM appoach n nonlnea acouscs V. Mallado and M. H. Alabad Depamen of Achecue, Unvesy of Feaa, Ialy, mlv@unfe. cuenly a he Impeal College London as eseach assocae Depamen of Aeonaucs, Impeal College London, UK, m.h.alabad@mpeal.ac.uk Keywods: Nonlnea acouscs, negal equaons, dual ecpocy. Absac. The pesen pape deals wh a novel applcaon of he Bounday Elemen Mehod (BEM) o wo-dmensonal (D) nonlnea acouscs. The acousc waves ae supposed o be of fne-amplude and he analyss s pefomed n he fequency doman. By applyng he peubaon echnque, he govenng dffeenal equaons ae ansfomed no a sysem of wo Helmholz equaons, one homogeneous and he ohe one nhomogeneous. The Dual Recpocy Bounday Elemen Mehod (DRBEM) s used o ansfe he doman negal o he bounday. The pocedue s valdaed by compason wh an ad-hoc analycal soluon and esed fo dffeen bass funcons. Inoducon The neacon of an acousc sgnal wh mae s sad o be lnea f he esponse of he maeal and he sengh of he oupu sgnal vay lnealy wh he sengh of he npu sgnal. Fo nsance, n one dmenson (D) he equaon of moon educes o he lnea wave equaon only f he acousc Mach numbe M s neglgble n compason wh uny. Fuhemoe, any wave wll become dsoed, no mae how small M s, f can popagae a suffcen dsance (see []). Fo hgh npu sgnal senghs, o fo maeals wh some specal popees, some nonlnea effecs may occu. Some of hese effecs ae nceasngly used fo nondesucve chaacezaon of maeals and damage deecon n ndusal poducs. A evew of he nonlnea acousc applcaons fo maeal chaacezaon can be found n []. The nonlnea phenomena whch ae lnked o fneamplude waves,.e. when he acousc Mach numbe s no neglgble n compason wh he uny, ae nvolved n he mechansms whch deemne a gea numbe of paccal applcaons. Fo nsance n solds, plasc and meal weldng, machnng and cung, maeal fomng. In fluds, pacle flaon, defoamng, dyng. The BEM s a numecal appoach fo solvng feld poblems based on he bounday negal equaon (BIE) fomulaons. The BEM has been used o solve exeo and neo lnea acousc poblems fo many yeas (see fo nsance [3]) because of s bounday only dscesaon and auomacally sasfacon of he adaon condon a nfny. A evew on he applcaons n elasosacs, hemoelascy, elasoplascy, conac and facue mechancs can be found n [4]. The nees of exendng such a ool o ealsc modelng of he nonlnea acousc feld geneaed by fne-amplude acousc waves s evden. Ths s he man pupose of he pesen wok whch s lmed o he D analyss. The govenng dffeenal equaons ae heaen by a peubave appoach, hen ansfomed no homogeneous/nhomogeneous negal equaons and fnally numecally solved by couplng he convenonal BEM wh he DRBEM. Some numecal examples ae pesened n ode o demonsae he effcency of he poposed pocedue. The govenng equaons The equaons whch goven he geneal moon of an unvscous flud ae mass consevaon, momenum consevaon and hemodynamc sae (see fo nsance []): ρ ρ ρ = u (a)

124 44 Eds: R Abascal and M H Alabad ρ u + P = (b) ( ) ρ γ P = A Q (c) ρ whee ρ and ρ ae he acual and he no-peubaon mass densy, especvely, u s he flud dsplacemen veco, P s he hemodynamc pessue, γ = c p /c v s he ao of he specfc heas a consan pessue (c p ) and consan volume (c v )andq s a consan o be deemned fom expemenal daa. Dsspaon mechansms ae negleced. By Taylo expandng he densy of he flud ρ up o he second ode em of he acousc dsubance p = P P, he followng govenng wave equaon can be obaned: P P, c = β ρ c 4 (p ), () whee comma devave noaon s adoped, c s he sound speed n lnea acouscs and β =(+γ)/ s efeed o as he coeffcen of nonlneay. Fo nsance, he wae a Chasβ =3.5. By adopng he followng appoxmaon: p = p l + p c (3) whee ba ndcaes he dependance on he me vaable, p l epesens he fs ode appoxmaon of he acousc dsubance and p c funshes s second ode coecon, and by assumng me-hamonc waves,.e. p l = p l e ω and p c = p c e ω, he dffeenal Eq.() funshes he followng nonlnea sysem of dffeenal equaons (see [6] fo deals): p l + k l p l = p c + kc p c = 4ω β ρ c 4 p l (4a) (4b) whee k l = ω/c and k c =k l. Thd o hghe ode ems ae negleced. Ths means ha fne bu of modeae amplude waves ae consdeed. In concluson, he oal acousc pessue s obaned by solvng he above sysem of dffeenal equaons n ems of p l and p c,.e.: p(x,)=p l e ω + p c e ω (5) The sysem of Eqs.(4) can be solved analycally only n vey smple cases. Fo complex geomees numecal echnques mus be nvolved. The numecal mplemenaon Fne elemen mehods (FEM) fo me-hamonc acouscs govened by he Helmholz equaon have been an acve eseach aea fo nealy 4 yeas. The BEM has demonsaed o be effcen especally fo scaeng poblems: n no ohe feld s he BEM used so nensvely by ndusy. Fuhemoe, ecen advances such as he fas mulpole and he panel cluseng mehods have emendously mpoved s effcency boh fo pulsang and fo nenal poblems. The nonlnea acousc bounday negal equaons (NABIEs) can be deved by applyng he weghed esdual echnque ogehe wh he Geen s denes (see [3]) o he Eqs.(4) o gve: c(ξ)p l (ξ)+ q (ξ, x)p l (x)dγ(x) p (ξ, x)q l (x)dγ(x) = (6a) Γ Γ c(ξ)p c (ξ)+ q (ξ, x)p c (x)dγ(x) p (ξ, x)q c (x)dγ(x) = p (ξ, x)b(p l (x))dω(x) (6b) Γ Γ Ω

125 Advances n Bounday Elemen Technques IX 44 whee b(p l (x)) = 4ω β ρ c 4 p l (x) =c β p l (x) (7) The negal on he lef hand sde s o be nepeed n he sense of Cauchy pncpal value and he fee em c(ξ) s equal o.5 f he angen lne a ξ s connuous. The symbols ξ and x denoe he souce and he feld pons, especvely, Γ s he bounday of he doman Ω unde analyss. The fundamenal soluons p, q ae gven n any BEM book (see fo nsance [5]). In D hey ae expessed n ems of he modfed zeo and fs ode Bessel funcons of he second knd. In he convenonal BEM he bounday s dvded no NE elemens (quadac n he pesen pape) and he Eq.(6a) s collocaed n each bounday node o funsh a dscee sysem of equaons n ems of he acousc ehe pessue o flux n he bounday nodes. The fnal sysem s solved by any numecal echnque afe applyng he bounday condons. Such a pocedue canno be appled decly o Eq.(6b) f a bounday-only fomulaon s equed. The doman negal can be ansfomed no he sum of bounday negals by he DRBEM (see [7]). The keypon s he appoxmaon of he gh hand sde of Eq.(6b),.e. b = b(p l (x)), by a fne sees of bass funcons fo whch a pacula soluon s avalable. I can be wen as: N+L N+L b(p l (x)) f j α j = f(x, η j )α j (8) j= whee f j s funcon of he dsance beween he feld pon x and he dual collocaon pon η j.theα j coeffcens ae unknown and hey ae deemned by collocang Eq.(8) a N + L (N on he bounday and L n he doman) abay pons. Mos applcaons concen elasosacs and elasodynamcs as well as he Posson equaon. So fa no applcaons have been poposed concenng he Helmholz equaon even f much effo has been made o deemne vaous pacula soluons [8-9]. In he pesen pape vaous appoxmang funcons f j ae compaed,.e. he well odden + along wh he +, he hn plae splne (TPS) Log and he augmened hn plae splne (ATPS) Log + α N+L+ + α N+L+ x + α N+L+3 x.the coespondng pacula soluons can be found n [6], [8-9]. On he bass of he above consdeaons, he NABIEs govenng he popagaon of fne bu modeae amplude acousc waves can be wen: c(ξ)p l (ξ)+ q (ξ, x)p l (x)dγ(x) p (ξ, x)q l (x)dγ(x) = ξ Ω (9a) Γ Γ c(ξ)p l (ξ)+ q (ξ, x)p l (x)dγ(x) p (ξ, x)q l (x)dγ(x) = ξ Ω (9b) Γ Γ c(ξ)p c (ξ)+ q (ξ, x)p c (x)dγ(x) p (ξ, x)q c (x)dγ(x) = p (ξ, x)b(p l (x))dω(x) ξ Ω (9c) Γ Γ The Eq.(9b) s ncluded n he sysem of equaons when some (le s say L) nenal pons ae consdeed o bee evaluae α j n he applcaon of he Dual Recpocy (DR) appoach o Eq.(9c). In he numecal pocedue he modfed Bessel funcons needs o be compued a he negaon pons. The accuae compuaon s vey mpoan. Two dffeen sees expansons, as suggesed n [5], ae adoped,.e. one fo small agumens and he ohe fo lage agumens. The numecal scheme s a sandad collocaon one. The bounday s dscesed no quadac, sopaamec elemens. The dscee sysem of equaons s solved by he LU decomposon. Numecal esuls In ode o demonsae he effcency of he poposed pocedue, some numecal examples ae acquaned. They all efe o he wave popagaon nsde he cylnde of adus R = depced n Fg. (a) and wh he followng paamees: ρ =, c =, β =3.5, all n compable uns. j= Ω 3

126 44 Eds: R Abascal and M H Alabad Fs of all he numecal soluon s compaed o he analycal one. In such a smple geomey, n fac, an analycal soluon can be obaned n ems of a powe sees. The compason s pefomed fo k l R = and k l R =. p = s mposed on he whole bounday. D θ C B A (a) (b) (c) Fgue : (a) Geomey. Adoped nenal pons: (b) k l R =,(c)k l R =. Fg. convey he behavo of p l, see Fg. (a), and p c, see Fg. (b), n he case of k l R =. The ageemen s excellen. Twelve quadac bounday elemens and 5 nenal pons, as shown n Fg. (b), ae suffcen o oban a elave eo of less han.5%, bu mus be undelned ha 9 nenal pons would funsh an eo of less han %. 4 3 Analycal soluon BEM soluon Analycal soluon BEM soluon Pessue p l Pessue p c Dsance fom he cene (a) Dsance fom he cene (b) Fgue : Compason beween analycal and numecal esuls (a) p l and (b) p c. k l R = Fg. 3 daw he behavo of he pessue fo hghe fequency,.e. n he case k l R =. A fne bounday dscesaon and moe nenal pons ae necessay due o he lowe value of he wavelengh. In fac, 44 bounday elemens and 84 nenal pons, as depced n Fg. (c), ae necessay o convege o he analycal soluon. The vey good ageemen can be noced n hs case oo. In ode o acquan he effcency of he DR appoach, he pocedue s esed wh efeence o wo specal expessons of he souce em fo whch an analycal soluon s easly obaned. Fo boh k =. The fs equaon s: p(x)+p(x) =x () wh soluon: p A =snx +snx + x () The second dffeenal equaon s: p(x)+p(x) =4x +4x +x x +3x 3 x +x x x x 3 () 4

127 Advances n Bounday Elemen Technques IX 443 wh soluon: p A (x) =3x 3 x +x x x x 3 (3) Seveny-wo bounday elemens ae used n ode o cancel any eo souce due o he bounday dscesaon. The value of he flux coespondng o boh he analycal soluons a he bounday nodes llusaed n Fg. (a) s epoed n Table and n Table. Pessue p l Analycal soluon -4 BEM soluon Dsance fom he cene (a) Pessue p c Analycal soluon BEM soluon Dsance fom he cene (b) Fgue 3: Compason beween analycal and numecal esuls (a) p l and (b) p c. k l R = The las column of each able funshes he hghes value, among he fou pons A,..,D, of he elave eo fo each bass funcon. Concenng he soluon A, he eo esuls o be less han % fo all he bass funcons, bu s educed o.% fo he one whch bes fs,.e. AT P S. The eo s slghly hghe wh efeence o he analycal soluon A. The las column does no consde he eo a he node A whee he exac value s zeo. A B C D e(%) Analyc TPS AT P S Table : Flux a some bounday pons fo dffeen bass funcon. Fs analycal soluon. nenal pons A B C D e(%) Analyc e TPS -.99e AT P S.8e Table : Flux a some bounday pons fo dffeen bass funcon. Second analycal soluon. 5 nenal pons I mus be undelned ha + fals n funshng an accepable soluon fo he analycal soluon A. The eason s well epoed n leaue and s elaed o he ll-condoned feaue of 5

128 444 Eds: R Abascal and M H Alabad he max nvolved n Eq.(8). Fuhemoe, he convegence ceon of he adal bass funcon n n D s no suppoed by a mahemacal poof and, hence, s use s no ecommended. Conclusons A numecal mehod fo sudyng he nonlnea D popagaon of hgh-nensy acousc waves has been pesened. A peubaon heoy up o he second-ode appoxmaon has been fs appled. The nonlneay povokes a doman negal whch has been ansfomed no bounday negals by he DRBEM. The effcency of he pocedue has been vefed n a case fo whch an analycal soluon can be obaned. Dffeen bass funcons have been esed fo some specal cases and he esuls have been compaed and dscussed. Fne-amplude waves ae decly nvolved n he mechansms ha deemne a gea numbe of pacal applcaons n ndusal pocessng. Numecal models ae clealy advanageous ove ohe appoaches as gvng soluons fo a lage numbe of dffeen cases and fo any egula geomey o specal bounday condons. Thus, he mehod above pesened can be an excellen ool o deemne he acousc feld n he ndusal pocessng sysem and o asss he expemenal ess. The man advanage ove he FE appoach sands n he possbly no o dscese he doman unde analyss, bu o lm he dscesaon o he bounday only. Fuhemoe, funshes vey accuae esuls n ems of boh pessues and fluxes. Refeences [] R. T. Beye. Nonlnea acouscs. InPhyscal Acouscs Eded by W.P. Mason, Academc, New Yok, 965. [] Y. Zheng, R. G. Maev, I. Y. Solodov. Nonlnea acousc applcaons fo maeal chaacezaon: A evew. Can. J. Phys., 77:97 967, 999. [3]L.C.Wobel. The Bounday Elemen Mehod Volume : Applcaons n Themo-Fluds and Acouscs. Wley, Chchese, Wes Sussex,. [4] M. H. Alabad. The Bounday Elemen Mehod Volume : Applcaons n Solds and Sucues. Wley, Chchese, Wes Sussex,. [5] J. Domnguez. Bounday Elemens n Dynamcs. Compuaonal Mechancs Publcaons, Souhampon, Boson, 993. [6] V. Mallado, M. H. Alabad. The Dual Recpocy Bounday Elemen Mehod (DRBEM) n nonlnea acousc wave popagaon. Compue and Expemenal Smulaons n Engneeng and Scence CESES, n pn, 8. [7] P. W. Padge, C. A. Bebba, L. C. Wobel. The Dual Recpocy Bounday Elemen Mehod. Compuaonal Mechancs Publcaons, Souhampon, London & New Yok, 99. [8] S. Zhu. Pacula soluons assocaed wh he Helmholz opeaos used n DRBEM. Bound. Elem. Absacs, 4(6):3 33, 993. [9] A. H. D. Cheng. Pacula soluons of Laplacan, Helmholz-ype, and polyhamonc opeaos nvolvng hghe ode adal bass funcons. Engneeng Analyss wh Bounday Elemen, 4:53 538,. 6

129 Advances n Bounday Elemen Technques IX 445 TIME CONVOLUTED DYNAMIC KERNELS FOR 3D SATURATED POROELASTIC MEDIA WITH INCOMPRESSIBLE CONSTITUENTS M. Jyae Shaah, M. Kamalan and M.K. Jafa 3 Inenaonal Insue of Eahquake Engneeng and Sesmology (IIEES), Tehan, Ian, jyae@ees.ac. Inenaonal Insue of Eahquake Engneeng and Sesmology (IIEES), Tehan, Ian, kamalan@ees.ac. 3 Inenaonal Insue of Eahquake Engneeng and Sesmology (IIEES), Tehan, Ian, jafa@ees.ac. Keywods: bounday elemen, dynamc pooelascy, me convolued dynamc kenels Absac. Ths pape pesens he explc and smple analycal me doman convolued kenels ha appea n he dscezed govenng BIE of he hee-dmensonal well known u-p fomulaon of sauaed poous meda wh ncompessble flud and sold pacles. A fs, he coespondng bounday negal equaons ae obaned fo he govenng dffeenal equaons whch ae esablshed n ems of sold dsplacemens and flud pessue. Subsequenly, he analycal me doman convolued kenels ha appea n he BIE ae deved. Fnally, a se of numecal esuls ae pesened whch demonsae he accuaces and some salen feaues of he poposed soluons. INTRODUCTION The dynamc analyss of sauaed poous meda, s of nees n vaous felds, such as geophyscs, acouscs, sol dynamcs and many eahquake engneeng poblems. Fom a macoscopcal pon of vew, sauaed sol s a wo-phase medum consued of sold skeleon and flud. Dynamc behavous of each phase as well as ha of he whole mxue ae govened by he basc pncples of connuum mechancs. In phenomena wh medum speeds, such as eahquake poblems, s easonable o neglec he flud neal effecs, and o educe he complee dynamc govenng dffeenal equaons o he smple commonly called u-p fomulaon [,,3]. The govenng dffeenal equaons could be fuhe smplfed by neglecng he compessbly of he sold pacles and flud, whch could be easonably assumed ncompessble compaed o he sol skeleon [4,5]. The BEM s one of he mos effcen numecal mehods fo solvng wave popagaon poblems n elasc meda, because of s effcency n dealng wh sem-nfne o nfne doman poblems ha has long been ecognzed. Pedeleanu [6], Manols & Beskos [7] and Webe & Anes [8] wee among he fss who developed bounday negal equaons and fundamenal soluons govenng he dynamcs of pooelasc meda, n ems of sold skeleon dsplacemen and flud dsplacemens componens. Lae, Cheng e al. [9], Domnguez [], Chen & Dagush [] and ecenly Schanz [] developed anohe foms of bounday negal equaons and fundamenal soluons of dynamc pooelascy n ems of less ndependen vaables. Bu he algohms wee based on ansfomed doman fundamenal soluons. Obvously, me doman BEM fo modelng he ansen behavou of meda s pefeed han he ansfomed doman BEM, because fomulang he numecal pocedue enely n me doman and combnng wh he FEM, povdes he bass fo solvng nonlnea wave popagaon poblems. Pope dsplacemen and acon fundamenal soluons ae one of he key ngedens equed fo solvng wave popagaon poblems n sauaed poous meda by he BEM.Consdeng he ndependen paccal vaables of sold skeleon dsplacemen and flud pessue, Kayna [3] was he fs who pesened

130 446 Eds: R Abascal and M H Alabad appoxmae ansen 3D dsplacemen fundamenal soluons fo he specal case of sho-me. Chen [4, 5]poposed anohe appoxmae ansen D and 3D dsplacemen soluons fo he specal case of sho me as well as he geneal case, whch wee oo complcaed o be appled n BE algohms. Gam & Kamalan [6] showed ha Chen's appoxmaon could no be used n he smplfed case of u-p fomulaon. They deved anohe appoxmae ansen D dsplacemen fundamenal soluons fo he u-p fomulaon whch wee sll oo complcaed o be used n BE algohms. Lae Gam & Nguyen [5] poposed much less complcaed ansen D fundamenal soluons fo he u-p fomulaon of sauaed poous meda conssed of ncompessble consuens. Recenly Kamalan e al[7,8] deved he ansen dsplacemen and acon fundamenal soluons fo he smplfed u-p fomulaon of 3D pooelasc meda wh ncompessble consuens. Pesenaon of he analycal me doman convolued dynamc kenels ha appea n he dscezed BIE fo he u-p fomulaon of 3D sauaed poous meda wh ncompessble consuens, consues he man essence of hs pape. Some numecal esuls ae ploed o show he accuaces and some salen feaues of he poposed soluons. GOVERNING EQUATIONS The govenng equaons of dynamc pooelascy wee fs deved by Bo [9] usng he concep of vaaonal fomulaon. These equaons wee lae ecas by usng ohe heoes such as he heoy of mxues (Pevos []) and he pncples of connuum mechancs (Zenkewcz and Shom [], Gam [3], ec.). Followng he pocedue oulned by Zenkewcz and Shom [], one can we he equaons descbng, especvely, he consevaon of oal momenum, he consuve equaon of he sold skeleon, he flow consevaon fo he flud phase and he genealzed Dacy s law as follows: Equlbum equaon: () j, j f u f Consuve elaon: u ) j, ( u, j u j, p j () Flow consevaon fo flud phase: Genealzed Dacy s law: whee p w p u (3) Q,, w u mw k, f (4) m f n, ( n) s n f, Q n K f ( n) K s, K K s u epesens he dsplacemen of he sold skeleon, p denoes he excessve flud poe pessue and w epesens he aveage dsplacemens of he flud elave o he sold. j epesens he oal sess, he elasc consans and µ denoe he daned Lame consans and =k/ s he pemeably coeffcen, wh and k

131 Advances n Bounday Elemen Technques IX 447 denong he flud dynamc vscosy and he nnsc pemeably of he sold skeleon, especvely. s s he sold densy, f denoes he flud densy, epesens he densy of sold-flud mxue, m denoes he mass paamee and n s he poosy. In addon, and Q ae maeal paamees whch descbe he elave compessbly of he consuens. K s and K f denoe he bulk modulus of he sold gans and he flud whle K epesens he bulk modulus of he sold skeleon. Fnally, f and denoe he body foce and he ae of flud njecon no he meda, especvely. Omng all ems of flud acceleaon n equaon () as well as all dynamc ems n equaon (4) and elmnang w fom equaons (3) and (4), he well known govenng u-p fomulaon of a pooelasc meda wh ncompessble sold pacles and flud, n whch all coeffcens /K s, /K f as well as /Q end owads zeo, could be easly obaned n he Laplace ansfom doman as follows: ~ ~ ~ ~ ( ) ~ u u s u p f (5), jj j, j, k ~ p ~ ~, u, (6) he lde denoes he Laplace ansfom and s demonsaes he Laplace ansfom paamee. In equaons (5) and (6), he conbuons due o nal condons ae negleced. BOUNDARY INTEGRAL EQUATIONS The govenng bounday negal equaons wll be deduced sang fom he equlbum equaon and usng he well known weghed esdual mehod. Weghng equaon (5) by he dsplacemen ype funcon u', negang ove he body, usng negaons by pas wce and fnally goupng he coespondng ems ogehe, one fnds he followng expesson: ~ ~ ~ ~ ~~ ~ u u ~ d f u f u d ~~ p ~ p~ d (7) Also weghng equaon (6) by he pessue ype funcon p', negang ove he body, usng negaons by pas wce and fnally goupng he coespondng ems ogehe, one fnds he followng expesson: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ k ~ p p p p d p - p d s p - pd n,n, (8) Elmnang he common em fom equaons (8) and (9) and eunng o he me doman, one obans he govenng bounday negal equaon as follows: ~ ~ ~ ~ u u d p q - pq d f u f u d p - p d (9) whee p q k n () and q denoe, especvely, he acon veco on and he flux nomal o he bounday ().

132 By Assumng zeo body foces (f and ) and assgnng pope un Heavsde pon foces and supplemenay mpulse scala souces o f' and as ) ( ) ( ) ( ) ( ), ( 3 H x x x x f () ) ( ) ( ) ( ) ( ), ( 3 x x x x () we can oban he Somglana ype negal equaons: d F p q G d u F G x u x c j j j j j ). * * ( ) * * ( ), ( ). ( 4 4 (3) d F p q G d u F G x p x c ). * * ( ) * * ( ), ( ). ( (4) x s he souce pon; x s feld pon; c (x ) s a max of consans, depend only upon he local geomey of he bounday a x ;, =,, 3, 4. G and F ae he me doman dsplacemen and acon fundamenal soluons ae deved by Kamalan e al.[6, 7] as d d j d j j j j v H e v C e v B e A e A G,,, 3 6 s j s j s j j v H D v C e v B e A 5 4,, (5) e v H g v e G d d 8,, 7, 4 4, 4, 4 (6) e v H e v e G d d 9,,, 4 4, 4, 4 (7) e v H k G d exp, (8) m mj m j j k kj j n G G n G G F ) ( ) (,, 4, (9) m m m k k n G G n G G F ) ( ) (, 4 4, 44 4, 4 () m m j j n G k F ) (, 4 4 () m n m G k F ) (, () whee 448 Eds: R Abascal and M H Alabad

133 Advances n Bounday Elemen Technques IX 449 (3) k v d (4) v s (5) x, j,,3, x x,, (6) Aj 4 B j 4 C j (3 ) 3,, j j (3 ),, j j (,, j ) 4 (7) (8) (9) D j j 4 (3) Funcons e (,) and g (,) ae gven n Appendx. TIME DOMAIN CONVOLUTED DYNAMIC KERNELS Implemenaon of bounday negal equaons needs appoxmaon n empoal vaaons of he feld vaables. Fo empoal negaon he me axs s dvded no N equal seps and he feld vaables ae assumed o eman consan dung a me sep, so ha hey can be aken ou of he convoluon negal, hus he me negaon nvolves only he kenels and s expessed by N n N n Gjd Gj n * d (3) G n N n j G j d (3) n hen by T N : G N n j N n G j T N n (33) by smla way

134 n N n N j n N j T G G 4 4 (34) n N n N n N T G G 4 4 (35) n N n N n N T G G (36) n N n N j n N j T F F (37) n N n N n N T F F 4 4 (38) n N n N m m j n N j n T kg F 4, 4 (39) n N n N m m n N n T kg F 44, 44 (4) wh he empoal dscezaon descbed above, equaons (3) and (4) ansfoms no: N n n n n N n N n N j n N j n n n N n N n N j n N j N N j d p u F F F F q G G G G p c u c (4) BEHAVIOUR OF TRANSIENT KENRNELS AT LARGE TIME STEP One of he mpoan popees of he ansen kenels s ha a a vey lage me sep he convolued kenels should educe o he coespondng seady sae kenels. A he fs me sep N= so ha T T : G G j j lm lm j ss j j j G x x x x (4) Smlaly lm 4 G (43) lm 4 F (44) lm G k G ss (45) 45 Eds: R Abascal and M H Alabad

135 Advances n Bounday Elemen Technques IX 45 lm n m, m ss F F (46) x j lm G4 j (47) 8k lm n m m,, m F 4 j (48) 8 ( ) ss ss whee G and F ae seady sae fundamenal soluons. as can be seen, all he convolued ansen kenels educe o coespondng elasosac kenels a a vey lage me sep. NUMERICAL RESULTS A se of numecal esuls ae pesened n hs secon o demonsae he accuacy and some salen feaues of he poposed ansen convolued kenels. A sauaed sof sol wh ncompessble sold gans and poe wae was consdeed n whch he maeal popees wee defned n he mec sysem as follows: =.5 MPa, µ=8.33 MPa, = kg/m 3, =, = -7 m 4 /Ns. The pon foce (o flud souce) s appled a he coodnae (,,) a me = and he eceve s locaed a coodnae (.,.,.3). Fgues -4 depc he pesened analycal closed fom kenels of G, G 4, G 4, andg 44 componens especvely. As can be seen, dynamc kenels gadually move owad sac kenels a a vey lage me sep. I s also neesng o noe he aval mes of he pessue (vp=), dffusve (vd=7.3 m/s) and shea (vs=6.7 m/s) waves, whch could be deeced by sudden changes appeang n he dynamc kenels. The noable nal values of he pessue componens G 4 and G 44 dffe fom zeo, because he pessue wave wh s wave popagaon velocy of nfny aves mmedaely and affecs he meda's esponse. CONCLUSION Bounday negal equaon, empoal dscezaon of BIE and analycal closed fom expessons fo ansen dynamc kenels ae pesened fo he well known u-p fomulaon of 3D sauaed poous meda wh ncompessble consuens, n ems of he paccal vaables of sold skeleon dsplacemen and flud pessue. A se of numecal esuls ae pesened ha demonsae some salen feaues of he dynamc kenels. The deved kenels could be smply mplemened n me doman BEM fo modelng he ansen behavou of sauaed poous meda and povdes he bass o develop moe effecve numecal hybd BE/FE mehods fo solvng 3D nonlnea wave popagaon poblems n he nea fuue.

136 45 Eds: R Abascal and M H Alabad..5 KERNEL G (µm) Tme (ms) Dynamc Sac Fgue. dynamc and sac kenels of G a (.,.,.3) KERNEL G 4(pa) Fgue. dynamc and sac kenels of Tme (ms) Dynamc Sac G 4 a (.,.,.3) Dynamc Sac KERNEL G 4(m) Tme (ms) Fgue 3. dynamc and sac kenels of G 4 a (.,.,.3)

137 Tme (ms) KERNEL G 44(Mpa) Dynamc Sac Fgue 4. dynamc and sac kenels of 44 G a (.,.,.3) APPENDIX: FUNCTIONS e, AND, g d g e v d, exp, d g v e v d d,, d g v e v d d,, 3 g v g v e d d,,, 3 4 e exp, 5 d g e v d, exp, 6 d g e v d, exp, 7 d d d d v v I v v g exp, exp, d v I g Advances n Bounday Elemen Technques IX 453

138 454 Eds: R Abascal and M H Alabad REFERENCES. Zenkewcz OC and Shom T. Dynamc behavo of sauaed poous meda, he genealzed Bo fomulaon and s numecal soluon. In. J. Nume. Anal. Mehods Geomech 984; 8: Pevos JH. Dynamcs of poous meda. Geoechncal Modelng And Applcaons. S M Sayed, ed. Gulf Publshng Company 987; Gam B. A smplfed fne elemen analyss of wave-nduced effecve sesses and poe pessues n pemeable sea beds. Geoechnque 989; 4: Gam B and Nguyen KV. Tme D fundamenal soluons fo sauaed poous meda wh ncompessble flud. Commun. Nume. Meh. Engng. 4; (3): Schanz M., Pyl D., Dynamc fundamenal soluons fo compessble and ncompessble modeled pooelasc connua, Inenaonal Jounal of Solds and Sucues, 4: , Pedeleanu M., Developmen of Bounday Elemen Mehod o Dynamc Poblems fo Poous Meda, Appl. Mah. Modellng, 8: , Manols G.D. & Beskos D.E., Inegal Fomulaon and Fundamenal Soluons of Dynamc Pooelascy and Themoelascy, Aca Mechanca, 76: 89-4, Webe T.H. and Anes H., A Tme Doman Inegal Fomulaon of Dynamc Pooelascy, Aca Mechanca 9: 5-37, Cheng A.H.D. and Badmus T., Inegal Equaon fo Dynamc Pooelascy n Fequency Doman wh BEM Soluon, Jounal of Engneeng Mechancs-ASCE, 7(5): 36-57, 99.. Domnguez J., Bounday Elemen Apoach fo Dynamc Pooelascy Poblems, J. of Nume. Meh. Engng, 35: 37-34, 99.. Dagush G F and Banejee P K. A me doman bounday elemen mehod fo pooelascy. In. J. Nume. Mehods In Eng. 989; 8: Schanz, M., pooelasodynamc Bounday Elemen Fomulaon,Wave Popagaon n Vscoelasc and Pooelasc Connua: A Bounday Elemen Appoach, Spnge-Velag publcaon, 77-98,. 3. Kayna A.M., Tansen geens funcons of flud-sauaed poous meda, Compues & Sucues, 44: 9-7, Chen J. Tme Doman Fundamenal Soluon To Bo s Equaons Of Dynamc Pooelascy. PaI: Two-Dmensonal Soluon. In. J. of Solds & Sucues 994; 3(): Chen J. Tme Doman Fundamenal Soluon To Bo s Equaons Of Dynamc Pooelascy. PaII: Thee-Dmensonal Soluon. In. J. of Solds & Sucues 994; 3(): Gam B., Kamalan M., On he Fundamenal Soluon of Dynamc Pooelasc Bounday Inegal Equaons n Tme Doman, ASCE; The Inenaonal Jounal of Geomechancs, (4): ,. 7. Kamalan M., Gam B. and Jyaee Shaah M., Tme doman 3D fundamenal soluons fo sauaed poelasc meda wh ncompessble consuens, Poc. of he 7 h Inenaonal Confeence on Bounday Elemen Technques (BeTeq6-Pas), Kamalan M. and Jyaee Shaah M., Tacon ansen fundamenal soluons fo 3D sauaed poelasc meda wh ncompessble consuens, Poc. of he 4 h Inenaonal Confeence on Eahquake Geoechncal Engneeng (4ICEGE-Thessalonk), Bo M.A., Theoy of popagaon of elasc waves n a flud-sauaed poous sold: I. Low-fequency ange, II. hghe fequency ange, J. Acous. Soc. Am., 8: 68-9, 956.

139 Advances n Bounday Elemen Technques IX 455 Sably analyss of compose plaes by he bounday elemen mehod E. L. Albuqueque,P.M.Baz, and M. H. Alabad 3 Faculy of Mechancal Engneeng, Sae Unvesy of Campnas Campnas, Bazl, edelma@fem.uncamp.b Cuenly a Impeal College London as an academc vso. Depamen of Aeonaucs, Impeal College London London, UK, p.m.baz@mpeal.ac.uk 3 Depamen of Aeonaucs, Impeal College London London, UK, m.h.alabad@mpeal.ac.uk Keywods: Sably of sucues, lnea bucklng, compose plaes, adal negaon mehod. Absac. Ths pape pesens a bounday elemen fomulaon fo he sably analyss of symmec lamnae compose plaes whee only he bounday s dscezed. Body foces ae wen as a sum of appoxmaon funcons mulpled by coeffcens. Doman negals whch ase n he fomulaon ae ansfomed no bounday negals by he adal negaon mehod. Plae bucklng equaons ae wen as a sandad egenvalue poblem. The accuacy of he poposed fomulaon s assessed by compason wh esuls fom leaue. Bucklng coeffcens and bucklng modes ae obaned usng hs fomulaon. Inoducon Demand by an accuae sably analyss of ansoopc maeals has ncease wh he nceasng use of compose maeals n engneeng pojecs. In geneal, composes panels ae vey lgh sucues ha pesen hgh sffness and sengh. Howeve, due o he slendeness, bucklng s one of he man concen dung he desgn. The bounday elemen mehod (BEM) has povded a poweful soluon o he feld of plae bucklng. Syngellaks and Elzen [] pesened a bounday elemen soluon of he plae bucklng based on Kchhoff heoy unde any combnaon of loadngs and suppo condons. Neanzak and Kasdelaks [] developed a bounday elemen mehod fo bucklng analyss of plaes wh vaable hckness. Elasc bucklng analyss of plaes usng bounday elemens can also be found n [3]. Bucklng analyss of shea defomable soopc plaes was pesened n [4]. To he bes of auho s knowledge, he only wok ha pesens a bounday elemen fomulaon appled o non-soopc plaes s due o [5] who pesened an ohoopc fomulaon wh a doman dscezaon. In hs pape, a bounday elemen fomulaon fo he sably analyss of geneal ansoopc plaes wh no doman dscezaon s pesened. Classcal plae bendng and plane elascy fomulaons ae used and he doman negals due o body foces ae ansfomed no bounday negals usng he adal negaon mehod. Numecal esuls ae pesened o assess he accuacy of he mehod. Bucklng coeffcens compued usng he poposed fomulaon ae n good ageemen wh esuls avalable n leaue. Bounday negal equaons In he absence of body foces, he govenng equaon of he ansoopc hn plae bucklng s gven by: N j,j =, ()

140 456 Eds: R Abascal and M H Alabad D u 3, +4D 6 u 3, +(D + D 66 )u 3, +4D 6 u 3, + D u 3, = N j u 3,j, () whee, j, k =, ; u k s he dsplacemen n decons x and x, u 3 sands fo he dsplacemen n he nomal decon of he plae suface; N j ae he n-plane sess componens, D, D, D 66, D, D 6,andD 6 ae he ansoopc hn plae sffness consans. The bounday negal equaon fo n-plane dsplacemens, obaned by applyng ecpocy and Geen heoems a equaon (), s gven by [6]: c j u j (Q)+ k (Q, P )u k(p )dγ(p )= u k (Q, P ) k(p )dγ(p ) (3) Γ Γ whee = N j n j s he acon n he bounday of he plae n he plane x x,andn j s he nomal a he bounday pon; P s he feld pon; Q s he souce pon; and asesks denoe fundamenal soluons. The ansoopc plane elascy fundamenal soluons can be found, fo example, n [7]. The consan c j s noduced n ode o ake no accoun he possbly ha he pon Q can be placed n he doman, on he bounday, o ousde he doman. The n-plane sess esulans a a pon Q Ω ae wen as: c k N kj (Q)+ Skj(Q, P )u k (P )dγ(p )= Djk(Q, P ) k (P )dγ(p ) (4) Γ Γ whee D kj and S kj ae lnea combnaons of he plane-elascy fundamenal soluons. The negal equaon fo he plae bucklng fomulaon, obaned by applyng ecpocy and Geen heoems a equaon (), s gven by: [ Ku 3 (Q)+ Vn (Q, P )w(p ) m n(q, P ) w(p ) ] N c dγ(p )+ Rc Γ n (Q, P )u 3c (P ) = N c [ ] = R c (P )u 3 c (Q, P )+ V n (P )u 3 (Q, P ) m n(p ) u 3 = Γ n (Q, P ) dγ(p ) [ ( ) ] +λ N j u 3,j dω+ u 3u 3, u 3 u 3, dγ, (5) Ω Γ whee () n s he devave n he decon of he ouwad veco n ha s nomal o he bounday Γ; m n and V n ae, especvely, he nomal bendng momen and he Kchhoff equvalen shea foce on he bounday Γ; R c s he hn-plae eacon of cones; u 3 c s he ansvese dsplacemen of cones; λ s he ccal load faco; he consan K s noduced n ode o ake no accoun he possbly ha he pon Q can be placed n he doman, on he bounday, o ousde he doman. As n he pevous equaon, an asesk denoes a fundamenal soluon. Fundamenal soluons fo ansoopc hn plaes can be found, fo example, n [8]. A second negal equaon s necessay n ode o oban he hn plae bucklng bounday elemen fomulaon. Ths equaon s gven by: K u [ 3 V m Γ (Q)+ n m (Q, P )w(p ) M n m [ N c = R c (P ) u 3 c m (Q, P )+ = Γ [ u ( 3,j +λ u 3 N j Ω m Γ dω+ u 3 ] ) (Q, P ) w(p n V n (P ) u 3 (Q, P ) m u 3, m u 3, u 3 m N c dγ(p )+ = m n (P ) u 3 n m (Q, P ) ) ] dγ R c m (Q, P )u 3 c (P ) ] dγ(p ), (6)

141 Advances n Bounday Elemen Technques IX 457 whee () m s he devave n he decon of he ouwad veco m ha s nomal o he bounday ΓahesouceponQ. As can be seen n equaons (5) and (6), doman negals ase n he fomulaon owng o he conbuon of n-plane sesses o he ou of plane decon. In ode o ansfom hese negals no bounday negals, consde ha a body foce b s appoxmaed ove he doman Ω as a sum of M poducs beween appoxmaon funcons f m and unknown coeffcens γ m,has: The appoxmaon funcon used n hs wok s: b(p ) M = γ m f m. (7) m= f m =+R, (8) Equaon (7) can be wen n a max fom, consdeng all bounday and doman souce pons, as: Thus, γ can be compued as: b = Fγ (9) γ = F b () Body foces of negal equaons (5) and (6) depend on dsplacemens. So, usng equaon () and followng he pocedue pesened by Albuqueque e al. [9], doman negals ha come fom hese body foces can be ansfomed no bounday negals. Max Equaons Afe he dscezaon of equaons (5) and (6) no bounday elemens and collocaon of he souce pons n all bounday nodes, a lnea sysem s geneaed. I s woh noce ha he only loads consdeed n he lnea bucklng equaons ae ha elaed o he n-plane sess N j and acons ha ae mulpled by he ccal load faco λ. Ths means ha all he known values of u 3, u 3 / n, M n, V n, w c, R c (bounday condons) ae se o zeo. Dvdng he bounday no Γ and Γ (Fgue ), hs lnea sysem can be wen as: Γ : u 3 = u 3 n = Ω Γ : V n = M n = Fgue : Doman wh consaned and fee degees of feedom. [ ]{ } [ ]{ } [ ]{ } H H w G G V M M = λ w, () H H w G G V M M w whee Γ sands fo sands fo he pa of he bounday whee dsplacemens and oaons ae zeo and Γ sands fo he pa of he bounday whee bendng momen and acons ae zeo. Indces and sand fo boundaes Γ and Γ, especvely. Maces H, G, andm ae nfluence maces of he bounday elemen mehod due o negal ems of equaons (5) and (6).

142 458 Eds: R Abascal and M H Alabad As w = and V =, equaon () can be wen as: H w G V = λm w, H w G V = λm w () o Ĥw = λ ˆMw, (3) whee Ĥ and ˆM ae gven by: Ĥ = H G G H, ˆM = M G G M. (4) The max equaon (3) can be ewen as an egen veco poblem whee Aw = λ w, (5) A = Ĥ ˆM. (6) Povded ha A s non-symmec, egenvalues and egenvecos of equaon (5) can be found usng sandad numecal pocedues fo non symmec maces. Numecal esuls The numecal esuls ae pesened n ems of he dmensonless paamee K c whch s gven by: K c = N ca D (7) whee N c s he ccal load (N c = λ he appled load) and a s he edge lengh of he squae plae. Consde a squae gaphe/epoxy plae unde dffeen bounday condons. The hckness of he plae s h =. m. The maeal popees ae: elasc modul E = 8 GPa and E =.3 GPa, Posson ao ν =.8, and shea modulus G =7.7 GPa. The mesh used has 8 quadac dsconnuous bounday elemens of he same lengh (7 pe edge) and 49 (7 7) unfomly dsbued nenal pons. The plae s unde unfomly unaxal compesson and he ccal load paamee K c s compued consdeng all edges smply-suppoed (SSSS), all edges clamped (CCCC), and wo edges clamped and wo edges smply suppoed (CSCS). In he las case, he wo edges whee he load s appled ae smply suppoed and he wo emanng edges ae clamped. The esuls ae shown n Table ogehe wh esuls obaned by [5] usng a bounday elemen fomulaon wh doman dscezaon and he analycal soluon pesened by []. As can be seen, hee s a good ageemen beween he esuls obaned n hs wok and hose pesened n leaue. Ccal bucklng modes fo each case ae shown n fgues, 3, and 4. Conclusons Ths pape pesened a bounday elemen fomulaon fo he sably analyss of symmec lamnaed compose plaes whee doman negals ae ansfomed no bounday negals by he adal negaon mehod. As he adal negaon mehod doesn demand pacula soluons, s ease o mplemen han he dual ecpocy bounday elemen mehod. Resuls obaned wh he poposed fomulaon ae n good ageemen wh esuls pesened n leaue.

143 Advances n Bounday Elemen Technques IX 459 Fgue : Ccal bucklng mode of cases, 3 and 5. Fgue 3: Ccal bucklng mode of case. Fgue 4: Ccal bucklng mode of cases 4 and 6.

144 46 Eds: R Abascal and M H Alabad Table : Ccal load paamee K c fo a gaphe/epoxy plae wh dffeen bounday condons. Case Bounday condons Loadngs Ths wok Refeence [5] Refeence [] SSSS N SSSS N CCCC N CCCC N CSCS N CSCS N Acknowledgmen The fs auho would lke o hank he CNPq (The Naonal Councl fo Scenfc and Technologcal Developmen, Bazl), AFOSR (A Foce Offce of Scenfc Reseach, USA), and FAPESP (he Sae of São Paulo Reseach Foundaon, Bazl) fo fnancal suppo fo hs wok. Refeences [] S. Syngellaks and E. Elzen. Plae bucklng loads by he bounday elemen mehod. Inenaonal Jounal fo Numecal Mehods n Engneeng, 37: , 994. [] M. S. Neanzak and J. T. Kaskadels. Bucklng of plaes wh vaable hckness an analog equaon soluon. Engneeng Analyss wh Bounday Elemen, 8:49 54, 996. [3] J. Ln, R. C. Duffeld, and H. Shh. Bucklng analyss of elasc plaes by bounday elemen mehod. Engneeng Analyss wh Bounday Elemen, 3:3 37, 999. [4] J. Pubolaksono and M. H. Alabad. Bucklng analyss of shea defomable plaes by bounday elemen mehod. Inenaonal Jounal fo Numecal Mehods n Engneeng, 6: , 5. [5] G. Sh. Flexual vbaon and bucklng analyss of ohoopc plaes by he bounday elemen mehod. J. of Solds and Sucues, 6:35 37, 99. [6] M. H. Alabad. Bounday elemen mehod, he applcaon n solds and sucues. John Wley and Sons Ld, New Yok,. [7] P. Solleo and M. H. Alabad. Facue mechancs analyss of ansoopc plaes by he bounday elemen mehod. In. J. of Facue, 64:69 84, 993. [8] E. L. Albuqueque, P. Solleo, W. Venun, and M. H. Alabad. Bounday elemen analyss of ansoopc kchhoff plaes. Inenaonal Jounal of Solds and Sucues, 43:49 446, 6. [9] E. L. Albuqueque, P. Solleo, and W. P. Pava. The adal negaon mehod appled o dynamc poblems of ansoopc plaes. Communcaons n Numecal Mehods n Engneeng, 3:85 88, 7. [] S. G. Lekhnsk. Ansoopc plaes. Godon and Beach, New Yok, 968.

145 Advances n Bounday Elemen Technques IX 46 BEM model of mode I cack popagaon along a weak neface appled o he nelamna facue oughness es of composes L. Távaa, V. Man, E. Gacan, J. Cañas, F. País Gupo de Elascdad y Ressenca de Maeales, Escuela Técnca Supeo de Ingeneos, Unvesdad de Sevlla, Camno de los Descubmenos s/n, 49 Sevlla, España lavaa@es.us.es, manc@es.us.es, gacan@es.us.es, canas@es.us.es, pas@es.us.es Keywods: composes, nelamna facue, weak neface, lnea elasc-ble law, BEM Absac. A numecal sudy of damage popagaon n compose lamnaes s pesened. Inelamna facue oughness (G Ic ) es of wo undeconal cabon fbe lamnaes bonded by an adhesve laye s suded. Dsplacemen conol s used n he numecal es smulaon o ensue sable cack popagaon. The adhesve laye s modelled n he D Bounday Elemen Mehod (BEM) code developed as a weak neface by means of a connuous dsbuon of spngs govened by a lnea elasc-ble law. In hs law, he nomal sesses acoss he neface ae popoonal o he elave nomal dsplacemens (openng) up o a cean maxmum sess value. I s shown ha hs appoach povdes a good epesenaon of he acual adhesve behavou. An mpoan feaue of he BEM appoach developed s ha he paamees govenng he spngs ae ndependen of he bounday elemen mesh,.e. dsances beween spngs and elemen ypes used. Ths fac allows us o pefom an easy mesh efnemen f equed. I s shown ha he local popees of he numecal soluon obaned nea he cack p agee wh he pedcons obaned wh he weak neface heoy. The pesen model pems he sudy of boh cack popagaon and cack naon. An excellen ageemen s obseved beween he load dsplacemen dagams obaned n he BEM analyss and n he laboaoy ess. The compuaonal pocedue developed can be used o esmae he maxmum allowed load of a sucue ncludng smla adhesve bonded jons of lamnaes. Inoducon Tadonally, he mehods ha smulaed cack popagaon wee based on Lnea Elasc Facue Mechancs (LEFM) assumng he pesence of a cack, whch made dffcul he sudy of damage and/o cack naon occung n he fs sep of facue pocess. Recenly, ohe models have been nensvely developed as cohesve cack model whch assumes hypoheses dffeen o hose adoped n LEFM avodng he pesence of a sess sngulay a he cack p. These models ae suable o sudy boh cack naon and cack popagaon, and also o esmae he facue enegy and he maxmum allowable load of a sucue. In many paccal suaons, he behavo of adhesve jons can be descbed modelng he hn adhesve laye as a connuum spng dsbuon [] wh an appopae sffness paamee. Ths neface model s usually called weak neface o elasc neface [,3]. In he pesen wok lnea elasc ble consuve law of hese spngs s adoped n ode o allow an easy modelng of cack popagaon along a weak neface. In he pesen wok he above descbed weak neface behavo has been mplemened n a D BEM code [4,5], whose ognal veson allowed modelng of plane elasc poblems, ncludng seveal lnea elasc ansoopc solds wh song nefaces o conac zones beween hem. The new feaue ncopoaed o hs code s he ncopoaon of he possbly of defnng weak nefaces beween he elasc solds whee equed. Anohe feaue of he code s ha he equlbum and compably condons, along conac zones and song o weak nefaces, ae mposed usng a weak fomulaon allowng an easy use of non-confomng dscezaons [4,5,6]. The undesandng of he adhesve laye behavo s vey mpoan n he qualy evaluaon of hs knd of jons, and paculaly n deemnng he paamees ha chaaceze s essance o facue and falue. These paamees can hen be used n desgn and qualy conol of he poducve pocess. The qualy of an adhesve jon beween compose lamnaes s usually evaluaed by an nelamna facue es, whee an esmaon of he ccal nelamna facue enegy (G Ic ) s obaned. An exensve expemenal sudy and a numecal sudy by Fne Elemen Mehod of hs es and of dffeen adhesves wee ecenly caed ou by he pesen auhos and he co-wokes [7,8].

146 46 Eds: R Abascal and M H Alabad Weak neface Accodng o Lenc and co-wokes [,3], a weak neface s consdeed as a model of a hn lnea elasc adhesve laye beween wo sufaces. In he pesen wok, adhesve damage and/o upue ae modeled as a fee sepaaon of boh sufaces. Thus, he spngs ha smulae an adhesve laye ae govened by he followng lnea elasc-ble law, shown also n Fg. : k f c, and f c () whee s he nomal sess n a spng, s he elave openng of he exemes of he spng (sepaaon beween sufaces), k s a sffness paamee, c and c ae, especvely, he ccal nomal sess and he ccal elave nomal dsplacemen leadng o he spng upue. Facue mechancs s ndecly nvolved hough he aea unde he lnea law lne n Fg. gven by G Ic value, G Ic. cc c G Ic c Fgue. Lnea elasc-ble law of a spng. Accodng o he weak neface heoy [,3], neface acons ae bounded a he p of a cack suaed along a weak neface, wheeas hese acons ae sngula (unbounded) a he p of an neface cack suaed along a pefec neface (called also song neface, whee no elave dsplacemens of bonded sufaces ae allowed). Thus, dung cack gowh along a weak neface hese acons ae kep bounded. I appeas ha local nomal acons n he zone close o he neface cack p follow he law []: ln( ) () whee and ae consans and x = a, whee x s he dsance fom he cack p o a pon (n he bonded pa of he neface) whee hese acons ae evaluaed and a s a chaacesc lengh, usually he cack lengh o semlengh. Mode I weak neface mplemenaon n he D BEM code Incemenal fomulaon. The numecal soluon of he non-lnea poblem fomulaed s based on a gadual applcaon, by means of a load faco,, of he loads and dsplacemens mposed. The soluon pocedue s gven by a sees of lneal sages, load seps. A he begnnng of each load sep an acual adhesvely bonded zone s defned, whch defnes he acual lnea sysem of equaons. By solvng hs sysem he coespondng elasc soluon s obaned. Ths soluon fulflls all he condons of he weak neface fomulaon up o a cean maxmum value of he load faco assocaed o hs load sep. A fuhe ncemen of he load faco leads o upue of some spngs. Thus, he soluon of he poblem wll be dvded no a numbe M (a po unknown) of load seps whee he values of he poblem vaables vay lnealy: ( x, ) m( x ) (3) wh m- m, m =,..., M, and =, and whee ( x, ) s he value of any poblem vaable a a pon x afe facon of load s appled, m( x ) beng he vaable value obaned n he soluon of he lnea sysem coespondng o he m-h load sep.

147 Advances n Bounday Elemen Technques IX 463

148 464 Eds: R Abascal and M H Alabad Ths pocedue can be epeaed as many mes as necessay o each he equlbum afe he whole load s appled. Neveheless, changng he condons node by node can make he cack popagaon o be vey smooh (especally fo fne meshes), n oppose o he expemenal evdence fo some ndusal adhesves ha show cack gowng by small jumps. Tha s why fo a specfc case of adhesve, lke ha smulaed n he pesen wok, he end of a load sep can be defned by a suaon whee a fxed numbe of consecuve nodes (numbe s chosen n he pesen sudy) do no fulfll he condon (7). Laboaoy es Tes descpon. The es used n he aeonaucal ndusy o evaluae he nelamna facue oughness n compose-compose jons s pefomed followng AITM.5 [9] and/o I+D-E 9 [] sandads. The specmen used s he Double Canleve Beam (DCB) shown n Fg. 3(a). The DCB specmen s fomed by wo lamnaes joned by a hn adhesve laye. The lamnaes ae pocessed accodng o EN 565 sandad, and he specmens ae cu afe he panel has been cued. The specmen s fxed o he gps of he unvesal esng machne hough small abs bonded o lamnaes as shown n Fg. 3(b). Dung cack popagaon he load (P) and he dsplacemen (d) of he wedge gps ae connuously egseed. w P º L = 5 ± 5 mm L = 5 ± mm w = 5. ±. mm = 3. ±. mm L l d Fgue 3. (a) Scheme of he DCB specmen, (b) Tes confguaon. Adhesve ype. In a sudy of expemenal esuls obaned fom G Ic ess fo dffeen knds of adhesve [7], was obseved ha he adhesves FM 3K.5 and EA 9695 K.5 pesen falls n he expemenal load dsplacemen cuve. Ths behavo was explaned by he pesence of a polyese suppo n hese adhesves. Evaluaon of he adhesve model paamees. As he paamees of he adhesve model adoped hee ae a po unknown, hey ae adjused by fng he expemenal and numecal load-dsplacemen cuves. Expemenal esuls povde esmaons of G Ic values, he cack lengh fo some load values and he load dsplacemen cuves. Wh hese daa, and usng equaon (9), a fs esmaon of he ccal dsplacemen ( c ) and he slope k s obaned. Afe a compason beween he numecal and expemenal esuls hese values can be adjused bee. G Ic k c (9) Ohe al and eo mehods o oban an esmaon of K adh value ae pesened n [7], he esmaed values of K adh obaned heen and of k obaned hee ae n a good ageemen accodng o (4). Numecal Resuls In he pesen numecal sudy, a D model has been solved usng he BEM code descbed above, whee he plane san and lnea elasc behavo hypoheses have been assumed. The lamnae consdeed s a 855/AS4 cabon fbe epoxy compose (º ples), wh he followng ohoopc popees: E x =35GPa, E y =GPa, E z =GPa, G xy =5GPa, G xz =5GPa, xy =.3, yz =.4 and xz =.3. The adhesve used s EA 9695

149 Advances n Bounday Elemen Technques IX 465 K.5, an epoxy adhesve wh a polyese suppo. The esmaed popees of he adhesve spng model ae: k=54gpa/m and c =54MPa. A maxmum dsplacemen of 5 mm was pogessvely appled n he decon nomal o he specmen bounday a a 5 mm dsance fom he specmen exeme whee he nal cack s suaed. The nomal sesses along he bonded zone obaned n he las load sep, ae shown n Fg. 4(a). The nal longude of he adhesve laye 5 mm s dscezed by 468 o 936 spngs placed beween he nodes of he confomng bounday elemen meshes on A and B sdes of he weak neface. Nomal sess (MPa) nodes 936 nodes Dsance o he cack p (m) / c Fgue 4. (a) Nomal sesses nea he cack p, (b) Fng of he nomalzed local sess soluon by an analyc expesson (x = a) (). I s noewohy ha he local sess soluon nea he cack p agees vey well wh he pedcons of he weak neface heoy (). In Fg. 4(b), he nomalzed sesses / c epesened as a funcon of (he nal adhesve laye beng modeled by 468 spngs) ae compaed wh he cuve of expesson (), obaned fom () by applyng he leas squae mehod ln. () c Compason beween he expemenal and numecal load - dsplacemen dagams. As can be obseved n Fg. 5, he numecal esuls obaned povde a good appoxmaon of he expemenal esuls. Theefoe, he use of he weak neface fomulaon seems o be a pomsng appoach o model compose adhesve jons. X Y a P numecal expemenal Load P (N) Dsplacemen d (m) Fgue 5. Compason beween he expemenal and numecal load - dsplacemen dagams, and a deal of he polyese suppo of he adhesve used.