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Title Development of the Boundary Element Method Utilizing the DiscreteIntegral Method

Author(s) Huang, Chao

Citation Nagasaki University (長崎大学) (2001-03-30)

Issue Date 2001-03-30

URL http://hdl.handle.net/10069/22075

Right

NAOSITE: Nagasaki University's Academic Output SITE

http://naosite.lb.nagasaki-u.ac.jp

DevelopmengoftheBoggndaryEllementMethod

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xxx

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ew

cmfftererEen

GENEffcakrwRODWCTRON

fieR'wnODWewXONi-

OnceamathefxgaticalmodegofanykipmdofepmgineeriRgprobgemhasbeenconstructed,

efforksaredgrectedtowardsobtainingasoiwtionofequations.TheregioRisoftenavery

complgcatedshapeandcomposedofzoncsofdifferentmaterialswithconipiexproperties.

Sincetftxegeverx}ingeq"atioRhasdifferentforrwaitCkvariousconditioftsarespecifgredlon

tke bouadarges,it"ffnericaEmethodsbecometheeffectivegneansofobtainingadequately

preeisea#ddetailedres"lts.

Tkeboundexyeeementmethod(BEM)[1--1]-lg-2]isawellestablishedtechnique

fox theaitagysisofengiiteeringpifobEems,papticeeEaflytheseinvolvinglinearanalysis.The

' maiRadvaRtageofBEMeverotherteehRiqueisthepossibilityofdisefetizingongythe

bouitdafyoftheprobleminsteadofthewhoiedomain,asffequiredfoxinstaneebyFinite

ElemeRts.AnotherattractivefeatureofBEMteehniquesishighaccaxracy.gfdoffnaiRinte-

gfagsoccurductobodyforces,heatsource,noniimearityoftime-depeRdenteffects,genef-

aggy, theseiRtegfalsareeagrgriedoug"singeelRelewteRtsEi-3].

Encgassieaibeifxtdaryegemaentgxtethod,althoughthevobumaeRRtegralsafeperformed

by discweaizillgthedomainintoceRRsandnofurtker"itknownsareneeded,itiswswaZlyto

feqwtffeaiwtegwationoverthewhogebody.inthisease,theeaassieagboaxwadaryelement

gmieehodbecegnesratherexmbefseaxkeandinotongyEosestkeadvantageofabo"ndaryeie-

mentscheme,betaUsoneedsmofecowtputefspaeeandtitwe.Severalmethodshavebeen

1

deveRopedtotakedomainintegragstotheboufldaryiftordereoeSiminatetkeneedfor

interpmagceg]s.X'ft}eseifKietkitodswiggaEsobedeseribedfintkxeseceion1.:2.

Tkemaaimeobjectiveoftkisthesisistoestablishaptewapproach,devefiopedbymy

advgsorProfessorKisu,inwhichdiseribxge.ioneftkedeetafunctionisused.Xnthisap-

proaehthefuitctioitisapproxignatedwtilizingdistributionofthedekafuptctioitandthe

disereteimtegragmaethodEi-4]•-[i-Siisdevelopedaxsifigthisfunctionaiapproxirr}ation.

Thediscreteintegragmethodisexnployedinthedomainintegral,generalintegral,bend-

ingprobgemofbeamandsteady-stateheatconductioitpffoblem.

g.2BAÅëKGRouNDeFRESEARÅëmeSONTREMMENTOWDOMAKNKNMG}RAkSgNBOeLJNmaewMEMENTMETHOD

BEMexpressesthesolgtionofaboundaryvalueprobiemintermsofanintegrai

equation,whichisthensogvednufnericalXy.Formanyengineeringproblemsthisequation

eowtainsbo"ndaryintegraisonly.Thismeansthatoniytheboundaryneedstobediscretized.

However,fersomeproblems,e.g.describedbythePoissontypedifferewtiagequation,

bothdetwainandbo"ndaryintegralsappearintheintegralequatiofi.Akheughthedornain

integragdoesnotintrodnceanynewunknownsitrequiresadditionalefforttogenerate

internaieelisneededfornugnericagintegratiormoverawhoiedomaifi.Thisapproachbe-

eomesptumericalgyineffieient.

SeveralmethodshavebeenproposedtoconvertdomainintegraisoccurringinBEM

intoequivalentboundaryform.ThefirstpublieationonthissubjectwasduetoT.A.Cruse

El-6]ini97S.rgrhearticlewasconcernedwiththeproblemofeRasticfractwremechanics

wkiehbodyfofeeexistsanddeveiopedatechnique,eagEedGaierkin'ff7ensoxMethod.When

bodyfefeeisaconstantandiineawgoads,'agoodRumemiicalresultswereobtained.Etwas

foeendehaathevariationoftheconstantinthef"ndaieckenta1solutionaffeetsthenumemical

valuesgivenbyGaaerkiRTensowMethod.Later,differenta"thersif1-3],[a--7]-\(1-a3]em-

pgoyedshismethodinegastieprobEeirxts.

2

En1981Kirchhoff'stransfosmatioRwaspresentedbyYu.N.AkkuratovV.N.

Mikhailov\(g-a4]andR.Bialeeki,A.Nowak[i-gS].ThesteadystatepmonEinearheatcon-

duetionequatienwithtemperatgredependeneeofgherrnageond"ctivitywasdescribed.

TheaxseofKirchhoff'stransformationwasshowntobeeonvenientwhenthethermai

condaxctivitydependencesehetempefature.ThistransformationtransferstheRoitliRearity

frotwdifferentiagequationandbouptdawyconditionsonRyintobo"ndaryeoRditionsofthe

thifdkiRd.Whentheprescribedboundaryconditionsareoftheffirstandsecondkindoniy,

theprobiembecomeslinearandcanbesolvedbyusingtheclassicaXBEMformulation.

Generaily,asystemofnonlineafequationshastobesoived.Thisappfoaehpresentssev-

efaladvantagesoverwtethodswhereeffectsofnonlinearityaretreatedasadditionaaheat

seaxrces.Heagradiationendtemperaturedependentkeattransfefcoeffacientwerealsodis-

cussed.Ktwasfo"ndthatdifectiterationproeeduresdivergewhenheatradiationpiaysa

eonsiderabieifoge.AvariantoftheiRcrewtentagmethodtosogvesuchproblemaLswasdevel--

eped.

Oneyearaatter,C.A.BrebbiaandD.NardiniE1-i6]proposedamoregeneraltech-

niqaxewhichtakesdomai"iRtegralstotheboundaryaRdisutowcailedduaireciprocity

rcethod(DRM).TheyfirstappEiedittothestudyofdymallkicalprobXems.Thenoniinear

termoftheeguationcanbeexpressedusimgaseriesofapproximatingfunctions.The

approximatingfunctioRswerechosenasthefunctionofthedistancebetweenpre-speci-

fiedfErxedcoilocationpointsandafieXdpoint.'rcoredueethedomainintegralintheequa-

tioittoeguivaientboundaryintegfals,anewauxiliarynon-homogenceusILaplacianfield

isdefined.ina986L.C.Wifobel,]D.NardiniandC.A.Brebbia[1-17]appliedthisap-

pfoachtothepfoblemokransientheatcoRduction.In1993B.A.Davis,andP.J.Gfamman,

etag{1-18]ansedthisDRMfortheprobleififxsofsimglatiRgfiowandheattransfef.inthis

research,ehesoEutioRoftheequatPeenofenergyfoxfiowpxobgemaswaspresented.'rcosolve

veEoeitiesandvegocitygradientsineeitveetionaRdviscousdissipationproblems,thebound--

arytwtegralequationfocereepingfiowwerepfesentedasweM.Thenonlineartermswere

greatedwithDRMwithrandewtlydistributedinterffkalnodes.Algorithmsfoffheatgenera-

tRondiastingexothermaecurereaetRen,vaseo"sdftssipation,ÅíeRveetion,awadvisceusdissi-

3

pagioncejmbincdwithceemvectioptweredevegoped.Osemaationsinthecoitvectionprob -

gegxkswereavofidedbyimcreasipmgtheitupmkberofeliementsonthebouendary.Effeattransfer

probXemsinvogvingxnorecoffifxpaicatedfiwidflowwerecomp"tedwsingaflexistingbound -

gnX9gg,anecaxcienttechniqueforreducingdomainintegralstothebogwadarywas

devePopedby3.P.S.AzevedoftndC,A.Brebbia[X-g9].inthispaper,gheintegraRcorre-

spofidiRgtoanysourcesactingonaninternaRregioncanbereplacedbyequivalentsources

dgstrgb"tedontheboufldaryofthatregionbyusingparticularsoiutions.Theeffectof

theseequivaientsourcesistakentotheboundarybytheusuaitypeofinfiigencecoeff• 1-

cilewtswitke"tiwtrod"cingRewaxnknewnstotheprobgems.Theapproachconsistsoftwo

differentsteps.Thefirststepistoexpressthesourcedistributionintermsofaginear

combinationoff"nctioAsforwhichparticuiarsoiutioRsareknown.Xnmanycasesthisis

straightforwardastheexpressiomsofthesourcetermsaregivenexactiyintermsof

pogyptomialsorFourierseries.ThisstepamouRtstocagcuiatingasetofequivagentpoten-

tiagancifguxso"rcedistribatioRsontheboundaryoftheregionwithsourcedistribution.

rif'hesecoRdstepistotaketheiRflueRceofthenewsetofequivaientsourcestothebound-

affY.

XRi9g9,A.3.NowakandC.A.BrebbiaEX-20]developedanewtechnique,eailed

theMultipaeReciprocityMethod(MRM),andappliedittosolvePoissonandHelmholtz

eqgations.lvthismethod,thebehaviorofasourcefunctioninsidethedornainisrepre-

sentedbyaseriesofitsderivatives,ormoreprecisely,bytheseriesofsubseq"eRthaplaeians

ofsourcefuRctioRcalculatedoRtheboundary.TheseriesofLaplaciansisdefTmedbythe

ffeeurrenceformula.MRMleadstoanexactintegralequation.Thesimplificationsare

iittfoducedasthestageofdiscretizationoftheboundaryintoboundaryelementsanddue

gotxuneationofMRMseries.WheRtheseriesconvergesitsvaauecanbecalcmplatedwith

highacegraeyandasaconsequence,resultsofMRMarealsoveryaecurate.in1991,A.

C.NevesandC.A.BffebbiaE1-2a]exxtployedinthepfoblemofelasticity.Xnthisffesearch,

ithasbeenextextdedgosolveacomapletelydifferentpreblexxx,i.e.theNaviefequations of

eRastieiey.ThegevemaingequatioRsaremuehmorecoffxipaexthanfortheeaseofthegener-

4

alkzediPegsg#ee'seqwaashegeapmdiggkaslbeewaemeeessewygedievegopgkigkfteweffdierjfeextdigtwent3g

$o' gwtiowaswwrkgevk&ifeeegEavwagkabge•geeshegageeckgagere.gwaesddiasa"wa,skere•smpgggeng•Rgeeegrggsweife

eoexRpeetedieeee3ixgericgggywstwgkgkEydigrdeerpeEyeeg#xtiagsecamesformae'iggeapmdiagegeÅëggvemag-

#xxeexieagiwategf3tioanseketwec.

kogeq6,Kkffykitya,N.pa-23idievekgpedirktwetkedibyCkeeesecefekeewaeewva#rgkbfteawadi

appttediRgeegheskeadiysgkgeneeegtwekwkkeategitdiagctgoeeeagagagRoenwiellkgegifxpewkkljifediepen-

dieveeeeofgkermrkkeeitdieecggvggy.Tkegtwekrgovemaiptgeageeksaoitofthegeewvkecgmebgewas

ebaatwediwarkewatheirffgkalc"asdieeetEnygtywasdescwhbedbyNwaeaw,eexgeoeeewtiaggwadipewexfune-

gigee$tweemasefSemegeefatwre.

g.$wmegemssENr]ffgwoax

gnewdie.rgeav"idiee$twggkehigkereifderfwaeedaifKxentaisokwtgoptsargdghederSvaeivesof

kkekekgsekgrceiktwgRM,gkedgseifeSeftptegvaggxgetkedisdeveggpedfofftkedoxKkageetwge--

gvaE.AkfgffskSkefiapteSggmegsasgepscoxggxkatedwtkXztwgdgssrcgba#tgoptoftkedegtftftwacsgeme.rifken

kesSgegtkisffeswactieitggappff"xggkxagSoittkedisereteiitSegraXyxRetkodisestabgisked."ffkksdis-

eyekegwtegraitwethodEseexgpgeyediwabeptdiitgprobRemofbegrwawadsteadiy--staSekeatcofi--

deectgeptprobgeKxit.gpttkeueeptRgftecewrkeaSco*dieectgowaprobgegxRthedSscffeteimkegwagggkeehodi

asveidiskkecegxkpgextwversetgeagesfoifrwkxsimgtkeKgrchkeff$trca*sfoffirrftatiome,apmdtheligx]i-

it.akgepmofktweeckseseftkekkerggiggcepmductivSkyusimgaxxgetkedbytkewseofkkeptew

vssxStsbEe.

R.$.gAgegeeenmeeasgetwffgkkwaecee*jgktwdiakeedigseciipee$eetwtwgegmaesbeagkctgwwtwggkggeeisSgeSbeeeeeeewa

ewffabeeedieeBimbeeweceeasss

gngkg$gkesSs,kppffoxgjixgagioge"fffeeeectiegeengggiztwgdigskifgbwtSexxoftkedegtaf$eectieit

SpmBewgsptgepe$edi.TkeefeeptcggopmgsappfexigxRakedibydistrgbmaioenoftkedegtafeemeetioem.

gLY$geegGreept'$gdieengiky,gbeefmpwacggopmiscewffgedeeekbyekeboasgedikjrycondiieigauaswadstrewagtk

"fdig$kffgbwtSemeijfkbeedieggafaseeÅëggeme.Aeceefacy"febeg$kkeowygsexesgxkwtedbyigkekwaewue

.

s

fuitction.UsingthisfynctiextaXapproxiixiationthediscreteintegfaimethodutigizingdis-

wibutiomaofghedeitafunctionisobtained.ThedisereteixttegralmethodffepXaeesdomain

integra#sbyequivakentbo"ndaryequaeionaRdsumofsemevaiuesatsogxkepoiRtswithin

thedornain.

Thediscreteintegralrckiethodisaxsedforthetwo-dimensionalheatcondactionprob-

gemwithaheatsource.ThedognaiRintegraiisperforgnednotusingtheinterrkalceils.

WhenaRexteritaXforeeexists,thedomaiaintegralistreatedbythediscreteintegrag

rne{hodutilizingthedeltafunctionforbendingproblemsefcontinuousandinhomoge-

neousbeam.Thisapproachbringsabouthigheffxciencyonthecaieulations.

Thesteady-statextonKnearkeateonduetionequationwithtempefaturedependence

ofthermalcoRductivityisdiseussedbythedisereteintegraimethodatiaizingdekafune-

tieR.Thisappifoachffeplacesthegoverningequationbyanewform.Thisformissuitable

foranykindoftemaperaturedependenceofheateonductivity.Thetheoreticalviewofthe

diseretei"tegralapproachanditsvaiiditythroughone-dimensioflagexampgeisinvesti-

gated.

k.3.2TkeaswarkRRwae*ffakEsetkesfis

Thethesisiseomposedofsixchapterswhichafeasfollows.

Enchapter1,ageneralintxoductionofbackgroundofresearchesoRtfeatrneRtof

domaiRiRtegragsinBEMispresented..Atlast,thepresentworkisdescribed.

kRehapter2,atfirst,thekindandcharacteristicofBEMarepresented.Thenone-

dimensionalaAdtwodimensionalPoissonequationsareconvertediRtointegraaequations

assiRgthedirectmethod.ByapplyingfundameRtalsolution,thecompleteformulationare

obtaiRed.ASgast,thenumericaliffxiplementationisdiscussedindetail.

Inchapter3,thebasictheofyofthefunctionaEappxoximationutilizingdistrib"tion

ofthedegtafunetionispifoposed.Fffomthisappfoxiwtationoffunetionthediscyeteinte-

gwaRmaethodisdevegoped.SomeRumerkcaXexagnplesaresubmittedtoverifythevaiidity

ofthiszzewxxiethod.

Erechapter4,anewawalysisgxtethodforbendingprobXemofbeagxkisproposed.A

6

sckeffif\)ewgehe"tanyvarkabgesatintexKif\)ediatepointsisestabfiished.Ageneralizedsogution

schemefoffaxxinhoxif\)ogeneousbeagxkisobtaancd.Tkeedemaaipmintegragisevaluaeedbythe

dfiscreteXwtegralgxtethodwtiAgzingtkedekafgnctgofi.

EftclaapterS,tkesteadystateftoiteinearheateondvctioneq"atioftwSthtegnperature

depermdieptceofthergT]agcondasctavityisseavedbythediscreteintegragmethod.Thisap-

koehapter6,thecoRcX"sieftsefthisworkarcesumamarized.t

REmaRENÅëES

E1-a]C.A.Bfebbia,TheBoundaryEaementMethodfofEngineers,(1978),Pentechpfess,

komedoen.

ifg-2]C.A.BrebbiaandS.WaEkef,BowndaryEiementTeehRiquesimEngineeriitg,(g9gO),

Butterwgrths.

[1-3\)C.A.Brebbia,J.C.F.TeNsag]dL.C.Wrobeg,BoundajryHementTechniques;Theory

andAppgiendonsinimgineerikg,(i9&4),Springeff-VerSag,Berlin.

(X4]Kisas,Hgroyuki,DeveXopmentaptdappiicatioptofthediseretegntegragrxkethodutigiz-

ingthedekafunctioit.ProceedingsofMechaniealEngineeringNo.OO-a,VoNg,(2ooO),

4S.

Eg-S]jKiseqk{iroy"okiandHwng,Chao,ANewSegutionforNoft-XinearHeatCoRduction

Pfobgegrts,Proc.JSME,(iftjapanese)No.oog-X,3,(2am),9i--ee.

{S-6iT.A.Cra]se,BouadaryKxttegralEq"ationMethodforThreeDimensionagERasticFracture

MechGEwticsAnagysis,AFOSRTR-7S(}gg3Repog't,(gonS).

{g-7jT.A.Cmse,]D.W.SnowandR.B.Wgisoit,NwneicalSogwionsinAxisymxnetricEXas-

ticiily,Cegnp.Stwct.7,(gon7),44rscSg.

Ea--8}T.A.Crgise,MathexifiatieagFoaxitdatie*seftheBoasndary-iwtegralEq"atioftMethedin

SoRidMeckanics,AFSORTR-77-ftCPeeReport,(ISi}:r7).

[g-9]D.J.DagksonABeundrayffementformulationofProbiemsirelinearXsotropieenastie-

7

itywi&kBodyForces,inC.A.Bfebbia(ed.),BouRdaryEftegxxeRtMethods,(198X),Spriager-

Veflag,Berii#.

[g-ge]D.J.Danson,LiRearXsotropicffasticitywithBedyForce,imC.A.Brebbia(ed.),Prosorgress.

inBoEgRdaryewenxentMethods,Veg.2,(g993),PewtechPress,konden.

[a--1X]J.C.F.Te}gs,TwoDimeRsgoRalEIRastostaticAxtaayslisUsingBoundaryMexnents,(E9g6),

CompkxtatioemalMechaniesPi}blicatioit,Sowthampton.

[g-g2]A.ER-Zafrany,R.A.CooksonandM.ffqbaf,BekmdaryElementStressAnalysiswith

DomainTypeEA\)ading,Advag]cesintheUseoftheBoundaryElementMethodforStressAnaly-

sis,(aSrg6),g534,MecharkicafithgiffeeringPubXicatioxts,Londofl.

[1-X3]C.A.BrebbiaandJ.Dominguez,BoundaryNements--Angntroductoryeourse,(g9g9),

CompwtationalmaechanicsjPgbRieatioitandMcGraw-Higll.

[X-a4iY".N.AkkufatovandV.N.Mikhaigov,Themetkodofboundaryintrgntequabonsfor

soEvingftonXipteexheattrgmsxx}issioitprobiems,USSRCompwt.Maths.Math.Phys.xeO,(XSrgO),

XV•b-a2S.

{1-iS]R.BialeckiandA.Nowak,Boundaryvaiueproblentsinheatconduetionwith

RonlineafmateriaXandnonXiRearboundaryconditions,Appl.Math.Modelling,S,(g981),4X7

-421.

[1-i6]D.NagrdiniandC.A.Brebbia,ANewApproachtoFreeVkbratienAnalysisusingBcund-

ayewejtwewts,4thint.Conf.onB]EIVi.,SouthampteftURiversity,(i9g::t),Springer-Verlag,Ber-

gin.

El-WijL.C.Wrobeg,D.NardmiandC.A.Brebbia,TheD"alReciprocityBoundaryffement

FormulabofisforTrarRsientHeatCondgction,FiniteERementsinWaterReso"rces,Vog.6,(i986)

Springer-Veriag,Bergin.

[a--ag]B.A.Davgs,P.J.Gramgnag\),et.aE.SimulatingFgowandHeatTransferiitPoayxner

ProcessingusingBEM/g\)RM.BoasitdaryffejrxkewtTechnologyVgEg.CoxifxputatioftalMechan-

icsPgsbgicatioits,SouthamptefiBosteit.

[g-g9jJ.P.S.AzevedoandC.A.Brebbia,An]EffxcfientTechniqueforReducingDogx\)aiptEnte-

gragstetheBousdary,inC.A.Brebbia(ed.),Pffoc.gOthBEMconference,Voi.g,(aSng8),as3-

2ay,SeaxtkkamptoR,Spxinger-VerXag,Bregin.

8

EX-20]A.J.NowakandC.A.BTebbia,MeMgltipae-recipffecityMethod.ANewApproach for

TwansformingBEMDomainkotegragstotheBoaxndary,Eng.Anal.BoundaryEgeinflents,6,

\(i{ngg\),i\(}`4-gos.

Eg--2g]A.C.NevesandC.A.Brebbia,TheMuitipleReciprocityBoundevryewegnentMethod in

Ellasticity;ANewApproachforTransform[}ingDomainimegralstotheBoundary.ffntj.Numer.

Eptg.3a,(iopi),7op-ew27.

[g--22]Ochiai,Y.Three-DixnensionaRThermaiStressAnaiysis"nderSteadyStatewithHeat

GenerationbyBEM.JSMEInternationalJournaXA.Vol.37,(1994),No.4.

[if-:;B]Kamiya,N.andXu,S.Q.,AnaRternativeEineafizedformulationforq"asi--harmonic .

noptXinearequation,Proc.ofCenf.onBjEma,(inJapanese),No.96-i42e,(X996),B.

9

ÅëwaEYffER2

enSRCTffasonYopTffesEBOmoARYEMEmaNTmsTmoDg

2.flrwiRODIgL]ÅëTKON

Thechapterisprimiiarigyilltendedteshowxvvhatistkechavactesistieofthebouad -

aryegegwaentmetkodandhowthegoverningequatioftwithpffescribedboljndarycondi-

tienscaftbecoRveptedintoasuitabXeiwtegralequatioaswsiRgtkedireetboundaryeXe-

ewentgyiethod.Next,thecorwpaeteboundafyformulatienusingthefuRdamentalsolution

isobtagmeed.Thenggneriealimpgemaentationisdisoussedindetaii.

NopfixMericaiaxitethodsfofengineeringhavebeeRinvestigatedbymanyresearchers

formameyyeafs.rifHhesemaethedseanbeelassifriediRthreegxeaincategories,fiptitediffef-

enees,fixtiteelements,andiboaxRdaryelementwaetheds.rÅëhefinitediffereiteeE2-1]-[2-2]

isahefifstsueeessfuliyappgiedtoiteemerieagmethodaRdisusgalgyderivedbydirect

appgicatiomeofadiffereRceopevatefeorrespondingtoghegovemiRgdifferenkialequa-

tion.Thisoperatieniscawtedoutatseriesofnodeswithinthedomainefthebody.Th is

approach,however,possessessemedrawbaekswhichareimmediaeelyapparentwhen

cogxkpgicatedbothndarygeomaeteriesawadffelativeRyaocasifatesolatiomsaffeattempted.Th e

fiwakteeKegxftentffxeethodff2-3]t-R-4]isthegxaostpeputarptu#ptericaEmaethod.Thedoaxuaipm of

thebedyissmbdividediittoacolReÅítioitofcemnectedsubdomains,calgedfiniteeleinents.

PoEyenemaiaEfunctionsarethenchosentoEeeaUayapproxfimatetheaetualbehaviorof the

so#utioge.A"bestfit"fertheapproximationisahemeebtainedghrompghthevairiatioRaE

priwaeipge.rifhemethodisgRkereeffkeientehamethefwtitedifferenceapproaehandisappEEed

ae

.teaverywSdeffauegeoffineaffanditomagiwaeaffpifobgeffxRs.Tkereafe,however,manycllasses

efpffebllemsfoffwhgchfiwiteelleix#eftksdewhoifbekavesag.isfactoffiXyandthiskasgedffe-

seeweherseogookfoffaEeerpmktiveeechnfiquessnckasthosebasedoniR&egrageqwatioft.

gnboasfidaryeXexgxenetechxtiagaxe,eckegovegrg]XitgdEfferewtialeqmpatieptist.ransfergxMed

intobekandgkg'yintegraXeqgatkon.Kfthebe"itdaryequationisintegratedinciosedforffn,

thgssegwtiopmwggfibeexact.Baggthisisvartk}aiMyigxkpessibgeinpracticagprebgegitsaRdlap-

proxixi?atiomshavetobeintrod"ced.Tkeerefeffe,thebo"ndarymaybediscretizedifitoa

pt"gxkberofeieEg]entseverwkichpoSyftosuxiallfunctioitareintrodueedtoimterpokatethe

vaE"esoftheapproximatedsogutioitbetweeftthenodagpoints.Irhisailowsfortheevalua-

tiollofeheregevapttintegrags,asasa]lybysoffxien"Kif\)ergcalprocess,ffesuEtingiptafifla]sys-

tewtofeq}xatioms.

gnthefoggowgngsthedirectbouitdaryegeewentmethodisusedfortheoretiealanaly-

sis.Xnthgsmethod,theunknowRfuitctionsappeariftgiRtheiemtegfaleqeeatRoitsarethe

act"alphysicalvariablesoftheproblegftk.TheiRternagpotentiagandfiuxaredifeetiyeom-

putedafterwardsusiwagtheboundaryvaguesobtaimedkhregghthesoletionofthesystemof

equations.

ZeZTmamoSmowwCmmCTEpmi]STMCSOWBEMif2e"$Iirw[2wt6]

AEtho"ghaElBEMkaveacegxkgiteptorigiittheyaredividedmaturakEyintethreedif--

ferewtbgtciosegycategories:

a.'Thedirectmethod:the"whknowftfunctioRsappeaffingipttheintegragequationsarethe

actualphysica:variabgesoftheprebEeyxgsuchaspeteittialanditsfaaxx.'rhus,ferexample,

imaaitpotewtialprobfiegg]s"ckanimtegrageqaxationsoiwtioeewo"gdyfieSdal]thefigxesand

potefttialsefithebouadew'ydiSrectayandthosewithinthedemaincaRbederivedfrogxktke

beeepmdaryval"esbynangxtericalfintegratgopt.

2.Segxgfi-dgrectgxxetkod:tkeSptegrageqanatioascaptbeformauEatedinterxxRsofeepmknown

feeitctfiogeanalogoasstostg'essfwawhcegoitsgneXastgcityoffstreamfvitetSoitsieepoteittiaXfiow.

ig

WkegeekeseggtiofihasbeeRebtainedintheseterifits,sigxltpeedifferentiatiopmwigllyiegd.

3.Tkeindigffeckgnethod:theintegraleq"atiomasareexpressedentiffegyixtg.ermofaunit

sgptg"garseRntioitoftheorigiemagdiffereRtialleqthatiowasdfistrib"tedataspecgficdeptsity

overtheR\)ompfidariesoftheffegfiegeofinterest.ThecEensigyfgnctionsthemi}seRveskkavexto

specSficpkysica]sigwaificaeceb"toncetkeyh3vebeenobtaifiedfroffnaitaxgx]ericaisoi"-

tfioftoftkeintegragequktioms,tkevafuesoftkesoRutioRpaffagxxetersanywkerewithiftthe

bodycanbecaieagatedfrofiitkegnbysimpEeintegrationprocesses.

Tkeadvantageoftkebo"ndaryegegnentffnethodissufngnarizedasfoigows:

X.BEMtreatsprobiexnsbysingEeboundarydiseretization.Thgsgeadstoaverymuch

sgxRagXersystegxgofsirrR"itafteoaseq"atieitsthaptanyseheffReefwhoge--bodydiscfetiort

"sedinFEMandFDM.SoBEMcanffed"cethedimaensioRaEityofpracticagprobgems.

FoxexaixkpEe,fortwe-dSgnemsiomagpifobgems,tkegxgethodcangenerateaone-dimenskonaP

bouitdaryintegfaXequatgoftandfoffthree-dimensioflalproblemsoptlytwo-dimeftsiofial

surfaceintegragequationwEgEarise.Beeauseofthered"ctieitimedkmensiopt,BEMcan

savetheeompwter--xif}emoryvogume,thecalcugatingtimeandtheworkofdataprepara-

kiofl.Tbesforthegfeatmajorityofpfacticalproblems,BEIVicanoffervefysubstantial

advantagesevertheothernvgxiericagmetheds.

2.Bywsinginfluencesoletioms,thesoggtioRefasRbou"dedproblemsdoesnotreqwifeany

specialtfeataxkent.ThisimpgiesthatagargeRumbefofcaseswhefethedeaxxainundereonsid-

etfzxtienextendstoinfgnitycanbeso#vedwRthoutreseufcetolargewteshesaitdartificiaXbody

coRditieRs.

3.gRBEM,thenumericalfesulthashighdegfeeofaeeevaey.Thismethodissuitablefof

probgemssuehasstressconcentfatieRandthosewithhightempeffaturegvadientregion.

4.Contvarytodomaknteehiiiques,boandaryelemeRtaRalysiscodesareeasytoiRterface

withstandaxdCAEsystegRxs.

g2

as TW\(P-DmeNSEONAUeP$msONEQWasB@Nif2-W]rw\(:Z-a2i

2.3.gTkefikkitegeEeqeetiee

Thegevergftieegdiffeffentiaa equakieRiRtwedimaensionsisgiven as

Kg72vaf.,<\)ing \(l2Z.E\)

wherefisutow agivemediswibutienofsoureestfengthsoverth edemaaiwasu.Thefiuxat

oit

aRypointwiNbe9=on"

Tkeboaudtwyconditionsf orEq.\(2.basskowptinFfig.2.iaffewrittenas

es=rtoflrl \(:;2e2Z\)

Otse-.q= onptgoitr2 \(2.3\)

]rhesoeereepoEwtisaliexpressedby p(orP,ifo*thebo"ndary) andtheintegralpoiRt

(observaciofipoiftt)bye•

inordertoinvestigate thepossibiKtyofactualgyintegratingjEq .(2.g),overtheraptge

su,af"netieptne" (x,p>wi ]Xbeintrodueed,whiehfts,asyet,uitde

pt

rl

finedexceptthatitis

mes=es

lr2opntan

g=g

Fggxgre2.a:Potentiaiprobgejnmsifttwodigneneiens

a3

s"fficieptgycowtirkano"sgobedifferefltiabkasofteueas req"ired.fffbothsidesef]Eq.\(2.D

aregKRaxgtgpgiedbyes*<x,p>amediwtegratedbyparts, ekefe ggowimgeqsuatieftgsobgagned

'(l;l2)e4)

Thefefofe,

\(w2es@+f@\)ec*\(e,Ddg.diOee@es*\(ep\)dr-\(Wes\(oj\)Wes*d9

On st

+f@us*\(ep>dg@sk

.Oes\(oj*<e,ptdptth.@eee*<epp>Tes

=o \(2.S\)

Eq.(2.4)eanbewrittenas

Oes\(ojee*\(ep>di-es\(thOes*\(2P>r

\(II2e6\)

Nowwres*(x,p>RsspecgfiedgobeasoRwtioitof

gKg

w2esS

Here,6(e-p>isthedeEtafupmetion.(Teseeappendgxindetaig).:if'hefundamefttalsogwtfiopm

whEchsakigfiestheEq.(2.7)isebtaiRedas'gies$\(e,p>=ho---

wheferisthedistancebetweeReandp.ByElqs(2.6)aRd(2.7)thefoliowingequ3tionis

obtaiRed,whewawffitteito"tfgggy,

.lp>.Oes(ptx(ep>dr-es@of(e'P)x"f@es*(e,p)dg(th

gfthefeMowiitgfogmugaisdefified

*x(Q,p>.bes(e•p)

theREq.(2.9)eanbewrittenmoreconciseiyas

eslp>=es*(e,p>g(ptdT(oj-g$(e,p>ee@dr@+f@gg*(ep>dg@su(2oll)rr

EfthepointpisiKxkaginedtoapproaehthebompndaryrfrominsideg,Eq.(2.II)be-

cemes

clv>eelp)--as*(ep>g@dl<a-g$(e,p>es(thdr@rr

+f@estw<e,p>dsu@c.g2)Åí

gs

gfthepoiautpgsaocatedSemsidedogxkaiit,C<p>=a.Eq.(2.g2)cag)cexgculatethepotefttia]at

afi ypoiwtpfroffK\)akwaowledgeefbetkthepoteneiagaftdthefigxatalEpointsafoafidthe

boepsudafyxandehespecifiedien{emaalsoufeediswibution.

2.ge.2NeetwemicaEjixgNptee]waegetwwhasge

inthSssectionageneralnmpgnericagproced"reforthesolutionofboundaryvalue

prebgejfkswilibedescribed.EnsteadofattemptingciosedforjmasoiutionstoEq.(2.12),

whichisadifficult,as"itabgeit"xxRericajapproachishereeKxRpgoyed.Thebasicstepsare

\(ffere,Nistheitumberefbeeeuedaryitodes.\)

g .TheboufldaryisdiscffetizedintoaseraesofeXegnentsoverwhichthepetentialaptdthe

fi"xarechosefttobepiecewiseinterpogatedbetweentheegegnentnodalpoiRts;

l:z\) .E(g.(2.a2)isappgiedimedgscretizedfoffmtoeachfloda]pogntoftheboundaryandthe

integraEsarecorgxp"tedovereackbovitdaryelenv\)ent.AsystegifRofNliinearaggebraic

eq"atioreimvogviRgthesetofNthenedalfi"xaitdNitodaSpotefttia}isthereforeob-

taiRedl•'

3 .BoundajryycoRditionsareignposedaptdconseqEaentlyNpmodaivakxesarepfescribed.

ThesystemcfNeq"at.ieftscanthereforebesoivedbystandaffdmethodstoobtainthe

regxkainingbo"Rdarydata.

gf theboeundaryisdividediittoNEcegRsandthedGmainisdividediRtoMeegls,Eq.

\(2.a2\)canbewrattenas

clp>eslp>+Eq*<ep>es@dr@-2ee*(ep)g<ptdr@

iclril':1r,M":Kthes"(e,p>d9(th(2.13)i=1g,

gn gkispaperqwadratgcskapefunctgonsasshowninFig.2.2areegxkployed.rifhecoordi-

g6

meatesofpoieetsaocagedwithipteac helegxftentriareexpressed ingergifxsofimterpollatioit

f$mectioees aRdikkeitedagcoorcdiwaatesof theegegxsewtbyekefegiowingffeXatiopt '.

x(g)= Åë1 ({Il;>xionI.Åë2(g})x2+Åë3(ig)x3 \(2.a4\)

wheifeigSs- animtriitsiccoordiiflktedefined oftboundaryeXemewt, 'whichvafiesbetweeit

-- gandi. epi (g>,Åë2(ig)andÅë3(ig)are expfessedas

Åëi (ig ):-ill-g-(ig-g>,Åë2(ig>-(a-&)( g.g.),Åë3(ig)-Sig.(ig.+i >

Xitsimigar wayboandafypotentialand fiuxareapproximatedovereXementthroughinter-

polationfunctions

es(ig-)= Åë1(ig)es;+Åë2(ig.)esi+Åë3(41;)ecja\(2ei5\)

g\(k.>- dii (&)gji+Åë2(ij>g,2."Åë3(ig.>gi \(2.a6\)

wkereeeji,esf. ',es//,qji,gJ2•andg//cofttaift thenodalpotentialand fiaxrespectively.inorder

tocale"late Eq.(2.B),itisRecessayto transformth eboundaryeiemeittdrfromthe

gXobalsystemtethisinwiRsiesysteKnof eoordinates

dTuadu,2+dx22

\(2.g7\)

Thebe"medarygpttegralofEcg.(2.B)for i-thso"rcepoiRtcanbe calculatedbyEcgs(2.IS)-

"erq uorqasorq

'

x2

xl

x3

ig`--1&-O

3g=1

Fggofe2.2:TheqwadrEktieskape faxnctionsandth ehomogemouscoerdinate

-e K

X7

\(1;2eW\).Thetwtegvalsinequatioit eanberepgacedbysuxKxffK}ations of gheform.

i

9* (ig>of(ig->ueÅé(&-)de- \(2.X8\)-1

1

.$ (g)Åëk(k->gkG(ag)dlvix. \(1:EeX9\)

rik=.rl -1

Thediseretizatioitofthe bouftdaryintegrai shasbeendisc"ssedand next,emphasis

iwigx begiventothe domaiftintegragsofthesourcestrengthsby the sameproceduxe.Xnthis

step thequadratieskgrfaceegement asshowflinFig.23is used to wtodekhegeognetxryand

the variationsoftkes"rfaeepotentia;andff asxfespeetivegy.For the domaifidiseretizations

ofEq. \(2.i3\)thecoofdinatesof pointsXoeatedwithineack eegS sti canberepressedbythe

foXllowiftg eeqasatxoit.

\(2eII2\)O\)

where ig(lg;i.,ig2)isangntriwsic coordimatedefigeedondomain. xk (k =g,2eee,g\)Theceor-

dinatesofsomespecialpoiittswh -rckdefinethegeometry ofthe cell. Åëk\(e\)\(k:X,2eee,8\)

representstheinterpogatioftfunetgons.1

3

8

ag2ig-i

6

2

FxgRkxe2.3:Thequadffagics"rface efiegcrxewt

t"

ftg

'

di`-.,S-<a" &i)(i-ig•2>(- &idig2Mi\)

7Rdi-=-

4\(a+ig-,\)\(a--g.2\)\(ig-i-&,2--x\)

.3mo-pmk\(x+&- i)(1ÅÄig-2>(ig-i+g.2--g)

Nts

.4. k\(i- igi)(i+&2)(" &i+ig2-i>

.5. g\(i. gi)(g+igi)(i- &2\)

.6. g\(g+&.i)(R-e.g2>(- i+ig-2\)

7gcb=--

2\(a-\(.llli\)\(g+\(.ll;i\)\(S-i-&- 2\)

.

.8. g\(i- igi)(g'"ag2)(ft+ig2)

KnasimiEewway,thepotentiaEandffuxafewrkten as

8es\(el-k=1

]ZÅëk(ig).k\(2.21\)

.

sq(g)-k=i

]:XÅëk(ig)gk\(2.22\)

Ktisconvefigent tocolmpute theeelliRte.qralsby"sing s"itableitumericalquadratuxeschefne.

ThedomaEnintegraaofEq. \(2.B\)eanbeeval"ated.

sti

f\(thec*\(e,mbd9\(th CXiti5-\)dig,dig-,a23\)

rwG(ig.)=wkere

eig-,oig-2.Erkthisgxxethod,the dexnainneedstobediseretizedtocom-

p"gethevoSumegwtegralsandthesasrfaceofthe domaaiptisdividedintoaseriesofeie-

gxRexets.'g"keis BEMiscafiged ehecllassicaXeffcemeventioitagbo"pmd aryeSexxientftxteShod.

p

gfEqs \(:}.aSl\)-\(2.1':7\) aRd ]l!kg. \(l22\)e23\) are seebstit"ted iitto E\(igte

\(2.13\),ehefoXgowgngequa-

gRoncanbeobtaiRed as

NE 3 IVE 3 M 8

c,ee,+EE h41ut.IJJ :E kgijq 4J E: dsc \(2.24\)jE=1 k=l ju1 k=1 Al k=i

where>Åëk(ig\) G\(&>d.fr. k,gil•=

1-1

\(e-\)<pk\({l.; \)G\(k. )dig-

and

d4..ij .* (ig)Åëk(g) 6\(ig- ) digi dig7e---bl

]E\(Ie \(2.:as\)feri-th seurce poiwt eawaagsobe expressed aS.

Ul 91ee2 g2

ciesi+[hilh2ee "

hiN]ee

=[gil&2

eq .

giN] e.

+[bi](2.2S). e

esN qN

FregmetheappSieation ofEq.

\(12.:iE\)dg\)to alg boundary itodes, a finai systemofequationsfor

i-thso"reepoSntarkses

c1+h11 h12eee

]ZIN es1

h21 c2+h22 "ee hrw es2

=hNl hN2 e"e cN+hNN esN

gil g12 eee gIN gl bl

g21 g22 eee g2N 92+

b2\(lil!\)e26\)

gNl gN2 eee gNN 9N bN

ByappEyingthespecified boasRdafy cenditions ,Eq. \(lllEe26\) caRbeffeofderedaitdiasetof

20

siKKguttfgmeeo"sgipmeareqkgatioftsgsebtainedas.

EA]{X}={f7}(:2te2})'"i7)wkere[A]isafuglypepeegatedmatrix.Veetor{X}gsffergxkedbytbeuitknowrmpoteittiae

aitdffRgxftewdthecowtxibwtiowaefthepreseribedvabuesisineg"dedfintovectorf.

xe.3e$osecigndReetieewtgeffakedigagmpeesieermwstwgkkempdilgwtgymaede.

Tkediagoitalterg[RshiioftimecoeffgcientgxgatrixoftthegefthandoftheEq.(2.26)caxR

beevaguatedindirectXy"singtherigidbodymode.intherigidbodymode,thestressand

thebodyforceinthedomaiftareaawayszeroifonXytherigiddisplacementoccurs.We

suppeseaffigiddispSaeeexxewtisec.,Eq.(2.g2)becomes

C(P>es.xe-ee.g*te,edr(2(2.28)r

erC\(P>+g*\(e,bdl\(thes.=OT

BeeaaseEq.(2.28)iseorrectfofanyes.,thisequatiomahasnoreiationeobogRdaryeon-

ditioR.'Fhefoliowingequatiogecanbeobtainedas

C(P>=-g*(e,bdr@(2.29)r

DiseretiziRgtheEik}.(2.29)yields

c,"Ed(epi)dr@=c,+2h,,•a3e)ThediagonakermsinthematrtxonthegefthaRdefEq.(2.26)canbecalcuiatedaxsingthe

foggowingequation.

Nci+hii=-Ehiijc.3g\)Jt=t<itJ\)

jE(].(231)pyevkdewsefuEmeansofÅíoffxkpstingtkeieadingdiagonalsubffifftawices,avoiding

anaRyticaEevaauatioenoftheeeeffgcientsaRdithepriRcipaEvaEueintegrais.

21

lr-

Z.ajcoNÅëacWSgONS

koekischapter,tkebasictkeoryafidgKliagitcharacteristicofthedirectBEMaffeas

foggows:

g.SinceBEMtffeatsprebgegyxsbyghediscretizationofthesgrfacesofthebody,BEMhas

advantagesoverFDMandFEM.BEMffednceindixxkensioitssothatBEMcansavethe-

comp"teff-gnemoryvoiumeandtheworkofdatapreparation.Opttheotherhand,themu-

gxRericalresugthaskigkdegreeofaccuracy.-:2.gnthetwo-dRrritensioftagPogssonprobEemtExegoveffg}twgeq"atieRisconveff"tedintoboasRd-

aryintegraleq"atiext.ThecDtwpgeteiptegragforxx}"iatioitispresewtedusingthefuRdamen-

tagsotwtioit.Usiangtkeq"adratieboundaryegexxxewtsafidquadraticsurfaeeelerrxentthat

theformulktioitofthesystexxkmatricesisobtained.TkedoiEkainisdividediptoaseriesof

skarfaceeegasandoReaehoftheseeeRgsintegraXsareearriedoym.

3.Cageugationofthediagonaitermisperfofmedusingtherigidbodymede.

ms]giiiEmsNCES

(2-1]G.D.Smith,NumericaiselgtRoRofPartiaiMbilfferentialEqwation;FiniteMtifferenee

Methods.2nded.,CiarendoRPress,0xi;ord.

[2--2]R.V.Sowthwell,ReEaxatienMethodsiRTheereticaiPhysics.OxfordUniv.Press,

bonden.

{2-3]MJ.Tasrgker,R.W.Cgough,H.C.MartinandL.J.Topp,StiffnessaxxdDefiection

AnalysisofCompSexSeructwres.S.Areo.Sci.,as,(X9SS),swSgas.

[24]R.WÅëllo"gh,(g96(]b).TheFgrftiteffegitewtipmP}ax}eStressAnalysis.Proc.2ndA.S.C.

E.Conf.oftewectronicCojni]p"tation,Piksb"rg,Pa.

[2-S}C.A.BrebbSa,Tkeboun(SaryeXegxitentxx)eehodfereagineers,(g97g),Pentechpress,

bondOR.

m

(26]C.A.BrebbiaaJiRdS.Walker,FouadaboptoftheBowwdEayeweffncRentMethod2kxxdgts

Appgicatgon,Tmes#atedbyKamya,N.,Taxkaka,M.andwaptaka,Y.,(X98g),Bai fu-kan.

{2-7liC.A.BrebbiaandS.Walkef,TkeBeitgptdlaayeweffxftentMetkxodforExxgipmeers, Tffans-

gktedbyKamiya,M.andTanaka,Y.,(Xogg),Baifu-kan..

[2-g]Kamiyk,N.,FoundationoftheBouadaryElegiteittMethod,(19g2b),Science-sha.

E2-9]Taymaka,M.andkitaka,Y.,BeuadaryEIeftxitewtMetkod-Fowrt dakonandAppgications

G{i}g2\),Mamaxzen.

E2-iO]R.YwwkiandEE.Kisax,ElasticamalysisbytheBEM(inJapanese),(19g"7), Baifu-Kait.

(2-1g]Tanaka.M.,Matsurnoto,T.aRdNakamufa,M.,BoundaryElementMethod, \(1991\),

Baifu'"lluae

{2-12]Kamiya,M.,Bo"ndaryffementMethod,(1982),Saiensu-sha.

.

as

CffIfiAPTER3

AwwaoxunvoNosgmpNÅëwmoNmomsÅëwaeffiErwEG}jwtg,MSil"ffwaeelgLJV]I"M]MMZHNGDESeff"RMBIg"XXON

OWTmaosMmapmNCWroN

3.fimamoDWÅëWroN

AlafgerangeofpxebiegixseaRbesolvedusingthebeundaryeiementwtethod.BEM

weqagiresoniytkediseretiz3tie"ofghes"rfaeeofthebodytwtoaseriesofeaetwents.'g7his

chavaeteecistgÅíwaotoltEyreducesthe"eembefofuRkpmownsb"tagsoeonsidierahlysigxiplifies

trheaKxiovawtofdataxequiredtorunaproblem.However,sonMepxoblemss"chasthebend-

ingprebiemofthebeaffxR,eheheatcoRdectioRproblemandethers,usuallyrequixeghe

dogxiaintobedividediiwtointerEkaaeeNs.\(ThisBEMisusuaIIycaEledtheegassicalorcoit-

vextio"agboaxitdawyegementwtethod.\)"ff'hisgreatlyEnereasestheamountefdataprepaxa-

tionncededandcausesBEMtogoseitsKnaiRadvantageoverdomaiRtypemetkods.

inaneffeftteavoidtheinteifgkagdiscretization,agreatamountoffesegrchhasbeen

cawiedeuttofwtdageReraiaitdeffxcientKxaethodtotraRsformdoffitainintegralsiRtoequiva-

1eatboundaryintegrais.A.NowakandC.A.Brebbia[3-i]deveRopedancwtechnique,

eaEEedthemeiuttgpierecipteeiaygxiethod(MRM),andapplieditgosoivePoissonaRd

XegitxhoEtzequatioits.MRMeeasistsofpfoposingasequeneeofthehighexofdexfunda-

jtwewtaisogwtiowswhiehpermitaheappgicatieitofGreen'ssecoasdkdeRtitytoseceessive

deExftainantegr3aeermsinawaeffofttotransformtkxeKKEintogl}eboundafy.ARthoughikereas-

imgaheordefofgkefuRdaifi[ftewtagsokution,thefeswitiitgffeouffewaeeformuRareducesthe

orderoftheheatsoewee.Ontheotherhand,Y.OckiaiI3--2]hasaiso"sedthesiaxkigarway

fofasteadyghermaRseresspffobgeitw.

pa

Howevef,MRMReedshigheerosderfundkmaentaEsegutionsaedderftvativesofthe

heat sewwce.SiitceitRssonxeeimaediffiewigteevakuateghesevft1axes,Sheappgieatioanefthis

wteShodisaiwtited.

intkesectgowathenewfaspmctiepmaaapproxfifEkatioitisdeveioped"tiiizipmgdgstribuegon

ofg.he degtafunctfioR.Usingthisappsoxigxkationoffunctioeegkemet.hod,caggedthediscrete

iwtegraXExRetkod,kasbeepmpropesedgesellvegeReraNwt.egrRianddomainiwtegrai.Since

tke presenegxkethod"sestkeEewerorderfuitdapaentaEsog"tioft,itissixnpaerthanMRM.

$.z

'TmsmaT]alffteMEewCmeROÅëMT"WBOmoAewEmemaN'grmewaOD(3-S]

inthissectienMRMfortreatiRgdotwainiRtegragswilgbedescribed.Theprobgem

"fiderconsiderationisasteadiy-seRtepotewtialfieXdgoveffxkedbyPokssonequatioR

rvgWUva

whereesstandsforthepoteRtial,kisthecondEgetivitycoefficientandthefuftctionfffepre-

seRts tkeheatso"rce.

'echeregie"iseonsidexedgwithitsboundaryr.Theapplicationofthefeciproci{y

theofeKxkaaiewstotransformthepfobaeme(3.i)kptteghefoflowingiRtegfaleqanatioR[1],

(2]e

kClp>eslp>+qX(O)(e,p>es@dl@r

=es*(O>(eDq@dl@+f(O)@ts*(O>(e,p)d9@(3.2)

wkerethefaxitdagxRewtaXsoliutioRes*<O>andf(O)satisfythefoggewkRgdifferexttiageqmpatiofl

w2ggS(O>(ep>.ums(e-pt(3.3)

and theestw\(O>laastkefoffm:

2S

es$\(O>\(e,p>----

ThedeRtakenetioenaetsatpoiwtpandrisageomeaffieaadistancegxkeasuredfromthat

Peiwt. Thesagperscript(O)imehefuRdamentalsellutiewaawadeheheatsempfcehasbeeRim-

etwded fofgheptotatioRpwampose.

Theffgxgandg*<O>arediefgnedRstkefoglowRmagformula

oesg=-k

q--(3.7)On

wheremestaRdsfortheoutwafdptormalderivative.

Enadditioittobouitdaryintegrais,Eq.(3.2)eowtainsadomainintegrai,whichhasa

genefagformasfo11ews

D(o)..*<o>f(o)dsu(3.s)sk

rlhe wtuttipgefeciproeisywaetkedisamethodthattransformstkedomainiRtegwaE(3.8)iRte

the boundary.Theproeedureimtfedaeesasequeneeoftkehigherofdeffgndame"taisotw-

tio"s definedbytkefeosrgrenceformulae

v2esX<}"i>--ne*ewf"-rO,1,2e"e(3e9)*ei>--evkoec* e"+i\)

gp"'(3.10)On

asweNassequenceofthekeatsoaxrceLapgacians

fen=wv2fC-i)f-"-g,2,eee(3egl)cama-kOfO

W-`"

gkake termof]liqs(3.9)r-(3.a2),aseriesofbo"itdasygptegraXsareobt3iptedas

as

oo

D(O)=-2f2.,.(esgQ"'i>wM-qgCi"'>fO)dr(3.a3)

gegs wog4thpegntingo"ttkatE(g.(3.g3)ffstheexacgfosgxkefthepfignarydoKnaiewgntegrag

\(3.g\) asitosimpgificatieptskavebeeitxxkadesofar.

E;inaggyintrocitueingEq.(3.B)into(3.l22),tkkeboeeeedaryformauEationeftjlkeprobSeffx]

\(3.X\)fisobtainedasfoggows:JkClp>eslp)+g*(O>(e,p>ee(thdf(th"

r

oo

=.tsta(O>(ep`A)e?(edM-(th-:llrmxZ.(uXCi'i>wM"<7*Ci"'>fO)dr(3.i4)

[ifSkeeobtajlitedforgnulatieftdoesemotcontaiftafiydomaiRintegrai,soitsdiscretgzation

does flotinvefiveanyimterRaficelEs.RrhemainadvantageeftheBEMisfuilypreserved.

However,inthegnaEtipRe-reciproeitygxketliod,sincetR]ehigherorderfuncgameittagsol"-

tioits kitdthederivatgvesoftheheatsourceareappikedtocoitvergenceefsogution,this

approachisnotgemeraalyused.

3e3 APPROwwTroNOWk]'WNCITKON\(3-3]tw[3-ss]

inorderteavogd"singtkeehfigherorderfundamentalsoS"tfionsandthederivativesof

the keatse"rce,thepmewfgitctgonagapproxSgxRatiofigsproposed"tiiimingdXstributionofthe

degta functiopt.ThisapproachwSggbedescribedasfeliows.

3e3ek Agegekwmogfi$wageffffwawaceegeewaapmfizfigagkkaediegtaffeegeeceegewa

Tkefunctfiextfcaftbe3pproxgmatedbyehedegtafuRctiefi(ehepeiptsource)as

fopmows:

miVvef=:Egb(X-Xi>(3.iS)i:l

27

whereEifistkestffempgtkkefthepegpmtsource,xiistRMe posgtionofthepoSwtsoasrceaitd misthemaesgx}berofthepoiwtso"reeftptditsstrengeh.fff usiRgthepofintseurce\(n=2> ,the

appffoxiKgxatiomeefekefuftctiomefcaenbeobtaiftedas[R,2,3}

l?s,

v2f=2D,d<x-xi>ij1

\(3eil\(S\)

MultipRygitgtkebotksidesofEq.(3.X6)bya functionwhichhasaproperty of

x<;,2yX=nv6\(xbxi>,thefoRowingequatienisebtained:

\(3e17\)

c(s)isacomstant.rErhefunctionfatanypointeaRbe expressedbythequantitiesofbound-

aryconditionandmuakasowRpointsouffeeswiththe stfeRgthefDi.ifthefunctionf isa

kitowwafunction,theaeouvacyofehisappfoximatioff canbeexamiited.FffomEq.(3.W)

thefolaowingequagioftisobtaiRed

mEDiy*(xi,s>=(wff-fwy$>ndptc(s)f(s)i=1r

laghisease,asetofsimuata"eogsequatioRseagbe constfuetedwithfespecttoD iby

moniteriitgthekwaowmeknctioitfaempeints.UsingD iwhichisobtaimedbythisewketh od,

ehefuwaetienfeaftbecaRcegatedataityl>eiRtasfegiows

\(3.19\)

iftfisebvgo"sthatbythSsschegtr]e,ghefunctioitfcan beidentified.

3e$e2Agegeilatsipaswwhoptw*ffffeewaÅëwhopme ewimgkeec$waimeeopwsdiisggvifoeewhopwadyffgkeediegtaffeegeecee

givva

in]Eq.(3.iS)ehefuaeSionfisappffoxixnatedby thepointsourceandktsstffegegth .En

ll}8

ahe followgitgthefunctionfisa ppffoxigrrkatedbytheutiNziitg ghecontgnue"sdfistribwtion

offthe delltEitfueectiext\(thelliite soggrce\).ORthegineseurce, the streitgtkoftkelifteso"rcefis

tc eoRtimgowsfunctiept.Tkisfi ppffexirrRationisexpffessed byth efoREewiptgequaeienaitdthe

aine so"ffcegsshowueim]ffig.3.Ee

in

v2f:EL(s,)&L(s,>i=1

\(3e20\)

whereSiistheginesogrce.L\(S i\)isthestrefigthcftke gine so"rce.6L \(si\)denotesthe

cQwtinitgousdistrgbutioptoftk edegtafunetionontheline source.Efxgsistgtheiineso"rceas

shOWllinFige3e2$frOjtwE(ie(3e20) ,theapproxixxRationof the fuptctieitfcanberewrittenas

(3.2D

UsingtkesagnemethodaswtiEizimgth epointsoufce,L\(yi )can beevag"atedasthefoliow-

esitg eeq"atMofi

mi:L\(y,>:tJ"*du. \(Kewfst-fWY*\)ndI-c\(s\)f\(s\) \(3.I!ielilit\)\)

fr-l-1r

,

sm

t

l

l

'

,

s.l

l

l

l

,

e

{ Sl

Figue3.g:Tkegepmeffgallgine soufee.

29

Y

k=mag

k==ma-

:

i-a e ft X

k--2

k-a-i

Fggure3.2:A modeaforgine sogres

Tkerefore,Shefunctioftfatag]y poiwt canbeobtainedby fe ]Xowiitgeqaeation.

clp>fip)=(vfy*-fvst)ndff-

r

m2fr.-l

>Y*dx\(3as\)

where pisanypoints.

3e3e3 NwwemieeXemakwapRes

Xnthisseetieit,twoexanmtpies are

epreseemtedindicating

the effectiveltessofthefunc-

tionaK apptfoxRmatioitaitdthediserete integraE method.Theseexamplesaxe:

g. AppxoxigKftatioptoffunctienwtigiziitg thepoant.

soaxree.

2e Appfoxixtkatigitoffunceienwtigiziitg thegtwe soanffee.

30

EwagxRptee#

ingkisexampge,tke faspmcgiQwaagapprexixx\)atfioxtwt.gllizimgth epointseigrcedeseribed

fiittkeprevie"ssectieitsgsegxipX oyedinthe twe-digueeeesfiopmalprebX emasshowpmgeig.3e3e

TheEengthokhesquaeceregioniS2eThefuenctionf\(x, y)ischoseRassin(su2Y\).Irhatis

f\(x,y>=es<x,y>=sin\(su2Y\).Thenumberof gy]Loititor pointm<m=:2g9\)isdistributed

evenlyonthesquareregion.

Ffomthetable3.1(a),the wes"ksare ooxxfit paredwithexaetsolutionandtheyagree

with' eachethew4figures.B"eit is werthmeotiagthatwhen y-.o,thenumerieagresultsare

netgeedandehetwaxi#xigaxxeemrer gsli!e)`#•.:k991o. Fromixtabge 3eg(b)y=O.062SO,thediscrete

iwtegralaxxethodwtiNziRgehepeiwtsoeweedoesRots"it foxthisproblems.Theasgmerieal

wesuttsayecompafedwithkheexaetsoiution asshowgeinFigs3.4(a)r-3.4(h).Itisevideftt

thatthediscreteintegraRtwethod wtiMziasgthe pointsoaxwce isnotappaicabgetothegenefaE

probgexKftsiRtwodimensions,Althoughth is appfoachisempleyed inone-dimensiollalpffob-

Regnsuccessf"SMy.[33]-[3-9i

yg

--k e

-g

Figtwe33: Squaffeffegtono f thefuncgionaEappsoxigxkatioxt

31

Tablle 3.it(a) :Thefuitctgemealappffexigkktion

wkenf\(x,y> =siit\(X 2Y\)gwtdutillizgng tkkepointsoewce.

x .i,ir EXAer RESULTSv

o.s -O.5 -o.7e7i -O.7oni

o.s 05 O.7071 O.7on1

o o O.CXKX\) -- 2.TE-g6

e.s D.S A7071 -O.7071

as O.5 O.707l e.707l

TabEe3.i(b): Thefgnctionagappreximation

wheRf(x,.v) =:sirt\(sc 2Y\)andgtilizing thepointsoewce.

"Y=O.os2S

x EKAer ]RELgU]LTS

-1 O.ort\)8 O.X07S

e.s O.or/\)8 O.11d46

o O.098 O.12i8

o.s o.cpe O.1i46

k o.ogg O.kovS

32

X:'mO.Sa

Oe6

e.2

ge-e.2

-Oe6 Exact

tmuaesamanPresefit

-g

-X--O.6-•O.2O.:l!eY

e.6 i

\(a\)

X-rDg

e.6

Oe2#:ilj

a2-Oe6 Exact

mamampwwmopmPresent

-i

-I"Oe6'ee2Oe:ie\) O.6 i

y\(b\)

X::0eSg

e.6

O.2::\)

'"\(]\)eIi2

'Oe6 Exact

wx -e-whll\)resent-g

"i'd\(I>e6-Oe2 Oelile\)Oe6 g

Y\(c\)

Fggure3.4:The numeriealresugtsoftkge thefuftetioxtalapproximatioen

whenf<x,y>==sipt\( paintso"rce.

33

".

Exacg-e---pgresengl

-1--i()e6-()e2Oel;2ee61X(d)y=o

-o.s

-' Oess

Se66ge-e.74

-Oeew

-Oe9

O.a

O.eS

Exact

-e---Pgresent

toa-Oe6-ee2Oe2Oe6gX(e)Y:O.S

'

mpwaExact

-e- mp--PreseRt

whenf\(x,y>=sim\(Ill2;

-a-Oe6"Oe[:2\(I>e2Oe6gX\(bTheewmerieairesgfttsofthefunctiomaXappfoxgmatkon

scy\)awadwtiRaziwagthe1paiwtsowwce.

34

---

eqeqeeee pt$as

Cl3ssieal

-e---PreseRt

e.07

-o.gg

t::\)i

-e.gs

-O.g9

Oqi4

O.12

ge'Oei

OeO8

e.ca

W

-1-Oe6"O-2Oe2O•6[E-x(g)Y$.cal2e;S

ufG9paewGew

..Y

Exact

-- e--Pffesewt

henf\(x,y>=skn\(l:!l;

'" 1-Oe6'&2O"2Oe6gX(h)EFigufe3.4:rKheRuxnerieaEifesugtsofellxefunctioRagapprexiwtagioR

xy\)agxd"tigizingtg]epointsoaxrce.

3S

EmmptexThef$nc•tiowaa appffoximatiee wtigizRRgth eginesouweeisgsedforthesatwepffobgexifk

asgheexagRple1.TheNneg.oeerce isshown inFig.3.2.rErhe pteemixeriealresedtsareshewRin

SabRe3.2audFag.3.S.Table 3e[2 andFig. 3.Sshowghat ffeswhtsbythepresentxttwethod

Tabge3.2(a) : ThefunctioRaXapproximatione

when f\(x, y>=sin\(scy\)andwtilizing2

thelinesoeerce

x Y E/XAer RELqULTS

o.s e.s O.7071orss -o.'7o71ocig

o.s O.5 O.7oni\(I>68 O.7ovi\(I>68

o o o -2.ms-16

s.s e.s a7on'gcxsg O.7on1068d".s o.s O.7071\(Xi8 O.717l068

36

agfeesweggwithehat exaetsegution. lltiscleafthatthedasÅífeteintegralmethodwtigizing

theitine$oeercegs saxpetiortoghe discweteiittegraEffxxethodwtigizingthepoiRgsousceinthe

gwo-diimemsioitaftp robgexgi.

Table3.2(b) :

e'

ThefuRctionaXappifoximation

wheRf\(x,y> =

Y::Oeosl22rS

x EXAerRESULTS-1 o.orrse.cms

e.s o.ogge.o!}7so O.CE}8O.097S

o.s o.cDgo.cms1 O.098O.CD7S

37

x=-os

-

Exaet

wwanewmmavPgresewt

-g-Oe6-Oe2(])e2ZO•6Xy(a)Xsu

El>gact

--- ew--e-gbeesent

-- 1-bO.6•gO.2O.2O.6gy(b)x-dis

Eixact

--- e--Present

-g-O.6-O.2e.2e.6iY(c)FEguffe3.S:ThefunctionalapptoxEffxaatRon

whenf<x,y>=sin\(sc2Y

3g

)aemdwtigizingthe1inesoeewee

y::-os

Exaet

-- G\)mmua'Present

-i-Oe6diO•2O"2ee61X(d)y$

-- O.S

-Oe6

gi:iik-o.'r

-08

-e.g

o.xo

e.06

O.02p-"-(I)e02

-Odos

-OeiO

e.9

o.g

g::\)iO.7

O.6

O.S

Exaetetny- e--Prese-"t

-g'-Oe6-'Oe:2tOe2O•6gX(e)y$s

Exact

-- e--Present

-1-"O"6-Oo2Oe2Oe6aX\(bFxgure3.S:Thefunctionagapproxiffxpation

whenf(x,y)=sRit(sc 2Y\)aRdiutiaizingtheaiitesoaxree

39

Y:O.ee2SO.i2

O.no.g

geOeee

e.ogExact

O.o'7.

-e---mPresente.o6

-g--o.6 -o.2o.2X

Oe6 1

\(g\)

y=:--o.o\(ii2ts

-e.g2

-o.g1

-O.1

ge-e.oo

'" OeosExaet

-O.07-a- ew--ff\)firesent

'Oeee

-g--o.6 -Oe2eOe2x

Oe6 i

\(h\)

'

Figuffe3.S: Tkefunctionalapproxignatioxt

wheof(x,y)zz>7=sin()awad2

wtiRkzgngthe linesouifce

..

oo

3.ajpmESCmsTEmaGanmeTeeODK3-k@]rw\(3-fiag]

g$ofdefaeavoidwsiRgtftftekkighewerdieftwRdagxkewtaEsoautionsandglxederivativesof

gheheatsoewee,ghedisexegeiRteg"falixkeghodispfoposedgoetwployiRthedoiKxaiRintegral'

andthegeneralintegra.Thisapproachwiagbedeseribedasfogaows.

3.aj.gpmjkscwebeimbegmakKmketlaedKgmefizfiagtrlaeegegimgsgassce

rMhepeteRtiagesuRdertheheatsogrcefsatisfiesthePoissonequation.Thisequation

isfewritteit'as.

'iwldes+

wherekgsthethefrr}alcopduetivity.ThebouRdaryintegraEeq"ationforthepetentialin

caseoftheprobiemisgivenby

kClv)eclp)+g*(ep>es<thdX@r

=esX\(e,p\)g\(thdKth+f\(ptes*\(e,p>d9\(e\(3.24\)rsk

,

AsshewpminElq.(3.24),wkenthereexitstheheatsoewceiAtkedomain,thedemaininte-

gragbecognesitecessary.Therefore,'thediscreteintegragmethod"tilizingthepointsource

isappgiedtethedomaimeintegrag.Thefirst,apmewfuitctionz*isintrodueedanditis

defmedasfollews

wherees*\(e,p\)= dagnÅÄ.

Thesecond,gsimgthefwwctEonagappffoxiffKkagion(me=4)theheatsouifeefksappwoxixlirkated

bythefoiiowingequation

4E

wheregv4isdupEaeatedhaifxnonicoperator,BiisstyeRgthofthepointsGusce,xiis

positioitokhepoingsogrceandmisnumberefthepointsouree.

XnttermsofEqs(3.2S)and(3.26),thelasttermEq.(3.24)ÅíaRbewifittenas

f<ees*\(e,p>dsu\(th---fv4z*dst

=frw3z*-wfv2z"+w2fvzX-w3fz*\)medrr

}n

i=1

whefetheexpfessionofz*Åía"beevaguatedas

.j.4Z=(3-22Ll]er)(3.28)as6x

FrogxatheEq.(3.27),thedomaiRintegfaiisevaluatedbythebouadaffyintegfalandthe

sasmofthestreRgthofthepointsoufeeBiaRdthefuRctionz*attheinternalpoints.

kothefoEgowiRg,anothersimaplerapproxi]ixgatioptisdescribed.IfthefuRctionz*is

iRtredueedasfoSSews

wr2Z*-.--es*(e,p)(3.29)wherees*(esP>= igemÅÄ

andthekeatso"rceisappreximatedas

theaasttermEq.(3.24)canbederivedas

f\(ptua*\(ep>dsu\(e=fv2z*dsu

=(wz*f-z*Åé7Deedhv2fz"dsu

42

m

wkeeretheexpressionofz*caeebeobtaincdas

r2gzg(ep>-ln--+ls.r(332)Asfouedintheabove,thedomainintegralgRthePoissoRequationcanbeperformed

byehediseretegntegraffgmeeehod.lathefoggowing,genewaliRtegraXcanaXsebeevaXuatedby

tkxedEscregeintegralmetkoCE.Tkeintegragisgivenas.

whereKisknownfufictioR,fisafunetionwhiehiseithefunknownorkRowR."lirhefunc-

tionfisappfoxiwaated"siitgthepointsourceas

Feerther,afuitctioemz*imtrod"cediwtoeqgatioftwhichisdefinedbytheftextequatioft:

Wkeptn=2,fintegratfiftgE(g.(3.33)bypartandthens"bstitwtingEq.(3.34)intoit,the

eq"atienischargedintothefofiowingforgxk.

m=.(WZ"fZ*Wif)nth.i,Z*(xi)Di(336)

43

-t

3eaje2 pmfiscgeeegeetwaegkptgwaegketkewthkaggbeimesewerwe

itJstwg theEliptesourceasskowmefiitFig.3.athefunctioptfgsRpproxigKitatedbythe

foggowEmege

os<77=

ngasatxen

-.

kot.hecaseofn=2 ,TheiittegralcanbeevaEuatedfromEk}.(3.33)

7*WhZ

fKdsust

=\(wzptf-zptwDmedi+w2fz"dg

tn

=.(SVZ*f-Z*WTif)esdl"Z.,L(Si)6`(Si)Z"d9(3e39)se

Tkisappsoach gscalSedthediscreteintegra}twethodutiliziitgtheaimesource.

FregxMEqs(3.27),(3.3S),(3.36)aftd(3.39)thedognaiftintegragortkegeReraSinte-

grafiisev alasatedbythebouitdaryiwtegralaRdthesumoftheval"esattheinterixa]points.

inBEM,the domaaifiiRtegra]iscarriedoutitetusingtheintern31ceSis.

"#ts`#-

-

3eaj3 NwaNgiigerkwieexee]#xkLgeRee

Thediseffeteintegvag ffxRetkod watigizfigeggk etiResource is presewtedforthe keatceit-

dmpctiept eqkxktgonwitha keatso"ffee imetwodijtwepmskofisas skowfl iptFige3e6eFOr gkisstudy,

tkte heat cond"ctioftequatioitettit bedescribed as

wr2T+b=eift su \(3e"#'O\)

where T istegnperature amedbisahefttso"rceoverthe domain 9eXfithis case,theboundafy eopmditioRsarewrittextasfeglows.

T=O;y=-i \(3.41\)

T=gee;.y=g \(3e`4'1]Z\)

g=Oexr:--g' \(3e43\)

q=Oex::Es \(3e"4-"#'\)

YT=:aeoa

qS q=O

o x

-t

T-r=O

Figure3.6:Two--difxkesioitagheatcenductionprobgegxft withaheatsouree

.

4S

ffpmtftxisexampge,the heat se"reebisseRectedasb<x,y>= -cos(ww!2)cos(scy/2)andthe

fiimeso"reesareehosen astftkefoifgftofFig.3.2,wkichis dgstributedeveniyoitthedo-

magit.Tkereare4eeexmeewtsORtheeipteso"rceaptdtheftugxiber eftkegineso"fcegsW.The

res"gtsaifeshewptiRffrg.e. 3e7 aftdgabXe3.3ftfldarecogxkpared xwiththeeiassicagBEMte

skowthecorrespoftdence of tkesol"tiens.Themaxijfxft"xnerror isoniyg.g`7o.

Tabge3.3: Tetwperaturediswibwtionwhengheheatsource

b\(x,y>:= -cos(nv2)cos(xy12>andutilizisugthe ginesoufce

x yElxA\(Trr RESULTS

05 -O..g.:2:5L.ies7 24.801S

o.s O..S7S.lg37 74.go1s

o O.so..2SX]XS 49.6943

-o.s -O.S2S.1zz7 24.801S

D.S O..g.75.les7 74.801$

"

as

X=-e.S

ll

cgassicagtmdiewtmanPresenS

-i-Oe6-Cb•:21Oe2Oe6iY(a)X=D

geeeesw70oo

s:\)seng302010

Ciassical

-- ermesuaPresent

-' g-Oe6'-()e2O•2Oe61y(b)x=o.s

l

cgassical

tu""ewmaswPresent

-g'tiiOe6tuOe:IZOeli2Oe61Y(c)Fgguffe3.7:Tempeiratueediswibetionwheneheheatsource

b(x,y)=--eos(nv2)ees(suy/2>axkdwtigSptngtheRfiitesomprce

4ew

Ciassieag

rmanesmmutPresent o

t-

-X"-Oe6";2O":2O•6iX(d)Y$

qassicalpmmdew--e-Presewt

iooooge70ee

pseas302ego

-g-O.6--e.2O.2e.6iX(e)y-=.o.s

Classkca1

mmvaitew--mPresewt

'- g-Oe6-Oe2bOel:IZO"61X\(b3.7:Texxxperatwrediswibaxtioeewheneheheatseuree

-"pt-eos(rw12)cos(scy12>andwtiginteegeheRfiitesouwce

ng

3.SÅë@NC]gL,`WSEONS

ffanghgsekapteff,thebasietkeeryoftkefanpmcgfiopmalapproxiffnatfieftwtilizgmagthepoint

soagrceaxtdtkegtwesouffeeisdeseribed.VsinggkksfuitctSoRagappffoxSmaSiopmthediserete

iwtegraXgKRetkodgsdeveXoped.SogxRecenctwsionsareobtaiRedasfellXows:

X.Ameewappffoackiswsedfofftkefupmctionalappffoxignationwtggizingthepeiittseewceand

tkeginesoenrce.Theftgfictioitisexpifessedbythebound3ryconditioitandthestrengthof

tkepofintsoaxrceortheXinese"rce.BytheknowptfanctionthisfunctionalapproxgEy}atiofi

isexaminedusingthepointseurce<n=2>ertheEiResouree.UsiRgtheSinesource,the

wauitxergeairesutcsagreeswitktkatbytheexaetval"esiwathetwodigwaeRsgens.

2.inthediscreteiwtegragmethod,tkegencraiftwtegragcanbeexpressedbythebouftdary

ifitegralandthesuExlteftheval"esattkeiwteriofpoints.TheseforgxRulasarederivedusing

thepointsource\(n=2,me4>aitdtheXtweso"rce.

3."rcheellksereteintegraEgnethodusingtheaiptesoarceisappgiedtotheheatcDRdgetioit

equationwitkaheatso"feeiptthetwodigxxensionsbyB]EM.SincethedomainiRtegralis

evaggatedbygheboundaryi"tegragandthesugxkefthevaEgesattheinteriewpotwts,the

iRteffpta#eeElsafeaxotmpeeded.rgrhewaux]xeffieakesedtsbythisapproachagfeewithtkatbythe

ega$sieagBEM.Thismaethodwsesthelewerowdewfundaffnenta1sei"tionsothatitissiewx-

paexthagMRM.

mswamsNÅëES

[3--i]A.J.NowakagkdC.A.Brebbia,TheMuRtipae-ffeciprocfityMethod.ANewApproachfor

TffexgsformiagBEMIg\)ognaiengwtegralstoaiieBoundewy,jikxg.Anag.BogndgayEgegxaewts,6,

\(g{rsY\),a6e#pgos.

{3-2]Ockiai,Y.Three-DijtwensiewalThermalStressAftalysisuaderSteadyStatewithHeat

GenejwabomebyBEM.JSMEgwtetwabonalJo"jmaalA.Vog.37,(gep4),No.4.

E3-3]H.Kgsu,AncwsoRwatieesckegxReforirwerseprebllegxkswgtkboua\(kalryreEegxaeitemettkod,

49

]ffbeoceedimgsofMechetwicalEExgineeringNe.msg,(2Cmo),9S.

E34]Kisu,Hgroyuki,DeveXopfixkentagxdappXicktiemeofthedgsereteiwte.\(yffaggxRetkodwtigiz-

iwaggkedegtafuaxctiege,PffoeeediwagsofMeckawaicalliEitgiReeringNe.eO-a,VeXgE,(2eeO),

`i#TSe

E3--Siff.Ekisu,ProceedimgsefMeckagxicaiElngineeriRgNe.oo-X,VeigE,(2000)

E3-6]jlH{irgyski]KifSUamedCkaQHUANG,ANewSolwtgonforNon-XiteewffeatCoftduction

Probgeffxfts,Proc.JSME,No.OCkg-a,3,(200()),9g-92.

[3--7iG.Roitg,H.KiskxaRdC.ffuang,Axtewalgoridmforbendingproblemsofcontiitasous

andimhomogefieoasbeambyBEM,AdvancesinthgipteeringSoftware30,(1999),339-

3os

E3--8]ChaeHUANGardHifoyaxkiKXSU,ANewSowtiopmforNoRlinearHeatConduetion

ff\)geobaegKxs"singtheDisereteEntegralMethod,SthgnteffkationaiJointSymposiumoR' New

DeveiopmentsofMeehaffxicalEengneeringEntheBegienningofComingCentufy,August9-

gg,20oo,atEoujima.NagasakiJap2kgx.

{3--9]G.Rong,EX.KisuandC.Huang,Newaitalysksmethedforbendingproblemsoftmby

theBEM(2nd.Rep),Twaits.Jpa.Soe.Mech.ERg.(inJapanese),Vol.6S,No.633,A

(1999>,20

ff3-10]T.A.Crasse,D.W.SRowandR.B.Wilson,NuimeriealSolutionsiRAxisymrwetrie

]EEastieity,Cemp.Struct.7,(aon"i7),4d4scSi.

(3-ag]D.NardiniandC.A.Brebbi2k,ANewAppxoaclateFffeeVibrabofiAnalysisansingBoaRdary

]EEeiwtents,4thllwt.Conf.onBEM.,SouthaffnptonUniversity,(ieg2),Springer--Vexlag,BerXin.

E3-12]L.C.Wrobel,D.NafdiniandC.A.Brebbia,TheDualRecipffocityBoundaryElexKxent

Form"3adonsforTmeNgientee{eatConductien,Fim[iteEletwentsinWaterResources,Vol.6,(g9g6),

Springer-Verla.g,Berlin.

[3-g3}J.P.S.AzevedoandC.A.Brebbia,AnEiXrficieptTechniquefoffReducingE\)ojmaaingnte-

gngstotheBouadary,ifiC.A.Bjrgcebbia(ed.),Proc.XOthBEMconference,Vog.g,Southampeopt,

Spriauger-Vergag,BreggR,(1Srg8),as3--M7.

E3-g4}A.J.NewakandC.A.Bffebbia,TheMvLampge--ffeciprocityMethod.ANewAppffoachfor

TeraxasformingBEME\)effy]aXmeintegrdstotheBoaxndary,Errkg.Anal.BouRdguryEEements,6,

\(gs$Eilfb\),gecxos.

se

ÅëwagygrER4

.

ANvawMeegewff$MwoRBENx\)ffNewRoecEmsowcewsffNwowsmo

ffmaOMeGENEOWSBEAMBWTmaBEM

ajefi rwRODWÅë'ffKON

Thereaxemanyoecasionstodeaiwithbendingprobiemsofthecontimaogsbeamin

mechaniealandstf"eeuragdesign.Therefore,manysoiversaredevelopedforealculating

thewt.EtcouEdibesaidthattheboundafyelementmethod(BEM){4-1],{4-2]hasbeen'satisfaetorigyestablishedasoneofthebestsolvers.SincetotagquaRtityofcomputationis

ROtheavyindadypfoblemsapersomaXeomputerisenonghtoperformsuehdesignwofks.

Et seeEKks,iptelIxissense,thatthestwdyagainopmreforitx"gatgoftoftheBEManaiysismethod

fof thecontinuo"sbeajinhasEittgegxtaeargings.

Ontheetkerftxand,ithasbeenfo"itdtkattheceRveRtieRagalgorithrwsareineffxrcient

and retwaingxkanypgacestobeimproved.kistruethatsEkehdefectsmayrtotbecomea

serie"spreblegnasfarastheyareputtopffacticaegseforaftindividwaicagcuEatiofl.How-

ever,DneetheyareappEiedtoakindefoptimaldesigRwkthaeertainoptimizationaggo-

ffitkgx},forexample,thegeneeicaEgorithm(GA)E4-3],thingshavean"ndesixabSetuma.

BecausegreatxxxanyrepetktgvecagcwiatkeftsarerequgfedinsuchprobXems,sgxkalldisad-

vaxttagesiflthecoitvewtiopmalaggorithgnsareagixpXfiedgreatlyandthecostofdesigftwork

becomaesverykigh.

gerogg]thispointofview,fitisigetendedtodeveaepaemewalgorithgn.Maiftpogntsof

thgs se"dyewe(a)toimprovethecemapositgeitofthesigxkwitaneo"seq"ationsbyintroduc--

gftg aptewfoffjtw"gatioitprocess,(2)toestabgiskasckegxgewSthoutanyvariabgesatiwterjrxke-

Sl

-

diagepoixts;aud(3)teestabRiskageneraEizedsogutionscheexxefewaRi-homogeeeous

beaaxk.(4)gotreatthedogMiaiwaiRtegffralbythediscreteineegralmethodwtiEiziptgtkepoint

soeefeefgRcgiopm.ThedoaxkainiRtegraggsevaiuatedbytheboeendexyintegfagandthesumof

tkevaluesatsomeiRterRalpoiwts.rErlkesenewalgorilhgxxswglgredueegweatlythesizeof

twatrixaswe#gastheeogx?pwtingkiscerke,andtherefofe,wElgbringabouthighefficiencyon

therepetRtivecagcuEations.Asaifesult,igisweagizedgogegtftxegowcostofoptixxiaUdesign

indaigywork.

ag.ewOummaONOFBE2fiwsBYBEMI

Atfirst,outMncofthenewalgortthmdevelopediRthisstudyisdescrtbedineontrast

tegheeonventionalformaslation.TheyafesuantmedinTabge4.1.Theprimarypurposeof

devegoptwgthisnevvaggorithmistedecreasethesizeefmatrixandCPUeime,bearingin

mindthatitmaybeappliedtesnchasoptimagpfoblemswhiehdemandmanyiteratiRg

pfocesses.ThestrategyithasbeeRpaidatteutionkoistosignplifythesimultaneegsequa-

triowasaRdteestabKshaschegxtewhickallowstoperformthecageulationwithoutvaiuesat

anyintermediatepeint(Eetgscaagitt&keRondividingscheaxke).ThetermsiRrXabge4.ftwigl

bedescribedindetailinthefoEEowings.

aje2efiButeceeKlguawh$wsffewwa]ffxR`g'sgeax#1beam

Asthesystegnofsimukarkeouseqgationsfor"ni-spanbeam,fourequationssho"ld

besupptiedasgneittSonedliater.ffnthecoRventgonagformulatien[4-2],theyconsistof

deflectiemsandsgopes(angXesofdefiection)attwoeitdsefthebeagn.Contrarytothis,itis

noticed,inthisstudy,thatasgffg]ptersetofequationscanbegntroduced.Namelly,asetofh

eackeqaiatieeofdeflection,sgepe,bendiRggnomefttaeedshearforcegnakesthegn"pinthe

preseRtformggatioit.SincetkepresentfoffKxRglatioitisgKk"chsiExkpger,notongytheÅëomput-

inggetsqwieker,bwttheprogramrrRiitgfofftheftondividingaagorithgxnbecomesgx\)"ckmore

S2

Tabge4.ft:CogRparisonofthe aigorithgn

Sljbject CoptventioftaR Preseitt

ConsterEkctionoffour lkeofWp"o& Eq.of"W,-o&mL.EqeOf

six]kuttaneousequatioRs i,.t.]Egeofe

p--iptO&p}--pLL

ep-pao&p-,,L.EqeOfMp-.

o&paL.lkeofQi\)-ino&,P-L

Bea#nwithnintermediate Dividethebeatw Setnunk"ownswithout

siwtpiesupport 'IRton+ispans. division.Numberof indueed

NanmaNbefofindwced unkmowns=n

unknowns=4*n

Beamwighpsteps Divideghebeam Neunknowni"dueed

-gntop+gspalls.

Numberof

imedueed

"nknewns=4*p

Beaffsc}smbjectedtemaextetwal Dividethebeagxk Noaxnkptew"iitd"eed

gxaoxnentEeads 'xntom+gspafts.

N"mberof

induced

unknowms=4*m

Generalinhognogeousbeam Dividethebean DeeidetheweightfuRetion

iRtosevefalsteps dependingonthe distribution

ofsettheuaknowwas ofEZNounkRowninduced

ofWiRside

DeEyxaiRigetegral Dividethedognaiit Obterinthebouitdary

ipttocegEs integraSa#idthes"m of

thevakuesatsomaepoints

S3

sfigxxpleff. Moreover,the pffeseasgprcocessisipm (Xis pensabgeforforxKxgRatgoitofgheinhemoge--

ncogs beagxk.

ajva2eRefle owtgimee*ffewgeweemeti@ma#fomathdi$ge

Letusffeviewtke procedureofcoitveRtioftaE ferxxiegatgoit[42]first.Itgoeswithout

sayXpmgtkftg"wi-spapmp ffobieExkcafteasigybe sogvedbyhandwithogtexaggeratedproeess-

here.gt isj"stbasiccoftsx 'derationneeded fofgifkerecompRicatedprobiems.Deflectienofa

beamRikeipmFig.4.iis goverrkedbythefo Rgowifig differeptialequationwhenEandIare

cofistant:

d4wEId.,-g(x)

=o \(4.i\)

where Wisdeflection, EisYo"ng'smodulaxs,I issecofidtwomentofareaandgisexter-

fiaXforceactedozzthe beameEl\(ite \(4.g\)istraRsfermedstaxrttixtgfromthefogRewingweighted

resid"ftS formof

Ld4wEId.4pm9<X>

o

Wdxdi-O \(4.2\)

Xwaghe]ill`{IRe(4e2),L isleftgthofthespaft aptd w"Estkeweightfunctiondefinedasa

p9(x)

x

x:FieldPointP: SourcePoint

\(Moving\)

W\(Marked\)

Figure4.1:A axfti-span beaxsck

sc

fuaucgiowaoffr,ekedistaneebetweenaso"rcepoimtp andafiegdpointxwhiehtakesthe

ferffg]of

.rgw=:<rciJac-pD \(4e3\)

"

AfeergptegratifigIEq.(4.2)bypajrtfouftfigtr}es,thefe igowingequatieitisobtainedas:

.d2wdWX dwd2W"d3w+EfmWEI d3wMLEiW- -du3du2'dxdxdu2' du3o

Ld4wXL+EIWg(x)W*(x,p)=Odx4- \(4.4\)

wisatasfiesthefolgewingequatgoftintermofEq. \(4.3\)

d4w*M=-6(x-p)du4

\(4eS\)

Ef]Eq.(4.S)issasbstitwtedintojl!k}.(4.4),eq"ationof deflectioptisgotas

d2w*MdW*d3w*W<p>=EIW9+ew.Ldu3'- dx2- EIdu EIo

\(4.6\)

Hete,eissaepe,MisbeRdingxnowteAtandeis shearforee,whicharerelatedtothe

dertvativesofdefleetionasfoEiows:

dWe(.),a2WM<x>d3wpt-e(x>

dxdx2Ei' du3EI\(4.7\)

kcanbeseentkatEq.(4.6)isdescribed"siRg eightvaguesofw,o,twandeat

bothends\(x=QL>.Fourvag"esagKkongtheg[karedesignatedfromboundaryconditionsat

bothendsbuttherestareandetermined.So,fo"reqwatiofisarencededtodetermffnethem.

Twoefthegx)EkjregiveitfromEq.(4.6)bygettingp.:.=O aff\)dp=L.TheresteftwoeanaxsualXy

beobSaincdwiththehellpofequatioitgfsiope.Actually, bydifferewtiatingjEq.(4.S)with

ffespectgep,eq"atiopmofsgepeisebtainedas

ss

e<p>.ewed2'W"MdW*eW'aFLg(x)w(x,p)du

-dx2mdu+Heffe,theeigdemarkoasghevariablesdenggesdifferentiatioRwithfespeetgep.The

sagkkeoperatioRsforlettingppmr=.eandp.LintheaboveyiegdtwogyxoreequatioRs.Then,

theyeanbesogvedasasystemofsimultaneousequationsoffourunknowRs.Afterthat,

defiectioRandsXopeatanipterioecpeintpcanbecagc"latedfromEqs(4.6)and(4.8),

fespeetiveay.

Byaxsingthexelationshipinlk.(4.7),equationsofbendiRgmomentMandsheaxforcee

atanypointpafeobtaiRed:

Mlp>.-MdW*+eWe'g(x>W'<x,p)dx(4.9)

LNLNelp)=fiw*e+g(x)w"(x,p>dx(4.ao)o

ag.2.a.ZRefomaasffformgkgawh*lkpKworcess

Centfarytotheessua1proeedureintheabove,asitwpRersetoffozzrequationsisintro-

ducedfoggowingmorecoReisefQrmugatioRprocess.Atfirst,thefgnctioniseififapgoyed

arde=ev"sSgn(X-P>(4.il)astheweightfuactionandeonsiderthefoglowingweightedifesiduagformof

LEfd`W-g\(x>w*dx=o

o

Here,sgnisthesignfunction.Wheftx<p,sgn<x-pp>=-1;Whenx>p,sgn(x-2rp):ReEcge

\(d4.X12\)isexecutedgwtegxatioitbyptwtoftceasfoigGws:

Lewd3wwsLewd

du3 \::Slt\3dex-Lg<x)w*(x,p>duo

du3eo- \(4.13\)

e

SS6

ingeri[ifxofE(g.(4.1a),gkefoRgewimgequatioptisobtaiRedas

dede:=-b\(x-p>

du\(`:g.g`g\)

ffg ftsftotedthattkesecoftdtergtrxeRgeft-hkpmdsideefE(g.(4.ll3)isst3tedas

-ewda3,g,-4t!l3i-dex=ew-ftiXIYm&\(x-p>dxoo

-- elv\) \(`#'ei'7\)

ThepusingtheresultofEq.(ag.IS),eqifationofsheaffewceeatpisobtained 'kmmedi-

ateEyas

\(`:#.g6\)

Byaettillg p.eandp--eptL,thefoElowiRgequationsaregivenas

\("#'e17\)

\("#eX8\)

Simxiiiafgy ,byreg>laciRgtheweightecfifesiduagformandtheweightfuitctionRR Eqe(4e12)

withthe foilowing

Ldid4W-g<.\)wkdu.o,w*.tr-Pi

o

2 \(4.19\)

and

Lmd"Wq<.>wgd..o,wg\(pt-PS>2sgn\(x-p>

o

\(4el;bO\)

Fxoma]E<g.\(4.g9\)integntioitbypafttwace,ehefoagewingequationcaenbeobtained

S7

ewd3Wwtwd2wdWX,LLew

\(4.2g\)

UstwgE(}.(4g9),thefoglowtwgformalaisstatedas

d:W*M---6(x-p)

dx2\(4.22\)

EquatiofiofbendingmomentMis

nelp\).-madrdM.ew-.*L +Lq\(x>wM\(x,p>dx \(4.23\)

latheeasesofp--twoandp--twL,EkE.(dg.as)becegxRes

M\(o>=-MdWk+ewkL`g\(x>wk<x,o>du+dxoo \(4o24\)

maa\)----MdwiM+ewg,L+Lg(x>wfu(x,L)du \(4e:2S\)

E\(i.e \(4.20\)isSwtegratedbypRrtthreetimes

md3ww*-md2wdWg.mdwd2WgLLEfw.nt':MSduWeodu3

du3Bdx2'dududu2o+

\(4.26\)

Fromlk.(4.2e)thefogiowingre]atioflisderivedas

d3w",Ef=-6(x-p)

du3 \(4o27\)

eqmpatSopmofslopeeSitgerg]]ofE(g.(di<g.:26)andEq.(4.2}7)caRbeobtained

elp\).ewed2W*e.MdW*eewgLLq(.)w*,(.,p)du

dx2MduMo-o\(4ons\)

ss

e<o>agedff<L>satisfyeq"atiowsasfolliows:

ed2W}MdWeewzLLg<x>wg<x,o>duS\(e>=EId.2+diduopEf di'

s<L>.med2'W$enedWZew*,LLq(x)WSe(x,L)dx

dx2E]duew-'

intkefoagowiRg,twatrixfosgukefthispifoblewt wiifgdescribed.Xnordertogetthesigifkple

form,thesevariabkes3redefiRedas

dW" e" \(.\),d2WXM*\(x\)d3w*e*du3 -El(4.31)

UsingShesevariabSes,defiectioneq"atiemscafl beobtainedatanintemaagpointandboth

eRdS(p=O,L)frOM]E(]e(4e6)e

wlv>=[e(x)w*.(x,p>-M(x)e"w(x,p)ÅÄe(x>M W(x,p)-w(x)e*w(x,p)]8

\(4e3[;I!\)\)

w(o)=[e(x>w$.(x,o>-M(x>eW(x,o)+ e<x)MX.(x,o>-w(x)e'.(x,o>],L

\(4e33\)

w(L>=[e(x)w*.(x,L)-M(x)oW(x,L>+ff(x>MW(x,L) -w\(x>e*.\(x,L>]8

\(4o34\)

rff'heSaopeequatioRscaRberewritteniRtermofE\(} s(4.ms)rw(`:g•.30)

eip>=[e(x>wg(x,p>-ne(x)ekx,p>+e(x)ne}(x,p>] Lo

\(4e3S\)

e(o>-ua[e(x>wg(x,o>-ne(x>oZ(x,o)+e(x)Mg(x,o)] Lo

\(`#'e36\)

S9

s(L>=[e(x>wg<x,L)-ne<x>eb(x,L>+e(x)tw}(x,L>] Lo

+ \(4.37\)

ThebeRdtwgffxioenfeentequatio"sandshearfoseeequatiomsaredescribedasfolK ows:

Mlp> --[e<x>wk\(x,p>-M\(x>sfu<x,p>]8+ k\(x,p>dx \(4.3g\)

M(O)=[e<x>wha"(x,o>-ne(x)s*M(x,o)]8+ k<x,o>du \(439\)

M(L>=[e(x>wfu<x,L)-M(x>eX.(x,L>]8+ Xrf(X,L)du \(`\(#s.`i`gAO\)

dw> (4."l•D

g\(o> \(44e41i2L\)

[(L) \(4.43\)

oo

\(g\)Tkeeval"eofboundary peipts

in theeightval"esofw,e, M,Qatbeehends,fourval"esamoRgthegneitregiveRby

boumedaycofiditiemsatboth endsandtheotherfourvfi1"eseamebesogvedbythe fogfiowing

axxaS.ecixfeffgxx.

NU+\(}7=B \(4.as\)

Hexe,the gffxatriees[U],[T] a[g}d[B]areaÅ~4and[if], [G]are4Å~4insize.Theirfoirms

afeasfoXlows.

U=[w\(e>s\(o>w\(L>e<L>]` \(4.4S\)

T:[M(O)Q(O>ma(L>g(L)]t (`:#•.`#-6)

ff= ea+M*,<e,o\) e-M}<L,o> \(`#'e4'7\)

xeV(o,e)MW\(e,o\)e*.<L,o>-MW<L,o\)

G=

oi+w6<o,o>g-ek(o,o>wk(e,o)*-ee(o,o)wE(o,o) o-wb\(L,o>eX,\(L,o>-wina,e>6S\(L,o>-w}\(L,e>

\(4.48\)-e".\(o,o>wW\(e,o> e".<L,o>-wW\(L,e>

Lg(x)Wk(x,o)du

B= oL

\(4.49\)

\(2\)ThevaA"eofiwtemixa]peiRts

Usiptggnatrixeqaxagioit \(4.as\),gt.canbeseeitgkat vaguesofw(o>,w(L),e(o),e(L>,

6i

ne<O\),M<L>,Q(O>,<2(L)areagEkwnowge.SethatvaEeeesofau iRtexKxaipointofsheawfofeee

bendinggixoffwaent M,sgopsanddefieetionWcafi aEso iffxigtrkediateEyobtaiwaed.

Mlv\)--W*M(O,p>k<o>+wk(Lpt>g(L>+ek(o,p>M(o)-e*M(Lp>iif(L>

+oLg<x>We<x,p>du \(4.Si\)

*elv)=-W"e(O,p>Q(O)+W*,(Z.p)g(L)+ee(O,p)M(O)-e *e

-t

(L,p>M(L>

*-Mg

\("#'eSI22I\)

Wlv>=-W*.(o,p)fi(o>+w*.(L,p>fi(L>+e*rv(o,p>ne(e)-6 *W\(Lp\)M\(L\)-MW<o,p\)e\(o\)

*+MW g(X)W"w(XsP>du(4.53)

Cognposeeq"ations(4.se)-(ag.S3),thefollowingeqgation eaubeobtained:

Qlv\)

Mlp\)efe\)Wlv> = O-M*s\(0,p>OMX,

d.(o,p>-MW(o,p)-d.(L,p>M*w\(L,p\)

\(L,P\)

U

+

*OM$ee

o-wb(O,p>OWb<L,p)\(o,p>-wk\(o,p>-ek\(L,p>wM\(L,p\)*\(O,p\)-W*e\(O,p>-ee<L,p>W"e\(L,p>

T+BVtstuye

\(4.sc\)*ew\(e,p\)-w*.\(o,p\)-e"w\(L,p\)w*.\(L,p>

Here, rwBis an interna]pointgntegratingcoefficient.

62

NB=

\(4.SS\)

Theequatioitofdefiectionisthesameastheeq"ation(4.6).Inthischapter,asetof

fo"fequationsaffecoxKgposed bygettingegtherp=oerpscLgpttheeq"atioms(4.X6),(4.23),

\(4.xe8\)aeed, addimgtotkem, eq"ation(4.6).kisrecogitizedthatthepresepttsystemof

eq"ations kasbeconeetw"ck morecoxnpact.koaddStioen,thepreseptprocessisbasedon

tkedifferentconceptfromthe coRventionaioRe.Inotherwords,theweightfunctionsin

eackequationaredecide dimdepen\(EeRtlyofothers.kgsnodoubtthatthefinageq"atiopts

arethesameakkogghprocessisdgfferent.Fromathis,thereformofconceptmayseemto

betrivial .Ofitkecontrary, tkereformisvefyimportantanditwilibefo"ndthatthis

processfis

beam.

indispestsabgeintke foNowingformugatioms,especialgyfortheinhomogeneous

age2e2Newee<iiitii wiidwswRaswhoewsckoe#Nke

ag.2.Z.fiWmatmewtdeffassimpM:sr seegege*whggm[BwhtrEKII•-S][t(g-6]

Stweeffeaetionforeeat asig]xpEysuppoftingpoingbecorriesunknown,iteanbeitot

soivedby ehescheirckeforaxwag-spaRbeaixgiwatheaboveseetaoR.Xemtheconventienalforman-

aation,thisproblemis ffeseavedbydividgngthebeaeeiRtotwogni-spaitbeaxKEsatthat

peiRt.boehse gmaentisregafdedasaagni-spanbeamathoughexÅíesswnknewnsappearoA

63

ghetwoptewends.Actuaggy,onlyw.oisdesigmatedandtherestremainssundetermined

ghere.KRghisease,coutnectiveeoitditionsofeaptdMatthispointareutiaizedtoeoxnple-

xxkewttheaacktwgeegeeatigRs.'TkereactgopmforcesRxenRdbedecidedasthedifferenceef

shearforcesontketwoncweftds.Thas,foura]ftkitowns,e,MandtweofQwigiin-

creaseateveryseeppertingpoint.Then,itfoieowsthatabeamwithnintergxkediates"p -

portipmgpogntshastheitxpmkitowftsof4*n+4

EnthepresentformkxXation,theftondividingschegneisdeveEoped.Nagne]y,thereac-

tioRferceattkkes"pportiRgpoiittistakenasanuitkfiowftextexftajconcentratedforceand,

correspondingtoit,aReqkgatiewaofw=OfrorrRequation(4.6)iscomplegxkented.Thispro-

eesscanbecarffiede"twitho"tkddiingaRygfitermediatevariabEes.'rhe"mekftowRreaction

foreeiswrkterkfregxxthetergtrxofexteifxkaEforce,xigkthandsidegnequatgon(4.2)forex-

atwpEe,asfoElows:

g(x)W*(x,p>du=R(xo)&(x-xo)W*(x,p>duoo

WhefeRisreaetioitfofee,xoisEecatiowaofthesupportiRgpointaasd6\(x-xo>isthedelta

funetion.Knthisformeegation,abeamwithnintermediatesmpportingpeintshastheun-

kmownsofjustn+4.

Ifthisgxftannerisappgiedtothebothends,itbriwagsthechangeofvariabEesontwo

endsfreixisheewforeetofeactionfofee.inotherwords,reaetienforeeiswwk"ownwhen

theendissappertedauditiszefowhentheendisfree.lastead,shearforceeiszefoatthe

bothendsiRaRybouRdarycenditions.

ajdik.Z.kÅëopmeeceememeiSecig[tw$waetwakoeediE4-ew]Etg:gtsg"g]

ingweatimgacowacefttwatedaxxoxffiefitaoadlikeiwaRF;ig.4.:Zbytkxeeonvewtionalmanner,

thedividtwgschexEkewaththeeennectRveeeflditionhasaisobeenapplied.Then,fourun-

keewms,Wll,e,Mandeshoanftdbeaddedattkeactgftgpeintofeveryftxkementgoad.0n

ec

the other kands tke ptefidividimgscheKif]ek ogds goodagsogntkiskiitd oflloadwithtkehegp

ofthe epefffttfiepmof degtafagpmcgierm{44] as feg]ows.

gfi erder eocorwertaconeemetratedmaoxg\)entge3d,Mo,into q\(x>,distrib"tionofmo-

Etrxeitt ma<x>awadslaear ferce[<x>git d"cedby Moshougdbedetermained.The eyaregxveft

as

foggews:

M<x>=Mosgn<x-xo>+ex+es (`g•.S7)

Q\(x>= dM=du

Mob<x-xo>+a \(4.Sg\)

Wherexo is gocatiopt oftkegnomeRt goad, arecofistantsregatedtoboundary

coft ditkeft ThentheeqwivagewtexternaR force,g(x),areobtaiitedas

9(x) de.M did

= dxox(X-xo) \(4eS9\)

Aecoxdiagto thedifferentiation formugaofdeltafuRetionE`$kg], theintegfationterwxs

ofextemaal ferceare giveRaconefete fO1rmeForinstance,eheffight sideofeqeeation(4.2) is

"wrfteeeptas

q\(x>W*dx= W*du

Mo

x

,fiil;li\)>\)

to

WFftgure4.xe: Concentwatedtwonentgead.

6S

=-Me6\(x-xo>ptdW

o

2xo=--4M

Mosgn\(Xe-p>\(4.6cp\)

kskewidbeitotedthatth is gx}aniterdoes netykeXdapmyiptcreaseinadditioitag"xt-

kmeowits.

age2evZto3.AhamewtedkbegestagesE`;gs-$]g`:gnfim9]

latreatingabeam withnsteps asshowninFig.4e3, theusaalformuaationagso

fequirestkedividingscheme,inother wofds' additioRaifoux unknowensattheemodebe-

tweeneverytwosegmeRtsffwaustexise. The Rondividing schemecanbeappgiedeothis-

g\)xobgeKKgwktkkDwtaddinganyunknowns.

.E!, E212

x

LzL2 L

W

Figure4.3: Abeamwithtwosteps

Theweightedfesidgaiform arewritteft, iptthis pifoblem,with theweigktfunctioftwkas

Ld2d2wEi-du2.du2

o

q(x)Wsudu=o \(4.6g\)

Simcegtgspossibletecharkgeth ewegghtfgptction everysegmeftt, ekeaboveexpressioncagR

berexwrittepmasfoggows:

L,d4wW*1du+Elf1edu4 L2 d4wdx4

W*2dw+oee

bo

=g\(x>W",dx+-g\(x>W"2du"eee \(4.62\)

WhereLiiscoordimateefigh steppednede (1$i st\)andW*iistheundetermiftedweigh\(

f"Rckioitforekeithseggnentein asirwigajrwaywith theuni-spaNbeam,theweightfunctioft

istakeenindepeptdentay.inorder toderive theequatioftsof shearfoffceeandgnomenttwat

aninterioffpointpi,thefuRction W6ofeqxgktioit \(4.gg\)andWMofeqk}atiofi(4.g9)are

empgoyedastheweightfuftctrioms.S bIRce theyare comgxkenthroughaiMheseggx\)eRts,all

t.heintermediatevaXuese(Li) andM(Li) ftrecaftcelied.Fina Eiy,theycanbeexpressedin'u

tkegnatrixforgnasfoiiows:

ec\)i\)-[N]:g[21-,L q(x)w6(x,pi >du•( 4i$,pi$L i)(4e63)-

Mapi\)-[K]SE21+mM(O>+M(L)fu\(x,pi>dx, \(4FPi$Li\)\(4e6"#'\)

Wherethegnatrices[N],[K] &Rd[L]are coefficieRtsrelated totheweightfunctionsand

o"are1Å~2masxze.+

KfthewwreigktfunctionW,*• isreplaced bythe fupmction W"e,Ofequation(4.20)with

E]=Ei2itheequatgonofsiope ff ataftimtexiorePOitRtPiiS obtainedasfogiows:

M\(L,> e\(L,>ma(L2) e(L,)

eepi\)-[A]g[21+[B] ""

+[C]eo

e e

M(Ln-1) e(Ln-i)

M(O)-}-[E]+[D]M(L> e\(o>L,e<L>-

,

q(x)Wgi \(.x\),pi\)du

L2-g(x>Wg,(x,,pi)du-eeeLl

\(4.6S\)

67

Here,theec(xatriees[A],[D]andEE]axe1Å~2and[B]Eeg}d[C]afegÅ~ninsize.Asfognd

iiteheabove,theiflserxKkediatev3Xgesofe(Li)doftotappearduegocanceggatioptwhige

M(ZJi>3ndg(Li)areenotcancegReda"togKkatSealSy.However,gheycanbeeasigyreRxkoved

freggEtheabovewsingequatiowas(dag.6S)and(4.66)byEettingpi--mpLi.Aftefagl,aigtheinter-

gwaediatevaguesaredeletedandequagioR(`4.67)eanbefgnaglydescribedongybyvaguesat

thetwoendsasfogEows:"6(p,>-[A]Z((2))+[M(o)M(L>](B)+[e(o)e(L)](S)+[p](4.66).

rrchehutwtaxkdenotesahatsemealterationsafegivenand\(P]denotesiRtegfaiterms.The

egwatieRsefdeffeegionareobtainedinaentirelysi3meilafway.Nagskeay,theintermediate

valanesofW(Li)ande(Li)arecanceg]eda"tematicalEyandne(Li)and<2(Li)aredeietedby

axsiitgeq"ations(4.63)and(4.6`#).Fiitalforj(xkoftheequatioitcanbewrittenusingsimiiar

ceefficieRtmatricesasfoglows:

wevi>-[qIX[21+[G]g[B+wr.M\(\(B+m

Thus,asystemofsimultaneousequationscanbecomposedbygettingpi----Oinequations

\(4"63\),\(4o64\),\(4e66\)and\(4e67\)e

ajexee3FgrmNdawh$wafferasptim$wa*gewaeeeqgsimEdiajl-me]\(d:g-kN

intteatingabeamwithchangefu1rigidity\(iRhomogeneousbeamillcg"dingthenon-

prismatie\)fogitovvingcoitvenkionalformugation,aspeeialfundameRtalsolutioitcorrespond-

iengtoaspeeificproblemrashouidbewsedl4-2].Otherwise,thebeatw,ingeneragpfobgeifecks

oghefthantrhespecificprobgeffifE,hasmaainiybeentfeatedasthesteppedbeagndividedinto

severaRsegnencs.XnthisstudyageneralteehniqgewightheeneRdividingsckeffnehasbeen

devegeped.Thisschegxxeeepeswiththediscopmtinua"sehangeinbeRdingrigiditybychang-

os

imgthefuitdagxftentagsog"tigrkeveryseggneRgaccewdixtgtogtsstateofckange..

"ffheweigkfttedifesidasalfQrmforthispifobnemacanbewsiittenas.

Ld2 m\(.>d2W-g\(x>w*du=o

o

kotkgsstage,theweightfuftetiowai.si[gndetermipmed.inasifi]iiarwftywiththeprevi-'

e"ssect.goits,fiX]eweightfuitctieitisdecidediptdependewtiyiiteveryeqgation.Foiftrheeqkta-

tiemsofeaitdM,itispossibletodesignatethefuflctionsofequations(4.ig)aitd(4.i9),

respectivegy.Tken,thegdeittic31eqaatioptswithtkoseforthegni-spanbeaffRofequations

\(4.16\)and\(4.as\)eanbeobtaifted.

Ofttkeotkerhand,itgspmetsePf--evidentvvhatfunctionistobeemployedfiftorderto

derivetheeq"ationsofeandw.Effoweverthatmaaybe,]etusintegratetheabovebypart

thffeetgxxgesbyaettingW*=W*e.Thenitbecorxxes

,L :2Z2,m\(x>ttllYIw*edx

::dm<.>d2Ww*,"..m\(.>a2wdWgL

L+ew(.)dwd2wgLnvdwdm(.>d2W'ed.

dudx2odxdxdx2(4e69)o

FfogxEthiseqeeation,itisfoundthagthefuptctioutsftkouldbedecidedsothatRtsatisfTiesthe

foftEowingffegation

ad.di(x)d SngLb'=b<x"xp>(4"7o)of

ut(.)d2Wg..gmsgn(x-p>

69

Procedngrekeretrfterhasveochoicebutdepefidson tlaestateofckaftgeinghebepmding'

migidgtyE2\(x>.gnotherwofds,theweightfuitcSiextsk ouldbeehangeddependiRgonthe

cencreteforgg]efEf<x\).georexagxkple,gfthefunctioxt isgiveaubyaNemeafff"itctioRas

Ef<x>=Eolo<bx+c\),theweigkefugectioitisdecidedas

2bEoIo

X-S+x<lnlbx+ci-laibp+cl)-(x-p>\(4e"172\)

II'hen,aeompactexpfessRoitofesiwtiaarwiththe previousoRehasbeensuccessfuily

reachedasfeleows:

s<tv>..Ef<.\)ed2W*e.MdW$eew*,`

du2du- o

\(4.:il3\)

ksho"gdbeitetedthattheweightfuRetiofisforexkomentand sliearforcecanReverbe

pxodwcedfreftkeheweightfunctienofequation(4.72).

SizzkigarEy,theweightfunetioftWlw9fofdefiectionisseRectedtosatisfyth efellowingrela-

di

tion:

d2W"\(4.74\)

Tkten,tlaewagwtfuneciomeWWisciecidedas

w$ \(x,p>.Sgn\(X-p\)\(x-p>2cW28bEolo2Ept+P

Å~{ll-+x(1itibx+ci-1eniop+ci)-(x-p)\(4.7$\)

70

-

e]rkeeqmpak;iosuofWksgivewhythefoanowingeedittgPacteqws2knciea:-

w<p>nt\(.>wd3WWMm\(.>ed2WW-MdW*w

rffrkefuncSioRof(4.ewS)aXsohasRorelationtetheotherweightfunetionsefWg,sw"Mand

We.'Eherefore,itcanbesaidthatequatiopts(73)aRd(76)areowi"gtothepresentform"-

latioRpfocess.

Besidestheaboveli"earcaseefEf,ghesiaxftilawwayteffeachthefuitdamaentaksogutiopt

withaeEosedformisagsovagidsincetheeqaxatien(4.7X)eaasintegfatedameaEySicaliyand

sooRasfafasEfisexpressedbyapoXynomialfuRctioniRciudingquadraticandcubic

fuftetioax.Mofeevew,thereispmoseriousproblemiitthisformwEationsehemaeevewaifthe

seateofchangecaRnotbeexpfessedbyasolefunctionewoybyaniRtegraafunctioR.Be-

causetheweightfuRctioneanbeappgiedteoitiyanassigRedsegment,sevefaliRtegfal

fuRctiescseaftbeconllecteds"chtkattheyfitthegivenfunctionoftherigidity.Smalg

eenrows,pgssibgyincEgdediiteertaincirc"ftxastaRees,aretobeitegEigible.Ffoifxkthisfact,it

maybeeneughtopfepareseveragweigksfunctiensatmostasoeeasioRdemaands.inaRy

ease,tkeiwterxnediateterrrxsofWY(Li)ande(Li)arecancelEeda"togxxaticaX}ywhigethe

termsoftw(Li>andQ(Li)aredeEetedbysixRpges"bstkt"tion,whichissigxkigarwiththe

steppedbeaxent.

agca2eagbeuetmentgffdi"]ijkknimimgegwaLfl

gnEq.(`g.a8),(`4.:2S),(`#30)anCg(`434)thedemaiRiwaaegragiswrttteftasfelEows:

wkereg<x>isgkeexter"agforceandw*expffessesWb,WM,W*eorWW.kogermofEqq

71

\(3.ms\)gkeexteffg]aRferceisa pproxigxk3ted "tkllkzirtgthepointso"ree\(n=2>asfollPows:

\(4.78\)

The fuitctioRz*ksiptrodgeed asfoEaows:

w2z"=vvs \(4.79\)

The itextsgep,thedoKmtainintegraE ofElgx. \(4.77\)canbetraRsformedintothefoklowiemg

forxptinkermofEqs.(4.79)and \(4.78\)

g(x>W*du=g(x)Voe 2z*dx

*-$tila\(x>-z*ss LL+w2

*-SdS\(x>-z$SsmL+Ez*<xiOb-1

>ri(``4'.80)ffn ordeftogdewtifyth estrengtkofthedegtafuitction ri,thefoEgowiRgequatieitcanbe

ebtaincdfrerxaEq.(4.78)

m.

2*moewfdg*-`gltifisy<.\) Lc(s>g(s)

"-lriY<xi,S\)dx

wherev2y*=-b(x-xi)e

aje3 VERwaÅëmeONew[R]asPmeSENPffimbooRXTww

Severaganalysi$exai[Kkpgesaresubaxitiktedtoverifyth evalidityefaitewalgorithlli.

BeariitgtwKxxiitdghattheeogTa epwtixtg systemiseoitstryetedoftapersonageompvter,the

pfogwamiseodedwsingtheViseaagBasic onthegxxachinewitkWindowsNT.Ktdoesnot

KKRattereoameiitdividuageaiewiatio"pffacticaily ,eventhoughthefeisdiffereneeintheqaxal-

pt

xty ofaagerkhtw.However,Rn ceeniputing awaoptimoiaEpifobEemianwftkichmotanyiterative

72

Tabge4.2: ResuttsefcoReinenows'beaffxxiwaFig4.4

Mo/PÅí<]b/ICPSL*(Eofo /PL2)Ro.3L/)P'Ro.sLIP RL/P'1["iwte\(s>

CextveRkienaX -().038g69 a.a4sa- O.am327Ss:#ltg2.`::#i08-1.3opS -o.1%sxei61

Pgresewt-O.os8g69 1.a4Sl- o.ooo3:ef\)Ts`ig12.`tgeg-a.3ees- o.g%s2go

calculatiesuisreqaired'itsqualitybecomes animporta"tissaxebecauselowering ofeffi-

cieneyandincreasiRgin eostofsnchwofk sbecemenottobedisregarded. Thus,comput-

iRgeimeisgpteasuredth xeughiOeetimes Åëagcugatioit.

aseg.Åëdewadnwa*wsbeasjmpkKW$thSimptYsugeRpa)}mupsimts.

Thefirstexampge laxsacontin"o"s beamwithnincintermediatesimpgysuppoffting

peiwts\(divideinte teftequal pafts\)s"bjectedtoseveragconcewtratedforcesaftdoRemo-

mewtioadasshowptiasFige4.4.Res"ltsfrojtwboththeconventiomagandthe presentcom-

putgfigsystemsas weNastkeircomputipmg timaearegistediRTa bXe4.2.Sincebotksystems

gavethesameresults theitewalgoritkrr] isprovedtQbevalid.Coixk ""ep"tmagtEmaexscut

dewpmgftarateef 3bout E/i6inthekew system.inadditioen,red"ctien inmeumbersof

uitkftGwmsisaiso pfofitable.Act"ally,gt" gsi3inthepresentsystemwhiEe itgs44inthe

coftveptiopmaSsystegxk.

034L L=tmP=ZN'

O,14L M=PL,El=1e6Pa

O.45Lpm.2P

O.7sc-- ttop

L

Fggure4A:AcowtftnueusbeagxR.

73

4e3ekim*KXkageweewsbekwag

AwaaitaaysgsexatwpgeasskowitiitFgg.4.S.ftswtSgizedift our verificatioge eftheptew

a]gercitke gptfoifanininegxftogeneeusbeaffffx.Tkebendingffggidity is supposedto ckaRgeac-

cordinggo&hefogRewiitg gi"eafffeenction:

El(x)=E,l,(bx+c)

P

L

geiguwe4.S:Aitonprismadeeantilever

Ei\(x>=Eolo<bu"fec> \(4.g2\)

Whereb=-O.Sandie=1imehiscalcasEatiopm.Theeenventienag waywhiehapproximate

thebeamgoabeamwiththweestepsafetestedinaddition toth enewalgorithm.Their

reswitsandtixnefofcoorntpwtatioRareshoweminrff!abge4.3comeparedwiththeofeticaasolu-

tioms,whieh3feealougated byha"d.ThencwaEgorikhwtisquite exeellentinbothofpffeci-

sieftandspeed.Co"trary teahis,itm"sgbeffecognizedthat the conveRtional wayofffe-

piaegRgbysomestepsgivesinaec"rateresugts.

-

TabEe4.3:ResaStsoffton-pristwaticbeamiR Fig. 4'•Se

MoIPL <il)/PWL*(Eofo/PL3) eL"(Eoio/PL2) Tigne\(s>

Tkeoxetgcaig go.3gag o.6a3rg

Corwewtiowaig.OOOO MemmOe39as"#' Oe62ms 12

PresewtX.amO g.o\(moa.3g62g o.6a37a Oe6

74

TabAe4A:Reskxktsofxton-pmsmaatcbeaminFkg. 4.6

two/IPL e,!P WL*\(Eo2o/PL3\)WLt*\(EoiolPL3\) eL* \(Eeie!PL2\) eLl* \(EoioIPL2\)

Theeretscaalg l 1 2.74S3O.g3333 2.i760 i.s

jF"xesent 1.cooo 1.Crm 2.74S3O.g3333 2.I760 1.sooo

TheRextexaaxxple is asitwigarbeagxxwithchangefulri gidity .Thebeamiscomposed

oftwosegne"ts, a eonstantpartandafiopt-prtsxtxaticpaptas ShOWutiitFige4e6e

Thebendingmigidigy ifith enoit-prismaticpartehaRgeswith a Riiteaffuitetiell ef oequataon

\(4e82\)\(b=-2!3andc'--513\) .ItcoineideswiththecoxtstaRtpairt atthejoint.As previously

imentioned, itisReediess todividethebeamintotwopieees eveninsucha probiemac-

cording to thepifeseittformulatiowa.Th eresuksarelistediit Tabae4.4together withthe

aRalytieal solugion eaaeulatedbyhand.Theytestifythevagidity ofthenewalgorithm.-

EZ=EoioEi=Eolo(bX+C)P

i::xg:#L-`"mp--2t

Figure4.6:AcaittiReverwithdiscontinuous Ef"

'

;

7S

wabge4.S:Res"gtsofUni-spag\)beamiit]Fig.4.7

e\(o> e\(4 Q(e) Q\(L>

ConveRtiomaftO.4ki3g9-- O.i4722 e.9g667 -i.7SooO

PreseRtO.`#63g9-O.i4722 O.9i667 -ft.7SOOO

ag.3.37aLJkigg-sgethwabeewkwt[rkkkwaeeme]twagff"rce

EntExgsexampge,"mi-spanbeamwithkexterff]alfofceis skown iRFig.4.7axxdtkeexter-

ptalferceg\(x>isdeseribedasthefoggowing

g(x")=ex2+bx+c \(``ll'ess\)

wherees=g,b=1andc=2.gRthiscase,

&

L=a andEf=i•

'x

"Jt

.

Figure4.7:Uiti-spanbeaixgwith aRexternai ferce.

Thenasjneericalifesugtsarecomparedwith thecgassical BEMasshowninTabge4.S.

ag.ajÅëoNcawsroNs

NewaEgorithmswigftftoutthedividin g schemehavebeenestablishedforbendingprob-

t

76

Eegnsefgkecentim"owsbeaffifxiitckudingtheifthomogeneoasbeaxxRbytheBEM.Thisstgdy

is perfergKkedferthepanffposeofappllyiptgtheftrcktothepractgcalloptigifxaldesignwerksona

pex$enaXceggipwter.Sfincethecagcaxgataopmfofftheoptiffg]agdesigpbytheGA,ferinstance,

ffeqwtresggiaRyrepetitivec3llcedatiopmRwaalgorithgx}witkeexceglewtqk]aNtyiswweededinerder

t.o ffeducethecgstofsnchworks.TkepresentaXgorithgxksprodwcesomeadvantagesof

redaseiitgthematifixsizeasweggascogxlipwtingtgftxkeaitdofdeKxiaptdiftgeexaschfewerre-

soaxrces,suchasmeit]oryaftdcapabigity.Xrherefore,itisnaturaRtogmprovegreatRythe

effgciepmcyofnxgmere"sgyffepetitivecage"iatiollsinpracticaRworks.Mainpointsofthis

st."dyaresummarizedasfo!iews:

g. Ferm"iatieRprocessisgmpffovedt6deriNyethesysterwofsgmugtaneousequations.Asa

ifesutg,tkeyafewefermedtebetwachtworecompact.Thisfefermisofgreatbenefitto

configuriitgandsimplifyingthepfegwambasedopmtheRoit-dividingsehemae.Thisim-

pfeve#ikewtispmotawiviagrewritingbgtaindispemsablenotionforotherprobgegKEslikethe

inhomogeReousbeam,etc.

2e TheRon-dividingschexrkehasbeenesgablishedcontrafytothedividingscheme,which

is fieededintkeconventiofialfoffrnulatioftontheeccasionsefsimpiysupportiitgpoiRt,

steppedbeagxg,inhoxnogeReo"sbeam(inciudiRgdkseoRtineo"sgychaRgefuSrigidity)and

cenceRtxkgedmomentEoad.

3.Ageneragsehemefortreatingtheinhomogeneogsbeamincladingthosewithdiseon-

tinaneusEychakgefuigrigidityhasbeenesaablished.Thisschemeisalsobasedenthenondi-

vidimgscheffgie.

4'e rif"hedowtainintegxalistfeatedbythediscfeteiittegraEmethodwtiliziptgthepoiRtsogrce

aitdisexpfessedbythebouitdafyEwtegragandthesuxxtofthevalueatthesoixieinternag

poinks.TheinteriorcegXscanbeitotused.

77

R]EEi{ki;msNÅëES

E4-g]C.A.BrebbiRew]dS.Walkeff#980Bouadaffyeneffg)eptTechniquesinEifxgineeriitg,(i9g()),

[K#-2]]R.Butterfgeid,Newconceptsgg]ustwatedbyoEdprobgejees.E\)eveRopxnentsinBo"ndary

E#effif]entMethods-X,ed.byP.K.BEkiterjeeandR.Bkitterfgeid,Loitdon,(1979),Appg. Sci.

[`g-3]L.Davis,HaRdbookofGeReticAlgorkhxns,VanNosmandReinhogd,(1990),AD" "-xvlslon

efWa<Sswert.h,inc.

E`#k#]it.M.Geg'fandandG.E.Shilov,GenemagizedfgRetio"s,Vol.g.ff'ropertiesandOperations'

\(i96`#\),NewYockandbondeav:NewYerkandbondon.

(4-SIIT.A.Cxuse,MathematieaEFoundationsoftheBoundafy-integralEquationMeth odin

SogidMechanics,AFSORTR-'X'-ioo2Repert,(1on7).

l4-6]waRaka,M.andTanaka,Y.,BowadaryEEementMethod-Fo"ndabonandApplication'

\(19g2\),Maruzell.

E`#-7]R.Yww1kiandH.Kisu,EgastieaRalysisbytheBEM(inJapanese),(i9ew),Baifu-Kan.

if`aj-g]Kamiya.N.,FeuRdatioRofttheBoandaryE!ementMethod,(19g2),Seience-sha.

[di`#-9]Tewkaka,M.,Masljmaoto,T.andNakamagrai,M.,BoundasyffegneRtMethod,(geqk),Baf"-

kafl.

E4aO]C.A.BrebbkaagMdS.Walker,Fo"itdabonoftheBouadralryERemefttMeahodandits

ApplScation,TraxxsEatedbyKajntEya,N.,Tanak3,M.andTaxiaka,Y.,(g981),Baifu-kan.

[`#-IX]Kamiya,M.,BogitdaryElllemewtMethod,(g9g2),Saiensg-sha.

78

mmEgeS

ANMYws@WNeNecffNffIAkRiSEATÅëONEbWÅëTgONwaOBEEms

SRrwmeOpm1arÅëTXON

TheboagwadaryeaerrxkentgwaethodisaweiXestablishedRaxmeriealE\(kodeigingteehanique

fewtwaRytypesofgiftearpfebgemsiRengimeeringandappliedscience.Recent

developmeRtefBEMbynvrxkerowsreseafehers[S-g]t-\(S-S]kasdemonstratedthatBEM

cafiaEsobeaxsedtosogveitonEinearlheateonductioRprobgemswithtetwperature

dependenceeftherfi]aEcond"ctivity.inthefoiiowingtheseresukswillbedeseribed.

C.A.Brebbga,S.WalkerES--gjewdP.K.Bancrjee,R.Butterfgregd[S-2ihavefirst

sollvedehesteady-stateitext]iftearkeeatcondasctioitequatiofiwiththetemperat"re

dependenceofthethexmaEcon<gaxctkvitybythebouitdaryegementrx}ethod.Enthis

approaektheeerrespoptdimgdifferentiaEequatioitistreatedasthePoissomaeqgation.

Becausethegovermirtgequktieptisptonikfiear,thevoi"meiptegraaisincR"dedinthe

bothndaryintegrageq"ation.AEtheugkthevog"gxftegittegraEcanbecomputedbythe

cXassicagBEM,theadvaRtageofBEMisReggected.Ofttheotherhand,because

teKExperat"regradieRtisevaigatediittheinternagffegiofl,thisapptoachitotoniydemands

a\(E\(ggtgomeaEcaEeulatioensoftinevol"gxRekntegralbutagsogxkakesdegreeofaccuracy

decreased.

Yee.N.Akkaxffktov,V.N.Mikhaggov[S-3]andR.BiaAeckg,A.Nowak[S-4\)kave

appgiedijKlirchkoff'stransfoffmagioaxtothesteady-stateueoniimeenrineatcond"ctioiteqmpatioft

wfigketkeeeKgkperatwffedepermdenceoftketkergnalcoxed"ctivfityindepeRdentgy.The

79

goverwagftgeq"atioitistramsfermedgwtoLapXaceequ3gioan.TkistwansforjtwatioretramsforrrR

theitoitgfiwaeaffgtyoniygfttekkebompwhdarycoitditioaus.Tkerefore,itisanmermecessaryto

establlfisktkeeenkptowmequawtityaeeoggxpaniedbygkewaomhiiteaffgtyipmehedetwairk.Tke.

doxxiafiengntegrallfisfioemeeeded.However,sincetheinversetramsforffygaifioptissemetiKxkes

diffgcult,thisKKxethodcanptetbegenerkEgyused.

Oxtgkeotherhand,Kamfiy3,N.ee.aE[S-S]haveproposedanapproachby"seofthe

waewvaxiabge' insteadofthetemperatwre.ButitisfiitwgtedtosoffnespecialcasesoftheL.

eegxRperatEkffedependeneeofthethermagconductivgty.-Asstatedabove,it.isessewtiagtoseekforanewscheme,whichavoidsthe

disa{SvantagesofKgchhogff'stffansforewxaeiortaRdtkegxiethoduseeftheptewvariabEe.Xn

tkfissSudy,thedisereteiRtegralgifxethed[S-6]ksattegztptedtoappiytotheptonSinearheat

coemd"etgefiprobgeExkbyiitcempgratigegitiwtothebe"rtdaryegemepttxxkethed.Theiteftlinear

ecgasatiowagstransformedliptoaneq"ivaEeittforxxksethatitgxxaybecomepossibgeteapply

thediscreSeintegraggxftethod.Thgsapproachisftotginegtedbythetemperatuffedependence

ofkhetkermaagcomed"ctivity.inthisthesis,then"gxiericalaptagysisiscarriedo"tfertwo

exarcifipaestwoemediifKkensioRandthecorwpwtatieRairesugtsarecomparedwiththeexact

sogwtioen.Tkenoit1inearkeatcoRdactiowapffoblemswiththetetwperatuifedependeneeofthe

tkermalceandactivityinthetweaneEtft}ffeedigxxeRsEomswilgbecawiedo"thereaftef.

ss.ZOWmsRIMasmaODSFORSenWIilNGmsATÅëOmoWCTKONEQIaLIATMON

lathissectiowatwoapproachestotheRonlinearprobgefxisofheatconductienwillbe

descffibed.lathefirstmethodtheEincarizationefnoftlinearheatconductioitgoverniRg

equationea"beaccoNxapXishedbyeheKirchhoff'stfansformatioR\(S4],ThesecoRdtech-

waiqge\(S-scdefiResghethermea#cewadeeceivityandthevariabEefiy=\(k+a>Tasthepmewvari-

abaesgitseeadoftheeemperat"re.Wkeasthethergxxalcoptduetivityisliinear,expeftentiagor

pewerfufieegews,thegovernfigegeqasatiemsoftkeehermagcoeed"etivityapmdthevaffiablev

affegixteaffe

sw

sseuetefi Aeeethedasang]KliigeecRgkkasff9sgmeifitsffgemawhtw

Ceitsidewatwg-dimniensioitag xegionsuwith bowadaryr.Thegeverning eequatgoas

forahe sgeady-statekeaaeowaductiewawithghekegKupefatwre dependeneeeftheShermaEeon-

duetivityisdeseribedas.

eeTeeT=o e+oxaxoyoy (gitsu ) \(Se1\)

k-f\(pt (inst ) \(S.2\)

wkere Tisthetexnperat"re, K-isthethermagcon ductivityaptdfisthefaxnctioftthat ex-

pressestketewtperaturedependeRceofthetherg]}al eon\(Euctivity.1if"hebouRcilary conditions

ewegivenasfoggews:

DSscichgetfseogeditioit: TzzTo \(onL\) 63\)

0T

k=Ne"maanit'seeptdiSgopt: 9oOn

\(opmr2\) \(S.4\)

OTk=Robim"scopmditioit: \(onr3\) \(s.s\)

ThegovemaingEl<g.(S.a) gspteRlifiearbeca"seof thetegxxperaturedependence ofthe

thergnalcoitd"etivity. However,itcanbegincargzed"simgtheKirehhoff'stransformadon

\(Se6\)

Tke orkgikallftoniinearEq. \(S.e\)isred"cedtothe fe g]owillgeq"ation

'VNt::xO \(S.7\)

Tbeboabeedarycoptditionsfor Elq.(S.7)areobtained

pt:U\(To> \(enL\) \(Se8\)

dv0T=:k= goOre3me

\(onX2\) \(Se9\)

gg

ot--

whefe Tf=U-i [tf].Theinversettfausformatienisexpressedasfoggows:

XRthisprobgewttheptonlgRearityofeheheateonducgionequationcanbetransfowmed

t6ehe n6RginearboandarycendgtioeeswsingKirchhoff'stransfoexxxationandeheeresukiRg

govemingequatioRistheLaplaceequation.TheKirehhoff'stransfeffmaatieitstiglremains

the itoRlinearityofaboundafyeoRditienofthethirdkind.'ffRheonlynoniinearequationis

the bouRdaryconditionofthethirdkiRd."rhenongineafity,however,onlyiRvoEvesthe

boundaryRodesghathaveptenlimearbouadafyeenditioits.TkeNewton-Raphsen'sffnethod

iswsedgosogvetkeRonaiReawgmawixequatioR.However,gtiswaoteasytoobtaiRtheana-

EyticaisolutioRfromEq.(S.1i).

sse2e2 ARxkeetheddytheww$ffetXkewewwawhmbkee

ThisimethodbyuseoftheitewvariableisdiffereRtfromtheconventionaX

Kirckhoffstramsfermation.inthisappreach,thegovemeiitgeqaationsofthethermalcon-

dactivityaxdthevariablevislinew.Beeausethethermaleoptduetivityandthevartabgev

are funetiomsofthetempefatwxeT,ifthethermaEcoRductivityk\(T>aRdthevariablev

becogxketkekg]owptfunctioR,theteffffxperat"reTalisobecomaestheknowfi.Thismethodis

describedasfollows:

jllq.(S.a)istransfoffmedintothefoglowiRgeq"atioit

0kOTOk0T++kw2T•---o(s.g2)OxOxeyoy

WhenkheehermagÅíowaduetivityisdescribedbytheexponcwatialofthepowerfunetionsiR

terms oftketexxkpewatwre,theequagRomsoftheghermalcoemdgctivityarewrigtenas

wkeeffees andbareconstants.Now,eketherEx]aacend"ctivgtykisregardedasthencw

ee

vrkwhabEeaeedthegovemtwgeqasatierethatthevariabXeK-satisfiesispropesed.

BytakipmgtheLaplaeeoperaterofjE<;Rs.(S.g3)and(S.g4)ffespecgively,thefogiowiitg

eqaxatiowasaifegivegeas.

.ek3TOkOT++kffr2TW"k----b

-t

'

.OkOTOkOT+kv2Tgogt\)Vdek=

+(S.16)OKOKoyey'vv

EfEq. \(S.12\)issllbstitgtedifito]Eqs.\(S.iS\)aitd6.16\)respectively,Eq.\(S.gS\)isequiva--

aent to]Eiq.(S.a6).Thiseqgagioniswrittenas

TheappwopriatebewadaryeeRdkkionsfortheEq.(S.gS)aregiven

K-rm--aexp(bTo)wako(S.g8)Ok,3KT-- bgo=(s.gg)Oneno

3K'i=-h

o.:E:<aogk"'ioga)-Tf(s.2o)lv thesimilarway,theboundafyconditionsofEq.(S.16)areagsoobtainedas

Ok'3k==goiogb(S.2)OnOn

Okg=-h

WkefttketherggiagcoptdanetSvktyk\(7>isdeseribedbyaaipmearfunctiopminteri[nsefthe

eeKxkperatwfe,theeqaatioftoftkergtrxagcofldgctivityis

whereaandbareconstants.Etfavariabgevisdefgnc\(gasv=\(k+a>T,tftxegovernfing

es

eqgatSoflSsobtained

TkebowwdawycoitditgowsefthevariabEevawewrittewaasfoXUews:

v=To[k(To)"a]=vo(S.26)OgROv==2go(S.27)OnOno

=-2h-2hBeeewseShefunctionskandvaxeusedasthenewvariables,thegoverningequa-

tiomsareexptessedbytkeLapgaceeqi]atienwgthnoniimearbokxndarycortditions.Al-

tkoaxghthisgnethoddeesftetpteedtkeecoffifxpgicatediRveffsetransformatgon,itiss"itabXeto

tkreecasesofthethermagcoitductivEtyongy.

g.:swwo-pmmaNsgoNMwaoBmEM

EnordertoavoidthecompiexinversetraRsform"singtheKirchhoffstransfeffma-

tion,aptdthegimitatioitofthreecasescfthethermaaSconductivityaxsiftgftgxRethedbythe

"seofthenewvariable,thediscreteintegralmethodintheChapter3canbeusedfor

steady-statenoniineewheatcoRductioftequatioftES-6].inthisthesis,thegoverniengenon--

gipmearheatcoitductieftequationistransforxifiediRtoanewformtowhichthediscreteinte-

gragmethodlisapplicabEe.TkenewequatioRcanbesolvediitsteadoftft}eorigiRaEequa-

kiowa.Thevogumaekwtegratioftistfamsformedgpttethebeuptdftjryiwtegragaitdthes"moftke

gntemi}aEpegwts.[RrftxisnewapproachfiswaotffestrictedeotftMeteKgxg;beratewedepeitdlenceofthe

thermaEcopmdgctivity.gtsvalidgtytkffeugkone-dlimensionagexamplewiiRbedescribedin

sectioitS.S.

gr

J

-

Ste3.fiTbeegeee"wyg@weermgeegguaee$geeemedifiasboeeewdageyec*medifiwh*ees

Tkegxkaiwaobjeetiveoftkgsstageasko ebtaintkegovemixingeqaxaeioitfrom ]E(;ll.s.(S.X)

and(S.2).BytakgitgtheSLapgaceopefaterof Eq.(S.2),thefirststepgeadstoas fo1gows:

,v2k.f\(idyOTOT.OTOT.f,\(T> O"TO'"lz-•

OxoxOyOy o+oO"XOky\(Se29\)

Oittheotherhand,lk.(S.1)istreatedas

OTOTOTOT.f'(ipt)umrm-'-+kW-T=OOxOxoy.Oy\(S30\)

TakingtkeE(as.(S.29)aitd(S.30)iRtoaccoaxntwh eRf" (dy#Oandf'ptto,the foggowiRg

eqeeatieiteawabeobtaimedas

v2k.f' (T)lf'(7)q"f(T>f'ff(T>v2T.Åë(7>v2T

ft (T> \(S3X\)

TheEq.(S.3g)isthegoverniitgeq"atgopt 'whichwewanttoget.fffthe variabgeis

deseribedbythethermalcond"ctivity,the boundaryconditiomsfortkeEl\(g. (S.3bare

expressedbytheknewftfunctieft3sfolgows:

Dirichiet"sceitditiepm: k=f(Te)=ko \(S32\)

OkNeumanit"scoitditieit:

f' (o

on=ptf\(dyo \(S.33\)

okKi>--

Robin"seoRdition: \(S34\)

ThisgoverptingEq.(S.3i)isnon ]SnearwiththeRoniinearboundary conditions

(S.32)•-(S.34).inthefoEiowingthediscussion ofjEq.(S.3g)wiigbetEgrgkedto.

g.gff"(:]Sit=o,k(lll)=constexpt.

TlaegoveffgxingEq.(S.i)becomesgheLapgace equationas

wF2•T-.o \(S.36\)

TbelapEaceequatkoncanbereseEvedbythecgassiealBEMe

2.gff" \(Z>=O,k\(:l\)=a+bTe

gs

Here,aandbarearbitffawycoRtstanes.Knthiscase,thegeft-kandsgdeof]IEq.6.3ll\)canbe

rewitteftastkefegRowipmgforma

w2k---v2 \(dibT>=bscp2T\(S.36\)Opttheotherhand,theffight-handsideof]Eq.(S.3X)canbeexpressedks

Åë(Z)N<:72T=brev2T'ew(S.37)rifNlkterefox,EkR.(S.3a)beeetwestkeideRticaEfelatkon.llRerdewtoebgainkhesoEwtgowaofE<g.

\(S.g\),gheafiothervatiabUeKRusgbedefined.Enthethesis,thisvariabXeisthesagxkeasthe

vinEq.(S.2S).ksgoverRingequationandboundaryconditionsareEqs.(S.2S)-(S.28).

3.RfthetegxtpexaturedependeRceofthethermalconductivityisexponentgagorpower

feeasctiofts,di<7>:()}.

TheEl(g.63g)beceexkes

XtisobvioExsehatjEq.(S.3a)canbeempaoyedifial1sortsofthetemperataredependence

ofthethexmalcondgetivity.El(g.(S.3i)andboundarycondgtiems(S.32)t-(S.34)aregeneral

fortws.

$.$.2rff'kene'ewswkoexewgewsngdiiseec:eetetwwuptmethed[

gfbothsideseftheEkE.(S.3i)awemultipliedbythefundaffMxentaisolutiones*(e,p>andis

integratedoverthedotwain,thefoilowiRgequationcanbeebtaiRedas

rrcheleft-handsidecanbetfaRsfotwxediRtobogndaryiRte{pstiowaas

cta

lp>klp>+*<gp)Ok(th+k(QOts$(e'P)dr

r

as

Tkefxenctgonv2TofEq.6.as\)gsapproximatedgs egaugtE]ediseregeintegralgxxethed.inthe.

Chapter3]likft.(3.7)isgiveRas

(S.`#•g)

wkereriSsthestfengthofdeltafunctfionarid misthe ftumberofthesourcepoint.gfEq.(S.4X)

issgbsrk"tedgptojE(x.(S.`#e)wheitn= 2andf-wwT,tiIk ewhgkt--harkdsideischEks]gedasfoggews:

op(T(th)v2T(thee*(e,p)dsu(oj=

st

Åë(T@)

st

\(S.42\)

]Eq.(S.3I)caRbewrittenas

clp)klv>+*(gp)3k@+K.@Oes*(ep)dr\(thes

Onr

OP2

\(Se43\)

wkerepistheineernalpoimterghebo"nday poim•cip> isthepasitiertcoeffxeient.Efthepoint

pisgocatedonabouRdary,c<p>=S/2.Efthe ]POiRt,piS gocatedinsidethedept]ain,thenc<p\)=i.

Wft]eitn=2amdf=T,gfEqt.(S.4X)is mukipiied byfufldtrmepttaPsoiutioRua* \(e,p\)andis

fintegratediftthedomaipt,Elq.(S.4g)canbe ewpttenas

&(e-p,)ts*(e thd9@(Sq`44)Byputtinga2keintegratienterExRsoftheright-kandsideoitthebempday ,theequatgonisob-

taincd:

gy

clp)T<pp>+ee*(e,p)OT(oj-i-T(pt3us"(e'P>dr(e

r

2n.

::ries*(ppi,,pp>(S.`4S)t=1ll

Seaj

TheifExainproblemishowtrosolvethebouadaryintegn1Eqs.(S.43)and(S.4S).Becaaxse

itis agixxostiasftpossibletoobtaiRtheanaiyticalsogutioR,wedividethebeaxndaryintotwanyeg-

emaents, thenineegwateeachekemefitandexpressehebouasdaryintegralequatieitwithehesugxt

offanitetermofinte.qvations.IfthebouadaryisdividedintoNEegemeRtsandthedomaiutisdi-

videCE intomaeeiis,theEqs.(S.43)and(S.4S)canbewxitteRas

N::gE*\(eth,@-f\(T<Qb>dT,@':M:i;'*,.,r,g4 f(T(e)-.,.,q<e•p>t(T(th)dl-"i@-clp>f(Tlp))

.l,r,es*\(e'P>9\(th-f\(Ti\(e' \)dri\(th-.:i.i;.-,.,gX\(gp\)T<Qdri\(th-cip>T<p>

"x

--

The simuEtaneouseqeeationscoftsistofgheEq.(S.46)aRdtheEq.(S.47),letting

P=Pl,e-e ,p=p..]rhesimaxEtaneousequatioitscanbesogvedbytheNewton-Raphson

meahod\(S5-'\(R-"{S-a1]tkoughtheeq"atEoitisitoniinearwithfioniilleafbo"ndarycoRditions.

$seNE-DamSHONMvaOBkEMeSoimaetkime,itisnecessarygosoEveene--dimeRsioitaawaowhRineexftxeagcondnctkonpfobgerxx.

gg

laehissection,tkeone-diKxxewsiowaaap fobiemaasshowRinFig.S.1wiglbedeseribed.]rhegov-

effffgingequagionandtft}ebogndascycoit ditioRarewritteRasddTk\(dy=edxdx

k-f\(n

a$x$b(S.48)"

(Se49)Je•6

Thebewadaryconditienisgivenas

Tx=a=TaTx'=b"Th \(Sese\)

inasiixkiEag"waywiththetwo-digrr}emsioitagheatcondaxctioitp rebgem,tbefoXiowingequationis

obtainedas

d2•ka2T-Åë<n

du2du2 \(S.Si\)

f'

f' (T>eFffowt]Eq.(S.Sg)thefoigowingequatioitisgotas

b*b.dkduK'-klp)-Åëto

d2T. (x,p>dues- d.x2es(SeS2)wherees$\(XgP\)"-ppillmge-pH.Eittermofthe

discreteiittegralgK]etkodtemperatureisexpressedas

\(S.S3\)

Usinglk.(S.S3)Eq.(S.S2)beeomes thefollowiRgformm*b.dkkdes-kip\),.:

es--dxdxardop\(Ti>riee\(xi,p>

ByGreen'sidentityEq.(S.S3)isc hargedintothefoIRewingeqaationm*b*`ffTdu-T<p>E

duduai=1?gits(xi,p)

Thesimaskaneousequaaio"seonsist ofthe][lq.(S.sc)andahe]Eq.(S.SS).

aa,figure:S.ft Ogke-digtrxensiowaEprobgerre

gy

oX

Figute:S.2AnaaysisfnedeE

Se6wwwaCMEwwLwsAltho" ghtkgetwe-dignensieRalformulashavebeeftderived,kere theonlyopte-dimeptsional

probSegx\)sewesoived.Tkeftomhinearkeateond"ctiofiprobgemsipmtweandthree dimensionswiigbe

eanEedouthereafter.Nowas thefirstexampge,theone-dimemsioitai problejmiscoRsid eredas

ShOWnim]FigeSe2eX'heenc-dgmemsienagheatcondactienequatioRis writtefias

dd.k\(ng.=o 1$x$S \(S.S6\)

inthis case,tkebogwadary cofiditiomsareexpressedasi

Txr=1=2T.x=s=8 \(S.S7\)

inthis problemathetemperatwffedepeeedenceofthethermalconduÅítivityisstated as

k=T2 \(s.ss\)

TheanaEyticagsolutionfoif]Eq .(S.S6)isobtainedas

3T=g2\(ix--i18 \(SeS9\)

x ExA(rrr(T) PREL9ENT(T)

l.1 2.741 2.741

2.0 .5..1i7 5.117

3.0 6.3g3 6.3g3

4.0 7.:Z81 7.28!

4.9 7.934 7.9fut

rMabEeS.g:Theexactsogutieeeaxkdthewa"gxRerieaR res"gt

ee

ww8•ejr

fa

6s

'

4Mpresent

3 o exact,

l;l!\)

g 2 3x 4 s

FigureS.3 : Dgstributiortof Temperature

'TkeaftaEytgcagRrtdpt"merieaEres"lts are showRgftTabae 5.gand F"gg.S.3. Thefiumerieag

fesugtsaxecempared witk theexactsolutions.rfirabEe S.iand Fig. Se3 showthatthey

agreedwitheachothex morethan4fig"res.

Eitthesecopmd exampie,the goverxitiitg eqkiatiofl andbo"ptdarycofi ditionsafethe

sameastkefxrstexampge.

Thetheifmaalcond"ctivity iswrittept as

k=:r2-y3T+a \(S.60\)

x axACEff\) PRESENTff\)L

ki 2.S12 2.SIS

2.0 481i 4.8l3

3.0 6.Ig9 6.190

4.0 7.ICLI\)O 7.l91

4.9 7.92Lg 7.921Lg

kbge: S.2Cotwparison betweeitEwtalytical soletgons andBEMresuks

gg

8stsT6

ges"#'

-presewt3 eexact2

x4 S

FigeweS.4:Diswibutioitof Tempefatgre

Theexactsea"tSonSscalcKESatedas

2T3+gT2+6T-396sc+332=O \(Se6i\)

Theceffxkparisonofexaet aRdnggxtericatsesuttsafeske owRasTabgeS.2andFig.S.4.ThegifRaxi-

meemerrewisO.11%.Utcanbeseenthatthepresentres"itsareaceufate.

s.7ÅëoNcawsKoNs

gnthisckapter,the newdiscreteintegraRappfoaehk asbeeRpreposedtohandlethe

steady-statenofilinearheatcondnctioneqwatioRwithSegnperatured ependenceofthermalcon-

ductivitybyBEM. Themaiitconckusionsares"gxkmajrazed asfogEews:

E.'ThegeveifxximgptoitliRear heatcond"ctienequationEs transformedintoaitewformg.owhich

thediscreteintegraEffnethod isappgicable.Theptew equatioitshaig be solvedinsteadofthe

originaSeqEiation.Forthe vo}uwteintegration, nottheiwtem3gelement b"ttkeintex'naEdiscrete

poiwtsandtheboundary conditioitareputtouse.

2.Entkeefie-digxRensSofia# andtketwo-diExftensionalp robllem[fts,thebokxR daryiwtegraleqasation

ajredergved. AsetofsigximpEtgmaeouse\(g"ationaresogved byth eNewtoit-Raphsoitmethod.

3e ThisapperoaekisitotrestrictedbytbetemperatuLxed epeftdeneeofthe heatcond"ctgvity.

4.kotheepte-dimensional probgexExtftxepm"gxaericagresults kffeobtainedandcegx?paredwithexact

valaxes.Tkesefesugtsskew ellxattkeenumeericalsoivtgopms haveadeqa]ate accuracy.

on

eefiEasRRi:N\(ms

Is-a] C.A.BwebbiaandS.WaUkef,Bouad3ryEkegnentTecimquesiasEngQipteering,(19gO),

NewasesBwterwoptks,kondgn.

{S-2] P.K.BaneajeeandR.Bauerfxlegd.DeveiopgneRtsinBoundary]Eftegikentixkethods,

\(gong\), X,AppEiedSeience,London.

(S-3] Yax.N.AkkuratovandV.N.Mikhaigov,Themethodofboundafyinter.qralequations

forsoivingnonliReafheattransirmissionpfobie#ns,USSRCorrkp"t.Maths.Math.Phys.

20,(i980),1i7-12S.

[sc] R.BialecjkiandA.Nowak,Bo"ncXayvalgeprobgegnsiptkeateonductgeitwith

fleft]inearmatemialexkdpteptgineajrboEgitdaryeoxtdwieRs,Appg.Matk.Medeiling,S,

\(gogA\),`:gx:r4:21.

[sew Kamiy&N.andX",S.Q.,AmealtemaativeNiteagrizedforgxk"lationforquasi--harmonic

nonlineareqwation,Proc.ofConf.oitBEM,(inJapanese),No.96-a420,(X996),13.

(S6] HiroyukgKgSUandChaoffUANG,ANewSoiwionforNoit-Rinearg{eatCend"ction

Pecobgems,]li)eroc.jSME,No.ms1,3,(2am),9a-92.

($n C.L.Chan,AEcmaliterationschemefornoniineartwo-dimensienaesteady-state

heatcoRdgctioue:aBEMapproaeh,AppE.Math.Medeaging,12,(ieq3),6se-46S7.

ES-8] R.BiaieekiandG.Kgkn,Bogndaryeleitaentsolutioitefkreateond"ctioRproblemsin

wauttizoRebodiesofnoniinearmaaterial,KAt.J.Nasmer.MetkodsEng.,36,(i9os),799-

gco.

E591 J.P.S.AzevedoandL.C.Wrobei,NonlinearheatcoRd"ctionineompositebodies:A

boisj[kdaryefiemewtformugation,int.J.Numer.MethodsEng.,26,(E9ga),93g.

(S1O} M.S.]KhaderafldM.C.Hamaa,Afliteratgvebo"ndaryiptegralptumericalsolutionfor

gemeragsteadyft]eatcemeductiowaprobRems,Jl.Ne3tllrffansfer,g03,(A9gg),26-3a.

(sgx] ]Kik"o,K.,Ko"ski,Y.,Shigerg},A.ew]dMasam,S.,Boaxrdaryelementananlysisof

galvaniccogtsrosieitprobleKxg,R)ifoc.ofCenf.onBEma,(injapanese),2,(g9gS),g43-E`g8.

os

Cenffte1ffER$

ÅëoNcmeLismoNs

ERshisghesis,ancwappfoaehispifopesedforgkefwnetionagapproximation.The

fuficgiofiisappfoximatedbywtigizingdiskribwtio"ofShedeRtafuutctiowa.Usingthisfuxtc-

tiofiaiapprexiasxationthediscreteintegralmethodisdeveiopedforthegeneraiiRtegrai

awadthedomaiRintegral.ThisdisexeteintegraamaeghodisegwapgoyedinthebeRdingpyob-

gemagfbeaffifxawadthesteadystaeeheateonductioRprobgeiee.Themeaknworkandeencbe-

sionsaresummarizedasfo11ows:

inehapterft,affeviewofpubliskedaiteratufeandoutKneofthisthesisafepre-

sented.AkkoggheheGalerkintensowmaethodisappliedtoeXastacpfoblemsandgivesac-

eaxratefesutts,iecamebeusedoniyfofagimitedfaRgeefthecoRstantand1inearforoes.The

dualreeipffocgtyteehRiqueeansolveawiderangeofproblemsbut"suallyffequiresasig-

ftificantmampmeberofinternaipointsefpolestorepresewtthesolutionaecurately,Inoxderto

avoidthefequireevxewteftheaocagizedparticularsoaution,themaxkiplexecipfocieymethod

isdeveloped.'grhisappfoaehRotonEyisegxii>loyedindRffefeRttypesofbodyfofces,but

aRsoisaxsedfortkeotherdoenniainintegral.Howevew,thisdualreciprocityinetlxo\(}needsa

sequeneeofhigherorderfgndameRtalsoXutioms.Rnthediscreteintegfaimethod,thefun-

daffifgentaEsokwkgonsoflowefoxderisusedforthegreatKxxewtefthedomaiRintegral."rchis

appffoackkssignpaewthanMRM.Agthoughthesteadystate"oniiRearheatcondgctionequa-

aionwggkkaeexxperatuffedependenceofthermaaeoeedneegvityeanbesogvedbyKRrcftxhoff's

transformatRon,thEsinversetraRsformationRscegltpEicated.Whenthermaleenductivityis

deseribedeptEybygimeaff,expt>waeewtiagaRdpowerfuRegiomskntermsofteffRpefature,aeniethod

eq

bythe"seofthepteNyvvari3bgecaRbeexxRployedinekisprobgem.Ontheotherhand,the

discreteSntegralgxkethodcawtbeappgiedtoanykgndofthergx}agconductivityanddoesflot

needtkeEueverseKirehhoff'stifansfergxgatiofi.

ffgechapteff2,thekindandckaffacteristicoftheBEMaredescribed.Thebouftdary

egementgtr?ethodkasadvftntageovexotheritumericagtechnique.Thegoveriiingdifferen-

tgaSeq"atioitofPoissogepifebiegxxintkegwodiKKkemsiomsistfansfofmedintointegragequa-

tioR"stwgfapmdagxkentagsog"tion.TheftuxnericEsllimpgementationisdiscussedindetaii.

Tkeqgadratgcshapefunctioitsandtkequadratics"rfaceeiementareused.Thedomain

integralisevaguatedbyusingaseriesofsurfaceceEEs.

'inchapter3,thebasictheoryofthefunctioma1approximationutilizingthepoint

sourceaptdtrkegimeesetwceisdeseribed.inthistheerythefunctgoitisexpressedbythe

bo"ftdaryeeanditioptandthestreptgthofthepointsourceorthelinesource.Vsingtheline

seufee,theR"maerieagresuStsagreeswiththatbytheexactvalaxesimthetwodimensioms.

UsiRgtkisfuftetioitalapproxigxxatioeetkediscreteintegra]xnethodisdeveloped.Thegen-

eraiintegfalorthedemaiitiittegrageagibeexpressedbythebo"ndaryintegralandthesum

ofthevagexesattheinteriorpoints.Tkediscreteintegraftmethodusingthe1ineso"rceis

ftppEiedtethekektcoitd"ctiopmequatiortwithaheatsokgrceinthetwedimensiomsbyBEM.

TkeR"inerieaares"RtsbythisapproaehagfeewitkthatbythecgassicaXBEM.'rhismethod

"sesthegowererderfkendamentagsolutioftsothatitissimpierthanMRM.

Xitchapter4,ThefiewalgoffithmswithoutthedividiRgschemeareestabXishedfor

bendingproblemsofthecoRtiit"e"sbeamincg"dingtheinhomogeneoR}sbeambytheBEM.

TheconventiopagallgorithrrltsoftkeBEMforthebendingproblerrksofcontinvousbearn

areifteffxeieRtgmdhavesevefaXpeimtstebeimprovedupen.Mainpointsofthisstudyare

summarizedasfoXgows:

g.T'hefoesffxulatieitpfoeessisiaxipwovedtoderivethesystegxkefsignultaReo"seqaations.

Asafeswies,gkeyarereformedtobetwgchmoreeompact.

2.rff'henendivgdieegsehefixxegsesttkbEishedeowtrarytothedividingscheme,wkgehisneeded

inghecowavewtiomaEfermeeg3tRoRentftxeoceasionsofsiExxplysupportingpoint,steppedbeaxxg,

twkroenrsogempeoagsbeaxxk(imetudiEangdiseowtiitagousgychaRgefaxErtgidity)andeoeecepttrated

9S

maefixkentgoad.'3.AgenerkgschetwefortreatiRgtheinlaogxtogefteeusbeaffxxiitcl"diitgthosewithdiscon-

tipm"o"saychamegefulrigidityXsestabgished.

4.Wkextgmaexgeff-gkagfereeexists,tkedogMikiitintegralgsperfergKgedbythediscfeteEitgegral

gxftetkod.1ifhedoffxkaiftipttegraiisexpressedbytheboundarycenditionaRdtheskerifaofsome

vag"esattheiwternaXpoints.Thedom[itaiitisitoedividediRtoceggs.

Frogxkptuxtrxericagexanxpges,therightnessofthenewaigorithwtisvergfged.Th.esenew

algorithmsgreatXyred"cedshesizeofmatrixaswegEasthecognputingtimeand,therefore,'

broughtabo"thighefficiencyonthefepetitivecaic"iations.Asaresult,thiswiligivea

gewcostferoptimagdesigngitdaiiywork.

inchapterS,inorderteavoidtheeognpiexinversetransforrrxEgsingKirchhoff's.. .--

transform3tion,afiditheEgmitatiofiofthreeeasesoftheghermaEcoitdasetivityusingamethod

bykkeuseeftkefiewvasiabge,tkedisereteiwtegrakgxkethedisattemptedtoappgytethe

ftonMnearkeatconductioftprobaegn.ThenoitXinearequatioutistransformedkwtoaReq"iva-

iewtforwftsothatitispossibgetoappgythediscreteintegraigxkethodtothisform.Forthe

voiutweiRtegrai,theiwternaidiseregepointsarepaxttouseoRly.ThisRewapproachis

swttablefoifa"yformofheatc•optduc•tivity.rff"hismetkodissimpiewandhastheadvaR-

tagesofwetai*iwagthepriReipagehawaeteristicsofBEM.Theoumericalffesukscoxnpared

withexactfesuksshowthattherwmericagsogwtionshaveadequateaccuracyintheone-

digTxensio"aiprobletws.ThispffoblemwillbecarriedouthereafterintheSwoandthree

dimensi"mps.

Rnb#riief,thefunctionkasbeenappfoximatedbyutillizingdistributionofthedeltafuRc-

tioR.UsiAgthisfunctionaaapproximatgonthediscfeteintegfalgnethodhasbeenproposedto

tffeatthedoffnainiRtegwaiorthegeneragintegralaRdappliedtoehebendingprobaemofbeam

andtheftxeateonductioRpwobgemfi.NwamericaXcaEcutationhasbeencarifiedoutanditisshown

thatresuttsobtainedbythepropesedtwethodhavegoodce"vergencyandadequateaceuraey.

9{S

AswecNDMX

\(g\)wbeeeffeetwerwgeffdieefitaffee#meetgopge

Enekgstinesis,weasseeagky"segkedeatafasncgionSodea#withbempdkngproblempof

beaixxandgheotherproblegxks.imthisappendix,wewallexpgainthefeatureofdeltafunc-

tioitindeSaiR.FwogptChapter2,weobtaincheequationof:

d2ua" \(x,p\)=-6<x-p\)

Wkefeb\(x-p>istheDiracdeggafunetioitwhichismathemaigicaglyeqljivalenttothe

effecSefaagmitcowaeerktratedsoewÅíeapp#iedatghepeingp.rff"ftiefundamaewtagsoagtioRisa

p3ftaougawsoawtienoftheadjointfgrmoftgkediffereReialequation.UsiRgthefvndafi]gental

soautiowhiwatheformugatieft,igispessibaetggreatapfoblemaonlyontheboundary.The

fuptdiaxwaeitgagsotwtionlhasameimpoertaittreRationshiptoDirae'sdistrftlbutRgn.Diwac'sdistri-

buttioitisageftefalizedfgRctioRwhiehhasasharppeekatx=p,awadiszewoexceptthis

eP"geet:

Eeisdefiitedas

oo(x=p)b<x-p>=(A-g)e<x#p>

Wkeref(x)isafunctioptvvhgckrgscewtin"oeesatx.,,pgMi]dflp)gsthevagueoff(x>atthe

poimgx=p.fi\)ivac'sdfistgrfibatgeancanbeexpgaimedimphysicaExtrkeanings.ForexampEe,im

jmaeckewaics,fiecaaubeexpgaimedasacoitceptratedforceactedonapogntpofanSjnfimite

on

paage.Tkfispoiptiscaaaedsowwcepoiftt.Againsttkis,anypointxinthefiegd is cagged

ebservktiganpoSwt.Bywsimgehefek&eweefdegtafasptctgoftexpertay,wecawareadigy d .ergve

gkeineegralleqasatiogencededfiittheBEM.

\(2\)geedieerxgitEteggeseggeagfifiueecgeedifieegsfigwaffeeeecgfi$wa

sganisthesigfifuitctiopt.Wheitx<p,sgn<x-p)=-g;Whellx>p,s.qgk(x-p) =g .The

feaXowingiwtegagcafibeobtainedwiththesigflfuftctieftinsimpgestyge.

f(x)s.eqi(x-p>du=F(x)sgn<x-p)

-F<x>2b\(x-p>du

=F<x>sgn<x-p)-2F<p>6(x-p>du

={F<x)-Fip>}sgn(x-p>+e \(A-4\)

Wkereeisaceastant.kissktisfiedthefoiXowingeq"atioit.

b<x-p)du=-lig-sgn(x-p)+e\(A-S\)

Efttkee>euedimgprobgegx)sofbeaffff],gkecomstawtcisdefine(gaszero.

eg

AÅëeeNewrswGEMEN#

TkisthesSswasaccogxkpgisked"ndefthes"pervisgenofProfessorffgroyukiKisu.g

we"gdgSketoexpressgxgygfeatestgratitEsdetohimforhisengighteninggugdanceaitdgfeat

bee}p.

#wouidalsoMketoexpfessmaypapticglafaandsineexethankstoProfessorsNoriterg

Nishida,Kg"iyasuKanemafagandMinoruShugyofortheirenthusiasticsupportandcon-

stfactRveadvkcetotkket&kesks.Agoteftltseirgoodsljggestionshavebeeptadoptedinthis

thesis.

EwanatoahankktgfsKuigxiingRongfofherdiscllssioRandhelp,Mr.Toyofukufor

hRstechnicaXassistanceandMr.Kondoforhisdisc"ssionandsupport.

EawtaEseverythaRkfuRtoalgthewtembersinthefeseafchfoofiafoffheXpfuadise"s-

sieftaptdfrkendgysgppQft.

Finalay,ffwanttoexpressgnydeepestgratitwdetomyparents,sister,mywifearRdmy

childreptfertheixgove,coptkimeeewssgpport,encouragexxRentandaxitderstandipmgdkarimgrt}y

stwdyinjapapt.

eq