Development of a Nodeless and Consistent Finite Element Method force method forever
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- Arabella Mathews
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1 W Ffh Word ogre o opo echc J 7 e r d: H g FG Rerorfer J erhrdeer eveope of Nodee d oe Fe ee ehod force ehod forever Tdhko K Profeor er The Uver of Toko 7 Hogo kok Toko Jp e: k@ocejp Ke ord: dvergece heore vro prcpe dpcee d force ehod odee fe eee ehod rc Ug he geered poe hch cde ork doe de of eforced dpcee o he dpcee odr e vro foro eped h pper o f he prcpe of poe eerg d copeer eerg ho rodco of grge per o h he codo of he o eerg c e ered re deforo of ec ode I deored ovg er of he pe re d pe edg proe h I er ec proe he oer od oo c e oed he ehod propoed h pper h cree of NOF of he e fco ed To e preed h h reerch odee ehod propoed order o ke he fe eee cco epec force ehod ockg free d o redce rde of he eh geero proe prcce
2 Tdhko K Irodco Tod Force ehod o deced he F co Fro he rcr deg po of ve hoever he ffe evo of rcre ed o he pcee ehod o geer coervve d ccrc of he cced ree feror o h of he cced dpcee de o ecer dffereo of he er d herefore og dre of rcr egeer o eh e ehod hch he oer od oo c e oed ho f ec proe Tord ch o poe chege h ee de for he er codcg reerche og he foog o e: deveope of e vro foro hch fe he prcpe of poe d copeer eerg o h he oer od oo c e oed e er ec proe deveope of e odee ehod hch ke he fe eee eer d ore effecve fe erc epe o he pe re d pe edg proe re rodced for verfco of he propoed ehod h pper eveope of he fed eerg ehod ec Geero of he poe eerg [] oder rrr e of re copoe j d r copoe j The re copoe j re ed o f he foog eqo of eqr q d he echc odr codo o he re odr q he he r j ed o e derved fro he dpcee g q The dpcee o ed o f he dpcee odr codo jj p here p he od force vecor he voe of gve od j j S here he or dr ord o he re odr S å j j j o S here S S ó S ; S he dpcee odr The o oo dffc o derve he foog eqo g he dvergece heore: ó jåjd pd ds S ds ó S I hod e eoed here h h eqo re rrepecve of he rer reo d e of If j d j re reed he foog er reo:
3 W J 7 e r or j jk k k jk j here jk d jk re erc rce q epree he of eerg coervo I h eqo he rgh hd de pe ork doe de o gve od force rfce rco d eforced dpcee o he dpcee odr Therefore he cocep of poe eerg W c e geered foo: W pd ds ds 7 S S Propo of e fed prcpe of o eerg ec oder he foog o eerg of ec e defed : U 8 W here U d 9 j j c j jd jk j kd 9 c j jd jk j kd No q8 c e re foo: ó p c here p j jd pd c j I hod e eoed here h j ds S j jd ds S c j fco of ece c j er fco of v q d Therefore f he re oo he prcpe of poe d copeer eerg he foog coco c e dr: r overe coder he ce he ecoe h repec o Sce of o pove fco of herefore f e oe of he doe o ecoe he c o e Th c e cocded h e prcpe propoed h eco fe he prcpe of poe d copeer eerg Ne e coder he rog for of No gve foo: ó jåjd pd ds S ds S The fr vro of q h repec o ó gve j j j j d p d ds ds Só S Th eqo rfored g dvergece heore foo:
4 Tdhko K ds ds j j p d j jd S S The voe egr ecer er hch he he vro cco d herefore e deeed ds ds j j p d S S Th coco verfed fe erc epere de rece d e dced he er eco g orreo d of e vro eqo derved d oher eg prcpe [] I hod e oed here h he hor derved prevo he foog vro eqo geerg he prcpe of vr ork h he e of of grge per: ds ds j j p d S S q he rog for of he odfed HegerReer vro eqo: R j r j d grge per here p d ds ds R j j j jk j k S S Th q eqve o R he j d re reed o q d I ereg o oe h dfferece of q d o p or g of he ecod er of oh eqo I e cocded h q c gve he oer od he q c gve o or oo hogh re of erc epere de o fr re ed I hod e o ephed here h q derved ho rodcg grge per d herefore c er he proper of prcpe he o he or R j gh poe ehod of oo ec Ug q dced he rece hor pper [] h 8 dffere ehod of oo c e propoed ho he T fe coe re de o he T foo: Soo d Treff ehod re qe ehod here co of he eee e vecor o reqred pror o h he c e reed depede For he re ehod co of he eee e vecor reqred pror o h he e ced he geered fe eee ehod I he e eco geer pproch o corc he odee ehod e eped ref Soo re o ced qr ehodii Force ehod Srvv of II c e epeced g h ehod
5 W J 7 e r v Soo o pproe ehod c ehod Ideed e ced coperded c oo The pree hor eeve fre pc of h ehod o c cece d echoog eveope of he odee fe eee ehod I eco ehod of oo oher h he oo d reqre co of eee e vecor dpcee d odr rco pror o of he eee e vecor codered foo : Fr coder de of he foog o fco defed he rego ho Fg: f g Fro h fgre c e ee: f g Ne eq of o fco re re o er of dcree po If hee o fco re epreed cr ere repced : rrepecve of c d I prcce cr ere e pproed poo of fe order Th de c e ccef pped o foro of he odee fe eee ehod foo: For pc he pe re proe codered: Fg N re dpo of he de No he ecod order poo of ed for he eee dpcee fco foo: 7 v v v he rgd od dpcee vecor of he coorde org here he co r copoe of he eee d NOF of h eee Ug q7 he foog e of eqo c e derved r copoe v re copoe v 8
6 Tdhko K [ ] [ ] [ ] 9 c odr rco copoe here or dr ord o he odre of gve rgr eee ho Fg eog he dpo of rrr po P ke o ho Fg Pg he dce P he foog eqo c e e oed: q he eqo for coorde rforo of o o he de ppg q o q7 d he e vecor o he de re gve : v Thee co c e epreed he foog e of eqo: d d d d d d d d d d d d d d d d d d d d
7 W J 7 e r 7 r R Q P here R Q R Q P U T S here U T U S S 7 Sce NOF of h rgr eee d o copoe c e dred o ech de Therefore he foog hree ce e fee for h rgr eee pcee ode I I v ed ode I I v qr ode I I here I pe he ode hch doe o f he eqo of eqr he ode copoed g o he re fco re referred o II No ce of q d c e e p for ech de of he eee Therefore eee preer ; v c e rfored o e of rforo r d herefore f he eee ffe r K oed h repec o he oc coorde he ffe r K h repec o e eee preer k k k k k c e oc oed he foog fr for: K K T 8 here he percrp k pe de of d repecve Sr he eee fe r c e oed Geer pekg ode eog o cegor I gve re of he eer covergec d ccrc
8 Tdhko K Prcc deveope of odee fe eee for e e e ec de For epe coder he pe edg proe The e vecor co of copoe d I coveo fe eee ehod he cope ode c hrd gve ccre re I ce of he pree ehod hoever o reo c e ee hch prodce feror re o copre h he eg fe eee ehod I ce of he pe edg proe co codo of he e vecor co of for eqo The ref rodco of he pe edg e gve foo : The eqr eqo d odr codo ee Fg qr eqo: q oced odr codo: dpcee c: o re c: o here he or dr ord o he odr : dpcee c : re c : precred c precred c : precred c : precred c : Q Q Q Q Q Q prcpe of he o eerg for pe edg proe ä ä r here dd d d d Th eqo c e rfored o he foog eqo: qdd d 9 8
9 W J 7 e r 9 7 d d d d qdd dd 8 Ug he dvergece heore q9 c e rfored o he foog rog for: d d d d dd q Therefore d d d d dd q r The geer oo of c e oed foog he d eee he eqo of he pe re d pe edg proe foo: [ ] [ ] [ ] [ ] [ ] [ ] Q Q I Re Re Re Re Re No q he vro eqo for he pe edg proe ho oced odr codo d derved ho g grge per ehod g 8 poe ehod of oo c e propoed d ec he e dco c e pped o h proe de he pe re : ehod of oo d pree qe oo procedre for of dcree e here he eee e vecor c e ed depede I he oher ce co of eee e vecor for copoe re pr or f reqred o he eee odre efore of he o e geered fe eee ehod I he coveo fe eee ehod eree dffc o f eve he dpcee coe of d eo o he eee odre I hod e g ephed h he chrcerc eee rce for he ed ode of hpe d order c e corced ec he pree ehod
10 Tdhko K oder for epe he coe qdrer pe edg eee e f o of rd order d order order d order o ech de of he eee herefore o NOF of he eee oeqe he h order poo of d e ed for Nerc pe o 7 er go he hor h ed o chege o eh e vro foro hch he oer od oo c e oed ec proe v geero of he prcpe of vr ork fe er er he recoged h h e vro foro ee eqve o HegerReer Prcpe HR ó j ë f j d re eed g he rer d he oced ere codo d o or prcpe h repec o j j g o ch codero he cocded h order o reore he force ehod frher chege e coed o erch e vro prcpe hch f he prcpe of he poe d copeer eerg ho rodcg grge per I h foo ref rodco e gve o he fve erc epe of verfco de pe I of he re cocero proe of perfored qre pe jeced o eo he ed [] Th proe ed g Treff ehod ed o he geered prcpe of vr ork Oe eee oo e fe eee oo re ho he Fg ro of o he cro eco ho h fgre ccrc d good greee of cced re eee oe eee d fe eee oo re oerved I hod e epec eoed h free coo of eh per h dffere e d hpe c e e doe he pree ehod ece odee ehod ed o Treff oo procedre pe II Toro of ec r h qre cro eco g he ere og Toro proe of ec r h he cro eco of rrr hpe c e gve he foog odr ve proe of Poo eqo h erche odr codo: G S o here he re fco for oro he re of d he g oe gve : dd e o he ere og propoed Prd q eqve o he foog defeco proe of he ec ere der for er od q
11 W J 7 e r q T o here T he eo he ere No he oced odr codo gve : o erche codo The fco for h proe c e gve : T gve : T dd qdd d here The rog for of d T q dd I cor o e Therefore he proe redced o pe eerce proe for ReghR ehod Toro of ec r h qre cro eco ed g he foog defeco fco of he rd order poo of d : 7 To re of cco for he oro rgd K re ho Fg The crve he re of cco g he ecod order poo of d he he re oed g he rd poo re ho he crve I ereg o oe h he forer gve he pper od oo he he er gve he oer od oo for K I c e ee h oh cco he oooo covergec o he ec K Toheko g he de egh pe III [] Ipe edg of cever pe jeced o odr her Toheko ded edg of cever hvg rro recgr cro eco of dh e force pped he ed Fg To ove h proe he effec of force P pproed dred herg re cg he ed foo: P 8 c I Th proe ed rece g he fed eerg ehod propoed h pper Fe eee e ere codced g he foog oeqr dpcee fco NOF of hch foo: 9 v v For he eqr dpcee fco of he e NOF o derved g h order of he foog Gor re fco I ref poo of for d
12 Tdhko K r re fco F Re[ ] c d Re [ ] here [ ] G v oo oed g oeqr dpcee fco ho he crve he he oher oo g he eqr dpcee ho he crve h fgre Fg ho he covergec of he cced dpcee v d re repecve I c e ee h he crve gve he pper od oo for oh v d o he oher hd he crve gve he oer od oo cer The roe of he pree ehod d checked erc I hod e eoed h c oo gve Toheko o ec oo for he proe eore pproe oo here he cped edge codo pproed cpg he pe he org Ideed dfferece of defeco v oerved Fg pe I edg of qre pe jeced o for dred er odg h for edge cped [] Th proe horc eko dffc proe o hch chor hve cked he p og he Toheko ork phed 98 he o e ko Th proe ed g he e vro ehod propoed h pper The h order poo of ed for he eee defeco fco Re oed re red Fg7 pe edg of qre pe jeced o for dred er odg h for edge p ppored d egeve of he edg vro [] Fg8 ho e eh dvo d cced defeco he ceer pcee fco ed re poo of h order here o e er of he pe defeco codered The c of he covergec crve re o er of degree of freedo NOF of he eee ed er of eee I c e cer ee h oo oed re he oer od oo he pe edg The egeve eqo for he pe edg vro gve : g for he pe edg proe d kg h repec o ko preer for q of hoogeeo er eqo for c e oed fro egeve oed hch he chrcerc eqo for c e oed eg oer od oo re ho he e ched o Fg9
13 W J 7 e r 7 oco Ug dvergece heore ec e vro prcpe propoed o he codo of he o eerg of gve ec e Th ehod f he prcpe of poe d copeer eerg he er ec o h ccrc of pproe oo c e defe checked rckeg he he pper od d oer od oo I ereg oe h dfferece eee he propoed prcpe d he e ehed Heger Reer Prcpe o dfferece of he g of ork doe de o eforced dpcee o he dpcee odre d e he forer c er he of he o eerg he he er o or prcpe Hoever rgoro hec proof ef for fre d ckoedgee Fr of he hor od ke o epre h herf hk o Profeor o K of he Ngo No oege of Techoog for h devoed cooro o he pree reerch for he 7 er Ideed e h ch ehc cooro h ork od hve o ee ccophed He o hk r r K of Profeor K oror for h effor o oro cco d crp prepro of h pper He o gref o er Tohhko Ohk d Ko Ko of he dvced So Techoog of echc R o d for her hep o he reerch gee Th ork h ee codced der he poorhp of he Jp Scece d Techoog orporo Referece [] K T The Force ehod Reved I J Ner ech gg 7 78 [] Ggher RH Fe ee Fde PreceH: geood ff NJ 97 [] Wh K ro ehod c d Pc Pergo Pre: Ne York 9 [] Toheko SP Gooder JN Theor of c rd d cgrh: Ne York 98 [] Zekec O Tor R The Fe ee ehod h d o cgrh: UK 989 [] Sokokoff IS Spech R hec Theor of c d cgrh: Ne York 9
14 Tdhko K SO NO Te: 8 Poe ehod of oo derved fro he pree vro foro ro eqo j j p d S S ds ds or odo rerk Pr of odfed Heger Reer ehod S j j p d ds o S pcee ehod I I S j j d ds o S p qr ehod I I j j p d o S o S Gerk ehod I S S ds ds Treff ehod j j p S ds S 7 ds 8 j j p o S j j p o S j j p o S o S II II GII c oo
15 W J 7 e r f g O Fgre : q of o fco: g f fro he cooco ehod po of ve N P Fgre : eveope of odee fe eee O Fgre : oorde d copoe of he rco vecor for he pe edg proe
16 Tdhko K Treff ehod 7 eee : : Oe eee oo OF: : 9 : re epo Oe eee oo S S S R R S kgf kgf Fgre : Sre dro o eco of perfored qre pe der for odg K Kec 8 98 : d order poo : rd order poo OF OF eee eh d order v poo OF eee eh rd order v poo Toheko K Fgre : of oro rgd of ec r h he qre cro eco dvded qre eh
17 W J 7 e r v : verc dpcee he po P 9 8 h order Gor re fco ed d order dpcee fco ed Toheko OF eh v NOF re fco ed dpcee fco ed : re he po P 8 h order Gor re fco ed d order dpcee fco ed Toheko OF eh v NOF re fco ed dpcee fco ed Fgre : Ipe edg of cever pe jeced o odr her of proc dro dvded qre eh 7
18 Tdhko K er defeco ec eh v NOF W Wec OF 8
19 W J 7 e r defeco eh v NOF 8 8 Fgre 7: Fe eee edg of qre pe der for dred od g he e propoed vro ehod Noeqr h order poo of ere ed for 9
20 Tdhko K eh v NOF To OF defeco Toheko defeco OF Fgre 8: edg of p ppored qre pe dpcee fco ed: h order poo of NOF eh v To NOF OF egeve c
21 W J 7 e r egeve 8 OF Fgre 9: egeve of p ppored qre pe dpcee fco ed: h order poo of NOF
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