Recap Shape Functions
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- Beatrix Stewart
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1 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Rep Shpe Fnon Th good ple o op nd remnd orele where we re n he proe o ormlng nmerl olon ng ne elemen mehod. For omponen we re olng he glol ore dplemen eqon or dplemen.e. The ke o dong h ormlng he glol ne mr [K] properl nd ndng nere. One we he oled or he dplemen AT THE ODES we n nerpole dplemen ro he elemen hrogh he e o hpe non. For one dmenonl lne elemen where he oordne w hed o he le end o he elemen. d K F K F d d d L L d d L d d d ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
2 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The lner hpe non or he one dmenonl rod elemen re rele o lol oordne em hed o he le end. For wo dmenonl onn rn rngle one he nodl dplemen were deermned he dplemen ro he elemen n e nerpoled gn hrogh he e o hpe non nd hee hpe non re ormled ng glol oordne e. For he onn rn rngle he hpe non re lner n nd.e. A A A m m m m j j j j m m j j m m j j L ˆ L ˆ
3 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng For he lner rn rngle elemen dplemen were qdr non o poon. Rell h Here er he nodl dplemen re deermned rell h he oeen oe re deermned hrogh he ollowng epreon nd he dplemen re nerpoled ro he elemen ng where now he hpe non re dened d d d M M * * * M
4 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng For he lner rn rngle elemen he dplemen r qdrll ro he elemen nd he hpe non m e le o nerpole he nodl dplemen qdrll ro he elemen. Th or he emple prolem 5 6 h h h h h h h h h h oe rell h he hpe non re dependen pon oordne em whoe orgn w hed o he r orner node. Chnge he oordne em nd he hpe non hnge. Trkng he hpe non or eh nddl elemen rele o ngle oordne em now eome prolem. Th no omehng one n do hnd or rk el n omper owre. We wll egn he e lol oordne em we ormle hgher order elemen.
5 Whkewz College o Engneerng Seon : ISOPARAMETRIC FORMULATIO Geomer Inerpolon nd he Conep o Sndrdzed Elemen We j reewed how hpe non re ed o nerpole nodl dplemen eld qn ro n elemen one nodl dplemen were known. One dplemen nd how he r ro he elemen re known dere o dplemen n e ken o on rn. One rn omped we ompe re ro he elemen. I he nodl oordne re lle we n perorm he me or o nerpolon o dene he red ondr geomer o n elemen. Conder he l prolem o ple wh hole je o enle re ondr ondon. We old e qdrlerl elemen wh or orner node nd rgh de.e. B here lo n del long he rgh lne edge o he elemen. 5
6 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng We old deelop qdrlerl elemen wh md de node j we dd wh rnglr elemen. The ondr geomer eween orner node old e red. Th wold reqre qdr nerpolon o he geomer ed on he oordne o he orner nd he md de node: Here he oordne o node re pll loed nd qdr re nerpoled hrogh he node o he elemen he geomer nerpoled. Qdr nerpolon non llow md de node o e oe rele o rgh lne onneng he orner node h prodng red ondre. Th pe o elemen h mh more del n modelng he geomer o h omponen. 6
7 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Mong orwrd onep re deeloped reerenng he gre elow nd denng he mhem gong rom le o rgh.e. rom nrl oordne o glol oordne. Howeer nere e or hee rnormon h h one n dene he geomer on he rgh hnd de rnorm he prolem o he nrl oordne on he le hnd de nd perorm llon n he nrl oordne em. Cllon n he nrl oordne em re r eer. Aer orml denon o oprmer elemen he mhem oed wh he mppng nded elow preened. rl Coordne Glol Coordne 7
8 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng 8
9 Whkewz College o Engneerng Seon : ISOPARAMETRIC FORMULATIO Ioprmer Formlon Deelopng hpe non nd elemen ne mre or hgher order elemen n erm o glol oordne em dl. Ioprmer ormlon or ne elemen llee gre del o h omple. In ddon he oprmer ormlon llow he deelopmen o elemen h he red de n he l omponen ongron. A ne elemen d o e oprmer he me nerpolon non dene oh he dplemen hpe non nd he geomer hpe non. Geomer hpe non dene he rnormon ed o go k nd orh rom n - oordne em o n - oordne em or wo dmenonl elemen. I he geomer nerpolon non re o lower order hn he dplemen hpe non he elemen d o e prmer. I he reere hold hen he elemen reerred o perprmer. Ioprmer elemen n nd do he red ondre whh mke hem more le n prng l geomer Howeer or he hgher order elemen ondered neer o emplo nmerl negron o ele he elemen ne mr. Trnormon o n - oordne em nrl oordne em le negron. The mehod re nee or qdr elemen well hgher order elemen. In ommerl owre he oprmer ormlon ed or oh low order 9 nd hgher order elemen.
10 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The onep o n oprmer elemen rher lred n he ollowng gre. Here he oprmer elemen ompred r rele o perprmer elemen n elemen where here re ewer node or he ompon o dplemen ng he wek ormlon o he old mehn prolem hen here re o dere he geomer o he elemen. For prmer elemen here re more node or he ompon o dplemen hen here re o dere he geomer. Clerl or n oprmer elemen he me nerpolon non n e ed or eld qne nd geomer.
11 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Renglr Plne Sre Elemen For wo dmenonl elemen he nrl - oordne em dened elemen geomer nd he elemen orenon n he glol oordne em. There rnormon mppng or eh elemen nd h rnormon ed n elemen ormlon. The oprmer ormlon now wll e ded rele o mple -node qdrlerl elemen. The ormlon generl enogh o eend o hgher order elemen e.g. he 8-node qdrlerl elemen. The - oordne em hed o he ener o he elemen nd need no e prllel or orhogonl o he - oordne e:
12 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Th qdrlerl elemen h egh degree o reedom.e. wo dplemen eh node. The nknown nodl dplemen re dened Here d h h
13 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng I we ole or he oeen n he l mnner or he loon o he oordne e gen n he preo gre hen he epreon or he dplemen n he elemen re or ] [ h h h h h ] [ h h h h h d
14 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng where nd h h h h h h h h
15 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Agn he elemen rn or h wo dmenonl elemen re nd The {B} mr n e ond kng ppropre dere o he hpe non. The relng epreon or rn wll demonre h he rn n he -dreon onl dependen on he rn n he -dreon onl dependen on nd he her rn dependen on oh nd ll n lner hon. d B 5
16 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The hpe non denng dplemen whn he elemen he een dened n erm o he - oordne em. We lze he me hpe non o geomerll mp he elemen rom he nrl oordne - no he - oordne em. Th le ] [ Solng or he oeen ng he nodl oordne geomer n he gre eld ] [ 6
17 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Or n mr noon where 7
18 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng oe h or oh he - oordne em nd - oordne em. Th hek or rgd od moon. I eer node jeed o he n dplemen e.g. hen oo h eer pon n he omponen h he me dplemen. A lr mlple o one prode he me rel. Th one rgd od moon. Alo rell h rgd od moon wll prode zero rn hrogho he omponen. 8
19 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng We now rn or enon o ormlng he {B} mr or he qdrlerl elemen. Th ormlon old e rred o n he - oordne em he ompon re dl o nerl mpole. I edo o eee hee ompon n he - oordne em dole. To onr he elemen ne mr we m he epreon or rn whh re heorell dered n erm o dere o he dplemen wh repe o he - oordne em. I o e he - oordne em o nd dplemen he dplemen re non o nd nd no nd. Thereore we need o ppl he hn rle or derenon. Th men he dere o he dplemen re 9
20 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Fong on we ole h em o eqon or Smlrl we ole For
21 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Ung Crmer rle where he deermnn n he denomnor he deermnn o he Jon mr.e. J J
22 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng For he her rn Thee deermnn epreon led o J J
23 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng nd In mr orm J J
24 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Sng or dplemen eld or J d D d B
25 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng where we dene he operor mr The mr {B} now epreed non o nd.e. ee o J he rle nd pper n he nmeror nd he denomnor o he omponen o he {B} mr. Th omple negron o on he elemen ne mr reer o or Cll e ook or he negron o ronl polnoml. J D J B 5
26 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng A nl noe on ompng he deermnn o he Jon mr. Wh hen nd 6
27 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng J Th nd J 7
28 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng In l emple 8
29 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng One Dmenonl Ioprmer Mppng The erm oprmer em rom he h we e he me hpe non o nerpole he eld qne e.g. he dplemen h we e or he geomer o he lne elemen. The non ed o dere he loon n he rnormed pe o n pon on he lne elemen n rel pe. Here rnormon dened o ke he nrl oordne no glol oordne. Ioprmer elemen eqon re ormled ng nrl oordne em or lne elemen h dened elemen geomer nd no he glol oordne em. The l -oordne hed o he lne elemen nd remn dreed long he lne elemen no mer how eh nddl lne elemen orened wh repe o he glol oordne em. For qdr lne elemen he non wold ke he orm 9
30 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Shpe non Conder qdr lne elemen.e. lne elemen wh node. The -oordne hed o he ener o he elemen. Shpe non or h elemen re gen elow reerene Emple.6 n Logn e ook Ioprmer mppng `
31 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Gen pon n he nrl oordne em he orrepondng mpped pon n he glol oordne dened ng he oprmer mppng eqon The hpe non re dened n he nrl oordne nd he re polnoml he were eore. In he glol -oordne em he hpe non n generl re no polnoml. Conder he ollowng lne elemen dened n he glol oordne em 6
32 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The oprmer mppng or h emple whh mple polnoml. The nere mppng no mple 5 6
33 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The hpe non n he glol oordne Deelop epreon or nd homework eere
34 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Grphll he hpe non or node # plo ollow n he wo oordne em mple polnoml lghl more ompled 5 pnl o e wh more hn one elemen. Thnk o where o ple he orgn o he oordne em.
35 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Mre or Lne Elemen For lne elemen wh qdr hpe non The rn dplemen relonhp one gn d d B 5
36 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng E Here nd re one gn or lner elemen B The onl derene rom eore h he hpe non re ormled n he nrl oordne em nd he dere oe re epreed n he glol -oordne em. 6
37 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng We know h ee we e n oprmer mppng h nd we wll e h epreon o ormle he omponen o he {B} mr whh re dere o he hpe non. Ung he hn rle rom ll In El we dered he relonhp d d dv I J w greer hn we hd olme epnon eween zero nd orrepond o olme onron. Inerpreng h relonhp n erm o lne elemen we he J dv o d J d 7
38 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng or d d J From omponl ndpon d J d d d For lne elemen he llon mmedel oe mde nered nd ed n he ollowng epreon: d d d J d ow he dere o he hpe non re ormled n erm o he nrl oordne em. d d 8
39 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng For he -noded lne elemen nd he {B} mr n he nrl oordne em d d J J B 9
40 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng The elemen ne mr epreed T k B EB B The negrl oed wh AY elemen n he glol oordne rnormed o n negrl n he nrl oordne em where he negron wll e rom - o n he lol oordne. The Joen non o he -oordne n generl nd pper n he negrl. The pe orm o J deermned he le o nd. The omponen o he {B} mr re polnoml non n he -oordne em. In generl Gn qdrre ed o ele he ne mr. Moreoer now hnk o he l o he hpe non. Here her dere he {B} mr re ed o ormle he ne mr n ddon o her e n nerpolng geomer oprmer elemen dplemen hrogh he elemen rn hrogh he elemen nd ree hrogh he elemen. T E B Ad A J d
41 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng G Qdrre In er mple erm dene negrl dened where I F d df d F Aenon now gen o mehod h n ele h dene negrl nmerll. A hrer o grop o nmerl negron or qdrre ehnqe known ewon-coe eqon re h negrl eme re ed on eenl ped le o he non. Coneqenl he loon o he elon pon ed n hee pe o nmerl negron mehod re ed or predeermned.
42 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Conder he rpezodl rle whh he mple mehod o he grop. Th mehod ed on kng he re nder he rgh lne onneng he non le he end o he negron nerl. The orml or he rpezodl rle I d Bee he rpezodl rle m e end pon le o he non here re e where he error oed wh ompon gen oe rel n gnn error. The negron error qe noele wh he non preened n he gre elow.
43 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng e onder h he rern o ed e pon reled nd one ree o ele he re nder he rgh lne jonng n wo pon on he re. oe h he lengh o he e o he rpezod mnned. B poonng he wo pon on he re wel rgh lne old e pooned h wold lne he poe nd nege error. Th deped n he ollowng gre G qdrre he nme gen o one l o ehnqe h mplemen h pe o reg. Beore derng he pproh nd e n derng ne mre or oprmer elemen we how how nmerl negron orml h he rpezodl rle n e ond ng he mehod o ndeermned oeen. Th mehod wll hen e ed o deelop he G qdrre orml.
44 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng To llre he mehod o ndeermned oeen onder n lerne ormlon or he rpezodl rle where nd re onn. Relzng he rpezodl rle m eld e rel when he non eng negred onn or lner non o hen one n e hee rel o genere he rpezodl rle. Conder h or = n d I d Undeermned Coeen
45 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng And or = Thee wo negrl eld wo eqon or he wo nknown oeen. Solng hem mlneol eld whh when ed k no he orgnl ormlon or he negrl o he non ge k he rpezodl rle.e. d I 5
46 Whkewz College o Engneerng Seon : ISOPARAMETRIC FORMULATIO The ojee o he G qdrre pproh o deermne he nknown onn or he epreon I Howeer n onr o he rpezodl rle h ed ed end pon nd he non rgmen nd re no ed nd reed nknown. ow we need or negrl epreon o nd he or nknown nd. We on hee ondon mng he eqon oe prode he negrl le el or onn non nd lner non.e. eql hold or keepng n mnd h d d d d 6
47 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng We e he nonl orm o he negrl.e. he lm o negron re rom - o ee hee re he lm o negron we wll e or oprmer elemen. To rre wo ddonl ondon we me h he G qdrre ormlon eld e rel when he negrnd polnoml o degree or le. On he preo oerhed geomer rgmen were mde o mke he pon h eql hold el. Eql or hee wo epreon wll e demonred momenrl. I he wo epreon oe re e hen d d 7 d d
48 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Thee wo epreon long wh he wo preo epreon eld or eqon n erm o or nknown.e. whh led o homework gnmen
49 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Conder he ollowng negrl wh polnoml negrnd I Ung he G qdrre or h negrl eld I d One onlde rom h re emple h he G qdrre e or polnoml h re or le. 9
50 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Beond he wo pon orml dered preol hree or e nd pon eron o he G qdrre pproh he een ed. The generl orm I n where n he nmer o qdrre pon. Vle o nd re mmrzed n he le o he rgh: n n mer o g pon Inegron loon Wegh = = = = = = = = = = = = =.866 =.7858 =.998 =.655 = =.655 = = = = =.5869 = =. = = = = = =.969 =.75 = =.6766 = =.6799 = = = = = =.75 5
51 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng I we wn o eend h pproh o negron oer n re or n oprmer elemen hen In generl we do no he o e he me nmer o G pon n eh dreon.e. doe no he o eql j n ne elemen nl h pll done. n j j n j n j n j j n d d d I 5
52 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Conder or pon G negron whh hown n ollowng gre where or n rrr non o nd where ll mplng pon re or nd ll oeen re eql o one. Hene he dole negrl dole mmon ehnll rell ngle mmon oer or pon n he elemen.e. he G pon. d d I n j j n j 5
53 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng For olme elemen we n el eend he onep ollow: n k k j k j n j n z dz d d z I 5
54 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng Elemen Sne Mr In generl or wo dmenonl elemen we he hown h For qdrlerl oprmer elemen I we e G qdrre o ele he negrl A T T d d B D B k d d J B D B k T T J B D B J B D B J B D B J B D B d d J B D B k T T T T T T T T T T T T T T 5
55 Seon : ISOPARAMETRIC FORMULATIO Whkewz College o Engneerng In Cl Emple 55
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