Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Inroducon We ne conder hpe funcon for hgher order eleen. To do h n n orderl fhon we nroduce he concep of re coordne. Conder ere of rngulr eleen depced n he fgure below
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS The nuber of node n ech eber of rngulr eleen uch h coplee polnol epnon wll enure dplceen copbl. Conder Pcl rngle. oe for he hree node rngulr eleen he polnol needed o defne dplceen of oe o e ee ode gu e e e e po o eeded o de e d p ce e o order one. For he node rngulr eleen he polnol needed o defne dplceen n he eleen of order wo nd for he en node eleen he polnol of order hree.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS For he hree node eleen ( u For he node eleen ( ( v 5 ( ( 0 9 8 7 5 v u For he en node eleen ( 5 u ( 5 0 9 8 7 v 0 9 8 7
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Wh needed ple ehod o genere hpe funcon for hgher order eleen. To do h we nroduce he concep of re coordne. We wn o defne he locon of pon whn he followng hree node rngulr eleen We could ue he followng wo epreon n er of nd Thee re nerpolng funcon nd he phcl nerpreon of nd re defned oenrl. We won be urpred o fnd
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Ung Crer rule Focung on he denonor h deernn ( ( ( If we e nd hen fro he Conn Srn Trngle noe ( ( ( ( ( ( ( ( ( ( ( (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Focung nll on ( ( ( ( ( ( c b
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Solvng he e of equon for ( ( ( ( ( ( c b
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Fnll for ( ( ( ( ( ( c b
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Here ( ( ( ( ( ( c b df d h ( ( ( ( ( ( c b c b nd f we e nd hen ( ( ( ( ( ( c b c b If we e α bβ nd cγ hen ( ( ( ( ( ( c b c b If we e α b β nd c γ hen ( ( ( ( ( ( γ β α γ β α ( ( ( ( ( ( γ β α γ β α
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS ( γ β α nd ( γ β α ( γ β α Thu nd re hpe funcon.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS We now ee phcl nerpreon of nd. Conder he followng rngulr eleen. If we focu n on pon P( hen P( Thu he ro of he ornge re o he re of he rngle. Slrl l l he ro of he pn re o he re of he rngle nd he ro of he blue re o he re of he rngle. Hence p p ( re( P re ( P
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Wh he phcl nerpreon of nd we now urn o he oprerc forulon of hpe funcon for he hree node conn rn rngulr eleen. In he - coordne e he eleen h vercl nd horzonl de equl o un. P (( P (( P ( ( P oe fro he geoer of he prevou pge h he nuber of he re egen correpond o he node nuber oppoe o he re egen.e. P P P
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Snce he re coordne re he ro of he re egen o he overll re hen P P P nd n urn he hpe funcon n he - coordne e re:
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS For he followng oprerc ppng ( ( ( he rnforon equon fro he - coordne e o he - coordne e re ( ( ( ( ( (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS ow le recp he forulon for he hpe funcon n he - coordne e for lne eleen: Two node lne eleen ( ( ( Three node lne eleen ( ( ( ( ( ( (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS ow le recp he forulon for he hpe funcon n he - coordne e wo denonl plnr eleen: Three node rngulr eleen ( ( Four node qudrlerl eleen ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
The precedng dcuon hould ndce h we re erchng for pern( o ulze n he forulon of he hpe funcon for ech pe of eleen. h pon here doe no ee o be nhng upng ou of wh we hve lred done. However le focu on qudrlerl eleen nd cegorze he hgher order eleen no everl fle. The re referred o he Serendp fl of eleen repreened b he eleen n he lef hnd de of he endng pcure nd he grngn fl of eleen repreened b he eleen n he rgh hnd de (wh cenrl node. Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Eleen Fle
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS grnge Inerpolon - Revew In d nl for engneerng degn we re frequenl preened wh ere of d vlue where he need re o nerpole vlue beween he gven d pon. Recll lner nerpolon ued eenvel o fnd nerede bulr vlue. noher coon pproch ung hgher order polnol o curve f funcon beween d vlue. The polnol uull e he for: n ( n n f 0 For (n d pon here onl one polnol of order n h pe hrough ll he vlue. For eple here onl one rgh lne ( fr order polnol h pe hrough hwo d pon. Slrl l l onl one prbol connec e of hree d pon. Polnol nerpolon con of deerng he unque n h -order polnol h f (n d pon. Th polnol hen provde forul o copue nerede vlue.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS We hve been dong h ll eeer whou he forl defnon gven on he prevou lde. lhough here onl one n h -order polnol h f (n d pon here re vre of ehod h cn be ulzed o obn he fnl for of he nerpolng polnol. Thee ehod nclude (bu re no led o ewon Dvded Dfference pproch The Mehod of grnge Polnol Regreon nl (lner nd non-lner Splne Here we focu on grnge nerpolng polnol becue he ehod led drecl o he forulon of hpe funcon for hgher order eleen wh n ppropre nuber of nernl node. The grnge nerpolng polnol reforulon of he ewon polnol bu vod he copuon of dvded dfference. The grnge polnol h he for: f n n ( H ( f ( 0
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS f where H ( For eple he lner veron (n f ( n n ( ( ( n ( ( 0 ( ( ( ( 0 ( f f 0 f 0 0 0 0 nd he econd order veron ( ( ( ( ( ( ( 0 ( ( ( ( ( ( 0 ( ( ( f 0 f f 0 0 0 0 One cn begn o ee he uefulne of grnge polnol b relzng h ech er H ( wll be nd 0 ll oher d pon (eep n nd he nde r 0. Th he qul we re loong for n hpe funcon.e. he hpe funcon for prculr node he node nd zero ll oher node. (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS In wo denon he nerpolon funcon h he for where ( ( ( ( ( n p g V f H f 0 0 Three denonl grnge nerpolon h he for n p where ( ( ( ( ( ( ( l n p z q Q g V f H z f 0 0 0 where n l p
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS grngn Eleen grnge eleen hve n order of n wh (n node rrnged n qureerc pern. Thee eleen requre nernl node. Shpe funcon re produc of n h order polnol n ech drecon. The blner qudrlerl eleen (four node grngn eleen of order n.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS We cn el e ue of grnge nerpolng polnol b preenng wo forulon for - coordne e. The fr poll for (denfed H for h l horzonl ( ( ( ( ( ( ( ( ( ( ( ( ( ( H 0 0 0 nd econd polnol n (denfed V for vercl ( ( ( ( ( ( 0 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( V 0 0 0 In generl he hpe funcon n he - coordne e for gven node he produc of hee wo epreon ( ( ( [ ] ( ( [ ] V H where he order of nerpolon nd he node nuber. ( ( [ ] ( [ ] V H
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Ung h procedure conder he node qudrlerl eleen. We wh o develop n epreon for he hpe funcon node #. V ( ( ( ( 0 ( ( ( ( 0 H ( ( ( 5 ( 7 ( 8 ( ( H ( ( ( ( 5 7 8 ( [ ] [ V ( ]
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS grnge polnol for hpe funcon re coplee polnol epnon. Fro Pcl rngle we cn ee how n node re requred for he repreenon of dplceen feld of n order nd copleene: 5 5 Blner Qud ( node Qud (9 node 5 Qud ( node 5 Zero Order Fr Order Second Order Thrd Order Fourh Order Ffh Order Sh Order
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Serendp Eleen In generl erendp eleen onl ue boundr node. Inernl node re voded. Serendp eleen re no ccure grngn eleen. However he re ore effcen hn grngn eleen nd he vod cern pe of nble. The four node erendp eleen he e he four node grngn g eleen. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS For he egh node erendp eleen r b forulng he hpe funcon he d de node. Shpe funcon for d-de node re produc of 7 p p econd order polnol prllel o de nd lner funcon perpendculr o he de. ( ( 8 5 ( ( 5 ( ( ( ( ( ( 5 ( ( 7 ( ( ( ( Fold he coordne pcure ( 8 ( ( ( Fold he coordne pcure down nd over he hpe funcon fgure. 5 nd 7 re lner n where nd 8 re lner n.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Shpe funcon for corner node re odfcon of he hpe funcon of he blner qudrlerl eleen. Sep #: r wh ppropre blner qud hpe funcon ˆ ( ( Sep #: ubrc ou d-de hpe funcon 5 wh ppropre wegh.e. ½ ee fgure ne overhed Sep #: repe Sep # ung d-de hpe funcon 8 nd wegh ˆ 5 ˆ 5 8
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Hence ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
Grphcll he proce pper follow Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Once hpe funcon hve been denfed here re no procedurl dfference n he forulon of hgher order qudrlerl eleen nd he blner qud. P l l f h d d l l l Pcl rngle for he erendp qudrlerl eleen: Blner/Serendp Qud ( node Blner/Serendp Qud ( node Serendp Qud (8 node 5 5 Serendp Qud ( node 5 5 5 5
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Shpe Funcon for Trngulr Eleen We now urn our enon bc o rngulr -ode (α β γ (fr order eleen. Fr defne coordne e o # (00 denf node nd pecf locon of pon b # (00 denfng dnce eured perpendculr # (00 fro ech de of he rngle. The coordne of ech node depend on he order of he eleen. -ode (α β γ (econd order # (00 # (0 # (00 #5 (0 # (00 # (0 0-ode Eleen (α β γ (hrd order # (00 (00 # (0 (0 #5 (0 # (00 # (0 #7 (0 # (00 #8 (0 #9 (0 #0 (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Slveer (99 eblhed he followng pproch o defne hpe funcon for rngulr eleen ung re coordne nd η η η p p ( ( ( ( q α β γ Here n he order of he eleen p he node nuber nd q he nuber of node n rngulr eleen nd η α ( α n α α 0 If we wn he nerpolon funcon (hpe funcon for node # n he node rngulr eleen hen he nerpolon funcon wll be qudrc nd 00 0 η ( ( η α α n n (
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS For node # ( ( ( 0 γ α ( ( ( n n γ α γ α The wo hpe funcon re depced grphcll follow:
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS The nerpolon epreon gven for he hpe funcon do no pper o be grngn polnol nerpolng epreon. However he dcuon bed on he fc h he re coordne re ndeed coordne. oe h he re coordne ( re coordne n he e ene h ( re coordne n Cren wo pce. Boh e of coordne denf he poon of pon. P P* conn Here ( P( ( P* ( P P * lo noce h lne prllel o he de of he rngulr eleen denoe lne of conn or. Th clerl hown n he ne fgure.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS In h fgure lne of conn vlue of he re coordne re depced for wo vlue zero nd one: P 0 0 0
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Hlfw beween he lne of conn vlue 0 nd hould be lne of conn vlue equl o / (d-de node dded for clr 0 / / 0 5 0 /
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS We now hve funconl vlue of ech re coordne long he de of rngulr eleen equl o one ( one hlf (/ nd zero (0. We hould be ble o conruc grngn nerpolng funcon wh h nforon nd we cn. Conder gn he hpe funcon for node # n he node rngulr eleen. Wh hree d vlue we hould be ble o eblh qudrc grngn coeffcen uch h l ( ( ( 0 ( ( ( ( 0 0 where 0 0
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Th eld l ( ( ( ( 0 ( 0 ( 0.5 ( 0 ( 0.5 ( 0 ( before. The proble h he publhed veron of he grnge polnol do no wor well n denfng he vrou vlue of he re coordne. So he prevou nerpolon funcon ued o eblh hpe funcon for vrou node n rngulr eleen wor well.e. robu. nd for of grngn nerpolon deonred bove.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS The dcuon bove nd he generl nerpolon funcon cn be repreened follow: p ( η ( η ( η ( p q α β γ ech node: Here cler h poon cn repreened b re coordne nd h re coordne wll e on dfferen vlue vrou node.
Secon 9: HIGHER ORDER TWO DIMESIO SHPE FUCTIOS Inegron Whn Trngulr Don