Outline. Review Quadrilateral Equation. Review Linear φ i Quadrilateral. Review x and y Derivatives. Review φ Derivatives

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1 E 5 Engneerng nlss ore on Fnte Eleents n ore on Fnte Eleents n Two Densons Two Densons Lrr Cretto echncl Engneerng 5 Senr n Engneerng nlss prl Otlne Revew lst lectre Qrtc ss nctons n two ensons orer ntegrl ters Trnglr eleents trl coorntes re coorntes Lner ss nctons Hgher orer ss nctons onr ters Revew Qrlterl Eqton φ n n γ γ Select ss nctons n n Revew Lner φ Qrlterl ote: φ δ 5 Revew φ Dervtves 6 Revew n Dervtves Hve ll norton to evlte or eleent

2 onr Ters Eleent onres le long lne o constnt ± or constnt ± onr ntegrl s on long these lnes Two cses to conser Hve grent n or r n onr conton to ncle n solton Copte grents ro solton or Drchlet onr conton 7 onr Integrl Fn ntegrl elow whenever one se o eleent s on eternl onr Γ û s n n Derentl stnce s S ± / or S ± / where S s length o se E.g. S en se strt 8 s onr s Eple ˆ s S n Γ s S φ n φ re zero long onr Length o se S s S Derentl stnce s S / 9 onr Eple II Evlte ntegrl or φ n φ t Γ ˆ s n S n ˆ s S n n Γ Se reslt or φ / onr Eqtons onr ter n S Eqtons wth not neee or Drchlet onr contons Use ter solton to copte grents onr Eqtons onr ter n Eqtons wth not neee or Drchlet onr contons Use ter solton to copte grents S E 5 Engneerng nlss

3 onr Eqtons onr S ter n Se sgn on s n ter Otwr cng norl ervtve s n opposte recton onr Eqtons onr S ter n Se sgn on s n ter Otwr cng norl ervtve s n opposte recton Revew Eleent Eqtons onr L ter n on R R R on R R noe s not on onr R on_ noe s on onr n s on_ on_ or noe on two onres 5 Locl vs. glol ners Revew ssel ssel loos t ll eleents tht contn gven noe Get eleent eqton or locl noe ner oe 5 s noe n eleent 7 noe n 9 Use eqton n 7 where noe s ltple eqton n 9 where noe s ltple 6 Revew ssel Reslt ll eleent eqtons wth glol nces or noe [ ] [ ] [ ] [ ] [ ] Revew prl 7 Hoewor Solve Lplce s Eqton or the regon n nte eleent gr shown Detls n ssgnent onr vles re t ll noes ecept notch where t s zero Eleents re Three nnown rhoos wth noes ses o h. Use locl coornte o sste to get spce ervtves 8 E 5 Engneerng nlss

4 Revew prl 7 Hoewor II Locl coornte sste h. - - > θ o - > h h - > h sn θ h cos θ > h h cos θ h cos θ o 9 Revew prl 7 Hoewor III h h h hsn θ hsn θ hsn θ Revew prl 7 Hoewor IV hcosθ hcosθ h cosθ h hcosθ hsn θ h cosθ hsn θ h cosθ h θ θ sn cos h cosθ h cosθ cos θ hsn θ h h cos θ sn θ h cosθ cosθ Revew prl 7 Hoewor V Use one-pont Gss qrtre or setrc ro prl - lectre φ Gss qrtre Revew prl 7 Hoewor VI h h cos θ cos θ sn θ cos θ cos θ For q o snθ ½ n cosθ / / cos sn θ θ cos θ sn θ sn θ cos θ Revew prl 7 Hoewor VII Copte ntegrls sng Gss qrtre wth one Gss pont See generl or o shpe ncton on net chrt or copttonl ese ssele eqtons or three nnown noes n gr Ssttte onr vles n solve resltng sste o three eqtons or the three nnowns E 5 Engneerng nlss

5 Revew prl 7 Hoewor VIII We cn wrte ll shpe nctons s Soltons se these eqtons to get ntegrls Revew prl 7 Hoewor IX Set h to get or Gss qrtre 6 6 or vros 5 6 Revew prl 7 Hoewor X or zero-pont Gss Revew prl 7 Hoewor XI ll eleent eqtons wth glol nces or noe - - UL LL UR LR [ ] UL LL UR UL LL LR [ ] [ ] UR LR UR [ ] UR UL UR [ ] -- LL LL LR LR Revew prl 7 Hoewor XII Ssttte nercl vles se or ll eleents - - UL LL UR LR [ ] 6 [ ] [ 6 6] [ ] 6 [ ] Hgher orer Shpe Fnctons Eqton on prevos pge s vl or n shpe nctons n qrlterl Isopreterc eleents se the se orer shpe nctons or oth the geoetr n the epenent vrle Col se lner nctons or geoetr hgher orer or epenent vrle Hgher orer nctons or geoetr wol llow eleents wth crve ses 9 E 5 Engneerng nlss 5

6 Hgher Orer Shpe Fnctons Use notton t rght or wrtng hgher orer shpe nctons Ths ws se n ervtons lst te lner shpe nctons sng ths notton re shown t the rght Qrtc Lgrngn Corner noes Se noes where n ± Se noes where n ± 6 Centrl noe 9 Serenpt Qrtc Corner noes Se noes where n ± Se noes where n ± 6 Other ss Fnctons Lner shpe nctons h or noes per eleent n qrtc nctons h nne Lgrngn or ten serenpt Cc shpe nctons hve 6 or noes or Lgrngn or serenpt Herte cc shpe nctons se rst ervtves s nnowns ll nvolve evlton o ntegrls n onr ntegrls Retrn to sc Reslt φ Hve generl two-ensonl reslt ˆ s n Γ K ppl ths eqton to n D eleents Lte to eple PDE Conser trnglr eleents net Use erent coornte sste 5 Integrl or Trngles Reslt o ervton n ters o trngle re coorntes In ths ntegrl re o trngle n n epen on verte coorntes 6 E 5 Engneerng nlss 6

7 Trnglr Coorntes / Vertces hve coorntes trl re coorntes strt t zero t se opposte verte re perpenclr to tht se n go to one t verte Three sch coorntes wth / 7 Trnglr Coorntes II / Trngle re seheght/ / or ths trngle Inner re hs heght whch s tes totl heght.e. h so tht [ ]/ Dvng gves / 8 Trnglr Coorntes III Trnglr Coorntes IV Hve three seprte re coornte vrles n Ech goes ro zero to one Ech / so tht / / / / 9 Lnes ro pont where ll ntersect to ech verte enes three sres Prevos chrt showe tht ech re heght se / or / Snce Trnglr Coorntes V t n pont n the trngle the n coorntes re relte s ollows Cn wrte tr eqton to relte n to pls constrnt tht Coornte Trnsortons Use orl or the coponents o - Det s nor eternnt E 5 Engneerng nlss 7

8 E 5 Engneerng nlss 8 Fnng the Inverse Det Det Detls or re t en o presentton Gettng ro n Ths eqton gves ll coorntes or n Crtesn coornte Cn epress shpe nctons o n orer n ters o coorntes 5 Trngle Shpe Fnctons φ Lner shpe nctons: Three noes t trngle vertces Sts sc reslt tht δ Qrtc shpe nctons oes n t vertces n noes 5 n 6 t ponts o ege Shpe Fncton Dervtves Fnte eleent coecents hve ervtves n ntegrls o shpe nctons wth respect to n Hve to get ervtves n ntegrls wth respect to the Onl two o the re nepenent Cn pc n s nepenent vles o Hve sl eqtons or trnsors 7 Shpe Fncton Dervtves II Trnsor ervtve o n ncton ψ ψ ψ ψ ψ ψ ψ 8 Shpe Fncton Dervtves III Sr o prtl ervtves reqre or ntrl coornte ervtve trnsortons Dentons o n gven here

9 E 5 Engneerng nlss 9 9 c to scs Chnge notton or shpe nctons ro φ to n se sc Glern eqton or Helholtz eqton s n Γ ˆ K Trnglr eleent wth e noes Trnsor ervtves n ntegrl ro to 5 Dervtve Trnsortons Use prevos eqtons or ψ/ etc. 5 Dervtve Trnsortons II Cone eqtons ro lst chrt 5 Trnsor Integrl to Use con eternnt / etls t en o presentton 5 Integrl or Trngles Orgnl eqton ter trnsortons n con Dervton o ntegrl lts on chrts t en o presentton 5 Trnglr Eleent Eqtons For lner shpe nctons Integrls t en o presentton δ Rght-hn ses ecept or onr noes

10 orer Ters Conser se - s L Others wll e slr ˆ ˆ s n n ˆ n s L e Γ long se - we onl hve to evlte ths ntegrl or n L n ˆ L ˆ n L ˆ n 55 orer Ters II To get orer ntegrl or wth se so snce s constnt long On se goes ro to zero s we ntegrte n conterclocwse recton L n ˆ ˆ L n Slr reslts or other orers Ters pper n two o three eleent eqtons L ˆ n 56 orer Ters III Eleent eqtons or orer on se L ˆ n Eleent eqtons or orer on se L ˆ n 57 orer Ters III Eleent eqtons or orer on se L ˆ n Dene R or noes not on eternl onr n R otherwse R R R 58 ssel w ro onr tpcl noe s prt o ve or s trngles st conser eleent eqtons or tht noe ro ll trngles Ech eleent hs three eqtons To ssele eqtons or one noe pc the one eqton ro three ro ech eleent where noe ppers pc eqton where the nol vle s ltple coecent whose sscrpts re the se 59 esh Trnglr esh ro TL PDE toolo 5 or 6 per noe 6 E 5 Engneerng nlss

11 ssel or S Trngles Centrl noe g s prt o s eleents lele ro α to ζ sse noe g n glol sste s noe or ll trngles n locl sste α β γ α β γ c R α g β g γ g R R α β γ δ c ε ς e e c δ ε γ g β ζ α δ ε e ς δ g ε g ς g R R R 6 δ ε ς ssel or S Trngles II ll eleent eqtons or noe g Other sstes wll not hve ll noe g coecents s α β γ β e c δ ε γ g β ζ α γ δ c δ ε ε ς ς α e α β γ δ ε ς R α R β R γ R δ R ε R ς g 6 Trngle Qlt For ccrc wnt ll trngles to e s close to eqlterl trngles s possle re o eqlterl trngle wth se h Qlt o trngle wth ses n Q Keep Q > Hoewor Two proles Repet nte eleent prole ro lst wee sng grent onres Trnglr nte eleents Get eleent eqtons ro hoewor solton One grent onr noe o Zero grent on these eleents 65 Hoewor II Recll qrlterl shpe ncton or Use ect reslt or ntegrls cosθ snθ 66 E 5 Engneerng nlss

12 E 5 Engneerng nlss 67 Hoewor III Trnglr eleent prole ll trngles eqlterl Ech se s h. Drchlet onr contons Dentons o n shown elow Coorntes or one trngle: ; h ; h/ / h/ re / h / nnowns 68 Hoewor IV Integrls or ro prl 7-9 lectres or Lplce s eqton Hve three eqtons or ech eleent ne setrc vles ssele eqtons ro s onng eleents to get two eqtons or two nnowns n overll gr δ 69 tonl Chrts Dervtons not covere n clss Chrts 7 n 7: the re o trngle n ters o the coorntes o ts vertces Chrts 7 n 7: con or convertng re to re s Chrts 7 n 75: gettng n verng lts or trngle ntegrton lts Chrts 76 to 78: etls o ntegrton or lner shpe nctons 7 re Clclton Detls Crete rectngle ron trngle wth rtrr orentton Trngle re s rectngle re ns re o three trngles / / / 7 re Clclton Detls II ltpl ot n cncel ters wth cptl letters or lels F E G D C F D E D C G Fnl reslt 7 con ro to Use con eternnt

13 E 5 Engneerng nlss 7 con Detls ltpl n show tht nertor gves prevos epresson or n ters o trngle vertces coorntes 7 Integrtng Trngle re Use ollowng lts or ntegrton over entre re o trngle Length 75 Conr Integrton Lts I we shol get re [ ] Conrs correct lts or ntegrtng coplete re o trngle sng the ntrl re coorntes lso shows ntegrl vle / / 76 Lner Shpe Fnctons Strt wth generl two-ensonl ntegrl or Helholtz eqton eple For lner shpe nctons 77 Lner Shpe Fnctons II Use ollowng generl reslt we hve shown t s correct or!!!! 78 Lner Shpe Fnctons III Use generl or st stte to get rener o ntegrl or δ 6!!!!!!!!