MMJ 1113 FINITE ELEMENT METHOD Introduction to PART I

Transcription

1 MMJ FINITE EEMENT METHOD Cotut requremets Assume that the fuctos appearg uder the tegral the elemet equatos cota up to (r) th order To esure covergece N must satsf Compatblt requremet the fuctos must have C r cotut at elemet terface Completeess requremet The fuctos must have C r cotut wth a elemet Deftos: C 0 - feld varable s cotuous at elemet terface C the frst dervatves are cotuous C the secod dervatves are also cotuous

2 MMJ FINITE EEMENT METHOD Polomals as terpolato fuctos depedet varable () Ρ ( ) Τ No of terms: T () Eamples: T () () 0 Ρ ( ) 0 depedet varables ( ) ( ) j ( ) j Τ Ρ No of terms: Τ ( )( ) ( ) T () Ρ ( ) 0

3 MMJ FINITE EEMENT METHOD depedet varables ( z) No of terms: Eamples: N T () N T () 0 ( ) ( ) j z z l j l Ρ Τ ( ) ( )( )( ) 6 Τ ( ) z z Ρ ( ) z z z z z Ρ

4 MMJ FINITE EEMENT METHOD Geometrc Isotrop for Iterpolato Fuctos Geometrc sotrop the polomal epaso for the elemet must remas uchaged uder a lear trasformato from oe Cartesa coordate sstem to aother 7 s 8 6 r 5 Global coordate sstem ( ) ocal coordate sstem (r s) ) P wth all terms have geometrc sotrop ) P that are complete et cota approprate terms to preserve smmetr have geometrc sotrop Terms smmetrc par ca be omtted Eg ( ) ( )

5 MMJ FINITE EEMENT METHOD Eample: I order to use the polomal but wth less terms we ca drop the par (eg ) ( ) P Cubc 0 terms ( ) P

6 MMJ FINITE EEMENT METHOD Tragular elemets for C 0 problems C 0 problems requre ol cotut of at elemet terfaces Number of odes alog elemet sde (hece umber of odal values of ) must be suffcet to uquel determe varato of alog that sde 6- ode tragle quadratc varato of ( ) ode tragle ear varato of ( )

7 MMJ FINITE EEMENT METHOD 0- ode tragle Cubc varato of ( ) Polomals of order greater tha are rarel used eg 5-ode (quartc) -ode (qutc)

8 MMJ FINITE EEMENT METHOD Dervg Iterpolato Fuctos To epress the geeralzed coordates (coeffcets s) terms of odal dof Cocept: ( ) ( ) ( ) ( ) Feld varable Assg oe value of to each ode Elemet has dof Select a -term polomal as terpolato fucto ( ) p { }

9 MMJ FINITE EEMENT METHOD Evaluate the coeffcets { } [ ]{ } { } [ ] {} G G { } [ ] {} {} Ρ Ρ G Thus Recall

10 MMJ FINITE EEMENT METHOD - s a terpolato fucto that apples to the whole elemet Epress feld varable behavor terms of geeralzed coordates N - refer to dvdual ode ad dvdual dof Collectvel represet feld varable behavor Issues: 0 [ ] at ode at other odes Sometmes G do ot est for all oretato of elemets global coordate sstem Cost of computg [ G] s prohbtve Tr to determe N b specto usg atural coordates

11 MMJ FINITE EEMENT METHOD ( ) ( ) ( ) ( ) Natural Coordates (-) () (--) (-) Global Cartesa coordate ( ) Natural coordate ( ) A local coordate sstem Reles o the elemet geometr for ts defto Coordates rages from zero to ut A fucto of the global Cartesa coordate Purpose: To descrbe locato of a pot sde a elemet terms of odal coordates

12 MMJ FINITE EEMENT METHOD Natural Coordates -D p - - p p - Ρ ( are atural coordates or legth coordates) Solvg for ad () () ( X ) ( X )

13 MMJ FINITE EEMENT METHOD The feld varable ca be epressed as Use table 5 to evaluate the tegral of legth coordates: ( ) d d ( ) ( )!!! β β β d

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15 MMJ FINITE EEMENT METHOD Solvg for : Natural Coordates -D ( ) ( ) ( ) ( p p ) Tas: To choose to descrbe locato of a pot ( p p ) ( ) ( ) ( ) ( ) ( ) ( ) c b a c b a c b a

16 MMJ FINITE EEMENT METHOD where the feld varables ca be epressed as c c c b b b a a a tragle area of ( ) we have c b

17 MMJ FINITE EEMENT METHOD Natural Coordates -D (Cotued) 05 0 A A A 0 0 Α Α Α Natural coordate ( ) are also called area coordates Α If ( p p ) s located o the boudar sa alog boudar - the sce A 0 use table 5 evaluatg tegrals of area coordates 0 ( e Α ) β γ dα ( e)! β! γ! ( β γ )!

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19 MMJ FINITE EEMENT METHOD Rectagular elemets for C 0 problems Classcal terpolato fucto usg agrage polomal To wrte the terpolato fucto drectl usg agrage polomals atural coordates ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) m m m m Π b a c c (--) (-) () (-) c c b a a b

20 MMJ FINITE EEMENT METHOD varables learl betwee odes ad ( ) ad ( ) ( ) ( ) 0 etc Wrtg N () eplctl ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

21 MMJ FINITE EEMENT METHOD N ( ) are also called blear ( ) ( ) ( ) ( ) ( ) ( ) total o of odes for the elemet For the lear elemet (- ode) N ( ) ¼ ( )( )