T AFSM

INTRODUCTION TO THE FINITE ELEMENT METHOD

TAYFUN E. TEZDUYAR

James F. Barbour Professor Mechanical Engineering Rice University, Houston, Texas

tezduyar@rice.edu

http://www.mems.rice.edu/~tezduyar/ Part of the Class Notes for MECH 454/554 Computational Fluid Mechanics Also listed as BIOE 454/554, CEVE 454/554

l .FrNtrE ELETqENT MtrrroDs ( eem)

FOR

L|NEAR STATIG, ANALYSIS

PR,ELI'Y I NAR,IE S

I{AIN cONSTITUENTS oF FEM FoE T}I€

3OLT'TION OF A BOUNDARY- UILUE PEOBLE}4 3

( BvP)

O. A sTeorrf(r ( cuasstclu) roRFI oF Tne 6lutEN

BourrueC'- \fALut FroBL€tl

- DIFFESEIfrIAL €truA'lloNs

,-. BouNoAcy CoNDrTtorrt

, ( nrrrrAt co}rDtTror r ) ( rp tr ts AN

|r|rrfAL/ BourrDAQY-rArue Dtott€n )(wEGUTED BE:ToITAL)

l. A hrEAR ( vlerATtoNlt ) forril oF Tu€

PsoBL€JY ' l . (? )

Z. A 6Atclk'|rf FoEH oF T]fC PfsEt€n'r

( = FINFrE DlHcr.r$o$AL \rEtstoN OF TllC

hfEAr, Fo:H ) BY ustN6 Flrlrr€

crcrlENr Fulrs'rrsNs (?)

3 . A F|ATstI noRH oF Tlt€ PEoBrEr-l ( ? )

z.

EXArutpLE : SOLVE, TllE toLtoWlNg Olf E-

D|F|ENSIONAL BOUTTDAR Y -VALU€

PBoBL€t\t

+ + + = o F a E o < x < ld x 1

'

DIFFEEEUTIAL I h, F \ DOmetU OF aUCFquATloN \u'E') ptoB\Er.t nctuDtN6

Tllc ErfD DollfTsg . , r uD l .

u(t) a ? I B*)NDABYv l : : : : : : : ._ (e.c.)

-jlto)= t J couDrrrolrs

4r 0rl : tsrrhrN Fut{cTlgrl AND corrsTA}lT:

IT : UNtsTFh|N VA3IABTC

ReuARr,s

tr'(l)=t ess€rtTlALtsoulrDASt corrDlTlox

NATUSAL

BourlorRY coNDlTloN

.- **t"= tr

3 .E xAr{F\€ P$YS|CAL P?oB\SrS:

.) tl€AT CoND$CnoN PEoB\Etr

DlFFEtenlA( EQUATI0N : ll€tfl ColrurGflott

6:quATlox

+ : H€A? SouBGE ( or srNt )

U '. TEMP€:arueC,

gssEriTlAL B_C: a _ -- TE1v1DG?ATUTE E.l'B.c.: ?,

TEt"f DCrATurE B.!.

NAlttlA\ B.C. , L: H€AT ftUX B. C .

-t Elssrlc 3oD P:,olttFr'l

-;l gTP'lNg PBOBIEH

ttOTAllotl S :

*t = urx

d3. = tlrxxd x a -

r F o < x < l r T l s s A t D r l r A T X € - S L

5-= F, ll

lF o< x < I rr ts sAtD rllAT xe-sl.

JL= Jo, r[-n, , G\ostllgn$\ F--'1 -.r E,, lJ?- : opErrl thmBVAL

( ootn Aril )H r { f , t

STEONS F€E,FT OF TIIC BVP :

R ( eesrur^r )-

Ur r r + f ro oN .SL

u ( r ) r t- u)r(o) = 4

( Vx€sr)af

\For ALL xr N t L i

a'

NOTE AtL DIFFIEB,ENNAL R\IAIIAN S HAUE

Tllg SAtrg GEN€EAL Fol'r-l

R = ot--- gtBrDuAL

rx rrrE AIov€ {ASE R= Urr.*f

TltE ttlAgr soLuTlolr tS TRlul^L

.TO OBTAIN :

x 9u t r ) =

t * A t - r ) 1 t

{ [ t 1 d c d yt r o l

NorE Tf{AT t

^o,O = AE€ DuHorY

VARIABLES.

6 .rrrear ( rrrerc,ril€p BBIDVAL)( Uee|Arrsunl)

re.Fr oF T}Ig DROBL€}I :

B r r . t { r o

wcElntr03RrNc1t0fr

Ita

I *gH5. dx

wc t6tflgD

: o

RCI3IDVA L

WN

BC}aTE,ES !

+ tNsrElD oE SA'tlSF tlrc Tll€ D.E. ATSlq:,v trgtNT lN -fi|, , TltE D.E. lS

sA.ttsFrEO lN A CtoBAL SRISE.

-) Tlill g\olAL CltlrJA'lloN llAt To BE

sATlr,FrS frF. AtJ- */'s ( r^rs,etrflu6

rr)lrcrroN i )'+ A g€lst:,KTtON oN w I w tllrTo

sA'IlsFt 'rut

oF tlrt sslfrtflAL B.<. '!

Fo:ra

T.ErauDLS3 :

u( I ) - I €

u ( I ) = o = E

u ( f ) * bee { )

u(o ) - 5o :D

EXAHDL€-r-

eSrun$ THAT

w(r) r ' o

w t l ) = o

ul ( l ) = o

w(o) a o

ru€ < LAss ot FuucttoNs wlllclf sAllsFT

THF tblro6€\lO\rs fionn oF TB€

BS€NflAL B.c.'5 13 c ALLEI)

Ttrs CLASS OF VAR, lA_fl0N S :

u ( j ) = &THe\J TUF CLABI oF VAElfrlos5

coNslsTs otr FuNcrloul wlrlctfSArlsFY W(f )=o

'(u)

/ V \ - / 1 ' o - l '

t ?ArL BetoNG To tf Dolrr Ber'oll6 To 1I

8.

tF t*€

anrr)

6LotA\ gQuAtlo$ ( ttneEAL

ls sAllsFlED fi)g A\L W I N 'tf

t ?

\ w ( \ , , r { ) d r = og J

T PIi\I TFIS 610IA\ RiJAT ION IS

Vwel|

€ulvAL€\sr To TltE sl6tNAL D.F

Se.ausg Tug G\oBA! R^JA.rfoNt r l l

ls TEST€D Fora F\\ TFs\ ( wner?il6)

FiructtqN: )

FOR FOl,tlrr AL FIOO F

FrRrr DTF|NE Tl^|o FuxcTtoil ctcstgs ( snecgsl

l . TR,IAL SouurroN SPAC,€

= [" t.":tj Y t "' or*G't csNDrrto.'. ]

Z. T€IT

f =

R€-trr|BITE TTC STRoNe FoRr'l :

FrrlD ue h srrtrTHAT

ulrr + f

u ( r )-urr(o)

o

?h

--

a

3

!q.

( s )

(w)

WC NIEt TO SI|OW TNAT A$Cig TO GO9HS

Ars EeurvArcNT_ ( (S)e+ tw) )

Frrtt itrow T]IAT CS) =)(\^r)

f rnrs la Atso rus DGrtvATbN oF (\lr|) ]

l [ *( u;rrrl) l ' = o yw e"lf|trtqgtrnG 3t PArrS ;

( *u , , ) I I - I [ " r , ,u , . l r . t " [w f { rB o

O W :

- u r . ( o ) = { A N D w ( r ) e o

Vwe 1/

\^,EAL Foltrt :

$w€1f

dr + w(.)lt

r r r rD ueA sucnThAT

lS-,,u,. rr = l5 -t

( wu. f ? )

@)(€esrC

T}rgI€ FctC

l[*,. ur, lr B ll*f tt + r^r(o) t

Ntrr tlrsrr r$Ar (w) =+ (S)w€ N$Ft TO Srsyp Tutta rt

u lt A Sotultop oF (t-l) TlrrN

ALSo A So1 r t? ;g11 OC (_ . ) .

r r , ; ' A € u t r ) = A ' /

1b.

Vw elf

lT ti

thr€?f:

sATtsFt ut (rl - o r

w(.)t:,:,1JJ=o

STqD

Tug ts?Y lt rrf,g crN DcFtNtout Duttotg oF

f t

Ar Lor|B AS lxE,IT A}IT r|AY WE

w€ 6tlrt xcrEt To trb\r TIrAT

U r r t r f I ourr (c)+ [ G o

.TO DO TTIA1 r I}ftg6fATF IT I' DArTTlN tC\FllSE I wE e et :

LI*(u, '1 t+) Jr +a o ?

lHow ?

DIooF. B9CAUTE

urAt{' To ltrvg

sor wg tGFnrG w At , "'

\ l r s + ( u r r r * t ).$lFrr ?nG orllT corrDlal$Ut su + !

t. {trf a o L t.rrurr w(r), o]

z . E > o3 . Q ( o ) = o

-A2 | r r . h t-

: \ O ( u r r r * * ) J ' s o $ s r r , + f " o

if ow Tnrr btF ?to\f eD TrrAT Ulrr + I e O

Tlrg ufAt "oth Dsrc,Fl -ro

wt0 [ u.,r to)+ t 1. t tJ =o Vw c,1/

wg cA$ DFFt$g $l ArrJ rrrrY $€ uAlfT :

.rrto) * |

= t [ r r t ) + l r so , /

D OF PToOF

lF u( I ) =

EXAI.IPTFI tog U A N D W

lob.

Lt\tsa,R'SrrNCTloNf

tl --

W =

tA --

\Af 3

l.A a

<- rrOt€ :

+ MrrE It

o|e-

I

T

cl t

u + (t-vr.)x l( r-x) J

o

I

H1I

I

o

\N =.o

l l.NOTATIQN

fl.t w, u) W r t { r * d r{ r t r

= o l

frexp[€ll I

" 5 v" ur* olxl c

u ) u r r \ r d r

a(vru)e a(urv)

II

s " \ w v d rl 1= . \ vw c r

I (wrV)' (v., W)

(sY rrHFTrv )

A r \C l t V . l I =- - \ - J

J

A r \d ( u r Y J =

oBvfouslY ,

(w,+) = ltEXfrnrP\€I

( * r v )

(v, \Al )

oBVtoustY

( sYHu€rE y )

w+ dr

l , , ^

le

= o\

*,. (o u + bv)"* olv

, r=u \ \ t ( ou ) , ' J x

t l ^

+"J wrr (Lv),r etll / ^

= oo)*, l \ * cv

{ - t ' ( * r v r

J rO J

3 a a ( w , u ) + b a ( w , v )

( Lr NEAETTY )

a(w'Hil]

(wro , lA+bv) B

o t*,;;-r L,;j;' Ir€!5

( LlNlEhR lrY )

Tll€t\t TllE \mf rro?H.' (afsm6cf ftr"r)

a (\^l u) = (w, +) + w(o) {. fweU

r3

EXAMPUE : BERNq,LLI- EuuEI, B€AT.I TH€ORY :

; EFT FOTFTUTATION

" (

A .THIT' GSAFI ( BESNu.,LLI - €UL€: TIIGORV )

F ITED AI ONC €}|D A\lD SUBJECTED TO A S}IEAR

FOE3E AND }|O}IENT AT TTIE OTHER END

sTeox6 Fo:Fl oe Tlr€ Pb$fFr :

u

6t\l€N :---

FINDir-D

tFurrlrorlr TM errl Q,

Arrrb lmt

, tr*t| THAT

(rRlrrrcetc eur'lltrun )oN lt = ]o, lLu, rltr = {

l+

BouNDAey coNDtTlo}rS

es:rur fE c . { u ( r ) = O ( z r i n o T G A T B Y s s s G D T S R A G E H G $ ; )

(, u' (r ) - o (z:ro s\oFr ).rrnruf urrr (o) = M ( p:sliccrrED ltoxesr)

t 'c' l"rrrl

(o) = E ( pE€scerBED s]rsAR- )

U'€AT, FOEi.| OB AH€ FTOBLEH

Ftesr DeFtr|g alrc ctAst53 oF TRIAI souuTloNs

AND UATTIATIONS

TLtt cLAlt O? Teltl FuNcrrorrt 1

J> l THt cr.Aslt oF Frn,cnorrS wrfrcH

SAalSF? TllG q1siq;f lAL B.C.'3

u ( r ) = o

U r r ( r ) = O

Tus CrAgS oF UA3ter|o}rl ( tre te rrru6 ?u$cTloruf )

tt : T$G cr.Ass oG Fuxc'rrqoS hrlrrll

sA'rl3F7 Tltg ]lo}|nG3\l9ut FoPFr

oB TltE Fsstilst|AL 8.G'.'t.

I S

lrcnC-

r x ( r ) = o

wrr ( t ) - o

stlrc5 rnG llss€rirnAt B.G.'s AtE

AttlE, ADY t*

$ot og€r'us FoGr :

ru€ Turo CLAttFj AIC

1f = -/)'TltaT 13,

toclfTrcAL

TO oBartN ttr€ wgAra FoEl{ ao

= O

II,TC6'ATE BY FASTS CI3€

j l w( \*r* -{) Jsr- = o if*r eV

( * ' * r r - f )R.TTS IDUA L

l6

(* , * , ** r ) l l - : l wrr urur dr -

' \ * f ar =o

6 t

ao a

=)

Ir.trCerArt lY FnqTS orrc€ moP€

( u, srrr') l: - ( wrr u,r, ) lj - j! w,*r u,rrdr

- " ! * f d x = o

g rw ( r ) = o w r r ( r ) - o

u r " r ( o ) - A U l r r ( o ) 3 M

e +wrr(o) M - !*+{x

=o) 1

oI *tr ur' dl - w(o)

t (

o\ *r'r urtt d x =

I

" l * fa* +w(o) q -\frr(o) M

t?

DEFINE

a( u,v) = :5 {,xrY'r cx

(u.,v) = lt uv Jr ( As B.bee)

+a(w,u) B (wrf) + w(o)q - wr' (o) u

WEIg FO:FI

Q: wHA't oTllEg coNDlTtolrs Do trrEi llAv€

olr rn€ cLAsslF oF R NcTtor|S

couTAtrsrs6 \A, ArrD t

A: ttnn \{r, a}|b \.r )lAug To Bg'f S,quARJE ntrge:'rtLE

n

l c . - l ' r . a

L.e, \ (wr. . )d* <6 J 9 J

Plculouslv ( i.e . Foe llr.. -+ s g )}IAD

\^rg I1rrf To llA\G $l ALID tlrx To EC

squARl htr€gE,AtL€.

I B

6ALER Krrf's AtrpgoxfMATtoN HgrtloD

\^'E SITPIY CANNCTT AFFOR.D ALL T}I€

FuNerroNs fN A At.rD 'lf.

L€T J BG A Fn*rE DrlfENSrouAt

T lltBGFollG uJg N€ED To APPIOXf T'fATCi

/) AND 1' BY FIuItT DIT.IENSIONAL

s\rrstrs oF .,/) AN'Lf .

w€ uss orfrY A PAET oF A AND V,r^,}IIGil W€ gAN AFFP D .

l f n

N!f9 h SuDer$trDT P€F€rrs To HFtr oR--

DtsclETlzATlor| . r.lN

on€OllrENslON,AL

FE HE:uE:, h P,€F€ns To rtlE

€u, }|€rir Lslb llT .

suBter ot At€T 1f

h 8€ A Fnirs DlNEt StoxAt SuBsg oF V

-zJ c, A (i... uh€)h+,^t€ A)ta

V (i... ",t€lrhs*t€ 1f)

l?

FoE €rA}1p1g

( essernrnl

( se"rneF|9tr

PltBlwt3l gruGAl FTNS FoR, tlh '

tNs.rEAtr oF U3fN6 -/) ( THs crASS oF ALL

FTTNcTIONS Fe t^rHtGll

q( t ) - t ( ssosnAr B.c. )

.e (Ti.l:'ffi,ilLtl

usEh

A ( THg c't'Ags oF Ptgc€hrfsE LI$AP

I A'�S

. I,*,)r d *

FuNsTroNt ht wrf tC$f

,rt(,) E ?

jt (*1,)'d' <B.C- )

TTTEBT ATTF

DFPIV}TI\'IE ) )t f

2o

stHltARLY :

tr0sTEAlr OF UEIN6

tUNs.llOul FeP

w ( t ) = Q

l " 1

o ) (* ' t) dr

U ( rrrt ctrss oF &LwHtct*

( ?br.roecuoirs Eis6rflAL B.C)

FfPtT UEeluAll\ft )

crAss oF plEc€ rrJts€ssg

LTNqAR,.-

w t ( r ) = el ? e

, \ (* t ,*) -J* ( , co

Furrc'TfoNs FOe UlUtcT

( rrorrooEilorrl €SsErfrat B C.)

( scuers rrJr€iB:ABr,rFregr DEEtvATTtE

)

FtrtGwtrs LlNerA: FurrGTlurl FoB ,nrh:

Uh ( rFE

z l

qullTloN i cAN wt uic 9\lRcwtt€ UNRt

FrNgllet.rs |!af. ru€ 1'tt1.1e U\Ll -E\ILER

Bgerl trQ oqt€'H ?

AuswgR" .. No .

8€Ert|sE r^E l*ED Fuilfn$tr

trrtl€as lRorrg DEeNATIUFS Atf'

lfi, Fcn. P€C€ wftig u Ncre. R NcTlo ut

LW'rrr ls urrognrr€D AT SolfE

Po rllrs .

.'. W€ I,EFD AT L€t\i7 PlEl5trrtst CUBIC FulrGno$S .

SQUAEE IN.rBR, Et \€ .

t . e . l -

^ [(*,lrr)tJ*O J

ziL

( BuEuor\ eeutertx rrgtno

Assun€ THAT U t

ls 6lu€r$. h h

Lgr us (oNsrnvcr J) By g1rN6 U '

.

Drrrwg A Fuucrlo\f d

(a6lvsr$Furrrtail)-

iucrr ruAr 6htr)= t

GrAHP\G fi)e At

( plelc€wrSE Ur$af )

7

f€? grr€Ry

\^,R,IT€

hv c ,

u h =

huh hv * a

htF \lNow c*rcL ro sgE BRoN6s To J

uh(,) = Jg.*dUo as A / l E f J

2 3 .

+

wt'rrt''

:.:;;**

toR'

uhe y ) \

a(*luh) = (*1 i l + *t1o1 / v*IUh

sugstrrurg bQ \th h h' = v + A

VwheUh

2 3 .

Ti{C!3C EgF GATEEtsIN R|?HS

r --. .o)t^$.il--: r r-ALgRtsfN lbct.| ts FllflTg-Dll,lE}lstortA(-rrp t

uESSfoN o9 \$Ar peH.

tt .c. 6AtertN frrH ts AN ApglOIlrrATE

tbtr.rvuATlou OB TltE IrlEAlC frCtf

^ \ L ha ! r h- t v" lt rug uNENowU PABT oF U

3") THg R.H.s.

lNvoL\lEs ALt

( f f t { L )

2A

otr Tffc ABtirg EquATloxqruq'N QuANTt"rFl

REmarr

I F .rh f 'uh +

hTIfAT t$, tF V colrG: FloH * ctAss

wn lcH tt DIFFCB€f.rf THAN TLlg CTASS

TrfAT wh couGls ;EoH , ruE

roat.ru\A'nelr ts cntt€D PlETQov- qALE BtlN

Folrau t' efTIOt.f . Tttll, FtraH ULATI9N

lS uS€D tre FuutD DYNAf-|ICS ptot\Srs.

FOS{.{U\ATION