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T«AFSM
INTRODUCTION TO THE FINITE ELEMENT METHOD
TAYFUN E. TEZDUYAR
James F. Barbour Professor of Mechanical Engineering Rice University
http://www.tafsm.org/~tezduyar Part of the class notes for MECH 454/554 Computational Fluid Mechanics Also listed as BIOE 454/554, CEVE 454/554 Please view these class notes with Acrobat Reader.
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$, tF V colrG: FloH * ctAss
wn lcH tt DIFFCB€f.rf THAN TLlg CTASS
TrfAT wh couGls ;EoH , ruE
roat.ru\A'nelr ts cntt€D PlETQov- qALE BtlN
Folrau t' efTIOt.f . Tttll, FtraH ULATI9N
lS uS€D tre FuutD DYNAf-|ICS ptot\Srs.
FOS{.{U\ATION
ts.FrATBlr gguATtolf! : 6TfFFN6S-@ x
sucH T ltAT
wt(x) - Z cA NA tx)A=f
Ltr tft couslsr oF AtL Lrf,ttiAR
Co!4Bt$aTtsfs OF 6IUSN FUWCflONS
NA (x ) ( a. lr zt}t..., . )
TrrAT ts FoR 5acH *h rN lthrHEeS F|(NrS CoNsrA\trs
C , A ( A r f r ? r g r r . r r n )
= C,N,(r) +CaNr(x) +....+CnN"Ut
NA(x) : SHAD€ Fu$cTlar I
(= BAstt Futcrrous )
(= tNTeRPoLATToN Fultcrrour )
ls gcr$,|gF TIrAT
N^(1.) = o
THAT
wh(,) =
( e = l , 2 r . , . , . )
n
U c^N^(r) = oA--f
26.
\S S�TFFIFD
U' ts sAtD To HAVE DluE\lsloN n
rf l6q€e n =$ BFrTe: AFFRoXII{AT|0U
exelrPLE L€r rr= 3ugF prEtwt$, LtNaR, Fuf,icrtaNs
{t.
t/s
wb(r). cr N,(r) +c, Ne(x) +cr N.(u
BY sg\€ertu6 PEOP€e vAtuFS FoR-
Cr r Ctt CI , A\\ PttEr3Fwftt lruFAl?
hrNcrrous F€a xe Lo, r L wrrHEEINES AT X- Vl A1.1D l r Z/3,
t/3
r b .lt
( . , - Vz
lc z= z/C
t c! . t/el .
( . , 9 o
€ ) c a = tL cr= t /Q
24.
{ -1. -
( . , = + |
€ , 1 . . . oILcr s - I
-/)'\rohr, w5 ltAug
fiDe rr HtS wt
So TT{AT fiO..
TO D€HNE
HAu€ To sDrRrFy At
. l
vhe 1J n
V3
.*h= tn*At e -At
D€F|N€ A g
2 1 .
NOw WE CAN Untt€ :
vh= i d^ NA(x) (ae,nus vLuh)As l
At = Z Nr,+,(r)
Tl{ €lN
h h h , 'lts
'c t'*an - Z- dA k^(r) +
A NvH,(*)
A e l
TtfF coeFFtclEllt s
d ^ ' s ( l = ' l t z t . . o ) n )
AP,g UNENO\IN CottrhNTS.
t{srF TIAT
uh(, )= i d^!e(r ) + ANy,(r ;A r f V l
= A
tS SATISFIED.
30. 1 1 L . hh (F-
w = L co Nn ( Vwheu" G' ALL GoFrtnrf'A=r
-A A \ Tto$s oF cja )
nu^ =
I, d" Ng +
t *n+t ( dE'. AF,b- Trr€
uNrNowtf s )
h
B = l
v" t^SUBSTITUTE, rXESE INTO T}I€ GAtERKIN FORM:
a(*1 ..^t) = (*", () + *to)r{ Ywl1t
, h n
? a(f, .^N^,I ,aeNs *tNn*,) =
wh , h
h
tgc^Nr , ( ) * (*coN^," , l |-Ft
51 h *\oj - FoB AtL GolrBlNrtn$t
otr CA's .
u6fN6 THE ElrlNEARtrY OF a (*rr ) :
3 l=t
({{,*Na, f )---
y y h
(t,oNo(o)t
w\o )
h-a(fc^N^ ,t\,,r,)A=t
- - r
r wh t :::i11,*.,,r lF Cr t
, ha( E crNe
A= l-.-ir-.d
wh
n, : . d " N " )
B: f\--fra-..-,
vh
3ILINEA:FY OF
A N o ( * r * ) !
r( No, n") dB - ( *r^,{) - N^(ollt
+ a(Ne,N, ') t1 = 0J
-> FOt, ALt (oFttl$Aalo$s
o F C r ' s
cQulrtol lt aBu€ Fot" ALt c.or-lElNAAlONt
, wg CAN ALtO WBITE lT AS Fotlorrtt 3
UStNc rn€
a (*i* )
n f n
: c^ l IAs l I B= r
L
slf,||(€ Tlll:
oF Ce's
h
f a (u^ ,NE) d * - (Bgt
3Z
+a(No,N*)?=of , : fp2;. . . t f l
Nsf )- N^ (o) A-t For'
L€:T US DgFiNE
TH5N, ?lrE gQUAT|oN CAN BC WgrTTgN AS.'
oR r r{eg€ €rptrgtTtY :
h
Z K ,sd . = F fBg l
n
I Kr.o J"C=t
. t i t O a
!Il-
L K h B o B8 a t
= F r. . o a a
= F , .
EQuartous
uNtrnldltS
h
h
Fot
K^s
q
= a (NA,n " ) ) FoR ArB = l r?r . . , r h
= (ru^rf ) + NA(o) i-a (Ne,N,,r, )7-f Rlt A: lr?r ...1 l l
n
IB=t
( 1 . 1 1 2 r . . . ) n )-- FAKo" d"
03, eUeN xoBC EIPUIGITLY :
K r r d r + K r a d a * . . . , + K t r . . d n =
F2
Fn
tt
Tuls ts TugP9o BL€I-I
Lgr us AD0DT Tll€
NorATtONS
= t *^o l=Ess MArBu" )
K-
( rrsrrrrrr
) tr f l
^ l f
:
)tt\ llf
K1 d , + K"adr+ . . . . * E rndr . =
t a a ( ) l t a a a a a O a t l a
Kn,d r * Knad" + r , . , . f K r . r . dn =
1a t i a
h uNt,Nowlt :
F,IAT:f I +bEI.l oF "lf E
FOLLOWIN6 TYIATEII
f tt( ' t1 ' ' ' ' | \ ln
h l Z . . . . f , z n
: :t
o t. l^
n n ? , - - - - - ' \ h n
33
nEcuA'?rcNt
F= t t ^ l
K-
d = F- i ,
Fr l
?:E
I t
Ie l
Iv l
a
a
a
a
Iv n
=l
r I t lct = t c l r l =( *J
( 'r Forc€ v.,ror" )
(' urslrAc€H..rr vecroc" )
34.
Eq\fATtOr!
AB€ N
N UNr'NOWNS.
e Tttts fs A LINEA.'SYSTqFI . TU€B€
FqUATIOrIS FOR
Id IS T}IF UNTNOU,N UK-TOR,.
Tffg FrArclr K ANb rlrE vEGTot F AR€- - ,
rlhlollll I THEIS CoFfForrStfS CAlr BE cOr.rPurE'D- r - ' - - '
By u3tlt6 rg€ gIPPrBstoN t D€Rlv€'D DgNl0UsLt.
j 5 .
E IAT"TPLE S :
K r r = a ( N r r r u , ) = ' I
N , , , N , r , d x6 I
= ' (
d \ d ,u , drs J d x d r
l "Kgy = a(Ng,t l*) = ") N.,, Nr,, dx
K r S - K s r ( w H Y ? )
Ft
F7
= ( ru , ,1 ) + N , (o )A-a (Nr r l . r r , . , )p
= o! tl, f .tr + trr,(o) ,. -(j ! Nr,r Nn*r,r Oi t
= (r r r r , ( ) + Nr(o) / r -a (ut ,Nr* , )?
=.! *rf a, +N*p)h fiNr,,
Nn,,,{f
36
Tltg rtNgAg EqoATloN sysTg}4 cAN Bn soLvtD
BV US|N6 AN tQUATlOtl SOLVER-
oss E Tu€ srsr€rr ts soLvgD Fon d , l"
ls TRtutAL tQ 6€T Uh A\rD lrs D€PluATlug .
n
,t(*) = Z dA NA(r) * tNn*,(r) Vx€-n=to,,]
A = l
h . 's(x\r ,.>'doN^(tf, *? Nr*,(t\,A e l
R,EHABK :
K ts sYHr.rqrgrc, BEGAusE a ( Na, N r)
TS A AIHHFNITC. BILINEAN, FOEH :
KAB= fl(l-t^rNa) = a(rua-Ne ) = Ksa' v r t '
TTnAr tS K ' = K
3?.
E X A t r f L E . l : 1 - D E G E J E S . Q F - F B E E D O H
( i... 1 uNtsuorlu )
h = |
w h = c , N r
u h = r t * t t = d , N r * t N z
SHAtr€ FUN(IIONS :
N , ( r ) = l - X
N . ( x ) = X
? Do TI{EY sATtsFY tue
N , ( r ) 3 o
N e ( r ) 3 l
K r r
Ff
d f
xI , o,
Q.Qvttr t.tcNrs ?
Yes
\FS
g = L*,,1F = l t , ]
d = [d,J
38.
K * = a ( N r r x , ) =
I
-' l
r-L
a (NrrNr)y
t r
\ v',' tu,,o ,r-_-
- l + ' t
= ( t , ' r f ) + N , (o ) h -
j l t ,-x)ft*tdx + /t-
It N,,' N1,' Jr
" tn r ) ( - r )dr
tr = .!(r-x )fb)dx!il
NOtJ
uh(*)= Llt (r-t)f(e)c, +ft ril(r-r) + f xNr(r)
= l l , f-x l f tr) dx r lL * t
TltS LIN€AR, q\fATtON S?ST€H
K 1 1 d , = F l
d, = Fr /Kr r =
cofitT:uc? uh :
= 1
+A rt
d x t
,1 outy ox€\
Aro oG
, Nr(r)
EQUATIOU\
UN Ti,OUr|J
dl
; - - . - - e - - - - -7 - - - - - - - - - - - - - - - - -?1 '
i,.'t*r = fr * { (,-x) + [:f,,-t)lt*g](r-') |l - - - - - - - - - - - - - - - - -
dl2
uNtNolr|N
tsNOtrrN
rara5-
R€FrgF{BE? T}f€ FIACT SouvnoU ; ANo coFrtrAP€ ,
u(r) = t+ r { t , -x) * ' t j [ &q de o?
$orrgcCltou SiOLiJTIOr|
P Agrrc,uuaR3 0 L u T l 0 N
oBvtoustY, FEM SorurtoN 6GrS
PACT oF rrrS SOLUTION EXACTTY
Tltg lbrroeeloU)
- ( Vxe [q , ] J
THQQETQ€ r sE ONtY N€ED To CoFttrAEF Tl{E
PAsTlcutAE, SoUW|ONS, T]f€ ACCUIACY OF Tl l€
Fgt*l pAgTlcuteQ SOLuTION , OFCOUPST/ D€FC\DS
oN rHF ForH oF fCr: .
CAEI3 L
PASTICUIAB
ITA<T
Flt Ft :,
, f (x)= P =corrSraNt '
sDrurlolrs:
um.r(x) = ,
(r-x.)
, f * r (x )= ! t r - x )
4 0 .
( w r s l )
( rrueaq ! )
I
)
+ huPl l t = sp^r t X = O A N D X e
AT Tufo [9DgS
AT
( i .e .
l{ow ABour
( * o^"r)r,
( *n ^. r)r,
S0LUrlON
x= iIS EIASf
T}I€ DEPI\IATIUCT ?
't= - P x l + t ;
f A r- - D l' - . b I
z J
4 l<.As3
PASTKUI A 3
EXACT I
tr€t't :
SOLUTIONS :
u PAEr (r l
t'en:, lr I
( c u 6 r c !\ ---F\
( t , u€ 'D E
= { tr-xt)
= ? t,-r)ETACT AT
( i ' c '
fu) = ?x ( tweae )
)
I 1F€l{ 30LUr IAN tS
D€rlutTtvfi ..
r (eO AuO Ia I
A7 TWO |.0DF:
FE!}l soLunurl I S
ErAer AT x* tfts(* oo.r),,
("L,'),r= - * 1 '"1
Sui"ll-tACY
a .
b .
=-+
hU' l5 EaA'sT AT X=O euo )(= |
FOg €ACII CA:C TIICCS IJ A.r L€D:T
oN€ eotNr rN TllE tll(€\rAL orl A1
wrtrcH *1* ls ExAcT.
4LEXA$A FLE 2- ; 2'Dg6R€g oF F Pgtbor-l
( i . . . L uuENowNs)
c r N r * c " N z
h hu " * t ' = d , N , +drNr * tNt
h = 2
* h =
h| - [ =
StlAP€ FuNcTloNs i
1 .
(.tr.r ,*)=[t;* j ; : : : ' : ]
nr,*r=f 2x 'o{"si} . -
[etr-r1 , *sr{r I
'l;,:;::::1.-l . X
N3(r)
? oo ru€Y sATtsFY
?N{, I ) - ' o.
nN r ( r ) + o .
?Nr ( r ) : I .
s s
"H€ aerplEgs4eNr: ?
't ES
YEI
rs
ST|FFNES YNSLXT FoCC€ VRToR
uif tr|onr$5
K
4 3 .
/ rrnwN
FIE
d-
=h::]=lil
d l
,d'
trl C
'z
E
= [
' I F F I'T
K a l = a ( N r r N . ) = lIn.'N..' dr
- l t t b
/ i - - / td , s
( A r s = l r 1 )
11, rK r r =
_ \ N , r ,q J
. ' . K t f = Z
Sfr.r[AeLY :
Krz = Ket
K a z s 4
!
N,r , dx = t ( ( -z ) ( -z )dx
o )
=, {,= L; ,il[ ] l+
= 2
= - 2
K = 2a
S.TTFFNSS ITS I
FrATt t rpotcE
ro = ( N^,f ) * N^(o) I -l (u^rNg)
= l! *^f J' + N^(o) A -]"[to,' Nr,, ,) f
lrs lNve c Ee
( A ' 1 , 2 )
*2r
case t , f t r )=O ( t toxoearouscesE)
45I
F, = it {.r-2'1f(x) dr + h
! l
Fe = 1 ( t,.f(r)ax *. I z( t-xl ft* ) dr' o J
L
d- r'E= L;i] [;} = ll:11, t(*) - (, * l) N, (r ) + (? * *l^,r
(r, +7 Ndr )
= f (Ar,(rlr Nr(r) +ur(r) ) + ( 1u,t4, ry)
= t + l(rx) <- exlr:t ,roHo6sirout
so,.u:/roN
QT'MARK FE t\{ ALWAYS 6€T3 TIIE STAG'T
uorroGeNpus soLuTloN B€cAus€ EXACT nOnOGStOUS
suwror.r ( urueae ) ls coN"AlN€D lN ltt€ cLAss
oF Tc.lAL soLurtoN5 ( plccgwts€ LlNCrte ) .
p-4
?-ra
:-
?t l
II tt -rb
uprnr , *(r_*")
S a*hr^."= Upoet
46t a
cAsg J(
F , =
F*=
d =
A6Atll, t^lE
souwrous
hu pe.r
ftrl= p = corssmN?-
\ * r + ? + A 1Lget"*? + Atzl
+ L
* 2 3
il \i;il,l =N€€D TO COFIPA:E TX€ PAgTTCqNQ
! x, + tt *, ( rlN€A: .! )
NL'w
Al X=O r
l .
f t ' Q ' A T
(elroeAT lc
* = f , X = I
rlr€ '*,ot
ei' )
: 7
lgQruartues ?DEQ,lvertuES Ar€ trF AT
x = Va AND x* 3/1 ( ExEscls€ :t tbw lT.
4?.
F , = g +' 2 a
F- = -3-*z _ f _
+ d= x l . tp- A -
c^agg cl's /C*l = g x (rrruene- )
,?
9leltcuuat, i, 'LuTte*,i,.
t'*", = !. r,r, * 73- rra (urrrrer | )
eper r = + ( r -x3 ) /6 ( tu8 rc f )
h[ t t- 'peq e UF^.t AT x:o , *=* rx* I
( r'.e . AT nrg rbDt.l )
DgsruATtvES ?
DrEErvrnu€l Ae.g EIAC'r AT Ttrb polNfs ; oNC
rg Tlr€ rrEgEEnAl, Eo,tl , olr€ tr t* , I I .
tL
lz= I 7 + 7 * J '
lztl+a +?+1,
48
Suttt{AE"f/
q,. Thlc Hor{oGti\l9u: soLuTloN 6tu€t\r Bv
TnG FCIYI lS tx{;T Fog At'L CAtts .
'rHls rS BeCnusE Tllrg e?(mT ttox()G€\rous
Sc>urntou ( rru*tcu rS urusre ) <.AN rl€
P.EtrtlSC\tflfD EI!$!!. By THC F"rr\rcTlo*rS
EErougllr6 Tg TrllS TC.|AL F$N('rrouS
CLASS ( wlrlcu tS glc'e€\^rrsF LtNne )
b. Ttttt trAnnrc(r"AB sotuTloN G\v€r{ Bv
TH|E FeI\r l$ €r.AcI AT THE
q}lg$ oF €Acu SuBtr'fi€E\rAL .
C. DtEe\UATru€ OF Tlr|E' F€r.t StlL\rnON
ts e:Acr ar (lr tEAsT) oNE PorNT tN
QrcH susrNTCR\IeL .
4 7 '
flEc|EWISg LIN€AR. FINTTE ELEMENT FUNCTTONS
trABTrTroN Tltts lorurrr ( rurcevru ) ll.lto
n suBDolnl$; ( suB-TNT!!RUALS )
L**" L x^, Xo*,]
( A r l r Z - r . , ) n )
X\ : NODT|L PolNfS
NODTS
JOTNTS
KNOTS
Xt = o. ( uertFrorr r.roDq )
Xn+ r= l . t c t eu rF {os NoDE)
L x^, X o*,J trlNtTE ELCHeT{T Dqpretxs
e LEF{ E$trs
eLEHE\JT r*€-N6Wr r ha = Xo*t - XA * COT.TSTAI\IT
50lF Tll€ eL$ttttST l;GNGTl{ | 5' (SrISTANT
TH€t * hn= l / n * h
3t{Apg FuNcTt0NS
F o R . T N T E R N A * N o D E S ( Z < A S n )
iltsouxrbAeY NGIIHS ( A= i A$r' As h+ i )J
N s ( r ) = ( x r - x ) / h t , X , { x s x a
N n r , ( x ) = ( x - x " ) / h n ) x n < x < X r . * ,
x - X ^ - r ) / h o - , r X A - r ( X <
X e * l - X ) / t * ^ , x A <
t, ELSe\JHgcg j
N^(x) =
5|.
rr xA + l xh
( a o - - o - - ! - f - + = - - - - {' xA-, xl xl*, x.r X"+r
Xn*r Xr! Yn+ r
x l '
Nrt r )
x^-,
' *o{ xe xr*t
€t--Ef}leeff LeNCTbtl lo*r? *Aqs T* *t
rn*,
? mte rHAT TllE S1*&i""
TtlESt liHr\pc Fut-lc'TloNs Ag€ c-euugD u
Fu*,cTroN3 r o* nR**'
RJNCTto$s . Tl{E t
sruFLE $ wTDELY usED.
AEFIABE,S :
'1. Na (Xe) = Eo'= l:
i.s. TlfE SFAPE EUrleTfON
C EBTAII| }\lOD€ IIAS A
tF A--B ?t
r F A + B J T oeoPR€SEo$DlUOf A
VALU€ OF I. AT
5Z
ltAT tt
ag€
LEFI qlQFErIt
Rt6tf" E\€i{ar
z .
THAT NODE 't HAS A VALUE OF O. AT
ALL OTH€E NObgS.
Tl*E s[AFg FUNeTTON COtRtSSPOtrDlN6 TO
A CligTAlN NOD! hAS A NoN-tClGl VALUE
oNry INTHF (eeN) suBDoHAlNs (etsrrElfG)
oNg To rt{€ \€FT A}tD o}l€ To TllE E,|6TT
6- aTA? rglC. tN o.rueB $oBDls, lT lrAs A
Nofrt€F rfALtt srLy tN TH€ (onetu) sugDotrAml
t€\SrG$S) T*A-t coNlrru THlr $oDe.
i .€ .
Na(*) f o ONIY FCT
X e J X^-r ,Xet (-
+re I Xr , xe*rf
53PRODGETTES OF tlG S'nFFNGSS F4ArElI s
h.tArn, ot THe, ENTnttns oB K AEs rcco.
l rKet =o\ No,t Nao dx
FOT. A 6I\IEN }FDC
\IIIAT VALUES OF B
Fr B= A- l
Ket *. o
Foa lrr A
l(er * O
A I LET US INsDFT ftI,
wE 6ET Kes * o
BS A-2 : KeB - O.
f -l A-l A-l
A-1 A-l
A-f A A+t
,,^.1
La/
stroR. B= Atl
(eB +o
B> A+2.
Krt =r O
r,r= Krz
K =-
A I L
\ \
+ 5 l s BAND€b.
55.
ssg:l. Tltc tt\frx€s oF E uJ$rcr wF Elnbt to B€ tteGto
lrr AutANtg ( rrrrflforrlf GALGI IATING K ), AEiE_,./r a
W i lN TH€ CgoATlort SowEI TllCt
AgG }FT OD€CATFb gN.
2. tT CA\t EE Sllobfir TnAT TnE STIFFhIESS FrA'rtlI
K ls FO3trlvE -DtEFllrrtc_'
' J l -
MA?II€F4ATKAL ABFECTS OF THg tr€7l SOLUTIO\I
l. rT €AN BE s$owN TIIAT THC F€Fr sotunotlrI
ts aAgT trt Tlf€ NobEs . l.c .
uh( xr) = U (Xe) ( e= ) ,2 r . . . , n+ l )
Z. |t CA\I 3€ ShOTTI}r T$T TTCEE II AT LEAT
oit Folrtr tN c\fEtY 3g6gr$ Ar br|tCH
THs pEEruATy€ oB rlfG Fst{ Sorunou ( "1. )
- f3 ErAgT.
% .
DCFTNTTIONS
t Ge*gG. rN @ AT Follt ot
e ( d ) = u h ( a ) - i l ( q )
t eetot ltl rlr€ lrEc fvATlvS Fr pofilt {
e,r r td) = r { r to) -urr
-) oRlee of Acutacy ( olDrc oF cotfvGee€$(G )
rr|SDTECjT ]low €rr (q ) t1g€tt To tC:O
A3 Tlrt Grgllcrlr tGtSrH hA GD|E.ll To
ircQo) . THlrt t3 r As
hA-r O
brrs err(a) = d(he) -t oo? gr (d) = o( hi) - r o
oBrfrousrr g(h: ) GoGts \tD z€Bo ruITIIAN g(hA
) . ttl €rNGEAt \E"
9rr(c) = g( ll I 3r k : oruEQ- oF AGG.tct
oF €rr(q).
57.P.CF|A:ES :
l . rT C^A}T GE STOUTs 'HAT
€rr(s) = g(ttil Foe
eri (d)
SECOND O:D'E?
s th^ ) Fo?
o( = L--Xnrr2
( gu€r.rFllr FrlDDolrr)
AeculeTE
s * I a lX r+ l
lfz
T
2 .
( t}| CCNtSAL )
!n F tes" oarGe ACCuIATE .
lr cAN B? strorrx T]IAT tF rrrG €rAc"
SoLuTtgN tt $rADtftlG, ?HCN lT€
ls:ruFtug oe Tlr€ FgM solurtorr
ls €t3Aet xr Tlf€ €u€mertf nrtDF!|ls
- - t l -
5E
TllE ttL€lr€NT CohtcgPr
MFSII :-- l l l
a. -a a
a
I 2 3 , ' g "'.-
\-1 hcl
xr Ir Xr 14 : XA xe*i: I.-, xo Y.*,\ t
o , o - - t '
€: €LCFTENa -iruFrQC;
nef ; \lurtlC: oF G\Gf\iertTS ( t <g S nel )A : NED? NUrtCqR
Foe Arr ELCmINT g
xe+t
F€3 TTl3 Slr"r?ta ql€-DtltCNStoNft CASE
Tt+ccc ts A slFtDLE gcvlrrcN tc"rt.r€cN TlrF
?gff.rCt'Sr hluirtER AwO rrE NoDE Nut.rBtES
oF A 6l\lQN gL€S't€l$r.
ICFT roOE, uus^lGt A e €
P.r3$" rrooc $uFrlct A+l = e'+l
xA
5q
Foa Gxarret5 Goe gt€l'r€flr NuFrQgC, 3
T]tg \Gtrr rrlolG Hur'ltlGR tS 3 ArlD lnt,
e,t3rrr silC lS I .
gE,FnA3E3
l. Fot A ClU€il t \€t{C}lr e, T$€rrroO€
A AlrD A+l AtE cA\t'EDNUFTBC3S
Ct€lf€r|" rroDE lluFrt€tS.
2 - Tlr€ Cn ffrcil Bcrwggu Tllg €\E|.crlr
NuF4tgt Arro guE+r€rSf uoDtF xorat€tS
t3 rulr so $Fr"!E Fo? T\uo (oe THrgE)
Drr.r€ltSrouAL FIC:$f lE:l .
Fot' A Orrx!,N ?LeFrgtt8? e' I wf tsrotrf rnC
t.CFr AND gtSrlT NODE r||XyrC€93 . Trt€a€For€ l|oc
trprrr TFrr cooSDrlJrfns Otr TTC LcFr Atn Bl5lf7
rus. r^rg tArt(l glg|:lEr, TO Tfl€ \TFT NODS A:
NolG t AuO TO TtrG RfShT lror, RS rNDe 2.
Cp
e
xi ;,e x;
xr€ s XA Le;T G.FaurDl't
xf t X r*t ctsrr (sotol$AtG
he = x;- xi q\G'ftErr LtarG.rrll
+ r{ob, DeFlr*, A LaAL Gseu}lAT€ q( A TnANsForHAtloN )
q (x ) = ( 2r- *i- rl )/he
+ Tlrrs rs A L!sA? R€affnoN
- ) t ,= E(x ,e) = - l
h "= h (x f ) = + l
61 .-a T$|F rN\fGEt€ e.|ElATrOu ( 'rEAllsFOrnantor
)
13 LINEE?- T€.
x ( \ ) = ( h t q + x i * x ; ) l z
-) nor€ TIIAT
x ( q , ) s x C , ) = x r e
x ( \ r ) = x ( + r ) = X " '
+ E\EH€rr (OoBDINATE
FPACTE
t t*r (Locnt cooeDtNAT€FtanG )
-) €tEHENr (OO?DllrATrE F!'AH€ 13 TFE SAF{€FeC AtL CFI€t{ENTS.
/
/
lr=-l O l. E
poDEgroBAr ^
gtEI.{ENr
A, B rc , . . o
c r b r c r . . .
N"(l) = i ( r+ ge t ) ( o = l , L )
Nur{BCPlN6
NoD€ NulrB€Fl$6
62.-t Tll|r'T3ArrsFolFrSltla,
g\!E|.{(?rr <ooeDts xB
Coo?Dt\lATtet UARfE:
gL€HENT.
S tlAF€ F\tNgTtONS :
v̂A
6LOBAL
xe*,
X(q) tT. rne,TO TTC GLOBAT
Feota Eteugrrr To
tf,rl
- ff.r-fLOCaI
N'(t ) = ? = !z ( t - r ) = *C r +q ,E )
N a ( l ) = ? = * t r + q ) = * ( r * t " t )
NG'TATIO$ I
!,(t)
63 '-t Nolf€ THAI Falr A 5|\IEN t'.GcAL
cooebtuATe q rtnEQE tg ACoe?€tiDollDllsS
6Lo3AL COO?DI}|A're x
x - x ( f )A N D
\(xf = ilh (t )
-) l{roul6lER T$lS lS hfr T?UE ftt T$C
DtglvA'nvtg OB TllE Stf ADg Fu$cTrot'ft
I* l . c .
Ss(x) *+(E)d x ' c l t
+ usg €.FA111 trut€
dN.- =
d x
€ Nor* = N..rt Er,
{3d x
d uq
4 .- l
Er" r
Err =
(x)
+x(r)IxrtlXri =
*x; )/rJddtt [t h't +x,o
_ he'T
I
r hQ '? - 'l - l =t - Z J
oa a
2-
h€
* [t.r- x,e-x.t)/ht]
t J tHF sArrg
oR.
r F d h / . , \a r x b S 5 t ^ ) =
( * * )
=
aO t Nor, = \lorl
ft
65.
€oeel\tA:iluEs oF THE STAD? FsNcrlON fN tOcAL
coC,E DINATC s?lrrE ..
No, r r 5 [Ne( r )a5 |L t tl+ i . t l
No,l G i t. (rourterr)
Nrr i =
i r t , t =
LOCAL -TO- STOBA\ COO3DIUAT€ TIIATTSFOIT-IATIOT
+- I
z+Jz
3Y SIIADC FuNC.tlONS z
Tll€ 'ttAt tFgLFrA'noN Ftot{ T}lC \OCAL
c.ooruxATG t
To Tllt etote\ co:DtilAT€ x
wHrc,lt wrt DgrBarilxED 'tA E€
x e x (t ) = ( h. q r-x,e+ x: ) /z
cA u AL30 Be €X PrEsseD llr T€eFrs oF
Thq Sh ADE FoucTl0ltl ArlD 'Tlg IDDAt
coocbtuATEs.
e 3 .
x(f ) = N,(t) x, '* N.(rt) Xf, L ae : No(l) x:q B l
v€elFrCATlOlr :
x ( t ) s * ( r - l ) x F
N,(q )
r - i ( r * l ) X.€
ue( l )
,o* ": ] /z= [t *.t- x,e) q + x- --_a,
hc
( h ' l + r , Q * x ; ) / z
(rus srrE AsBgFoA!' )
- - ' E -
6'ln i l
DISDLACC}T€TF F $$CTIONtr
cLoEAL PANTPErl
LO(AL FrCTtrCg
GLOBAL :
LO(/AL
\IERtFlCATlOt.r : Fl lr.rsDes'ftoN r'
d*t
d,e
xe+ttl
x;
uh1r1 = N^(r)d^ * N$) C^*,
uh(q; = Nr (\ tdi * Nr(t) dr'L
= :- No(t)d:Cls I
d;
xAI t
X,'
- l
t ,o t q
r.
|NT€BFoTATION otr TH€ !6erAcgt{Elr Furrrnorr:
aa
6g
R,EmAe'E,
T}Ig LOCAL INTEEOOIATIO}I OF THCIt tf
DtsDtAC€ r-rErtr FUI CTlOrl
rrAs Tltg setr€ Fol,H A3 Ttlg LgcAL-To- GUDlAt
ccor?DtNATC'T3AN !$ tsgpt ATTOX
,rh(h ) =?.z
q G l
No(E) d:
x(h) =2_z
cr=lN.(t) x:
Tlll3 ls THt DeFlNL"lql, oF AN
rSoPae,anefEfc e L€r.rENTt l
6l
ELSFIENT STIFFNESS T"TATR.I X AND
FORc'E VECTOR
R.EFrlFrqC. T}IC GLOBAL SrIFFNESS T'TAT:,IT
ANO Foteg \rCCrOR
? euoBAL lNTEstlTlgalS cAN
A3 STJWTS OF ITTTE3SAqONS
K r s = a ( N A r N s )
FA = (N^,t) *$g{ - a(ru^,N"*')/6o.
rh Csr
hIRI.ITEN
R. E,tElfChIrS
,C J-st-5st
L
rlcl,- s- z _
€=l
BC
o\f €
-
J
dL€LOBAL
DOI|IAIN
EtElilsrDorqet rt
7I).-rHeN, rNc CAN wR,rTc :
nela(t,x) = t ' a (*, *)u
€ = l
(x, x) = f (x,* )u€ a l
wlrtE€
a ( il, *)t = It*),, ( x I,. cxge
( * ,x ) = ! , * ) ( r ( ) dxje
DcFtlrE GtE F{Ea$ L€ltE L srlFFNE:s ltliJF4CC ugcToR
r i u = INo,r N"r, Jr G O(N^rNJ
Ff = (*^,ff + 5",6o, /t-l(N^,'jf
ur*EcE ? | '
(rue,{) '= I ru^{J:r-ge
-9'e = [x,trx;] (ewrrorrDo]rAN)
couslD€t T]lE FoLLowlNG €tt Hqflf I
f EtEHST NSHBCR'e-
e,t
,..* uoDENur{ B€f
FoL F^'
Q,+l
t-n t6HT NolC
NurqBgl
OFA- wtLL BE Fq\rAL Tg tcn,o uNttss
A fS Entt€P, Tlr€ L€91 o3't l lF E,l6lf l
ruoD€ ot rllAT eLelfealf .
i . e . rF ( A*e AND A t e+ r ) EN= ! 7 - - -
trl= ot A
' 13^ ._
?FrrE. K^-B
-
oKai wttt- B€ lGso oNlsss
t
(A tS "mHE? tll€ LCPT oe. ttlC
uoDE oF THAa g\€r'l Gtsr J
(g rS r . . r !D t t . - , . ,,,
t
L.C .
:tc|lT
AND-
IF-\,
Tlrr\frr-p
lr-
oR,
)
( t * * e( B t e
AND h* e+r
AND B * e+l
K;s r' o
)
)
n t
eK =|A.
e€+l
Fe'! r FI T
-
n I n FtA.|Brt o l l
73.
cof,r SILC?
euD rnc
Eue:YT]ltN6 Ci'ts€
ilc80.
TflC lruCrtYtATR lI
il* lerz)L " l
suB.\|ccrol,? . t
l * l' l r i ( z r t 2L !
lsKE-
I N)
ANo F-_
SUB. FI ATII I PCCAtL
TllA'1
THtS
IS
2r t
eq b =
iI-
2x1
N., Nr)a(
54c
a r b t lrL
Nor, Nbrr Jr
srlQ- ucgtot +'€ALL TIIE
TIIAT 13
_
[ 5 . . { i . = r 1I o i e. 22.-7tlel-l f
t- t l. i €ro3f Jf: = Ir.rJ& +
a,t lra g,C
74.At33ur-rtN6 TIIAT NE'U€ COr-rnrtgD
'r]tG gr'gHFNT LG.G\ t',a?Ptx [rr-l' - r ' r r r ' r L * t J
ArrU trt etc FrtrrT tc\rEt uec.ra
= ke'-
bl = f t,
}IOl^| E l^JC FTACE T}IEH IN T}I€
6LoBAt' HAT: U, K ANo Tlr€ etotAL-t
vecroR. F ?-
gXAFIPLE
€ I t\GHill? tntFrlc? r ?
ESIES rrobcNUit!€3 r Q,el
CTOBAL NODC--
NUFrB€l : A= 7
CTENETT' NOD€Irr-
NUFrt€R as 2.
6rotAL NoDg
i . iunr0CR , A= l0
ASistl|C
k Q =-
IIAUF
+',b
cOc{FufgD
3 lfflIXA? t^lE
f t.t, k.i 'l
L tl, ti' J
trr G- .F.Flo +
l -I
Ia l a
e
.rfi'it t o
f}K r . +
K g , ,o s
K , g g +
Klo,to +-
\rprri\
- t l
a t ,- - . . ^ r l -'-i ttl
'r- -.tl- l \- '
a '
o - 'i t \
€ t ,
\e,e
? 5 .
,( esuruDrrwlfcu K
i,
ts sY lt til
a a a
t?i"t a -
r t t \ a
) l
v - ^ l
l r l
a c
-a
r o
l(,':. .:'l'r {- r -t I 'L-. -! - l l - - a\ - -
\ - ! -
.
i d ' t
1t:,E- t o
aa
+
b -)
e
a , t J
l(.^q'l r f-( +I ! I r \ - - / f ^
\ r . - \ t\ -
\ Ot _
76.
|MPL€}'r€NTATION :
ASSEF|BLY OF
(g= lr-, Dct)
IK.!
F
K A N D F :- -
It
LM Ag3.A7
THE tfTFOBHA'TIOX
€ f,fT|tTES OF
A r o E .
Jf '-
k".t
IF-
Trf€ AREAY
NEgDED
ke ANo-
a t
LM CONTAI}IS
TO PLACE THE
fe rrrA
K.ry
ELc'mgNT suBR'ourlNe
Ltt I ASSE]"fBL.I, INFO--
L0CATION l'lATtlI= =
ASSEl,rgtY suBtoUrNE
77.l r l r
DIHEUSrcNS OF THE LiYI ABEAY .'
h e n x D e [rrutrBGf of J t- NulrEEB oFELEI.ISNT €L€Ff GNTgNODES
EIAFIFLE I- D FTOSL€I.I I-
hgn = 2.
n e [ r n
6tvEN eL€ttlirT Nt5.tCR g
6luFN E tCr.rcNT NOb€ NUI*rtgg A,
F|ND THC SLoBAL EC)uATlOtf NoFrt€R By
A = L M ( " r e )
Fot TUE l-D pgO g LCH
LM (ore) = { e rF (f'= | ( L€Fr NoDE)
\(. "*l rF 4=z ( ererr ""D9
J
LM (2rnel) € ? = O ( s€sAus€ to EQuATloilr-oc THA.I $otg )
\s'*ETTHEN?NODE I}
1
ET.EH€T$NoD€UtJr{ttG
Et EtjlEl[?*
EOBALrloD€ N 1
6IOEALEeutnorrB
I
(oNSEqUE\rCE
2
T}IE LF4
3 ' i .. llel-Z hcl-l Del
net ne{-l Oel a t.t
4 h'L fFl n h+l
h.a l\-l n
-f a.ATTAYF€3 ?HC l-D egggteH
l3 As FoLLOWi ..
z
2
32
3z
4sz o
a a , , t ' a . . a \ a o - l . - a a
d- -. . .>i ; i . . - .$:-- - :)i l . __ : . )
wTAT tS Ta€ OF LM (z,rs1)=o
.7?.
ACC }IOT
taa
oa
r ? n CK - A N D T ,A- ,g
. t - l \
I e t . .
:{lj,iii
E?i l t r ,a gg u
.526 3A
( mor
. . . t t t t t i t t . a . .
tr.,.'.:.tti.:.,111,{i:,t t r '
' ' " ' - - t c t " '
Keii;g'' "' | .1i:,1j.:,- ' l l !
\ r r r \ r r r . . - . r " "
Fs. + frit' l e . i \ r - - - - - - - - J . r o .
Fe
? E ('5 i l t tr'Z di t' rl
k [ ' , k I ' , k ] ' , +n ' lAss erlBLgD.
ASS€nnBtY OF TIIE TNTPIE] OF
K"r"*,
K.*,, e <E
IK*,c+f +- *E{,rfi-i
+- K':'.. /:<.'+fe i ' ie. ;l ^ - . 't . . i . i , - - c o t
i - - a o -
\ . t '
+
PE RbCl{ED IF K-
ts s.tHr{GrPc )a o - - l
o o e \" + k ; "_ j.fjr,ia.]a
s t ) o a
<-
8o.START
DATA INPUT
SET UP LM
INITIALIZE K AND F-
FoR,M ke AND f e. usE "LM'To ADD
r - -
b. AND f t rNTo AppRopRtATE
LocATloNs oF TllE E AND E
e + Q + l
YES
ASSEMBLY AL6OR.ITHM
e l}{CITATION i
EIEF,CISE I
r ^ € l CETE LTCTT COFTPUTATION OF K
- EXb T .
- - t -
pRELlfr,ltNARy FOEHULAS :
dx (tt
THE ASSrlrBtY ALGo:,ITU|{ CAN EE R€PBEStNTED
BY AN AESEHBLY OPIETCTOR At t
nc [K = A k e' - J l q '
€rg I
nelE = A j ?. - J l . : -
€ e l
1dr (;i:.,)
('.1i1 )
rz, Q..
, , I f ( r )dx =*,J+(x(r))
& f(x(t)) = dl(x(l) drdt
82 .COHFU?ATION OF pe
a t -
k l t = l4,9
* ) (= -,1d r
* l r - - . - l
= _, l .Nnt(t),nttt t l ( x,*At)
( - , ) t ( - r )b
2 Z
q,+b + |
l :g (crz h e - r l I
T
No,r(r) Nrrr(r) dx
dq
dl2aIIl
he
s.+b- ( - t )- !:rEr
he
=)
k " =-
!he ll :l
coFreuTATlON oF83.
BOUNDASY T€RlrS
PCsG LE}r TO
FO:, EACH
+ ; = tg,C
N.(rl ft*l dr +
gur f txl cHAN3rErr Fror-r
PTOBT,E}I. DO WE R.C PBO6:AH
f t x l ?
ft*l -:- f1', ( uxeerlrl.l€:,Fot tflorl )
a.
Ia3f
(n ,
t nt.
+ frN.
+.+- IT'TEEEOLATION
{
{,e
T}IE LINEAS
t3 LI$€AE,
s x i
NTEEDoLATICI ls EIAg? lF fttl
t . e . f t r l . a r +b
{f i*l
t4.
TTICN i
I N,,tr) fl'l lx-1e
.ge
= sNo
I N.(r) ft*t tr g
gc
heez
N,(h)N,( t )d1 = f tr-t) Jt
Ltr) Z Nr( '){s dx
b=t
q , - * l-
*, [ '*ott)Nb(t"{fu
?+ l
-,1 *(t-t)+ l
-, 1
N,(t) N.(i) J1 = .5 t(r-i)) t(r+il.r,1
zr-
3
J3
::I- t Nr(t) N.(i)d1 a
-'l
t(,*t[t+t)dt = 3
ES.
BouuuAt,yTgE f.l S
BoUNDAR,Y
TTStnS
slt ll llj+
fg Jt+' + +, I {FT t +, *z+.[ I
= D cr-
-} rrl r
-
ED- p C
+f g
At
- f l -
86 .FOET.TULATTON OF TWO OR, T}IN-EE
DIHCN�TONAL BoUNDAEY-VALU€ PBOBTEM S
xe! =
[::]
PROBL€M S}rAIN
BoUNDABY oT THg DoHAIN
PAgr oF TUC IOUNOAIV Wll€tE
EISQNAIAL GOTTNDAIV @Nolrloll lS
specrFrsD ( 6-
ryFE touNDABy coxD.)
Dlcr oF lllg buNDAr- trlllgRE NAIUIAL
BourDAEy coNornoN rt sfiFrFtgD ({-noe)
NG|fATION :
-sL
r
\
I
r t :
x =l I r? -- -?DINATE v€croR'
A?'
- (r.! : c'oosDlNATE vEcTroR'
R 3|:t l : uNrr NelrAL . lEcrce,'A tnu)
PSELIFITITRITS :
DNSR6IEI\|C€ TIICOEEM :
( $ . , d o - = ( f n i d F_ J
' t L - E - JJ L r
t .e . ?, r s r 5 g f d & = S f n d rJ L r
NITEEAATION BY PASTS
I ti g d& = - I+ g,iJ-s- + I+a ni dt'-O- .3L f
EE.
i . c .
ltyt)7d-4. = (c ,o,])ca + S+ardrO ) . ^ . t d
= q ' ' - - J r \ v ,
"
pnooF :
J {,.a drL = - j ?,rf csa + 5,*t),;crliL
_fL lL
( e:ooueT BuLE )
= - Ig;fe-a, +, I+a ni Jr3' rr
( Dl\er6€\lc€ ileeg! )
gl .
LIN€AE RSAT CONbUCTTON
NOTATIOI ;
tA : TEMFERATURE
9: r lfEA? FLut VGCIOR ( t )I t
9i r - *ij \j psurreei LAw oFIIEAT eoNOUC,lf 0N
Ii . C . I r - F ! U-
,3 : CoU! UC"IUITY f*lATE|l-
E ,E Syt rrST:rC
liosrLY wF nevF AN lsorEoptc llgDlutf
t^lT l r T N i $ e e I \ t D € N ? l t t i l A " r t r
cr A.i =, ^V PloNecrst D51"o
+ : HgAr SouGtF Pe3 uNfT VOtulrF
10.
DIFF€T-ENTIAL RUATIOTT'
ii,i B +
g ' q B +_
(-r i; %ilrr' = t
g.(-s 'gu) - f
]rEAr EquATf oN
( Polsso\rs tquA't lolf )
tN VEErol Fos|Yl
BourrDAsv c'oNDlTloNS B.c . )
t r= t
o N v TEi"|PEFATU:,S B.c.
( esseNrrAL B.c . )(onrrHt€T B.c . )( p-TYPg B.c . )
ft€AT Ftul B .c.
t xeru?AL 3.c. )( xculrAxN B.g. )( {- TtPg B.e. )
-1 in t- 9 ' 1 1
- l -
- l3 r ' )
9 tLINEAS TEAT CONDUCTION :
\\,EAK FORMUIATION I
l l , L - + = o
( qi,i -f ) dsr-l*$
REITOUAL.sLwEt6TTlN6 FUilC'TION
BY PASTS ..INTE63AT€
(- ! w i g i
SL .sL
+ !*f in idf 'Pt
oN frr
ot\f fr,
I'
!*tin;v(
) wl in;
It
E€M Eri3CQ. oL
FUltcTlON cr'.Als€'ll :
st!$l,At To l-D cAsg
ugA a u- t ox f 'qwe 1f =t w=% o* f.
Yw e\f
dsr- + t
w g;ni dr- I*f*=
o.
dr' - JT
THAT \At =
ginr =
f - ; - - - - r - - - r - . D - - - - - - - - - - - - - - . D - - - J
i - \ w , r j i d s L = ( * l J - s - + [ o [ d r ia J . ! r r
J J i. r L J L 1 . 1 IIL----- - --- ---- --- -- --- - ----- ---- - J
9 Z
THqBt FogC I
- 5.cL
\^r,iqi J.rL -- 5 w{ Ct' - Iwf dsa =e.
CJEAC3ANSI\Ig :
'rHrS lS TtlGi S|€AE "Oe,fYl Vrrelf
s\fBsnr\rnNe FoR, 9i= -r<ijuri
fq
+
f *,, rij urjd$,ilwfdsr
* 5*[cr.sL
VwelfWEAK FOR.M
ct3
NOTATTON :
a (w,u) = I*,
"i j urj Csr-JL
(w, +) = I wf d-sL.(L
(w, 1)., = I w[ dt-t l't
THctc An g ALL syHlt qg!_c . BILINqAR
FORlvls .
wtrlf Tltrs NSTATION lN FrtND, T]f€ w€AE
FOET.IULATION CAN B€ WRITTgN A3
FOLLOWS :
a(w ,u ) = (w , f ) + (w , l . ) rVwel/
t\, '?trlt FcSrrt, Tutg wgAE roarru\A1lo N
LAoES vgEy Slr{lLAP, TO T}fg TWO OnttR
wgAE s|trrs otreruED prcvtor,sty (ttr pp. lz al|rp)
j { .
LINSAR. HEAT CONDUSTIO N !
6ArgE,E,tN ?OErlUIfrtoN
INi.IEAD OF FUNGTTON €LASIS U AND A
w5 u3C FrNtTe DIHBIXSIONAt AFFTOXIT'IATIoNS
?th A*.D d
TIIAS ll t
h h hr F w e ? l l c \ t w l = O o N q
( o : * b i o ) o
rF tt. d ruGh ot = E
olr Pt
(os *t =a )
JulT L|EE l^|g DID tN Otl€-DlHErrSplrAL
C n l l , h
Assut{E rr{Ar 1f K :T*
AND ^ -h
L|Ft Vs colrE'r EucT JJ 8Y urlal6 v .
1 5 .
pg3!e A ruuernN 2'
(A 6lvGN Fulrcrrou)
sucu TIrAT t^
= ?
o\r fl
( o R 0 ' : A )
Fot' e,vt?Y tn- th
w3trg ,lh = art ,,t h, ?Now c,ttBE, TO SIFF ;F Uh BGtON6S'To -/5
h
o th. ' . d ' =
? O u | 6 , /
E '
Naw wQ,tt€ Tllg wEAs For,t l Fot
*te U'v h e U h
uh e, Ah
q 6 .
a(wru) = (w,f I + (w, 1)n VweV
w G - * h
u+ , *h
.n a(* ] , t ) = (*1f) + (* t r { )" vwLuh
uh r tt* A'
e a(w] .,'*7) = (*1 +l + ("1 {).. v*'euh
a(*1 vt)UNfril|l-f
= ( ",1 +) *(*l {),"-a(*] ilICNOt^f N
:+
( coFtgAlc te THq SALegtslu trc3rrr
ostArNGD Dtfr|teuttY ! nf aI) )
FINITS EL€F|CNT DTS3ESTI?ATION
DrSCI?Ertt€
E\.EtnENT
rooEi
Tuq DonAlN .St trfto
Doprarus -ge Q < e, S nel )
3? .
Drsccell2fTlohl
oF -St AND
r
ltAU€ A1 tg11gT Ttrg€g uoDEl.
rtoD€t5 .
4- NoDti RErr€rsn$ uortuv)
ELE$5NIS WILL
o E , 4 1 3 r , , , ,
( rlrc r^llLL ul3
Horf Do wE geuatg NoDE OBtrEErr6 TO
Ttrrg oCDGS|N6 Oc gquATtorr S 7
AP?Rqllt51q1
oF 3 t t rn
o N n B tt
Ltsgas
NTERFotATtoN
n-- 1 sgr oF oF- 'np J
, '6roBAL noo€S
hnD : NumtG?' oF
NODSS
LET tl =lt
v
I r r 2 r " ' t
g ?
l"
e1LET tla Bs TlfE srr oF NoDtEs AT wlftclf
h r l x* " = ? . t ' A - N o D € s ' .
rHE csretGrrcrr oF n(O rn nt ( i... T)rrr
gtr^Atlltr3 r.nuut ) t3 THE sFr oF
NoDGlS AT wtllclf Uh ls uuruotrtf '
wrlA't t3 Turg NumBGE oF gQvA'tloxr .?
= uurotFE oF \lCDlEls I N Tl.lg
sq" \-rb
= ..1 .
Tt{tx, JFclt AtL Wh C, tf b
tr.|lF cA\l uretrei
wh( l t= Z NA(I , )crAcl"lt
trlt{nl,E NA1I) ts rrr€ sFAp€ ?tlucTrorf
AStOCIATIID WrTlt NODS A.
ts wh=o oN f-. ?tl-
wlfY ?
loct.
NODA L VALUS!
OF \A
SiHILARLY !
vhcr )= f NA( r )dA- AQlne t- uNENo.^,l,
dA : Tn€ uNrrbNN (ro Eq FrrEEJlrrao )A'r uoDt A .
i.c , TtF}rPglA'tUtE At NODE A.
lf,lTEEPoLATlorr oF ol t r ) oN lta s 3 , c
A'rt)= I NAt!) ?*Ael
w lrEp,rE fle
s A (L^)
sulrrtrrr€ lFr. *b, V
6ALERr3tlf FoR t'l I
f-"I S t / u r l
h hAHD
0
lo t
rN rnC
: a(NA,t lc) dE = (N ̂ ,+) t (Nert ) r8e1-1'
-f a( gArNs) ?sBelt
Va e?t n7rlr€t€ AEF h-- tFqgATlg;ll ttggg--l -
Fos tlal \l!EW
wtrCgt h..t lt TlrF NUFrBge- oF
NGDE: rN Tlre sqt 1tb.
To DgrgEHlUg Tt*lF STIFFN|F�SS FrA.tPlI,
AhlD rnc hP3g uRTo:,, EQUATIOU lJurlB€II
t.tulT BF ESTABLISIIED.
frDft g\CC,Y NoDC tN ,tna
AN FqUATION NUIYIBCE.
Ft rne NoDet rN ltg
NunnIEg lt AitlSuGt . (
sDe,rFlE AT r)fAT NoDg )
,
Dg$.TTNATION AEEAY ; Ib
lol,
urtF ASSfeN
NO QUfrTIOTTI
hEEC,AUIE U, IS
: NUMi€P Otr NODALDot lrTl
: N01{EE3 oF EqJATnrt
Asst6NrlENt OF
uoDG tS ACH
A R R A v ! D .
I D ( A ) =+I
-teLoBAU troOE
tSorrA'Ttorrl rt t'lGEPS To
tsfEr) 0) Dtsnpqr|crl
e f oLoBAL EluAllss .8.
i F ,F Ae rl-rfl
\ o r F A e " ( l )
NumgeR.
NGTT' THfiT
t < A ( h " P
l < P S 6 c l
nnp
h=t
lo3
wrlAT tt Tfr€ trlHEfi$ox eF rxE
:E,D ARRAY ? . hnp ,
MATR,IX SUATlorf I
K d = F- r - i ,
K"u = a (NarNg)
P = fD (A )q = I D ( B )
F p = ( N e , { ) t ( N e r { ) r
Z a(NArnr.) eeec1?
utuee*. neff cOttDUCnON :
ELEFT€NT SIIFFNESS MATRU A}ID FOEC€
vEcaoR
ELC,;feN" STIFF NCSS t'lATBlI
ELCF|CNT FGl, \,ggToR,
tfLCI.I€NT
iloD€ xunaB€ R
n
f,lutrrtB€B OF
EIEF,I€NT NODES
S a r b ne
L
k'= L klr]
f"= tf: l
*;i*u,j d&-tge'
o-l(N o r
k : = Nor.a ( No, Ltr)' -
4t r-t c r - ( No,f )e t
Tg N " ) * ( g N u ) d s u
nr
i),.-F: klufi
t06
NcFr€ TllAa :
- c - |t [ = t ( IL) te b rs A ?-tuoDc
e O l F b l t N o t A ! l - N s l C ,- o
A COiI\,GNIGT|T FORT'I OF WR,TTIilO B. :
. G c , Tn . u = I ( g N . ) x ( g N b ) J - s
ge
DCFlrrC t
D= * ( corlutrlultt mattlr)
E.= Y N"
E = [E ' Err . . . , E. . r r lTHCII :-
k iuE I B :pEbJ-o .gc
B" =a[ g'P E J-sr
lo6
coFrgrraATtoll oF HEAI FLUI U€C"OI,
LEr d-'= [11"1wH€rE d: 3 uh( I: )
rlr€N q = . D B d C- f l e n
3 , - ' P r g o d :i -
a 3 t
fo?LINEAL }I€AT C.oNDTTTIOil :
DAaA PcocCSslNG ABEATS r grI9, !!!
ELEFTENT t|Ob€: ^*31" : lEN
A = X E N ( c . r e )t
' f-c.I.-EL€r'r€NT uurrB€R1 t-Loc.a1 (ererrGrr) rsOS {f
6tOBAL NOD€ {3
D€s'lrlNATloN lglfY I ID
P I rD ( t]- sroBAL NoD€ surrB€?f
euoStL Eq,r|rnON ll$}|t€R
LocAT\o$ gAfetl : LM
P = L M ( q r e )
t: Er€}tcr|r NrlFrBgrEL€FrErn rloDE Nr,P|B€P6r.o8AL €QITAT|oN NUF|S€:
S ,sNg
AND
6s"s uP I!:NTHCN OT}TAINS
l € .
ABI,ATt
?OLtotrll :
ArtD gD
LM AS
LM ( q , re ) = ID ( f EN (ore) )
E"IO AEQAV ; 6LoB,qL rloD€ t
I z a 4 s a 7 t t
rosExAlrtrL€ 1
E33t}JtTAL
B.c. DA?A
SPGCI?lco
( i .e.
€TCFI€NT NODCNul'rtCElNG
l = t 1'h= l' l
\-11-- I I
1'10 HAt NoDel
. ' . UNE,f\FWilt
EQVATIOU t
hntr=
7 -Nosct
lx .
l_xf
A :
2
2
P = E D ( A) 6LogAL SrATloll il .
q.l =
i loElAr'rgL€ 1
cS3tlrTtAL
B.c. bA?A
ggtFcl?lcD
( i .e.
. o. , ,2!
r o l
t t = [ t
t l l = l
, L r . . . o
t r { ,7 t
1-1t= [t't t 3'orl,rrtrlt
-Noscs
lx.
l_xl
ETCHCNT }TIDCNUTiSCEINcr
ID lrl,QAv r cuoBAL mo€ |l
S 6 . T t t l o l l l
ll-ln n.r E NoDel. a ,
.'. A uNE,Nowsl
A EeuATlot s
l.lntr= 12.
q . l = 8
E.A :
P = E D ( A) gLogAt SfATlOll * .
E(€HCN.r
t q 5
i l Ir€$ AER.AY :
LocagNODC
cr.|lutqttt
hen =
LM ACT.AY :
t\g,nE
\ruFl Et t
3
As fEn (cre )t atoBAL NooG
€LEFf ElfT Nut\. Cte
I Z t g B 6.csnL ( t
Irroot o I .
uu,rqtl l !I\.t
hel 3
usr.aB€R
h c l =
= ED(fer.rlerc))
Qvnnot.' uumB€R
P= Lt*1 (ore)
t'- Gtog4g
C\€t{CN.rT€N ASR,AY :
LocaluoDC
q.luuiet
h36 = '4
LM AC:,AY ..
hen= 4
As fEtr (qre )t eroBAL NgOG N$r,rt€lR.
Nut\i Ctt
hc l = 6
I ta
\Jut teC,l,
h e l = 6
= tb (f er.rlarc))
QVerrOu uUr.rB€R
r.ccAL ( tr.rooG " J
.tttl,ttCl
| !
\. '
3
6
P= LM (erc)
t'- €tog41
I L + 5 -l I2 3 5 6 g 35 6 E 7 f l l!
1 5 7 I l0 ff
€LCr{E}I?
o I o 3 o 5
I z 3 I 5 g
3 I 5 b 'l Io 3 o , o 7
EXAMPLE 2
COtlSlDEn, THE
i'|ATRTI ANb
f
FOLLOWIilg E,LEI{ENT
FOnCe v€croR :
l t3
STIFFTIESS
{.'
r lq '\ l
k t=-
o
{.:?:i
k;
k;
6f
$.
iit)
f,
ki
t 5
f==k!,
@. , - ^ la r -
- - ' r
ik i j tkir l\- -- l \ oJ
9
ASSUr.r E
E LEFTE N"
TIIAT TIIS LFT
15 6l\tc,N At
ARTAY FoQ, Tlll t
L]*1 ( t, €)
L l r (2 , €)
LFr (9, e)
LH (rrc)
5oo9
l |4\ rnAT 13 Tll|F
ELCt.t€ilT TO
*tu
K t t
K g r
collrr.lBuTloN
K A N D F ?i r - - '
n E
Ktp
K r,,
Fs
Fl
OF THTS
+
@@@
@@
+
+
<-
(
f,e = - ti,u:-[&;+ ; G
, - --'o" - iuiint\- - -
-$itt
f t5
BILINEAR, quADf,lLATERr\L €LEM€ NT
(-tr-t) (tr- l)
ll ttPARENT DOMAIN
:;ll(r;, g: IFt
(x;f)
(xi, g: )Dor{A111
lfPAR.€t|T DOTqAN ,
nIIATUR,AL CtrORDINATT1S
!=[;] =[;Bl1 I
5= [ ; ]
!=[;] =t;l
(ri, ei)
r ) I 1r,r1 3
I 2q
rf tr tl6TH€
--NIVSTCAL ceo:,Dlurffis oF A DotlT
rN .$l.,e :
x -- I xl -- I x(lrl )lR' L :J tU( l , l ) lAe€ 6luQ,N tY THC FottrOuftilo MAPplt{Gl :
r (i,1) =4z
C,=l
eN" ( 1,1) X.
e (q,1) =+I(l--l
No(t,l) g:
u'(q,1) =1z
lNo(\rtl ) d:
oEr tN v€cToR. FoBH I
+I ( I) = I N.(!) I :
C=t
|$TECDOT.SnON G A Fur|cTrct| HAS TTg
sAtlE Foc,ttl :
(OORDINATT
TRANSFoRItATtoN
I}ITIRPOLATION
ll?SHAPE FONCTIOT'IS :
ONE_DIH€NSION : ( R€,r,flNDeR)
5,=- |€ . = + t
N , ( t ) B * t t - q )
Nr(h)= t ( t * i )
ruro- DtH€NSloNS :
N.(t , r t ) = *( , *h. tXt+r lo1)
t ,= - f " l l= - lN,(i,'t)=*(rtx,-l ) t= *, 1tr = _,Nr(i,r)=t(rq) 0-q) l'=
*' t3 = +rl q 3 - l r l . t = + l
ur(t'l)= +(q)(r+q)^l(i,l) = *(t-rx,+q)
N,(h) = * t t+ i . t ):t
TTrs ts nouf A snAtrE ruucTfoNI rg
LOors ur,C :
NA
t r lA T€NT .Ftort NoDe
rf ,l
€F Tlfg .r€rn Eulttlrgst, ll
Por€Tlfg
A
nlutueaR TRtAtrlguLAn €L€FI€NT :
ItAN EXANiPT€ O? Dg6gN€EATIOil
-t LIN€AG, TR|ANGUIAR ELeH€NT
<^AN BC D€BU€D FEOT}i
+ BIL|N€AR QUADR[ATEIAL gL€T'r€rtT
t.| c,oAtFscttlg soDES 3 Arro { .
{
r N K = t N q I : r . , s s e r I l = I ; .C--f
TllErt :
! = 1,, I:r N.X: +(Nr+Nr)I:
= ui li *NiX: {r N:, I:
X B-
3 r -rlD - -
) N - XI
- s + ( l
o= l
COORDINATS
TP,AilIfrBT{ATKIX
= N r = ( r - t ) ( r -q )
s N l B
= NrtNr =
( t+t)(r-q )
(l +'1 1
I1
!4
I
z
,
Nf
lNr
lN3
taowHcrr Tllg srnD€ FulfsTloNs AtE:
t TTl3 wAY ) htll€TlfgQ trr€ lfAU€ A
quADEl LAT€e.AL €tgr.f€llT OP A
TR|ANGULAR €t€Frl€Nr Ul€ CaN
u5€ TUE SAH€ PAPtrlT DO]"IAIN.
-) rLElncr$r INT€RDorATtoN nA: Tlfg SAHE
tuElr AS COOEDINATE TEA[lSFoEttATlON. t.e.'
a^h = i Nld:Csl
N; d: + Nld;0 .e '
+ Nsd,
r! llOw STIAPC FUNCTION
lal
Loors 1rt€ :A
nri
ELEHTNT STAD€
FrrreTtOil
Pt?Atf lDII
EtoBAL $TAD€
FUTCTIOI'
t'i/ l
}f, 'eer-)
la2-
T}IR€,E. DITIEN SIONAL SIIAPIE FUNCTIOIIS "
TSILIN€A:, II€IA TITDPAL EL€}I€NT
tfH€lAlfEDSO N , BRI(,B
5= q ([)
TIf t tPAr'C i{r Do}lAlN
PT$|CAL Domerl
X = X ( I )r l
DArS\r D$rAtN , tlAlUSAL COOIbINATS: :
t q lq B I n l' - t l l
- t - o .
rz3rt tl
ru€ Pfl.rsKAL csoEDlhrA.ttl oF A DOINr
f N - Q e s
l r ? t r ( t r t r B ) Ix = { : t = 1 v ( \ , ' r , i ) t- r \ u ( l , I ) S ) I
Ar,t 6lulEN Br TH6' Fo,.tohfltf. '{APPIN6S :
x ( l " t r5 ) B6z
C3,
Irq 3 f
aI4 = l
No(q,l,q ) x:
9(t t,5) = ru.(t tI ,5) yi
z( \ r1 ,5) s No(\, t l r5) " :
a2 t frf vEsToS FO3M :
ax(1)= Z \ t :) [ l
e3l
(ou:ilNAr€
TRNININXOiI
Er€r.r€rlr nrERxttATlorfAS 'rhtc cooBDtNATE
12"4
llAS rt.;" SA]f€ Fo:t{
TEANSFoBt T ATION :
uh(trntr S ) =E
T No(\,t tr5) d:Cr3f
EtEruNT NTERTOIATNN
S}IAPE FUNCTIONS :
No(l ,I,9) = t(r+h.lxt *1"1 Xr+5"q)
v.lff gt€q, H. 1o 5oI - l - f - f
z + l - f - f
3 + l + f - f
q s f + l - f
5 - f - l + f
6 f - f - l + f'l r l + l + l
I - l { - , + t
t , l f
DE6ENEfI.ATION OF B?TCT. TOrr rl
weDCt,fz3
eoALESC'€ ItoD€5 AND +Ah|D
Axb E7
1s$ll:
I ( ! ) =
trf tfEF€
tNa ( : )
? ,+ Z 'N . .
creS
?.= t e2- Nq IoCl,3l
el .
Inutr)= l"'(:)t*+
(!l
, Q-lrlr5r6
tNr q) ,tNa(5) ,7
a=3
q = 7
jO O E O
- e - - o - -
n e
DGo€N€:ATlo$ oF t wEDo€tt To TgrEAn€DRort
cort.EseC ;1obEt 5 1 6 A N D 7
Tl€lf :
x( ! ) =
$HCR€ :
,,
N.(l) =
t ,
I:, + Ns(!li N:(r)a3l - x
t-
[* l ( 'D[ " ;
( : ) *
,
,
N. (I)
4 = l r a r 3
+ N'"(I) q,= 5
- r . G I - e ' - 1 - e O - - ' -
l2-?
lll6uER - oIDCR C\CHerlTS
IN ON€-DIHEN:ION "
Na
N , ( l )
N r ( l )
N3 ( t )
= * t ( t - r )
3 (r-q)= ( r+txr- t )
e.sralrl'E, :
h c n = 3
@
@
= tt ( f+ , )
$16$€l-oRD€l t\€MErlts lN
Trdo oR T$tsg Drn€NlloN:
C.Ar| BE OBTAINCD IvoCSUrsT BUU€
rrETlfls ts florr fi wotrs .'
i l(1+r;
@ ( l-rt.)
ETA'iPLS ..
@ ! I (1-')
* t (t-n) ( r-ht) t t(t+r)6) @
Ne(f,r ) = * t (l-t) (,-rl')
lz?
NU'TIERICAL INTI6SATION
TO CAl,cuLArg EL$Fr€NT STIFFN€33iiATntI' AND Fotc€ U€CTOR rrr€
N€€D TO , tor|gHohr, t\lT€5EA'rE
TCRltrl Lftg Tllls ..
,flf t1)ds -- ?
AAr{OL€ : FoR' tlgAf CONDUGTIOr|
?rlr3 COUID B€ AS ?ot1'ouf I :
ft l)= (gNjs (gNu)rscN, Tlr|E t\lTg6BATtoN
6tugs 'rrrE a- b cor'reoilg}lr
oF TtrE el€rrertT S{IFFNESI
l.rATEl |, .
PoR Flsd=l f j ( lnco;r^ i l )
| - / - . - t t
_\ f t*rdx = \ t t r ( I ) ) *r \ ( t ) d\d J - , J
130
Fot' n3g =L
^"lf(.,g)dgJr_J dfl?i'fh(rr\,rr),e(\r)) j(1'r )dt rl
O | - f t
det
t--fr..:
JA(oE|AN , j =
Jlar
H
f
(A) = detgan3-r?1
Foe nsd = 3
t
Jf(\sp)j*- =+ drdgde
+l r l r l
.!.[F(xm,5 ), 9(r,T,5 ) ra (t,?,b ) ) j(t,?,5)- t r r r f
{ q { d l
j ,B dct
?.I"5u-"5??
-2"3
?r?t?.:l2 t
a!"q
?r"teq
J '7f
??fra-a1
13ltN CAss Cel€ , TH€
TAis ts :
INT€6:ATION
-l ts(h) rt = ,
e o g t \ - t = L-lE
l : f i
]tlt ?(t,r) d1\ = ?FoR nSd =, $
= ?
,WC D6Tr WANT 10 TPY TO TTIVC
TH€sI INTSBATTONS €IAgTIY.
Tlfcc€eonc, w€ AgPeorlMATE ?lltM
BY rlut{tF9rtAL rNTE6lATto}l
-5 fr(r ,1t5 ) J1 J1 J5+ l
wk
tr int
t(E)dl * Z ?(qr.)k= t
lltf3z
NrrtrrB€9 oF lNle6larloN 0otUtSn irtt^,
H c
wL
kt*
at
ao
1 CoOBDTNATE
INTIEGIA-IION
WCTI6HT OF
Forrrr
OF TIIE
DOTTIT
rHE TNT€6}ATIOI
w, wa
Acuplr6
OATTOMIATS CTACTIY
Tllk
btAr{DL€ : TEAP€?OIDAI, IUU€
nir* =1av
l , = - fq " a * lw r 3 I
w t = I 2*D oRDGt-
loogt€g
: f s , + lr-1-rrlta/\-r,
trrT€crATgs
e,IAr"rpt€ : SfltpSoXi R$LGra3
W3
nint^,q ,^,q.d
I g
|NTE6e56=
EI lc tLT. rT
Ws = t l l
wr - 4lI
w l = t l t
RD3f 3
= 3e - l
= O
t r + fq
wa hft
ggDE:- DolvrroirllA Ls
4t* oEbER. AGcuGATE .
RCl.fAB:3
l .
z .
FEh|€3
cstT
IH€P,€
ACr|lEri
FCI^I€C,
INTE6:ATION polrlrs ]rgltN tgSS
ARe trnggHTlox ILULE! wntctl
Tlr€ sAlrc AsB$tec v wlTtl
r1n.r€ql5tt6N fotrrl
t34
6AUSSTAN QuADrAtul€ RULCt :
u TH€r ae€ oFllifAt lN ortc DtFrENltor
ii) \rrrlr ( nint) FotNTt/ TnGt Aclflcug
(z n int) oe,D€" oF AccutAcv AND
nn€orfiG (Z nirrt- l) orDEB FotYxoFrlntt
EXACILY.
l . hint = |^,
{ , = O , \ r J f = Z
Z*o OEbtS acaUpATG
I ET oc.Dtl DotYltoFulrr lllr€lb?ATgD et5tY
Z , n i n t - 2 -
aE|
l , = - t / { E , W . = lx ^ - -f - = 1 t l { 3 tr J
4"* oet€e eceu:nrG
3LD oRDeC. FOLrNoFilAtr n$Q3qaau EtrAsTtY
W . * f
133
3 . n i n t = 3rylq , = - \ / F , w f = s / 1-
T.= o , \= a/ l
i =*{1F ' W3 = 5/?
6T'r o3!GB fcuBAT€
STtf o:,o€t Do\yr.oFtfAts r$rc6PATtD tl GTty
Nulrc3lcal |rlTElSt A.Trc|| t\l Tw0-Dtwlsltlout ..
nn1 :I6(i,fr) wrL= l
rrr€clArrorr PorNrt ({.rtr) ArrD
TlEll:, (OSg|ESDONDInG l^rGlCTT$ wk CAu
8e oBTAINgD Florr oNQdt|rNsloNlt
6hfssr0rr q\ADcAa\nE l,ur,5 Bv PSotucteuLc
B6
Htlrg ls ttow lr woPtss '.
Ef lraD\t !-a-
2.x 2- r\rT€6rNrlox :
(2)| w ,n*i l l turlil I
^r(3) .
tr e - tllj*fit I
t:-r{Ibrl? |
1
t
(r)g = {
,.ut+ t/(ir j -
"l{
t
FrtIIItI- - -.rn- -a;-i t(d,{.): wltvt'i
:
t t '. - -€€.- t - - -
ti.ltl?iru!'r^rG ) t' ' r ' --?
:I
i(,il i:)l/ "Tw,"'a \
- - . j f O o - o . . .
i
t3?
fieSumnrlAtv oF 2;tL c,uLc:
Ir int = 4
(q, 1"., ) = (-i r-ft )(d,{.)(i,{ ')(s* ,i.)
NOTS THAT :
hi.t
l-
L=l
= (-t )-t)(*fr 2*+)t t , * t )
, wf = I
w a =
w r 3
w r c
( ArGr oF Tlf€
DAg€rIr Dometrr)t\OtE: :
stlrltAgtY , oNg cAN APDtV TlfE PlODrr?
o$\€ To TltrcE'Dlnslltp$AL rllFBEPffilorf.
|3A
C-ALCULAIIGN OF ICBIUATIUEI OF
'tlt€ 6lrAP€ FUNCTrcNI :
wE N€€:D -ro gALGU1AT€ trrc DEPTVATTIF'
oF Tlr€ sl|A9{g Fn4alort wfTtf BBDrEtr.TC GiloBAL CoogDrrr ATE!5.
Wg HAVE Ttig IIflVATt\llHl W'9;r' \ocAL
Cq;g;ROtXAlF.
h s d = I
Nor, = Norl l, * ( eest.l;l-.r-../ t
V
w€ neuc 2-TPt r he
hS{ = 2-
Norl = Nort €rr + Norl ltrr
ntorb s Nari qr? {- No,l nr7
r31
llr Frtrttll FonFl :
Ir,*( *o r t t Nor ; ) = ( r t . r t rN . [ )
|
H Ln,,t 'u]
na J'TtrGtE
-
?a
[ ; ; ]
xrrt
?nH-l; ]
, lqn= f IFU,r
= 3 37x
_
f z x f l- . f -- l -
\ ?1 )
- l
h s d - 3
T(esf )'-
Jcof
4-:-
J
140 .
?N.-r
?t-
?N.-- =
? x-,
a+.€-
? x-
?x.t" t
-
?N.E
-
-- ?Nr (Ef t-EE , l
coF rr =
cdrL =
Cof rl =
gofzr =
Cofet =,
cofa3 =
9rtl zr5 -
9r\ zrt
9r t? , I D
Z\ t r t i r '
?rs x4 '�-'
Zrt Xrtl rD
gr5 a,?
9r; ?r5
grl arq
3,i x,l
7t\ Xrs
zq x' l
14 l
d., = Xrl 9r5 - ag 9,rl
t o { r . 3 t S 9 q - x r + 9 r s
tof sr a Xrl gl '�- *rT 9t+
j = Y,i 4o$,r r xr1 cofru + Xf GFlr
\, = ( Nort 6f,, * Nc,, c4re +Nqri C"f ,r)/ j
"o,b B ({rt @far * uo{ Cofar + Nory coFzr)/i
Nqr? a (uo4 Cd' + N.rt (ofrr + Uerl Cfl )/i
r - - : l -
