Chapter 3
Literature Review
This chapler p!6ent3 litersrure reviry lor both rhe
FSI prcbl@d
ed llabiliad
AlE ruving m6I
genemiion {or
nixed finit€ elendnt n6ihod (MFEM) tor Ddcy
flN
used
a d ce srldy in ihe r!6b. li 3le p@nis difident n6h pdtitioling algorilhm,
rnich m uselnl 10 pdliiion ihe h6h lor both the static snd d}nMic load b8lecing
io a di.dbur€d meDory slddtr. M6h p{titioning is o ihportei lsk io $l€ ihe
probbn nr parauel usine FEMS.
The
&e bNed on lh€ Sroph ains nininum
cli
6.t
md sp6tiol ddia structu.F.
3.1 ALE Movins mesll
In ihc rccenl sMa of @npuldiional
is 6€d to
rcpreqt udolying
nehdics, n6!
oI the
iine ALE hme ol
physical tsSl probl€E due
b
multipl€
re{er
one
ini€riac &d
nonftEaity ilvolved lor ths conput6tion ol then solulion. Don€a ei ol. (1977)
431
rcs o@pi for intaf&ins fluid wiih ihe struct@ ed crmtel
ALE lomuldiion in ihe FEM, whiclt ws euli€r u€d in FDM Othd
inircduc€d the
the b$is oI
@lie!
tuid
120,
Mrl
e
$ith difercnt qpli@lion!
Hus!6 .,
ol
(1981)
c
be lound
i!
Belytschko a, dl. (1978) la4 45l
l2ll. ALE meihods include ihe stabiliarion
by mplovins
(cls) rilite
bv gush6 e, dl 14, a?l
StFumline UDwind/Petrd Galerrin (SUPC) dnd GalerkiD/Led Squms
etemcnt lornulstiond lor
difiere
now
prcbl€d dereloped
l5
3 tnedruF
Cnaprer
id6
The bdic
oI
neh
Reercv
16
of ALE fomul6tim is to
motioh, wlnch is different
&d
e
the e&rential
both the
dotui!
h$egi& sd
@
th€ sprli6l donain for the
n€thod is to litrk
th*
dediptio
a
d*ripiion
daripti@.
Eul€ri&
Laarangie approadr us! ihe m81€ial donain for the d$clipiion ol
6ppio.!
lor rle
nqh md EuleriM
of s m6h. Tle main
id6 of ALE
th@ donains uing apprcpri.te nappins lunctiom, so that
delolmaiion grsdiois of both the nst€risl6nd spaiisl dohain6 ca! be calcllaled lrom
doMii uinS e chain nle
Hugh6 el dl, (1987) 146l iomullted
lh€ Efe@tial
milarion of d ad@rionnifision
nonoon
is
ihe suPG t€chdque, wlich
ted thc
equation c6ued by rhe spaii6l iNtabilily. This phe
norn6lly o@r€d in Csl€rkin di*reiiration st€p 6ing linw 3h!pe lorcri@,
when the value of s
tnNn
!due. Th€ stsbilizetion
*6
nrcsh
pdmeiA (e,e. Pehl nmber)
a.hier€d by 5dditr8
*eds
a ihBhold
! disontinuds @igliing lmclion
is up{ind pdlurb5tion) 6lons wiih lhe shtrd.rd Coldkin
{eisl
elni
@islltilg ftnclion
(ui)
(th&r
The
he pdmeter d wiih a pNibiliry to @nilol the st&
hility snh difsent rdge of iis $lu$. The finar SUPC @idting tunction (oi) c& b!
rS
lunciion included ihe
(31)
ce
be
IluAh€
e,
{here a
ch64 4 Au.
dr
(1989) 144 injriaUy popoced the GLS i4nniq@ lo shbilire ihe
nonlined! algebrai. equ&tions {or
in@np@ible flow
doD.id wirh
a
lnFu in-iine appronnation ed cs
poblm. Tris i! !
suibblc
sp@ tine helhodologv
stlbiliario opddor
be appued on
dd u€
6red spaiial
to sisbilia rhe clNical Galdkin fomu_
l6tio. by adding lesl squaE tens of ihe residual, which should be midniz€d The
srdbili4tis t4!m
m
be oI the
foll@ils lorn:
sds(4! ,u/) = D@4j ,LLa)
1"
\3.2)
Tlis ws ihilisly derelop.d lor the aa€tioniifuion prcbl@ ed itr@np€ible
sd SUPG alabiliz the chsidl Col€rkin iornubtion,
bri CLS is nor t g€beralized cls and @wn 6 widd 6nse of prcble@ including fie
R@
probl€N. Boih the ClS
t7
chsPte 3. LtrQAtM Rzi@
@npresible iows. Tb€ bsic ide& of Gl,S ir to Nbdivide lh€ sp@ tine prcbl@ inio
difi@trt di$oniiouous spa.erin€ dondN/st€p6. This
mN
€a.I 6p@tine slep
ed
di@iiation b dientiluos
@niiluoue iNid€ ihe 3p@ti6e siep
i! lh.
probls. which na,k6 novinS nsh e lomidable lsk
I'r FSI problens lege d€fom.li@ is s chalen&ns &d hoi iopic 6pei.lly
freld of CFD dnd
nnltithysie
oun€icslsolq MiUs .l ol. (1981) inkoduced ite m@ins m6h 6nepi ror
ihe 6nii€ el€ments Ia8, 491 This mrt nainly coveled the th@relic6l sd onpuieliolal
spRts oI rD ploblen using Burg*! eqlsiion s s t*t equariotr in 1491. Ii v6 6l$
Ior s
dleDded io ihe slver of ioplicil 61if otdinsry diffeMlql equatiom (ODB) Ia€l A
Idl deoiption
for ih€
hotug ffnii€ elenmr (MFE) of Min€!'s mrk for tD to hiShd
s nnknM
panolet€rs s @niral to nornBl FEM!. Thie MFE atprca..! .indtamusly slB boih
ibe m6h equalioE and onsinal difi@trtial equatioro. ln this cee, s inldpohlion oI
depend€ni uiabl6 ftoh the previou m6h to ih€ cnrcnt n6! b not lmey due
dinmiore is p@nled
to th€
by B&iM
i!
1501.
MFE mntaiE addiiiolal
lod6
i!rclwd lioe dis!1imily It h4 ben c5l€d d intspolai'on fre novinE msh
T€zduF r! ol
(
1992) I51. 521
rdtinS equilibriun equations ol
prcpced 3 nes nwins n6h algorithm by in@rpe
oldticiiy ud the elen.nt Jeobis
trMeric"l calcul.tio6. Tn€ etminstion
enilg €fieci in the computstional
th€ remBhins
pr(xs.
ol
m
ddst J@bio i.irodlc4d
dons ed
reduced ihe calculaied
scluded in ihe
ihe wiable stid-
projeiior *rors D
Th€ initial rcrk included Ddom.bte Sp8iid Domin/Sbbilizel
Spee Tioe (DSD/SST) nniie el€m€ni fornuletion for bot[ tho @mpKible dnd in@n
p6ibk nh poblm lE
d on lhe ALE lomulation Th€ slabilized SUPC
sd
GI,s
sd inconpr6ibb now prcblem, lAp@lirely
io bo{h ihe 2D sd 3D poblem [53, 541.
s$e spplied io boih the mnpmible
TIis
@ .ls su.Gtully rpplied
Joh@n
dd T€zdws
(1904) l53l *lended the con@pt oI
.he DSD/SST fomuldio! for p.rsu€l
Stol{6 equ&lions ol ar
Bid pll){l$s
n&nprsibk nN ed
r€m. Tlis fomdalio
slso lefi ont the
elm@i
ttdbilized
eeh tDiior bad
on
Thi! mrk lcu€d on NNid
tle nod€l
by adding tho Gt S
Jebid psrm.ta 60r l@ dd(mario
t8
Cnapter 3r ,jter&tue nevieP
dd &rked or bolh the struciurcd
and un+lruciured
ndh$. Nmdical
simul6tions
rele @nducied lor 2D pobleh6 including a vbcous drcp fallins on the vi@uB fluid,
ei6tiondy snd
Msud
osillatiq .i!foil, &d Rd ihroud
and
uwbe
(1997) [19)
the sluic sate on CM-5 @epuier.
qieldod the sp@iime conepi ofTerdur!! .t
for compuiltion oI ih€ Navi.r-Stok6 equations for ihe mslFis of
.1.
Iie $.I4 prcblos.
Thd inrl!€d chusins spaiisl onngorariod uing ihe spMetine CLS 6rite el€@ni
formulstion tor ih€ &rbsis of
lie
sorf@ n@. This
timedigsti!@B Galokin
ot 6nile el€m€nt tunciions provided 6 nslural mechei6e for
r4m6hirg ALE oeilod w6
applied io dea.l with
method
ildporaiinS the a&piiw
iine dependmt fluid flN plobleD,
{hile chsDgins the spatial coMgubtiohe. This incoryor&ed &biirdily elDped donDins
dd s@nnodai€d the iridgdsr
movins m6h b6ed on ibe ALE
m6h
elmots. In
&rd quadrildeDl
IrMo of
rel€!€o@
@!ld
E
the €leneni
yins *slableinconpreibiliiy ener3
J&obie sit!
i!
rhe
oi thh ih6ic
d
ALE noving
vith ih€ itrlrcdu.lion oI
cmputstioml donah ed u!.<l
ilisrete pcitire trrshl tulctio! to
a
!6i
be iemed
This ALE nwins m6h mplo)€d s L5pb.e €qualion
sptriially
the
Ey
th€ stifi€nitg
e6ci
in rhc @mpul4tiooal donain Th€ €ulriDs FEM formnlstion includst the tusnented
Loglegie fomubtion for the moving merh Nundical 6dis mE
geieo.lizcd
niliorl .6idusl (CMRES) i&laiivd
(EBE) pr@nditioner lor lbe sohlion of lineor
fo! nuid
qe
flw
ploblens
v€rili€d Io! 6
dd
solrer and CauF&idel edge by 6dge
systn ol equations. This is the suitdble
stable lor a wide nnge ol
Pqool& nmber. The
lesultg
sliialy waw plop68aiiotr, nisile bunch fton
of cmplimt cosling with vi&ou duid md *ismic BpoE
noms
subndine, @uplins
obleined $ing
cyljnde!,
tel pobletu.
Joh@n ed Tednys
a
or
liquid-nued
1000
{1999) I5{l 5Lo
pt8ltd
sph€r6 ldllils in s liquid-fiIed tube, airnw
psins erG
moving thrcu8l ih€ ne6h. cube rcllins thiough lhe
pdiicl6.
This work
e6
the pdalel 3D sinuladon of
n6h sd
rhe extension of then pEvious
the pso.Iute, 6phse
penodic
nen
of
ffra,l
pdallel nesh moiion lor the 2D
problems. Nrnenc.l eeennentations @re sg6iD cdiiied out on ihe Thinldn8 Machine
CM-s 6nd sh@€d the
ustulns
oi nou sinulalions involvins th€ pqiodic 8@setri6
CLtp@t 3. Ltarcrure Fcy6w
&d
Suid pelicle
Farhar
19
nixrur6.
ud
io calc'$t€ rhe
G€urain€ (2004) l23l romutared ed applcd ihe ALE tihe iniesr6ior
$lnrid oI nNr€ldy 0@ Foblos otr ih€ h@ia8 sride ed inorporui.d
ihe lhrce-ponn bockw$d
diflffice
tqpercidal rule lor iime inr€grarion
I'rd
Th! AID 6cl6e dlended ttE fxed-grid
by sunins s nonoione fu tunction tor ihe non-
scttene.
stlb'lily eal'6b. Thi! fomuhri@
w
spptied |o Nsvie. SlolG equariN
ed
nunericdly dpqinenied @ Advilory Creup for A€rupe RAdch ed Developnmr
(AGARD) Wing in wios unlreldy now probler md
s *dl 4 a @mplere
F-16 @nfis_
uhnon rn
Thir
blt
treshic aiEilres,
onfihed
rhe siabiliry ot ibe nonli!€&
$luti@.
ALEtin
inres6ior p@n€d rhe duB., of Ederjd s.lEFe on rhe oovrns grids,
previouly ihai we noi lne,
Mdud
(2!06)
nN equalioN
And
which
ro
inBrisaied rle.ffere
156l
$lw
m@ing boundary
n@iq nsh on tbe stsbiliry of nuid
n@ prcbltu in lb€ ALE irMe of rcf€rene
or
enplotrd ilrc srsbilized [email protected] equsrion
deply oalrze the d6|abilizsii@ €ffet
novitrg
n6h
th€o siabiliad
i|.'rati@
FiEtly, th€ prcblo
b
uins s pGitive t@or.
s
a
nodel problm. This siudy
ot rhe nodel probtem, which
model€d
uing
Nu*icd
ee induced by rhe
stsrLd Caidhn foedai,on ed
slution ws obtued Ning CMRE5
$lEr sith
c6lseseidel EBE [email protected]. lr wd sugseded thar Elariw
v.leiry bei{@ sm6h ed Auid 0@shoutd b€ oinimi,jd b
bette. etts. rr
@
ser
Mhieved by employins rhe
KhuEn
peiou
and M&ud
(21Xr6)
tine etution in ihe
[86] expro&d lhe
lrsHi
be
mntind catoldlioD
AlE novins n6h
mDcepr in rhe
tulli4de
sbbiliz€d fornubrioh of ih€ inomp@ibl€ N6vie-Srot6
equario4. Shbility
vd calcul.ted on borh ihe 6ne ed core ssle s a pdori qiinarid, {hich
we s good
s ihe cl@lical SUPC ud cls merhods. This ws addcd
in rhe
lornulalion to prodr@
calculat€d
numbq
a
&d verif€d in
40 Hw.relj
Dltriral€ wi6iiomt fomdaaion.
th€
oudicd sinut.ti@
of
fte
now
Cood
$ed&d siaiiohai
stabiliry noru wm
(M a cytjdd !t
tuynotds
KMre vorrices &t F€yDolds nunber rOO
&d qN eupp@sed by inrmd!.iio! ot. ihin b@ st the MI€ Egion.
This fo@ulsiion
@ tunher applied io & elsiic b@ sirh rhe lese mplitude eilsrid s o€ of.
a squee ptoduced th€
Cbrpt€.
3 lrebiu( nevi.r
bsic FSI
i6l poblo.
20
Tte b€an
rs
Gcitlated uing ih€
h6h r%niDg @nep1 b6€d
ALE Iornulation 6ing rne fol@ils d€detior equarioG:
rir
slrde y(r, 4 is dE
r)=,a(r - rofdrn(2',r)
kq y-dilpl@enr
oi r!€ bem for r srw, ratue st
underorn€d b@m [email protected] al a given tine r, ,4 is thc
of ih€
bwr, u
is the sngllt& wlocily
MGud e, .r.
(2007)
(3 3)
dd ,0
is
x-tu
ndinuh drptiiude
or the
ar tbe
iip
m idtist displ&enent of rbe be@ al
odirs h6D olcept btEd on irle ALE
trme oI refer@e tof borh th€ situcrured ed u!-siructued m6n6.
n *e appli€d to
. @iety of 2D problm. Thi! sbo pr{et€d ite pEoDdirioned @rj!g!re ghdienr
ll4
en@ded the
r@thod lor snShented La3rmgie fohula.iio!
oi rhe ALE eoving h6h. The e6h
rczor'ns schene wa alm oDbedded in theiL no{
solwf to
E Gdis oI dow6 oE
2D novirs nahe. A vejety of i6i cses
ihcluding a piichng 6i oit, stora sepsraiion,
chal Mre propssation, ptrlserile norion in disiensible eterjs, penodicaly
oscjllaiing
wi6,
be@ ed hulriple m@ins cylindeN we sueeinly t6idt
tor
Kaichi
ed Mdud (200?) ltsl applied rhe.ddpiw
rhe evotvin8 boundary
Ar,E mvins
n6h t .Inique lor
probl@ t[dr inwlvrd mei.s nuid boundMe
od fuid/snid inrerlsG.
q\tla hl@ny ffeld ot rhe und€rtyrhg
@ntinuuh *s ato $lvtd u addtion to comlitutiw
equaiioN This mrk focNed on the motion dd detorharon
or 3D h6h6
3D nuid nos
mepdsod
ot linear tenshedlal and h*ahedht elenent
for boih rhe skuciur€d and uh,rruciu&d
m€h*. T6i poblahs
$hse, qodytrhic
a'd
hdt
qe
dqibte dofo@lioa of mulripto cyljnde^, f@ killarion
of
yFstDdsdon of
| ? jet, lotariond moton ol s subDs.ine propeller
b@ring simutalion.
Tne
sppliB.iotu sh@
,E .eB€ ol appti@bility
ot
ddbg n€h fo! boil ilp itulully od [email protected] d€lG h@ivjn8 niilioro
o,
nNlruciioM in FSI pbblene. This rerhodoloey
@ siry be €xlend€d !o the hisher
ihe ALE
caldd.r and Msud
(2010) [E?] ptsletrted tne v&aronal
nuuiscale stsbili?ed
Chdpret
3
Lt@aturc Revev
a
IDM fu
incoBpr6ible Navier,Siotre equsliom uing th€ ALE lomulsliotu
rrdn8 bounddi6
4'que
2l
The bsic
$gssred by Msnd
ide ot rhis
€r d1.
The multisca.le siobilirsrion
wLid
*s
paper ro 6pply rhe
11? 191
sth
AlE
dovinS
si$
n6h tah_
6 nultilcate siabitized lomdatioD.
0.triev€d by pNjedins a 6ne-ksl€ sotution
olio
ihe
spe tor both ihe rebcny freld od veighiilg tunclioa Bsckward diflererce foftula ss liilized fo! the rime di&,€iizarion ih.! wa
6[M to l,e a smnd orde.
coa^$cale
&cuaie *hene. r"ulEncd sinulstioB
Be
hrtq ehedding eound
snd fffl eund s detoning
@nducled fo, ihe
cyiids ard a cylind* art!.lEd to s lin€& sprins,
beM. Tbis Eialiondl mltli*,le ALE fomdslio! w found ro be 6m,sieni
wilh fie
A 6x€d
3.2
Stabilized MFEM for Darcy flow
Decy
(1856) l58l qp€dDdr!"d |ne
nukai€d tlE Dqcyis
1o
Te
qler tnbush
b€ds oI
M.l
.nd
pc
l.*
&d 6
However,
difdent ,pplic"iiotr
Thei hsin
of
of @Eriluliw €qtalion. M.jor obteh@ of
Dorq,.3 law is
equsiion for rhe fld of o vi@u luld D a ppM
mlw the Poien
n's! onFvation
howndr
rug*
Lepl&iu ed p&sbolic Dodeb cm
dilferent appt@h6
@
be inorrera|ed tor
troln grouDd wstd.lt@ io peimhun elSDelins
f@us is to siudy both
th. pwuft dnd velocriy
rields
ftl s
136,
34
ofnuid ib the ppM.
to lolre the probtem rn finfte elemenr toreutaiionl
nar one eDploys . sin8le 6etd lorhulsrion lor the p@ure
neu only ed thc seond
us€d
one
us.
dixed 6om ation for both rh€ p@rc &d releity
nelds. Mixed 6nii€ eloenr
o€thods (MFEM3) simdb!tuly *ire boih
ihe poteral ed Elairy 6eld5 to obtajn
the appuihst€ sluriotr.
Brelzi
.J. (198r) l59l inhoduc€d a novcl @rep| ro obtsin
apprqimart slutiob
of sond ordd ellipiic eqnsrio@. The
nunericat sotuiio! ws @nputed D rw spe
valiabl6! for iwodifiereni tsmilis ofmixed
6Dihelemert, Obe,mity oI FEM ws bs€d
on h6sles dnd orher w6 bos€d on
@tsnsles The swernins fo@ulaiion ws node
on
,i(o)
a,
6nd H(div, O) spo@. The colurion of rhe atseblar
equalions
ws sinplified
J
(.t''p@t
Lit.tstue
by ucorporuring the
Review
22
lasonse mdtipliq for
prcdu@ a hybnd b.m oI hix€d neihod.
pdine
elemeht
errcft
rhe @ntinujty of
fie
nornst @nponenrs. Thi.
bsic idd b€hind
is ro
sinulteduly aF
p6nre sd !€t4iiy tietds uiag difiercnt spse ltong sith ihe 6niie
sp&6 to slabitt. fDte olprdinstioE for ihe @Etructiotr. The Nlmprotic
botn |he
*e*
abo srimaied otr ihe sbow
6pee &d @nspodihs to R viet Tho@
8pees. These disdet sp&es hare bee! suc@Iulty apptied in nany appticatiohs
{or
better eurMy lor borh ibe @tocity and prelule neus. h !.lieks
boih the l@Nl od
global
@Nmiion
Bt€zi
e,
,t
oI
h6,
(1987) 160l
dtdded the olcepr of rm r@ri6 or nix€d FEM io
lh@ sp@ wiablG tor tund onter ellipric pobl€tu Tle dsi'€i!
rc @nduded
on srhplice &d cnlJ6. ThG qreDion $d done
by *tending 1m s!@ d.tg io rh@
spMe daia usina addiiional
vetor polynooia.ls. Ah Altdnalins Diretion Itardion (ADI)
nerhod s& !.lso iniroduced lor a r"pid converssn@
oI the sorunon due io soddte poinr
ror a better
ecurmy ADI loh€d
XdimensioD
od ?diDdsion, rsp4li€tyj
rhe sy6ten of equ6iioN
shich
m
atiemdtiEty in x-dinemion,
41endfrt hom a frnte difie@nca
*hen€ to s nred fiDite elehent foollation.
Ire@
io!
er
or
gd€bl oobimlioG
includirg
pop6€d s lew silbilizrd Dned frniie et€medl 6mem.t
si6 &d dbpl@lMr in&rpojari@. Ttie w s.li€v€d by
(1988) 16ll
the.lNi.,]
of
H€llinSer
tuiqcr
fomularion, which
b
eorhq r6idn l
lorm ol
u! equilibdun equarion in rhe pctrov C.lelkin 6nite
elehot h€rho.l. Here, ,, ,p&e ws
u€d for strses ed u(0) c used for displenq|s.
This nethod is atso rntrn 6
3 mi*ed P€k@ Calerki! finire cl6e nelhod
Nunericat linut.rione @re conduct.d
2D prcbtem usins rh€ Eidlat s 10-. ed
@nfirh€d ih€ nuDeDcalerimats with th€
Meud
latr
@
e, ,.1. (2002) ligl
proFcd
s new sbbilizld MFDM for D4cy
be sol!€d
n@ Ddcy!
uinS the *istins nuhericd oerhods (like fni!€ diffden@
hethodologis) with e adequsto @urdcy. HNerer,
ihe ns co@rvalD!
anie€d du€ to the
16
oa
ih€ @u!&y in rhe
formularion of FEIVI €quatioN
uing
!
pr@*
pnod eror
So
sd
elen@r
is .dr
g@-
ihE beftod stdbitia ibe reak
€timtion
i!
ihe srability nolb io
Ctapts J. L',prature Rpview
23
iDp!@ ihe eurmy ol lhe pt1}6 This .@nnodsted t@
!q
wi.iisal
sttbiliz€d
cMs tbe continuos €locity sd p@te inleDobtiod, sd !@nd
6@ ihe onlinuos wlditie ud di*ontinuou p@nr€. The [email protected] r6uli3 we
d$ rerified using the numcricdl dimulstion lor ,z norm of the €locity 6eld, md ,r nom
&d llr 6eninom of the pt*@ 6eld for boih th€ lin@r quadrilateral ed iriusuld
fomulatioo: nst
elenenis. Nurcica.l conwryence i6rs also
for
rll lhe cs. Sbbiliz€d MFEM8
snd inrprde ibe
mrls
flN
!edu@ rhe ompdtibitity i$u€ monS difl€Enl 6elds
a&u.ey ed reli6b ny by 6ddiq lhe penurbaiion !erm. Thn metlod
for the no!-sJmneidcd siabiuz4d nixed 6niic €l@eni
by addins the a4ioint
Ayub
dd Msud
fonnulotion lor
im sd
in tbe
6i6ed the etinat€d shbility_norn errcB
!6id!a1{otb.
dioded ihe st6bilizsiion of mt€d fniie elendt
&nreciiediflisi* h4t tldNler eqnstioN. Tltb st8bilid ihe temper&
(2003) [62]
ihe empemtuE
clsic4l
fonul6tio6 for D*cy
nu
lor
&
equsl order ol
inl€ryolstis,
Galerkin lornuiation. The new siabiliz€d n€thod
wi.tionsl nulris..le (HVM) hd€wL
which
m
ws idned
nol31able
s
Hugh6
tor FEM io ioprcw the st5biliry uing a pnor
eror etindiion Thi! HVM m€tbod iniroduced the siabihy norn by including ihe
{14e ud
ihe
fin€ *dle ol
reidtins lunciio.
pobld uing
@n be
appropiste
sp@ e g oNe dd
6ne
sde oI
s*n a:
(3.4)
vhm
liJ
indiclts tle @@ *ale (@pon€nl dld
/
indic
e
the 6n€
srle
onpddt
The followiq equaiion shoffi on€ of ihe 6n6l torm ol lhe HVM siabilitv:
LdvNtu,ct=
where
r
Ltw.et
+
rt,tN
u.+)
(3.s)
o
the ulder
sque problen with the ih@eiicsl prediciion Fi!!.llv. ii wN
applied on
is a bubbl€ iunction. Nun.ricat conrersmce rot6 vere vsif€d
sludy biunit
1D corvetion doninated problm using Pelet oumber
Maud and Xia
the @nprdibt€
(2005) [63]
prMted
sd inonplBible
a
03lor
a dehiled studv of the
nd€l mdtiscale .i6bilized MFEM fof borh
elmliciiy prcblem Th€ slebilitv in MFEM
@
Chapt€.
& lite.atu.€ Reviry
'24
!8sin obtained by introduction of boih ihe
6me
6nd 6ne esle deconpsition oI ihe
dispbadsi deld uine s priori e@r etimstior ol ihe srllen, {hich *a btsd on
the HVM lrmewrk. Th€ 6ulian! siabilizario! *s ensured sd velifred eEn for lhe
{,@
m6h by in@rpd6iing tle
|he displ&elnents
BE i .l
(2002) lor
di.
Dmy
ol rh€ ilispolaiion tunction of
sbiilqy @bimtioN
ad ltlgs. It ws dlier wtsbilized
in boih th€ SUPC
(2005) l40l applied iL€ srabiliz€d MFEM deElop€d by
fltr
io a total disconiinuos eleneni. Thi6
involviDg jump slabilizalion
*a
ed
CLS
Mald .l rl.
inili8lly lolltd by
Galertin method Thb wa mniei€d by
h disltinuou
> 1 inste€d ol u€ib8 6ddnional
jump ter@. Tbe probles und€r study {s dsntinmu Caldkil D€thod for Dscy
applying polrnomi.l inieDoldtion tunction of deg@
p6diy nethod, lle eiabiliation lornul6tioo wa
flw.
INlead o{ sdding the siMddd
done
vithoui Ning du wiable by induding the spproprisl€ly wiSht€d Bidual ol ih€
Decy
6
lsw. Th@relicd equaiions
mE
also
@li&ted uinS boundngs &d consisiencv
reE min€d sing the nunaicd dpdim€ntstion of !
sqle lor thc disnriluou pr*ure ed \€l@ily @r itiagle ed quarlilat€rdls
of $abiliraiion, {hich lo!!€rly
biuDit
Hugh€r e,
oi
(2006)
l4ll dielded the stsbilired MFEM lor ih€ di@niinuou
Catelkin n€thod in wLich mnlormins v€l@ity 6elds
re
tequi.ed lor 3D
poblds
i! o volumelric, 6 residua.l-bsd siabiliation
ps@d$. It is indep€ndmr of tte saeh psmets ud 6 {qk e@ difismt laleB
The key ingrediehr in this fornulsiion
oI dimntinu.u! naielia.l parsneiers This lonulation
|anSedtiat
oniinuily ofa
sd
bv applvina ihe
oEtxnlt &rc t[e materitl
rsle for both ih€ prdure ed wlocitv frel.b wd€ obtained
tlxitt
ldyeE. Optimai conrergene
ws mhie*d
freld @ non-ph'sicol
@ified tor rD,2D &d 3D smpl€ prcbhm
Lnra
€,
d
(2m6)
164, 651
sle dteded
lnmnditiondlly stsble MFEMS lor Ddcy
nd
the
linil8 srk, vhic! prelt.d
by intoducing
l6t
squ@ leridusl {oF
nulsiiom. This siabiliz€d Eethod is Gd ot n6h dep€nddt PdMete6 6ePt
Stok6 conpaiible mixed melhod,
nne
elmdt spaG
{hicl
tor borh ihe p@ure
employs the clssica,l @ntinuous
dd lelftiiv
lields
ihe
ror
l,egrlnere fi-
cing irispolation tunciioB
Cnalier
3 L,r€r.luF neiry
This merrod *lended rhe Mrk of
tinit
elemen!
MNd
e, d.t. (2002) l39l
6t s srrmetnc llabte hix€d
fomuldtion The crulch of ihe symnernc sbbiliaiioh is io
shbnny by srabiliziig D6!cy's lsw for the pl*ure neu,
ne
inplN
the
bald@ Ior the @ldiry
ed drl in il[ Daryt lav lor tle €nho@hdr ot,eleity u'lg a 16r sqne€
Eidul for the equ.iion ofI)N bdee.
Maud (2007) 166l q&lded sbbilized MFEM to! Ddcy-Srok@ equarions, which
6eld
deconped
the relocity 6eld
i.to
esM/@ltd *st* &d
ine/unleolwd sGt6 bsed
fil\l
lrMemrh. Multi$ate sislili?.d forn of ihe Dey,Sroks €qmtiom ws
dhbiliz€d by modeling of the unrBolved &a.16. prelihindy onbsen@ sildy 6
on ihe
dd i€ted for 1D md 2D tined €tenenis. Nuneicat @ulis shd.d rhe
a.d onv€lsenc Id wious conbimtions ol borh lh€ wtociiy ond pr66uD
conducted
$sbility
Xia
h
ad M6ud (2009) 16?l al$ ej{tqded rhe shbitizel
aiioD for tbe nixed
ans.r}6is
in
disptalhor ed prsure for
ssbehmiG
formulation
ws wriile!
€
eldent foF
nnite detolhstio! elsroplstic
Th€ nerhod w6 apptied to catculate the beanng cap&,ty
of el8sbplsric sormierisb ihtrt
lno prosr@jlely bqhe
nixed fin
iliiisly stosql 6Epr6ire ttunehic strdN &d
i!@mpr@ible due to high @mp@i@
usins
m
updat€d
Lasresie hethod
stlM.
due ro the
The orisrnat
nonlirNiiy
n'hltd. lt
cDt ol the
w6 ilrcn srabilized by ehptoyins the @ishld co&se snd rine 6cate 6n
Biational mntli*.le tomularion md iniesr6tal wjlh s th@ surf@ snmlh
elaloplsli.
csp nodel. Tt€ @Mrtutive equa.tiou @rc snfieh
sr6 by
equetio$ ss
uing Clucly
addins a pu6h lorsard opeleiion The nuEerica.l ilresralion otcomiiiutive
rhpreneni.d using an eletiepredicior &d a pl4lic<omcio! dlsornhn.
Tte 66| par
disard€d th€ pleiic ii&rMdon ed clet@d ihc elsric tn,t sta|e. It the
sLte Blu€
s6.dhisible rhd
the
the
send pei
b
i1€nr€d Bing Ne,ton,s nethod by ircludinS
plaiic corsislcn.y pdmeler.
Numerical sinulstiond
sb usin8
bq.tuddt
tor both
tle co@ ed
w€
conducl€d tor Ie8€ detolnarion of €lNtoptsiic ana.ly_
eleoenis, whicl confimed rhe stlbitity
fine srsined
h6h6.
&
predicted jn th€
This oei[odolosr
6
thsry
be applied qrh
Chsptet 3:
L
@nfrden@
toi the got{hnicsl enginering qheF spsiisl demity cheg6 due to lhe
enture Rrvtev
@nrpeiion ol ihe sBnuld @teri6ls,
3.3
The
Graph based m€sh partitioDiDg
peiitionins
oI
n6h/smph
is
u
NPomplete
pob[4.
slsorithru qist lor the sraph psnitoning to podue
oultila*l s.!ph pdtitionins is wU
Pmelis ed MelC librdi6 127, 29,321
rently.
Karypb
ud r'!@
Thdetore, di8€Ent h€uritlic
$lltid
of
rlifml 3i6.
CtrF
knosn alSoril[n lo pdiition th€ 3raph usins
(1995) 168l inlDduc€d the con@pt or
nultilewl petiiioning
dd sp@ nstdx ordelis alsorithn. Il us
thre $epr io pdiition ihe sraph usinS minimum edgeut set: graph @bih8, ini
tidl p&litionins, dd uncoNning,/rodnemdl slep. ll @Blrucis a cqlene of lnaller
o,llolithn wiih rauBiw Creph bb&iion
gruphs includins
n|3
fem rerticd,
phF md do6
Filally, ii
ro oriEinsl
s
!
by @llAp3inS adj4ent
w.li6
dunns the graph c@@n-
ihe ditial partiiioning $ing minimun edsecut
s.eedy approeh lor rh€
sisp[ hy
Nislils
bisti@ lkonlhn.
refi@me 6iep io pojet ba.l the mom Boph
the @llap6ed
6dj@i Eti6.
This skotubn
m
$ed
io find 3 nU leducin8 ordsing for a synneti. BpMe natrix thmtrgh !€cmiw d5lylis 6Dd ouiperfornea
lle
other {idely used nultiple mininun desE a.lsolithms, Thi6
ekornhn ir bded on ihe
nullilftl n6!.d
di$eclion (MLND) Thb redex sepdaior
dgornhn ninimized tne @rtex @rer uing the
ed KMd
edSe
epeaior.
nultil*l
graph p8nitioning and spN bttrn ord€ring algorilhm for the Btor pro(Kr They
uriliz€d the rm i}?e ol p&all€lim. ln the 6E! type sinsl€ pllN. rcuBiwly bisi
lhe grsph, iho im pri)lgs bi@t sd e on. In the wnd tr?e ot porall€lbn. all
Karypis
(1998) 169l
p'6eni.d s pdall€l fomulstion of the
pruo$6 si*i bisiion d p&ollel. Tte
lelod
Ptl)@s mre
6.hiercd on t28 prccss Crsy 3D puallel eopule! silg ih€ hypertrb€ a intmn!e1. Thb study ws condlcted tor wiou 3D Biifin6 nein6 md 3D n6h5.
in the
nee *hile, Kdrpis
sp""€dup
dd Kumd
(1998) [34]
of 56 on l2a
als
exiended ih€ @n@pt of
Craprer
3 lnehimRfle'
27
p{rtitiolins to mdiil€vtl l-w.y singl@drEinr sraph psrriioning
llgornhm T[ei G fdt€r ths 6 simpl€ nulrile!€t re@i@ bileciion a.tgorithh for the
drc Dd.il€vel graph
paiitioniq oI
a graph by the
f&rd
ot O(lg
t),
whidt is inh€rently pealet, Crsph
(rMins ed inilial p&dtimilg phs of ih€ l,sy p&iitioning ee bt sinph ed ih€
reult of l-*ay comedhs ot w i@ Dlodue rhe ilirial k pui ioff Bur rhe refineaent
srep I nNh nore ompl'cat.d ths rhe sibple oulrileEl gr.ph psrtilioning due to the
nolment of wnie alolg diff@r peliiioN. shich inc,e,g rhe sp@ @pldiiy.
Karlpis and KMu (1998) [?0] propGed the Dultihwl mnlri.comiDini sraph
p6ititionins for Ddtnphse eihul.ti6! usins th€ EnNtu bislio .ppl@h. pFiouly. tle obj@tiw tuncri@ for t say pddtnding @ to nininr€ lhe €dge@i lor
h-way paritionins unrS sdne ste ot pstiiioh s a onshaini, b this pape!
*leided
ihe pa.litiooing algoriihm by dploying e sbi1r6ry nusb* of bsl&cins
coBriajnts.
This w d@e b, i@rptraring ih€ new ir!€ of htuiltiG for
rhe @aing, iritiat
pdiilioniry, ed re6nehen1 phs siih the i rodudion of nuhiple reishiins
fa.tos
fo! ah Edd. ltorironrsl dd wnicat nutrioE rsjrr fomulaiioE
Ee appled for
the
difcmi
or qudnrriy!
for rhe
diffMr
algorithn. Atl
n6hs md
rh*
ptoduced
bistio,s
(1999)
,igtrtbn
lor
I&toF ree ohaihed frcn
rhe
rhe
tue
rype
veiely ol
conpd€d to the eiheltunshajnr
dsdrb@
e
i!@rpoBr€d
l?llqiended the nultil@t k_my sinsle @nstraint
Eior @npui6. Nunel@ €xpdlh€ntaiioE re
@nducied on 128 procBsrs Crey 3D pe6lel
to 35 for
s
dd
wer€ @lducied on a
s€q!@li6t @d6 for snph petitionins
ud KuEd
grapn p.rtitionin8
ron $
rype of €nri1i6
rsp@iirely Nundicat qpedheDtaiios
3D nniie elenenr
Karypis
Epllgialio!
reighis
r&s€ of
14
@pnie, 6ins l2&*&y.
The spe€dup
ro r? on rhe Crsy 3D for smaller graphs
bisso 86phs. This sp€dup
ed
iDp.o@flr @ t6 €ticidt E @Dpesd
io ihe sinple multitert gbph pdiitioninS alsoriihh dieuca€d in
[691.
Schloesel .r at. (2000) F2lpBented lhe pea el nuldftl buhi-onstlaini graph
Pa$iiionins alsoritlm. This
pdntionihg equired
a
*G
rhe
pu.pe toi nulri-pbe simrtsiioE, shft
nunber ot b6ls.e .oherBinie rhat sbonld
b€ €qusl
rhe
to the lunb€r
Cnap'e.
J
Lrtsralra nev,ew
oDpltation.l
ot
pha
slie s ud ndiniz
pN tr6
Th€
lhe
28
&iie.i6 ot rhG peltt€t sJgorithD
p@ibil
included lor sll the
y of
Erii@
Eins@l n@6
m
to
bst&e ,lt
ihe
on-
wiih *alsbilty. An sddirional
for 6ach le0nement srep. Ih
6sr st€p, n upddr6
th€ lo.ai .lah .nd rhah upd.t6 rhe slob6t dsia by calling a redlction op€rdlion, it all
sti!fied Nuhaical sinutariod {erc @ndu.ted on Crsy T3E
containinS 128 procqsoN sd s@d efrciency wd eporred for rhe ls8d graph, which
tused lbn 70% to 9070. This art qd aI pnor @& on ih€ gr.ph b.s€d p.rtirioni.g
b8lmce @rotraihh ae
locused on ibe
llstic l@d
bataDcing.
Schloegel et di. (2002)
&d extad€d ir
xr
1o
!
l?31
pFsented th€ siaric mulii.coctrsinl g!6pb pdiirioning
n€s p.$iiionins aisorirhn lor dyrMic nulri phM sinularions.
dyuni.
Boph parritioninS atsornhn 6n additional obj€tive au.iion is ehDlo}td ro
rhe ddta redilrributioo for ih€ ndi pslinion. Two spp,oach6
e ir@rporarftl
,rinin
?e
fo. the
dlrui.
I
repartnioning:
Muliiorsr€int (Mc) Soatcl R€nrp (SR): Us & tuqns ssrc
io conput€ ile nexr dyneic leparrilioning, which nrnnnE6 !h€
ohDuicarion, bur dG mr ninimid ihe dst! Fdi.tribuiioD
6t.
2 Multi{nsimint
(Mc) Loolly Mtrtd€d So6rch Remp (LMSR): rcdu6
rhe dais
lcdbhibutioh ci by implicii ndpping fu.iion hob e odgihrl psrrition
ro a !e{
pa itior
tion
and by nodifying ihe ennement algodrhD to ninihize dara
redishibu-
cb 6 a s&ndey objerire
the sraph
j<iire
sd
tuDction. This
ps ebi€vst
by locat
iDoDoratins ihe ad.liiional dsi& .ediski bltion
@Nnnrs
4 a s4nd{y oL
durins the rcfinehsr ph6e.
\nner'cal experibentatiotu
l@ ofttlcied
on 128 pr()@oB CR,{y T3E mehiDe
sins difereni ei oI probl€@ @trr.idns i@, rhre, four sd nw @clDjnrs. ti ws
obeNc{l rlar MC-LMSR atgornha conpured sinils battu@d perilionins
ro MeSR
*hmq
while hinimi.jns the dsb rediBrriburim fioE 50% io
tcrh ot lhe spe.dlp {ore obiained tbm
used
?0%
b
goyo.
n
Cood efrcienci6
Tlis slsorjrbm
m
be
ir
€d&.riwly
n! ihe problens vhere D6h b clesed in each ph@. Tn6e psdirionins
serc slso
Chaptd,
3 l,jreBture
appliai ro
p
Revr@
29
fbrtal inpacts wilb a wsll of an Audi dd a BMW for nuhlphse
c& cr6h siolla.ioG lor the dyndic load bsldcinS. The petnidilg turciioN e€
the
u@Dorded in Peh€lb for both fhe peallcl
Karypis (2q]3)
l29l applied rhe
sBph peiitioinS.
@h &d
mutdcosirsjlr paditionids approMh ior d@m-
psms ooni&r/mpet nuueri.&l sinularion in which the n6h elenenr6
witb ean orlFr by rcd&ing the onouni@rion
tre b8d o.
rhe gmDetnc intom6tiob.
mrh€ad It mplold
It ompni€d
come in
conr&i
d dtra dctuion
I ns n6h uihg rhe nulri
@miraint srsp! algo.iihh. Nuo6ic.t rellrs md pdrormee @e m*uod uing 25
ard 100 wey srepb pstiiioning u6ins evticidy-peallel lNrrncrion Conpriing (EPIC)
3.4
Spatial trces based m€€h partitioning
Sprial
lE
daia srtucturE litc
lmBir
o.ihoSonat
bjsnoD, kd_tE, quodr@ al{l
o.!@ de f.nons trhniqxe i. lhe rnlodlic n6h
s€neraiion, n6h E6nemo!, $lvins
N-body problcn
od
padirionins oI
m6h
13, SO, 31.
?4 ?Sl for 6nite difidence, 6nite
elchenr 6nd ffnft€ rctu&c t*hhique lor uhbatan@j
sids and neshe. Oct@ Feiho.l
@ ars. be u!€d io dirfibur€ rhe ultrucluied ns[6 ot FEMS.
Bqg€r
rtuciure
ad Borhai
s!
(1987) [7{] prcpc€d ihe
siaiic toad batucing
a RCB This slsoilhh
akdiiln
sjhp!6i r@ b,r.d sp.li.l
for noD-honoeebu grids, which ir
d6ra
&.@r
plae ihto iuo pt6he o{hosonary uDg cooldin6re dis.
The tudhd exrension of RCB can be vjeftd 6 quddt.*
ett ociEe thai simulie@ull
cuis ih€
cul the probl€h dohsi! inio fonr
wdren ard satbon
0993)
dd
€ishr subresiam,
I?51
6$t line dplorrd tho
Epeiiwl,
in an o.rhogonal
i<lca of
quadfa/c|ta
Ior nuneriml limulaiions oI ihe n-body probleo.
O.ne/quadrEe c& bc usdt tor the
parttioning ol in€ probldn doo6in dd lh.n ile
iraEBsl defines ihe odering oI dats.
Difie.ot |re |ra€6al define ap@ nllng curc &d sig.s
th€ global nunbdils io
elcndisi which @y imprcw ca.he perlorme@ of prcBoB
D in€ ooput rjon due
(lapter.l !r.r3tuP
10
neyFw
30
lt€ som€r c locdiiy ofsparirl da|a F9,801.
liundiol
siaulaiions @e onducted oh 512
slalled at Cahe.h
dd
pl@or
Inret Ibuchstone Deha in-
odqing ISll for rhe rek NiSnnerr
ftr diff@t proclls fte pbp6.d srrare6/ codd be apptied o. a wi€.y ot spptiodioN li}€ airoph}ri6 (s5ldy 6omaiion &d hus€ strucr@), @nputaiiohal biotos,,
urilized rhe Pe@GHilben
dy@ic6 oI b@ rele8ror,
chemidiry,
elecrrobdgDetic $siierinS, fluid ee.jhaniG
(p&el md brtex nErhod4, nol{ulsi dynsnics, molecutd srrucrue, thernodJmmi6 4d plM. physi€. C@rtuction of rhe b.lqc€d o.te for a dara sia 1al6 the
ron@ins tirtr€ (?(n)) for rle
i6k eisnme,l:
r0.):Br(;)+o(D)
so th€
h6t
onsfiuction
oi the
e.t
of
bsldcd
iE
d6t3 struciuG
6ibg @rre &d quadrE metlodr
sohe edljd sork by
ror rhe
spariat
Flatery
dyndic ldd balecilg itr
ar
(3 6)
da
t*e ,(,lg(nD rihe. h
[d ba
dohe
fd
ibe adspdw
(1997) l30l incoaolared the
rh€ adaptiE m6b
FEM,
n6h
@rrc lneihod
refinmdt for th€ AD @nst)@
lion l6m uing dilmnrinuos CslokiD nerhod. T\ro ad@iag6
or rh€ @he nethodr
rerd enpbycd siDulteeously ro pdtitio! and lor
the m€h rcffnqFnt The sequ€ntiat
octre pditioninS dlsoriitn ws exidd€d ro p&sllet eire p{ritiorirs (OCTPART)
dgo iE lor dynMtc load bli&cing. The o@uioiion @ s@ite io rle
@uEiE
This sork a.l$ prcp@ed the 3n@rhnes herhodology
lor dynamr losd ba.lacibg of
@tre p$tirior using 6re djfiereli air€ a ot e adj@ €thflr fsG io
other prc
(s6. Nuhqical sinulalions kE @ndmr.d o! eighr po.|e6 oI
fBM Sp2 ompnr€r
$ins MP!. The lun€icat rdullr ot OCTP FT @re tunDard
krh p&4lret $rr insiisl
r@Niw bi,{iion (PS!RB). Tbe 4suli6 showed ther OqIPART isd€rt
the
ro nove leqer
elenenrs
6
conpdred ro PSIRB md ouip€,fo@ed in borh rne rme
,tn rhe dtu imprmeor of iZ 26fo.
Flshery e, dt (1998) I?61 p@ented rhe oclr@ nelho.ls
in all the
ed
dara nowh€nt
ex.hpls
lor
dlhmic
load bal@c
Chsprer
3
Lttenttre tevrce
31
tr's in m adaptiw FEM lor ihe steady and hmneni soturions ot rhe 3D Euld €quaiionE
w
ofdeprcssibk nos Numdicat qperinenisiioE
tlrior Regeh Centd
n6n
(SCOREC)
t@1, *hich
n6h data.be, finitc ocl@ auioEslic msh
re6nene
.
palallel, it
siB . g@d indicsrion
A$ SCORTC
nsh
OCTPAm cocu€d twie th.
pFdu@
rdly
mMucied on Sci.ntinc Co&pu-
pdid6
gdera,ror, parallet
rhe turcii@atiiy of ih€
n6h
d
ab@
toot is 8en€rsiin8 ih€ meh using ociEe
nrr pssllel FEM code io inco,por&te
t6
iih€ d (mpeed ro PSIRB.
ed
&d
ctoring in
ocre
method!.
Morffi,
rhe
odrc
bale@ load @r a! pFj, shict b D/p' The nu€ricd sinubtioB
cohducled on eiShr
pru@o! tBM
MitcheU ( 1998)
l?81
dMlopot
rch
*r€
Sp2 madine
nsh rcfinehor nerhod
refinqenrre aeihod for the pe,Uel
rhe iisme@rk oi a<l6ptiw
thdt b 6 t-wdy resbn ot lhe euBiv€ bis@rion
&rutio. of PDB sing @ire m€rhod, DyDMic lod bsl@ins
.iodic6lly rep&iilionins ot the
Srids du€
10
lhe refinemenr of
G comid€red due ro
tre
lennemeni. EMb node ot irce had been eociared by peEonat
s
p+
lor th€ adlptire mBh
kishi
snd 6 subr,ee
@isbl The pesonsl reighi i! cslculat€d sins ihe conput,iion sekned to peiicule
4
nod€. Subtre Eighr b cdculared uins the bdec
Deamrar of s tre, Th,s heihod
drsiicllly Edu€d
rhe
@hnuni@iion
oqhed
in a peallel
ihpldoraiion ed sud-
4t@d to produ@ the conheted &d bd&ced pelitioG. Nuneical qpeinrorationg
w*e @nducred for ihe !€6neeent usibg hi6nsnla! elemeb6,
Bi.la& .,
'n
d.
(2005) l82l p,6ent.d a psnll€l
re:lisiic beiN lor .he lese *al€
per slotr€d ihat the rep@nialion
onpute
drtqlrre
@tre bsed FEIu,,hich is appli€d
grcund noiion sihuldiion. Thjs p&
of,robld i! ocrre
@Nurn€d the
ls nqory per
.nhdced ibe cele pedomece. The Bulis Me cstculsied by 6D.
p&ing a 6rite dif€rene clde wiih m unstructu€d
retrahed,at FEM cod€ ro sinubte
hode and
th.
1994
Northids€ E&rhq!6le. The e6h
Euclid sofiwse. The Dah
s6
@
g@d,t€d by.pptins ih€
disrributed o@ <tifl€rclt
er@ b.![od in
p.o@i!s in Mpl dvionn€nl
usiis Pdneiis librsry developed in rhe Kdypis t6b.
Mirch€u (2ff17) l3ll stso enhued tre a{hprtu m*n lefiD@enr Nins peslrel
te n€{hod of r[e rE Efndot n€thod. This orqd ih€ dmdt rednenent
@l'rs 3
LnersruP
ing the rriogl€
Rdr6
biRtio!,
32
kiaagl€ quodrieti@, qu.dritsierd bberion,
qtg&ihiq.t
quadriarion, r€inh€drc! bi*ctioq tetnh€d@ octa€tion,
hq5hedron
wight to
nd
$
octetiotr
a.ll
nethodoiosig,
ihe .od6 based on
e
hdd€dM bi@iion ud
The enndeni-r@ Delhod 6bo Ncisie.l the
eigaed turk ot rbe @rBponding s@iaied ele
. A3 st e..h siep Io! lhe refinen6rt ot h€h,
the
arE
oerhod
60r rhe
adapti* Deh
conshi @hpli.rioro mE equn€d,
Fjlmmr hs rhe fou@ilg
iio€ @hpldiry:
r(n)= s"(;)+ o(r)
Whic!
r*died in O(z) iihe @hpldiiy
fir@dt.
otb€r
odrs
Derhod tor
e
addDtive
nsh !s
This paper opolred rhe @hpdaiiw b€ er parriii@ins fo. adapliE m6h
rcfn@€nt uins the
oE
oI tbe
(3.7)
refirhoLrtu
(retar€d ro
spprlElB lite P&n€ris, ESF!,
iet r6ul& rere !.hiercd urng rhe
octe
octF .nd sp@ tltins @m) appt@h
ed ruA. H€ .how.d tt r |he be,
6 @hpded io t!€ mulritewl t-way
RCB
meihod
pafiitDnn,g ugnB Pemeiis, He also Epo ed thsi p&m€iis
drc mded feM. elemsnts
s @np&Ed io all orher appl@ha. Nuee cat oi&ltsiioM er€ condkred on AMD
Athlo! bs€d @mpurer ctulter cils MpI libnry fo. bort! ihe 2D ed JD
@hB.