Finite Volume/Element Discretization on Unstructured Meshes of the Multiscale Formulation of the Large Eddy Simulation Method and Application to Vortex Shedding

(1)

HAL Id: inria-00071864

https://hal.inria.fr/inria-00071864

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Finite Volume/Element Discretization on Unstructured Meshes of the Multiscale Formulation of the Large Eddy

Simulation Method and Application to Vortex Shedding

B. Koobus, C. Farhat

To cite this version:

B. Koobus, C. Farhat. Finite Volume/Element Discretization on Unstructured Meshes of the Multi- scale Formulation of the Large Eddy Simulation Method and Application to Vortex Shedding. RR- 4722, INRIA. 2003. �inria-00071864�

(2)

ISRN INRIA/RR--4722--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Finite Volume/Element Discretization on

Unstructured Meshes of the Multiscale Formulation of the Large Eddy Simulation Method and

Application to Vortex Shedding

B. Koobus - C. Farhat

N° 4722

February, 2003

(3)

(4)

01/ 2-/3546 2*70 *8 !9$*5 :*$% ;54< 2-7=>$5 ?5@A.*.*B

C

DE$% ;01 2*F.)$*.HGJI*IK9$L '!MN5! O

C

2-/..D-?

P*Q,R*SSUT"V"WYXFZ\[Q,],^`_Dab^dcfe

gihkjlnm%o\p;qsrtlukvxwfyzr|{~}Kmy/{yzr|lnrxwyrt{}

m5syjDlnmD/{~lnkv|mmD

bz{myqslFw~h

wk`{y mLzmDhkmDzhkm}U osp;kmDkzuwzLp'&wmY

*!D ¡f¢k£ ¤J¥§¦ }r¨yzm©{~v|uklnmfª«mvtmlnmD}~y rtzzmyzr|¬Ywfyrt{}n{~}nymyzzwhkm zwvklnmDhkmD{®Uyhkm©fwzrtwyrt{}wv

l&ukv|yrxDwvtmb®¯{zlukvxwfyzr|{~}L{®kvxwzm,m k Lr|l&ukvtwyrt{}rt!kz{`{~m ®¯{~yzukukv|mD}°y {~lnkmYrtkv|m,±{f²³

gihrt rxmyrt¬Dwfyzr|{~}F®¯mDwyukzmD w}-mD{~}k{lnrxwvdkz{m ukzmwm {}*w~v|{~l\mDzwyrt{}\®¯{imwzwyrt}k

wkzr|{~r~yhmizwv|mYfw{~zmDd{~} rt}k{mD8yz{®¯{"mvtr|lnrt}wfyzr|}yzhkm/lFwvtvDwvtmD®¯{~lyhkm/yzml

{®/mD´°uwfyzr|{~}D"w} w}µm¶`mY8yzr|©~m-{~}~yz{~v,{®/yzhkm-}suklnmDrxwv rxr|dwfyrt{}µrt} udm °·uks²/rt} rt}k³

gihmmYuv¨yzr|}k¹¸!ºq»lnmyzhk{ rt-©«wv|r wym ²/r|yhyzhkm¼yhmDm½ r|lnmD}rt{}dwv}suklnmzrxwvrtlukvxwfyzr|{~}

{®w·vt{f²i½¾dmDm ±{f²¿dwyFw>z´°uwzm¼sv|rt} mDÀwfy-Á6ÂÃÅÄkÆtÇ¼w} »È%ÉËÊ ~~kw} ©fwrt{u

{lnwrx{~}9²/r¨yzhmdmDrtlnm}°yzwv wfywk³

Ì¼Í°ÎUÏzÐÑ ¡fÒ ¤ ¸ºqU!{lnkzmDzr|kvtm&±{f²+©fwrxwfyzr|{~}wv lukv|yrxzwvtmuk}yzu8yzukm lnmYhkmY!w½

~v|{~lnmwfyrt{}°©~{ym-hkm k rt}k³

ÓiÔÕ«Ö×ØÙÛÚÛÖÜÞÝß«Øà%á8ÕÜÞâØãäãäÖäØÙ!åæå¾çè ØâYÜ àLézÜÞê«ÝëézÜÛÖäìDíYØÚç~îïîKð8ñYò8çóôð8õ8ñà5ö"÷øù`údû°ûså¯ú`üýîUØß«ØÿþLñYçfÙÛéÕØ8ç

ØÜå¯÷!üåDá8â«ê«Öé ÕÜÞÖäâá8ãäÖäÚ

è ØâféÙ¯ÜÛë/ØÕÜ,áØÙæá8ÚÞâféØ/úÕ8ÖäÕ«ØØÙÛÖäÕÖäØÕØÚç°Ô!Õ«Ö×ØÙæÚÞÖÜLáUîUá8ãäá8ÙÞéßYáézÜUá8í«ãäß«ØÙçsî+éë/â«í«ÚUáþ%ô8õYç

Uá8í«ãäß«ØÙçîö8ð8ó8ð8õ ¯ðô8õYçÔ

(5)

.Å$:4f :*$% &;:*8 /b2*/ .Å$M=> C $5IID $%

*/9 53$5/@. 55IIý3.*Å!&-D-

¤ }km rtzyzrtzwfyzr|{~}*ºbvlnmD}~y/ r|}krx8ªb{vtuklnmDi rt}krt/uk/lFwrtvtvtwmDyywj rx´°ukmD m

vxwL®¯{~l&ukvxwfyrt{}-lukv|yrDhkmDv|vtmD mvxw&qsrtlukvxwfyzr|{~} mYw} mD º,hkmDv|vtmDæ¸!ºq9mDyik{~d{° Dm`{uk

vtm5wvtukv "!D{ukvtmlnmD}~y{~lnkmYrtkv|mY9yukzkukvtm}°yzD³ #9myyzm rxyrxwyrt{}uyrtv|rxm5uk}kmLkz{ ukzm

Y{~}k{lnrx´~um5´~ur!zm`{~mukuk}ml$yzhk{ m "!w~vt{l$wfyzr|{~}-`{uk%wmD&%Fkzr|{~rv|mY&Yhkmvtv|mY

u}·{mD8yzmukL`{u'vtr|lnrt}kmv|mYLdmyr|ymYDhkmDv|vtmD uµyjlnm "!D´°uwyrt{}Dmy5uk}µ{~}~yz(~v|m m

vxw rxzrtwfyzr|{~}¼}sukl$zrt´°ukm ukm&wu Dm}°yw~m³/¸wÀl)yhk{ m\¸ºq¼Yukv|yzw}°ym5mYy©«wv|r Dm5wvtw

rtlukvxwfyzr|{~}ý}sukl$zrt´°ukm&yzr|½ rtlnm}rt{}k}kmDv|vtm "!uk}*Y{~ukv|mDlnm}°y5ukd{~}krt´°ukmnwuky{u +!uk}>svtr|} m

Dwz,%5Á6Â Ã ÄkÆtÇ/my È%ÉÊ ~~kfmy mD9{~l\dwwrx{~}"w«©mD mD {~}k}DmD,m-Drtlnm}°yzwv|mY,{~}°y

YmD}~ymY³

. Ñ Y Ï £0/ §¤ ¸ºq+1Y{uv|mDl\mD}°yzÀ{lnkzmDzrtkvtmDD,®¯{zluvtwyrt{}©fwzrtwyrt{}k}kmDv|vtmKlukv|yrDhmvtv|mY

lFwr|vtvtwmY9}{}KyzuyukDDkw~vt{l$wfyzr|{~}ï ©~mvt{k`mlnm}°y m%yz{ukzkrtv|vt{}D³

(6)

! "

! F.K- &

{%yzukukv|mD}°y5±{f²Dïvtw~mnm k ·rtluvtwyrt{} Û¸ºq LwzmFw}>r|}°yzmzmDyrt}k¼{lnkz{lnrxmn`my² mmD}

w~ukzwym5uy}{ywv|²iw«{~lnkuyzwyrt{}wv|vt®¯mDw~rtkvtm rtmY8y%}°ul\mDrxwv!r|l&ukvxwfyrt{}d #%$q 8ïw}

ln{~m mD{}k{~l\rxwvskuy,}k{ywvt² w« u&Àr|mD}~yzv|\wDukwfyzm /mDs}k{v bw«©mDzwm $w«©sr|mD½q°yz{('mYln{ ½

mDvtD³bgihkmlFwr|}*r mYw5`mhkrt} ¸ºqÀrty{\zmD{vt©miyhmvxwzm½¾zwv|mi®¯mYwfyumY{®!wLyzukukv|mD}°y9±{f²Lsw}

ln{ mDvyhkmukk~r ½ÿDwvtm yzmDzmD ²/hkrxh\zmmYmD}~y yhm/m¶UmD8y,{®dyzhkm/uk}kzmD{vt©m zwv|mY {}\yhkm/zm½

{vt©m {}kmY³"g{5yhkrxm¶`mY8yY~lFw}s r|¶`mDmD}~yi¸ºqFl\{ mDvthw«©m/`mmD}Àkz{`{~m rt}Fyhkmv|r|ymDzwyukzm³

gihm/{zr|~r|}wvqslFw~{zr|})'sLm k ©srtz{°r|y5ln{ mv*+,hwbyw r|yrt{}wv|vt5kvxw«m w {lnr|}dw}°ybwyD

mDr¨yzhkmwv|{~}km{~w wy/{®bw}{yhm/hmlnm³.-{f²9mD©mY~®¯{{lnm±{f²syzhkrx/ln{ mvïw}Kmhkrtkr¨y

{lnmhk{yz{lnrt}k~buhFw9w%²/{~}kLw°lnyz{yzrt/`mhdw«©°rt{b}mDw² wv|vx°w}nmkmDzrt©m rxzrtwfyzr|{~}

rt}yzhkm\kmYmD}m&{®,vxwzm{hkmDmD}°yyzu8yzukmY+w} yhm&rt}wr|vtr¨yKyz{*wv|vt{f²®¯{w'zwyymDD³g{

zmlnm

¼yhmDm\rxzukmY!mD©mwv ln{

r¦

wfyzr|{~}%{® yhmFqlnw{~rt})'°¼ln{

mDvbhw«©m\dmDm}>kz{`{~m ³

¥

ln{}kFyzhkmDmnyw}

{uyyhkm

s}wl\rxukk~r

½ÿDwvtmLln{

mv/*s+0,"rt}²/hkrxh¼yhkmFqslFw~{zr|}k½

)'s·{}yzw}°y½ÿ{sm1&Àrtm}°y5rx{}r

mzm

w&wK®¯uk}yrt{}¹{®w~m-w}

yzr|lnm³32ÿ}¹ln{~y&w~mYLkuy

w-®¯m² ±{f² {~} ¦ ~ukzwyrt{}Ddyhkrx s}wlnrxln{ mDv,wk`mDwzy{r|lnkz{f©myhkm ¦d mDv|r|y{®9¸ºq æmm

®¯{~5mkwl\v|m4*ok"5,®¯{~&wKzmDm}°y&wkv|rxwyrt{}-³6-{f²9mD©mDD®¯{5uk}dyzuyukzm lnmYhkmY"r|y&w}µdm

{lnkuywfyzr|{~}wvtv|FmdmD}rt©m~³

¥ }kmD²<wkkz{~why{L¸º9qdwm {~}\w©fwzrxwfyrt{}dwvslukv|yrxzwvtm7q!®¯wlnm² {8'%² w~"mYm}°yzv|

rt}°yz{ um s9-uhkmY w} hrt"{½Þ² {8'mDz.*ä5:;,¾³ gihkm%7ýq°½¾¸!ºq r¨¶Um!®¯{~lyhkm yzzw r¨yzr|{~}wv

¸!ºq»rt}w>}sukl&dmD-{®²iw«³<2ÿ}»yzhkrt-}km² wkkz{~wh{}m {smDF}k{y ¦ v|ym-yhkm4$w«©sr|mD½q°y{='mY

mY´°uwfyzr|{~}uy udmY9rt}ymYw w&©fwzrtwyrt{}wvkz{mY8yrt{}³ gihkrx rtiw}Àr|ln`{yzw}~y r|¶`mDmD}mdmYwum

w~ dmD®¯{~lnm r|}*yhmyzzw r|yrt{}wv`¸!ºqU ¦ v|ymDrt}k\²9{~)'9²9mDv|vU²/r|yhdmDrt{ rx`{u} wzÀ{~} r|yrt{}

uy!wrxmYUlFwfyhmlFwfyzrtDwvrxzukmYïrt}L² wv|v|½¾d{~uk} m ±d{f²³ gihkm,©fwzrtwyrt{}wvfkz{mDyrt{}5w«©~{r `yhkmYm

rxzukmY³kukyhmzl\{~m~kyzhkm 7q°½ÿ¸ºqlnmyh{ mDwwfyzmD/yzhkm&zwv|mY>+p'yhdwfy%rtDd`m®¯{~m

yzhkm/rtluvtwyrt{}rxbyzwym ³ ¥ } ln{~y rtlnd{~yw}°yvtr|ybln{ mvx yhkmim¶UmDy,{®kyzhkm/uk}kzmD{vt©m zwvtmD

{~}kvtFr|}*yhkm%mY´~udwfyrt{}d9zmmYmD}~yzr|}5yhm5lFwvtvtmDy zmD{vt©m

DwvtmDDw}

}k{yirt}-yzhkmLmD´°uwyrt{}®¯{

yzhkmLvxwzmLDwvtmDD³1#9{}mD´°ukmD}°yvtrt}*yzhkm'7q°½ÿ¸ºqUm}kmD~Àrtmsywym

®¯z{l yzhkm

¦

}kmmY{~v|©~m

zwv|mY °5wyzzw r¨yzr|{~}wvsln{ mvkuhnw!yhmqslFw{~rt})'sm k 5©srx{~r¨yLln{ mDvÛuyb}k{%m}kmD~Lrt

r|zmDyvt\msyw8yzm ®¯{~lyzhkm%vtw~myzu8yzukmYrt}*yhkm±{f²L³bk{~yzhkrt zmDw~{~}ï~{~}km%w}-mYw{}wkv|

h{`mFy{Ë{ywrt}¹w¼dmyyzm\dmDhw«©sr|{~5}kmDw5²iwvtvtDbw} vtmDz rxzrtwfyzr|{~}µr|}¹yhkmÀmYmD}mÀ{®/vtw~m

{hkmDmD}°yiyzu8yzukzmDD³

gihkmr|}kr|yrxwv m©~mvt{klnmD}~y{®"yhkm'7q°½ÿ¸ºq¼l\myhk{ ®¯{udm {}KmDukvxw/~r w} dmY8ywv

rtzzmyzr|¬Ywfyrt{}dw~rt}yzhkrxLwm5yzhkmFmDwwfyzr|{~}ýw*krt{zr {®,yzhkmFDwvtmDrtLr|lnkvtm&y{Kwhkrtm©~m`®¯{

mkwlnkvtms<r|}°yz{ ur|}k>wË®¯mY´°ukm}6uky{¶?*ä.:k.;,¾³Ak{~ ¦ }kr|ym¼mDv|mDl\mD}°yFwkkz{«r|lFwfyzr|{~}D

wKhr|mDzwzhkrxwvbwrx5wk{°wh@*+Y5, w} w}¹wv|ymD}dwfyrt©mnlnmyzhk{ wm {~}µmvtv9w~v|{~l\mDzwyrt{}

*A+(+1,%² mzmýzmDm}°yvtkz{`{~m ®¯{~mwzwyrt}k6w<kzr|{~ryhkmµ{°wmËw} ¦ }km·zwv|mY³B2ÿ}l\{°y

DwmDD`yhkm$%7ýq°½¾¸!ºqýl\myhk{ ² w~wkkvtr|m lFwrt}kvtKy{hk{ln{~m}m{udrx{yz{krx5rt}{lnkzmDzrtkvtm

yzukzkukvtm}m«w} zmDmD}°yvty{/rt}{lnkzmDzrtkvtm"yukzkukvtm}°yhw}k}kmv±{f²D®¯{²/hkrth%r|y mDl\{~}ywfym

w}*rtlnkz{f©mlnmD}~yi{f©mD yhkm%yzzw r¨yzr|{~}wvï¸!ºq*lnmyh{ *:C;D,Þ³

gihkm{mY8yzr|©~mD,{®+yhkrx9w`miwzm/y² {L®¯{v ³E2¾y9kz{`{~mDw5mD}kmwvdwkkz{~w~h\®¯{ rtzzmyzr|¬Dr|}

yzhkm%7qs½¾¸!ºqÀlnmyhk{ {}-u}yzud8yum ~r 9r|}{ m yz{&w«©~myzhkm%² w«n®¯{~iwkkvtrtDwfyzr|{~}²/r|yh

{lnkvtm~m{lnmyzr|mY³ngihkrxLwkkz{~w~hïU²/hkrxhË²iw

¦

y{~uyvtr|}m

r|}@*A+(+,¾rx%mD´°uwv|vtýwkv|rxwkvtm

(7)

yz{-yzhkm

¦

}kr|ymn©{~v|ul\m æw}

¼¦

}kr¨yzmnmvtmlnm}°y æ"º lnmyzhk{

Dïw}

y{r|}d{lnkzmDzr|kvtm&w~²9mDv|v

w~/{~l\mYr|v|m±{f²D³ 2¾y 'mDÀmvtmlnmD}~ywzm%w}KmD{}k{~l\rxwvUk{mDy{ ®¯{~/mvtr|lnrt}wfyzr|}knyzhkm

¦

}km

zwv|mY/®¯z{l yhm&symDlJ{®mY´~udwfyrt{}d²/hkrxhrxw~m

{}Ëw~v|{~lnmwfyrt{}w}m¶`mY8yzr|©~m&ywfymD

®¯{~{~}°yz{vtr|}kyzhkmÀ}suklnmzrtDwv rtzrtwyrt{}>r|} um sýuks²/rt} rt}k¼yhdwfyL®¯mDwyukzmDwr|syh½¾{ mD

©srx{~r¨y *A+DE+ :5,Þïw} mkv|rxr|yLw²9mDv|vbw~mlnr|½Þrtlnkv|rxr|y®¯zwl\mD²9{~)'®¯{yrtlnm½¾r|}°ymDwfyzr|}k-yhkm

~{f©mD}krt}kÀmlnr|½ rxzzmymmD´°uwfyzr|{~}D³gihkm\wdmDwvt{Àzm`{yz{~}yzhkm\wkkvtrtDwfyzr|{~}¼{®,yzhkm 7q°½

¸!ºq<lnmyh{ yz{>yhkm¼kzm rtyrt{}»{®LwË©~{ym6hkm k rt}k>±{f²¿®¯{F²/hkrxhmdmDrtlnm}°yzwv wfyw·rt

w«©fwr|vxwkvtm³"g!{&yzhkrx/m¶UmDyDyzhkmLmDlFwrt} m/{®yzhkrtw`m/rx/{z~w}krt¬m w ®¯{vtvt{f²³

2ÿ}&qmD8yzr|{~}&Dyhmluv¨yzr¨½ÿDwvtm,©fwzrtwyrt{}wvf®¯{zluvtwyrt{}Lrxwkv|rtm y{/yzhkm9{lnkzmDzrtkvtm.$w«©sr|mD½

qsy{('~mDLmY´°uwfyzr|{~}5w} yzhkm-mlnr|½ rxzzmyrt¬Dwyrt{}>lnmyzhk{ w {~ym rt}µyzhkrt²9{~)'ýrt{uyzv|rt}km ³ ¥

{m ukzm*dwm {~}»w~v|{~l\mDzwyrt{}>®¯{~ÀmDwwfyzr|}k·w·krt{zr yhkm¼{~wm*w} ¹¦ }mKzwv|mY\{®w

yzukzkukvtm}°y"±{f²¹{}&uk}yzu8yzukzm ymyzzwhkm zwv~lnmDhkmY"rt kzmDm}°ym rt}nqmD8yzr|{~}\³ 7K{ mDv|rt}k%{®dyhkm

ukk~r zwv|mY,rx rxzuzm rt}qsmY8yrt{}*o³ 2ÿ}qsmY8yzr|{~}*°yhmmlnr|½ rxzzmyrt¬Dwyrt{}Flnmyzhk{ hk{°mD}

®¯{~\yhkrxn²9{~)'>rx rxum r|}<®¯ukyhkmD mywrtvtD³ 2ÿ}»wyrxuvtwDw}6uks²/rt} ywkrtv|rt¬Dwyrt{}6ymzl

rt}s©{~v|©srt}kFwFr|syh½¾{ mwyrxwvï©srx{~r¨y*rt/rt}°yz{ um r|}¼{~ miy{Flnrt}krtl\rt¬myhkm&{ln`myzr¨yzr|{~}

`my² mmD}¹}suklnmzrtDwv w} ln{ mDv|m ¹ rxr|dwfyrt{}d³ýgirtlnm½ rxmyrt¬Dwfyzr|{~}µrt\w mYm rt}<qsmY8yzr|{~}

0³ýgihkm*mYukv|yrt}k*%7q°½ÿ¸º9q>lnmyh{ rt\wkv|rtm r|}<qmD8yzr|{~}»*yz{yhmr|l&ukvxwfyrt{}µ{®w¼©{~yzm

hkm k rt}k {~lnr|}wym ±{f²§w~y/wLyzhkzmm½ rtlnm}r|{~}wvUz´~udwzmsvtr|} mY³ gihm{~yzwr|}km }suklnmzrxwv

zmDukv|yzwm5{}°ywym ²/r|yhKyzhk{~m5kzm rtym s©«wrt{udir|lnkvtmlnm}°ywfyrt{}d{®"yzhkmvxwzrtDwvïw}

°}dwlnrtDwv`¸ºqnlnmyhk{ w} {~lnwzm ²/r|yhÀmdmDrtlnm}°yzwv wyzw³ rt}wvtv|~°{}v|u r|}zmlFw8'

wm{¶Umzm rt}¼qmD8yzr|{~} :k³

N$ $% &-$ËE $ 4f :*$% & 54µ 2-Å9&I* / K

Å$&À/ C !L/¿n*$% *

gihm%{}mz©fwfyzr|©~m®¯{~l {®!yzhkm5{~lnkmYrtkv|m$w«©srtm½ÿqsy{('~mD9mD´°uwyrt{}irxirt©mD}*s

Æ "! ÃAÄ

#

$

Æ%#'&"! Ã)(

+*,- Æ.

/

0 Æ213

/

-*

!45

Ã

Æ%6.

7!

Æ%98

:

!

²/hkmDm ;<7;

* ; . ;=/>; 8 ; w} :

wzm/zmDdmY8yrt©mDv|5yhkm m}r¨y~©~mvt{sr¨y~kzmDzum~©srtz{~ubyzmY

yzm}{Yyz{yzwvmD}kmz~yhkmDlFwv!{~} uyrt©sr¨y~dw} yzmln`mwfyum5{®"yhkmL±d{f²L³ gihkm©°rxz{udyzmY

yzm}{irx rt©m}*s?.A@CBÃEDF9GIH@CBJ(KL HNMOMQPR@SB ! ²/hkmDmTDýrx yhm©srx{~r¨yFw} HU@CBÃWV

K

YX[ZQ\

XR]Y^

X[ZQ^

XR] \ ! ³

gihmy{yzwvdmD}kmz / rx~r|©~m}Fs / Ã #_J`

:a

VK Æ ²/hkmDm _b` rxyhkm%`mDr¦ hkmYwfy/wfy {}yzw}°y

©~{vtuklnmUw} yhkm m}r|yUkzmDzukzm5w} ymDl\`mwfyzukzm&wzmw~uklnm yz{-`m&zmvxwfyzm syhkm\ywfym

vxw«²A®¯{~w\dmD®¯mY8y~w~

* Ã È :

³"k{~rtlnkvtrtr¨y~cD·w} 8¼wzmLwzuklnm yz{\`m5{~}yzw}°yY³

ukyzhkmzln{zmr|yrt/w~uklnm hkmDmyhdwfyRd

(8)

gihkm5`{uk}

m

±{f²

{lFwr|} rtiyz{Fdm

rtzzmyzr|¬Dm

s-w\ym

whkm

zwv+lnmDh*®¯z{l ²/hkrxhKw

uwvïlnmDh m ¦

}km

smvtvxi{{~}°yz{vU©{~v|uklnmY Dw}K`m

mzr|©~m ³

gihkm{~}s©mDyrt©m±umYFwzm*yz{µdmyzmDwfyzm s<wµ lnmyh{ rt}²/hkrxhw·©fwzrxwkvtm rt

¦ yKwkkz{«r|lFwym sw6{~}yzw}°y-rt}AmYwhA{~}°yz{v©~{vtuklnm³ gihkrx-Dw}A`mý²/zr|yymD} w

Ã

@

@b@ ²/hkmDm J@!rx9yhkmLhwzw~8yzmzrtyrxi®¯uk}yrt{}K{~zmDd{~} rt}kLyz{yzhkm{}°yz{v`©{~v¨½

uklnm _ @"w~{rxwfym ²/r|yh}k{ mw} @ m}k{ymYyzhkmL{~}yw}°y9©fwv|ukm{®rt}-yzhkrx {}°yz{v

©{~v|uklnm~³$msyD«yhrt ¦ y½Þ{~ m!wkkz{«r|lFwfyzr|{~}%rt!yw}d®¯{~lnm rt}°y{wr¨syhk½Þ{~ mD©srx{~r¨y

mDlnr¨½ rxmyrt¬Dwfyzr|{~}3*A+D + :5,ïs-w 7 q-# ¸ *A+o +Y ,wkkz{~whÀyhwyrt/dmYr¦ m rt}KqsmY8yzr|{~}

³

gihkm r|¶`udrt©m±umYFwzm*yz{>dmKyzmYwfym s6w>"º lnmyh{ rt}»²/hrthwË±{f²¿©fwzrxwkvtm

rtFwkkz{«r|lFwfyzm s¹w·{}°yzr|}suk{~ur|mYm²/rxm*vtr|}mDw\®¯uk}d8yrt{}³6gihkrxFw}6dmK²/zr¨yym}w

Ã

@

@J@²/hkmDmJ@`rt!yzhkm/ + hwdm9®¯uk}yrt{}nwz{rxwfyzm ²/r|yh&}k{ mw} @ mD}k{yzmD

yhkmL©fwv|ukmL{® wy yzhkrx/}k{ m³

-{f²9mD©mYsr¨y/rx/}k{ym yhdwfywvtv+lnmyh{ {vt{rtmD kzmDm}°yzm r|}*yhkrx/wdmD/mD´°uwv|vtÀwkv|Fyz{nyhkm

Dwm²/hkmDm&d{yhýyhkmn{~}°©~mDyrt©mw} r|¶`udrt©m&±umDwzm&wkkz{«r|lFwfyzm s¼w Ûywkrtv|rt¬m º

lnmyhk{ {/wFlnmyh{ ³

gihkm¹mlnr|½ rtzzmyzr|¬Ywfyrt{}N{®\yhm¹{lnkzmDzr|kvtm $w«©srtm½ÿq°yz{('~mD*mD´°uwfyzr|{~} Û Ks wlnr¨m

Lªf"º»l\myhk{ ²/r|yhKlFwziv|uklnkrt}kFvtmDw y{FwnymDl¿{® mD´°uwyrt{}i{®yhm%®¯{zl

@

; ! Ã Ä

? @ ; @ ; ! Ã

_ @ ; @ ; ! Ã Ä +

²/hmzm

Ã

;<7;

:

!

J@; ! Ã

J@

X"!$#&%&' \ Æ()J@ $*

@; @ ; ! Ã #

@ +

X,!$#-%&'A\

'& () @ *

X"!$#-%&'A\

*

() @ *

. @

_ @ ; @; ! Ã

/

@ +

X,!$#-%&'A\

/ * ! Æ.() @ *

. Æ @

8

: Æ @

0/)12 J@ mD}k{yzmD%yzhkmÀ`{uk} wzý{®9yzhkm-ukk`{y5{®3J@"w} (Art%yzhkmÀ{~uy² w }k{~lFwv"yz{¼yhrt

ukk`{yD³

(9)

¸my6w}

m}{ym%zmDdmY8yzr|©~mvtnyhkm5wmLw}k}km

sFyzhkm%yhkmLhwzw~8yzmzrtyrx®¯u}8½

yzr|{~}, M ~sw}

yhwyiw}k}km

°nyzhkm +hw`m®¯u}8yzr|{~}" M~³E2ÿ}*{

my{&mdwwfymw5kzr|{~r

yzhkm5{°wmw}

*¦

}m5DwvtmDD~yzhkmDm5wmDiwzm

mD{lnd{°m

w~ ®¯{vtv|{f²

µÃ

; Û0

w}

Ã

Æ æ

-mzm*w} yzhk{~ukh{uy5yzhkmmDlnwr|} m\{®yzhkrx&dw`mY"yhkm{f©mDvtr|}m mDrt}wymD\wý{~wzm-zwvtm

w} yzhkm

uk`mzzr|y mYrt}wymY&w ¦ }kmKDwvtm³ -m}m{}rtymD}°yvtµ²/r¨yzh»º,´sD³ Ûk 8 rt

mD{~lnd{°m rt}°y{Fwn{~wzm%Dwvtm w} w ¦ }km5zwv|m

Ã ÛÆ ¯o

{~l yhkm

mY{ln`{~r¨yzr|{~}Û w}

Û ,®¯{vtv|{f²9yzhkm

mD{~lnd{°r|yrt{}*{®kz{v|mDl +9rt}°y{&yhkmy² {

®¯{~v|vt{f²/rt}k\ukk{~kv|mDlF

?@; ! Ã Ä

??@; J@ ; ! Ã

_ @ ; @ ; ! Ã Ä

Û0

w}

@ ; ! Ã§Ä

@ ; @ ; ! Ã

_

@ ; @ ; !

Ã§Ä

0

²/hmzm

J@ ; ! Ã

J@; ! V

J@; ;

!<;

V

b@ ; ;

! Ã

J@ +

X"! #-%&' \ Æ.()b@$ *

X"!$#-% ' \ Æ()J@ $* ; Þ

?J@ ; J@; ! Ã J@; J@ ; ! V

J@ ; J@; ;

! K

J@; ;

!=;

V

J@; J@ ; ;

! Ã

#

J@

X"!$#-% ' \

J & () @ $*

X"!$#-% ' \ & () @ *

X"!$#-% ' \ & () @ $*

X"!$#-%&' \ * ( È : ! ( @ *

. @ ;

K

@ ; ;

! Ã

X"!$#-%&'A\

J & -() @ * ;

:

(10)

w}

_ J@ ; J@; ! Ã _ J@; J@; ! _ V

J@; J@; ;

! _ K

J@ ; J@; ;

!<;

_ V

@; @ ; ;

! Ã / (

#_

` : ( VK Æ

"!

@

X,!$#-%&'A\

/ (

#_

` : ( Ç

G Æ -*

( È : ! Æ.() @ *

X,!$#-%&'A\

_

`

:-

Ç

G Æ È :

!4

ÛÆ.() @ *

.

ÛÆ

@ +

. Æ @ +

8 :

æÆ

@ +

;

_ K

J@

;

J@

; ;

! Ã

X,!$#-%&'A\

/ (

#_J`

: ( Ç

G Æ -*

( È : ! Æ.(J@$ *

. Æ

J@ Æ

;

-m}mº,´sD³Ûk0 9Dw}dmLzm½¾²/zr¨yym}Kw

?@; ! V @ ; ; ! Ã Ä

@ ; @ ; ! V @ ; @; ; !

X"!$#-% ' \

J & ( @ $* Ã

_ @ ; J@; ! _ V @ ; J@ ; ; !

X,!$#-% 'F\

/

#_

` : ( ÇG Æ * ( È :

!4

Æ( @ *

. Æ @ Ã Ä +D

w}

@ ; ! V

@ ; ; ! ÃAÄ

?

@ ; @ ; ! V

@ ; @; ; !

X"!$#-%&'A\

J & ,()b@ * (

X"!$#&% '7\

&- -( >@$ * Ã

_

@ ; J@; ! _ V

@ ; @; ; !

X"!$#-%&'

\ / ( #_J`

: ( ÇG Æ -*

( È : ! Æ.(J@ $*

(

X"! #-% ' \ / ( #_J`

: ( Ç

G Æ -*

( È : ! Æ.( @ *

. Æ @ ÃAÄ

+(+

my{®mD´°uwyrt{}9wzm{ukkvtm ³E$msyDwl\mYhw}rtl rt9r|}°yz{ um ®¯{imDwwfyzr|}k5w&kzr|{~ryhkmYm

zwv|mY³

(11)

C /I$5 $% $I 5 >46"2-@ $/

¥ ir|} * ,¾yhmLlnwr|}r mDw\hkmDm%rxiy{ m ¦ }km w/w\kz{mDyrt{}*{® {~}~yz{ <{&yzhwfy

Ã * ! +Y0

w} dÃ)(

* ! ! Æ +D

*

{mY 's}k{f²/}¼w~w~vt{lnmwfyzr|{~}4*+ 05,¾³ gihkrx/kz{mY/rx/~zwkhkrxwvtvt mrtym rt} r|d³ +%®¯{~/yhkm

y² {½ rtlnm}r|{~}wv+Dwm³

Agglomeration

rtukzm + d }yzuyukzm l\mYh uwvUlnmDhïkw} w~v|{~l\mDzwyrt{}n{®"{~lnmmvtvt {®!yhkm udwvUlnmDh

rt}°y{Fw\lFwz{½ÿmDv|v ¯y²9{½ rtlnm}rt{}wvUDwm

yzhkm%®¯{vtvt{f²/r|}knm}m

Ã * M M M ! Ã M M M +o

²/hmzm

M Ã

B"!$#&%')(+*

_ B ! B

B"!$#%

')(+*

_ B ! ; +Y0

_

')(+*

_

B)0 _2143

M"5 ~kw}

_2143

M"5

mD}k{yzmDiyzhkmLlFwz{½ÿmDv|vU{~}°yzwrt}krt}k\yhkm5mvtv _ M ³

(12)

{~l¿º,´`³ +Do w}

º,´`³+Y09r|y/®¯{vtv|{f² yzhwfy

w}Kwvt{\dmL²/zr¨yymD}Kw~

Ã M M M + 0

²/hmzm

M Ã

')(+*

_ M !

B"!$#% ' (+*

_ B !

B"!$#&%

B Æ +«

}m}°y

rt"wkkz{«srtlFwfyzm hkmDm s5w{}°yrt}suk{ud®¯u}8yzr|{~}&²/hkrxhrx"{~}yw}°y"r|}\mDw~hLymywhkm {~}

zwlnmlnw~z{½ÿmDv|vÞ~w} rx9vtrt}kmDw9rt}Fyhmyzmywhm whwzm sFwfyivtmDwyy² {lFw~z{½ÿmvtvx³2ÿ}-{yzhkm

r|yz{}dyw}°yiyzwmYæmmL r|d³5®¯{~w\{}km½ rtlnm}rt{}wv`zmmYmD}~ywfyzr|{~}-8³

W

macro−cell i macro−cell j macro−cell k macro−cell l

: Nodes

!rtum5#d}km½ r|lnm}drt{}wv+mDkzmDm}°yzwyrt{}À{®"wn{°wm%zwvtm%{ln`{}kmD}°y

2¾ywvt{\®¯{vtv|{f²9yhdwfy

Ã ( Ã

M M MT( M ! Ã

M M M%Ã

M

M ( M ! MÆ + :

ukzm5{~uyvtr|}m

y{

m ¦

}km5yzhkm&{~wm5Dwvtm

{ln`{}kmD}~y

Ã * !

s JM³E-m}m

*

rx

m ¦

}km

s

* ! Ã

M ' (+*

_ M !

B"!$#% ' (+*

_ B !

B"!$#% B

M + ;