Multiscale Expansions for Linear Clamped Elliptic Shells

HAL Id: inria-00071623

https://hal.inria.fr/inria-00071623

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- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Multiscale Expansions for Linear Clamped Elliptic Shells

Erwan Faou

To cite this version:

Erwan Faou. Multiscale Expansions for Linear Clamped Elliptic Shells. [Research Report] RR-4956, INRIA. 2003. �inria-00071623�

ISRN INRIA/RR--4956--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

Multiscale Expansions for Linear Clamped Elliptic Shells

Erwan Faou

N° 4956

Octobre 2003

34/5687:9;6=<?>

@BADC/EF$G)HJILKMENDOQPRSKUTWVF/RXTZY[RSKUEKQ\]P^R]KMTZV

_ F\a`[\aR]CbEcFd\#eTWEcYfOUF/gLFd\

h;iST^jkF/R#lmOMP

_

KMV

nXP^YfY8TWiaR

_

FoiSFbepADFbiSepADF'V=qGWrts^u+HwvxeR]TZyfi]F$z^{W{Z|H}GW{cYfP^~WFb\

€ƒ‚/„b…‡†fˆ^„Z‰#Š F;KMVL‹ZFb\aR]KM~WP^R]FŒ\]TZOMN[R]KMTZVf\?TZWRSADFŒRkŽT^_ KMEcFbVf\]KUTWVfP^O^‘TZKUR]Fbi’EcT _ FbOZP^V _ TZLR]ADFƒRSADi]FbF_ KMEcFbVf\]KUTWVfP^O

OUKMVDFbPZiŒ\aADFbOUOLET _ F/O[KMV)R]ADF#e/PW\aF“Ž#ADF/iSF“R]ADF#\]ADF/OMOLKM\Œe/OMPZEcY=F _ P^V _ KURS\”EFbP^V\]NDia•PWeF“KM\ƒFbOUOMKUYDR]KQe^–ƒ—DTZiŒ\]ETtTZR]A

_ PRpPD˜“R]ADFd\aF™\]TZOMN[R]KMTZVf\2P _ EKšR2E)NDOšRSKM\Se/PZOUFF/gLY8P^Vf\]KUTWVf\KMV›Y=TŽ“FbiS\cTZ

ε^{1/2 Ž#ADFbi]F} ε ^_ F/VDTZR]Fd\cR]ADFœžA8P^OUŸ

R]AfKMep¡LVDFd\]\T^’RSADF$\aAfF/OMO¢–Œ£T^RSAFg[YfPZVf\aKMTZV8\“e/TZVtRSPZKUV2RSF/iSE2\¤KMV _ FbY8FbV _ F/VtR#T^

ε^{PZV _ y=TZNfV _} P^iS`OMP‡`WF/i¤R]F/iSE2\

Fg[Y8TWVDF/VtRSKMPZOUOM` _ Fbe/i]FdPZ\]KUVD~2Ž#KUR]A+i]Fd\aY=FbeRmRST

r/√

ε^˜?Ž#KšRSA r ^{R]AfF _} KQ\aRSP^V8eFRSTRSADF)y=TZNfV _ P^iS`T^ŒRSADF)EFdP^V

\aNDi]•PZe/FZ–@BAfF)F/gLY8P^Vf\]KUTWV¥T^“R]ADF)R]Afi]FbF_ KUEFbVf\aKMTZV8P^O _ KQ\aYDOQPZe/F/EF/VtR$eTWVWRpP^KMVf\x\aNDYfYDOUFbEF/VtRSPZi]`y=TZNDV _ P^iS`

OMP‡`WF/ip\/˜ŒF/gLY=TZVfF/VtR]KQP^OMOU` _ FbeiSFbPW\aKMVD~+Ž#KšRSA¦iSFb\]Y=FbeRcR]T

r/ε OMKM¡ZFžTZiYDOQPR]Fd\/–¦§m\]KUVf~+RSADFb\]FFg[YfP^V8\aKMTZVf\cŽF TZy[RpP^KMV\]AfP^iSYFb\aR]KMEP^R]Fd\By8F/RkŽ“FbF/VR]ADFoRkŽTcEcT _ FbOM\XKUV‹P^iSKUTWNf\VDTZiSE2\/–

¨™©Wª?«S¬­ …‡®”‚W‰°¯ NDOUR]KQ\]ebP^OMFoFg[YfP^V8\aKMTZV±˜D²KUVfFbP^iBFbOMPW\kRSKMe/KšRk`W˜fILADF/OMO’RSADF/TWi]`W˜D£“TWNDV _ PZi]`2OQP‡`ZFbiS\

#ƒ# #“ ’##

‚ ‰ TZN8\/R]N _ KUTWVf\oOUFd\o\aTWOUN[RSKUTWVf\ _ N EcT _ CbOUF _ F2‘TZKUR]Fbi‘F/R _ N+YDiSTZyfOUCbEcF _ FObOMPW\kRSKMe/KšR)RSi]K_ KU

EcFbVf\]KUTWVDVDF/OMOMF OMKUVdP^KMi]FW˜ _ P^V8\OMFebPZ\ _ NfVDFe/TWNfF F/V8e/PZ\aR]i/F _ TZVtR)OQP \aNDi]•PZe/FET`ZFbVDVDF Fd\kRcF/OMOUKMY[RSKtNDFW–

h”TWNDi _ Fd\ _ TWVDV/Fd\$i/~ZNfOUKMC/iSFb\b˜eFd\\]TZOMN[R]KMTZVf\$P _ EFR]R]F/VtR _ Fb\ _ b‹ZFbOUTWYDY8FbEF/VtRS\$E)NDOUR]KUdepADF/OMOUFd\oF/VYDNfKM\a

\]PZVfeFd\ _ F

ε^1/2˜DT ε^Fd\kR‘O /YfPZKM\S\]F/NDi _ F$OQP2eTtNDFW–"!“Fb\ _ F/N[g _ /‹WF/OMTZYDY=F/EF/VtRp\XeTWVWRSKUFbVDVDFbVWR _ Fb\BRSF/iSEcFd\

KUV _ /Y=F/V _ PZVtRS\ _ F

ε ^{F/R _} Fd\#R]Fbi]EFd\ _ Fe/TZNfepAfFb\#OMKUEKUR]Fd\XFg[Y=TZVDFbVtR]KMF/OMOUFbEcFbVtR _ be/i]TWKM\S\SP^VtRS\BFbV

r/√ ε^˜=T r

Fb\aR;OMP _ KQ\aRSP^V8eFmPZNcy=TZi __ F#OQP$\aNDi]•PZe/F#ET`ZF/VfVDFZ–#mFXYfOUNf\;OUF _ /YDOQPZe/F/EF/VtRŒR]iSK_ KUEFbVf\aKMTZVfVDF/Ofe/TZVtR]KMF/VtR _ Fd\

R]F/iSEFb\ _ F)e/TZNfepADFd\XOMKUEKUR]Fd\mF/gLY=TZVfF/VtR]KMF/OMOUFbEF/VtR _ deiSTZKQ\]\SP^VtRp\BF/V

r/ε^{˜’eTWEcEF _} P^Vf\mOUFd\mYDOQPtNDFd\/–%$xi'&Ze/F ( e/Fb\ _ b‹ZFbOUTWYDY8FbEF/VtRS\b˜[VDTZN8\#TZy[RSF/VDTWVf\ _ Fb\XFb\aR]KMEP^R]KMTZVf\BTWY[R]KME2P^OMFb\BFbVWRSi]FoOMFb\ _ FbN[gET _ CbOUFd\/–

) ­ „d‚ « ˆ* ‚ ‰ #/‹WF/OMTZYDY=F/EFbVWRp\2ENfOšRSKš+bepADFbOUOMFb\b˜",;OMPW\kRSKMe/KšR™OUKMVdP^KMi]FW˜@BAbTZiSKUF _ Fd\eTtNDFd\/˜-!“TZNfepAfFb\

OUKMEKšRSFb\

) *+,.-0/214365.,87/2+

@BADF~WTWPZOT^ƒRSADF9 :&";<& =$KQ\XRST?>fV _ P^V PZYDYDiST‡g[KUE2PRSKUTWVT^ŒR]AfFR]Afi]FbF_ KUEFbVf\aKMTZV8P^O±OMKUVfFbP^ixF/OQPZ\a

R]KQe \]ADF/OMO¤YDiSTZyfOUFbE yL`"PRkŽ“TZ

_

KMEF/Vf\]KMTZVfPZO;YDiSTZyDOMF/E Y8Tt\aF

_

TZV"RSADFEFbPZV \aNDi]•PZe/FZ–™l \]ADFbOUO“KM\)PR]ADiSF/F/

_

KUEFbVf\aKMTZV8P^OZTWy[jkFbeR

_

F>fVDF

_

yL`xP‘eTZEYfPWeRTWi]KMF/VtRSF

_

\]EcTLTZR]A\]NDia•PWeF

S^{F/Ey=F _f_ F _ KMV} R^{3P^V}

_

P#R]ADKQep¡LVDFb\S\

YfP^ipP^EFRSF/i

ε^–ƒ—DTWi ε≤ε0

\]N@eKMF/VtRSOU`2\]EPZOUO˜WR]ADFo\aAfF/OMOfKQ\;RSADF‘KME2P^~WF

Ω^εT^’RSADF‘E2PZVDKšžTWO

_

S×(−ε, ε)^‹LKQP

R]ADFPZYDYDOMKMebPR]KMTZV

Φ^ε:S×(−ε, ε)3(P, x3)7→P+x3n(P)∈R³, ^œ'AZ–BA  

Ž#ADF/iSF

n(P) KM\xPNfVDKšRoVDTWi]E2P^O”‹ZFdeRSTZiC>8F/O_ TZV

S–oItRpP^i]R]KMVD~žiSTZE RSADFRSADiSF/F_ KMEF/Vf\]KUTWVfP^OFtNfP^R]KMTZVf\‘T^

\kRpP^V _ PZi _ OMKMVDFbPZi)FbOMPW\kRSKMe/KšRk` žTZicP¥ADTWETZ~ZFbVDF/TWNf\PZV _ KQ\aTZR]iSTZYDKQe2E2PRSF/iSKMPZO¢˜ _ KD?F/iSF/VtRcET _ FbOM\)AfP‡‹ZF y8FbF/V

_ F/iSKU‹WF _ y8F/RkŽ“FbF/VEAbrts^rP^V _ AbrGFGAIHx\]F/F2KMV"YfP^i]R]KQeNDOQP^i2JKMLNPOQSRBAbr[˜zZ{[˜zAT˜U!VXWZYX[MLCRzWs\T˜^]_KZYX`SRA\abT˜

U4K#cMK^dYXL"eGK#cfRzgFhT– ¯ TW\aR)TZRSADF\]ADFbOUOET _ F/OQ\iSF/OM`+TWV P

3×3 \]`[\kRSF/E T^XKMVtR]iSKUVf\]KQe2FtNfP^R]KMTZVf\$TWV

S

_ F/Y=F/V _ KUVD~2TWV

ε^{˜fPZV _ Ž#i]KUR]F}

K(ε) :=M+ε²B ^œ'AZ–z

Ž#ADF/iSF

M KM\‘RSADFji6ik :lXTWY8FbiSP^R]TWimTZV

S ^{P^V _} B KM\xPmk#n"gocTZY=F/ipPRSTZid–Cp“PZOUOŒP^y=T‹ZF)P^N[RSADTZip\mPZ~ZiSF/F

Ž#KšRSA"R]ADF _ F>fVDKUR]KMTZVTZ“RSADFEF/E)yDipP^VDFTZY=F/ipPRSTZi

M^{˜ _} KD?F/iSF/VtRFg[YDiSFb\S\]KUTWVf\oT^

B e/P^V"y=F2žTZNfV _ KUV"RSADF OUKUR]FbiSP^R]NDiSFZ–—fTZi$~ZFbVDF/ipP^O;\]ADFbOUO;~WF/TWEcF/R]iS`Z˜’RSADF2ETW\aRY8TWYDNDOQP^iPZV _ VfPRSNDiSPZOŒET _ F/O;KQ\oR]AfF2TZVDF2YDiSTZY=TW\]F _

yt`qJ4KZLNPOQ”–@BADKQ\oET _ F/O _ Fd\]e/i]KMy=Fb\xR]AfF _ KM\]YDOQPZe/F/EF/VtRT^“R]ADF\]ADFbOUO;yL` RkŽ“TR]FbVf\aTWiS\oiSF/Yfi]Fd\aFbVWRSKUVf~ RSADF

epAfP^VD~WF)TZ“EFRSi]KQecP^V

_

epAfP^Vf~ZFcT^“e/NDi]‹P^R]NDiSF)TZ¤RSADF2\aNDi]•PZe/Fc\]NDyDEKURaRSF

_

R]T™P

_

KM\]YDOQPZe/F/EF/VtRb–

¯

TZiSF/T‹WF/i

R]ADKQ\#ET

_

F/O±KQ\#FbOUOMKUYDR]KQexžTZi

ε >0 ^{œ•\]F/F6RzbT}  –srmTŽ“Fb‹ZFbib˜tžTZi

ε= 0˜DR]ADF$VfP^R]NDiSFoT^”R]ADF$EFbEyDipP^VDFoTWY8FbiSP^R]TZi

_

F/Y=F/V

_

\TZV RSADF~ZFbTZEFRSi]` T^BR]ADF\aNDi]•PZe/FZ–jp V Y8P^i]R]KQeNDOQP^id˜

M KQ\FbOUOMKMY[R]KQe2TZVfOU` PR$R]AfFY8TWKUVtRp\Ž#ADFbi]F

S

KM\cF/OMOMKUY[RSKMeZ– @BADF™‘TZKUR]F/i)ET _ F/O#iSF/OMKUFd\YfPZiaRSOU`NfY8TWV›eTWEYDN[RSP^R]KMTZVf\cE2P _ FyL`t]_KZYX`KMVuRA\abT–›£“NDR)RSADF

WNfFb\aR]KMTZV TZ _ FRSF/iSEKUVDKMVD~R]ADFvk ƒET _ F/OƒŽBPZ\X‹WF/iS`™eTZVtRSi]T‹WF/ip\aKQP^O;œ¢\aFbF)KMV YfPZiaRSKMe/NDOQP^i‘R]ADFcKUVtRSi]T _ NfeR]KMTZV

KUVqR|TP^V __ KM\SeNf\S\]KUTWVf\KUVqRz^{[˜8zWs\T   –

p VwRA‡sbTxŽ“F¥~WKU‹WFR]ADF Fg[YDiSFb\S\aKMTZV T^oRSADF¥ETW\aR ~ZFbVDF/ipP^Omy8FbV _ KMVD~ TZY=F/ipPRSTZi PZYDY=FbP^iSKMVD~KUV RSADF+| #

FtNfPRSKUTWVf\b˜tPZV _ ŽF‘\]ADTŽADTŽ RSADFx‘TZKUR]Fbi¤y=F/V _ KUVD~cTZY=F/ipPRSTZi“KM\OMKUVD¡WF _ Ž#KUR]AR]AfKM\TZY=F/ipPRSTZiP^V _ RSNDiSVf\“R]T

y8F$RSADFETt\kRXVfP^R]NDipP^OP^V _ \]KUEYDOMFy=F/V _ KUVD~ TZY=F/ipPR]TWiXP^ETZVf~R]ADFTZR]ADFbiS\b–xpVyRrbT”Ž“F)P^OQ\aT2~WKU‹WFP~ZF/VfF/ipP^O

Fb\aR]KMEP^R]F“y8F/RkŽ“FbF/V$RSADF| #›\]TZOMN[RSKUTWVP^V _ P _ KQ\aYfOMPWeF/EFbVWR”i]FdeTWVf\kRSi]N8eR]F _ ži]TWE R]ADFBmTWKšRSF/iEcT _ FbOt\]TZOMN[RSKUTWV±–

@BADFmiSFb\]NDOUR“T^zRA‡s\T=iSF _ NfeFd\;RSADF‘| # YDiSTZyfOUFbE RST)PžTZiSEPZO=\]F/iSKMFb\“z# Yfi]TWyDOUFbE ‹ZF/iS`\aKMEKUOQP^i¤R]TRSADFomTWKšRSF/i

EcT _ FbO¢–

p VRSADFBe/PZ\]FT^fYDOQPRSFb\b˜‡R]ADF#‘TZKUR]FbiET

_

F/OD\]YDOMKšRp\”KUVtR]T‘R]AfFEFbEyDipP^VDF“TZY=F/ipPRSTZiƒPZeRSKUVf~‘TWVR]AfF\]NDi]•PZeKQe

eTZEY=TZVDFbVtRS\#TZRSADF

_

KQ\aYDOQPZe/F/EF/VtRmPZV

_

RSADFy=F/V

_

KUVD~2TWY8FbiSP^R]TWiBPWeRSKUVD~2TWVR]ADFoRSiSPZVf\a‹WF/ip\aF

_

KQ\aYfOMPWeF/EFbVWRd–

v‘VR]ADFTZR]ADFbiXAfPZV _ ˜DRSADFŽTZiS¡ KMVyRAd{[˜MAIA^˜_aT”\]ADTŽX\BR]AfFFg[KQ\kRSF/Vfe/F$T^;P^V™PZ\]`LEY[R]TZR]KQe$Fg[YfPZVf\aKMTZVT^ƒRSADF

|# _ KQ\]YDOMPWeFbEcFbVtRžTWi#YDOQPR]Fd\XeTWVtRSP^KMVDKMVD~cy8TWNDV _ PZi]`OQP‡`ZFbiR]F/iSE2\BT^Œ\]ebP^OMF

ε^{– Š} AfF/V™\aN8epAPZVPW\a`LEY[RST^R]KQe Fg[YfP^V8\aKMTZVcKM\“P‡‹P^KMOMPZyDOUFW˜^ŽFBebP^VFb\aR]KME2PRSF#R]ADF _ KD?F/iSF/Vfe/FXy=FRkŽF/F/VcRSADFm\]TZOMN[R]KMTZVcT^?R]AfFXmTWKšRSF/iŒET _ F/O=P^V _

R]ADF| # \aTWOUNDR]KMTZVKUVF/‹WF/iS`VDTZiSE–

p V RSADKQ\BŽ“TWi]¡?˜LŽ“FxžT[eN8\BTZNDi#PR]R]FbVWRSKUTWV RSTRSADF\]Y8FdeKQP^O’e/PW\aFxŽ#ADF/iSFxR]ADFoEFbPZV\]NDi]•PZe/F

S T^RSADF\]ADF/OMO KM\?"(#$/˜±R]A8PR$KM\oŽ#AfF/V+R]ADF $oP^N8\]\]KMPZV eNfi]‹PRSNDiSF)TZ

S KQ\$\aR]iSKMeR]OM`™Y8Tt\aKUR]KM‹ZFcTZioFtNDKM‹P^OMF/VtR]OM`™Ž#ADFbV RSADF YDi]KMVfe/KUYfPZOŒeNDiS‹PR]Nfi]Fd\xPZi]FcF/‹WF/iS`LŽ#ADF/iSFT^“R]ADF2\SP^EF\aKM~ZV±–%pV RSADKQ\$e/PZ\]FZ˜?RSADFEF/EyfiSPZVDF)TWY8FbiSP^R]TWi

M ^KQ\

F/OMOUKMY[R]KQe œ¢\aFbF{RBAbu[˜ƒz^r[˜GT

–™lm\R]ADF y8FbV _ KMVD~¥TZY=F/ipPRSTZi

B ^{KQ\TZBTWi _ F/i} 4 ^{Ž#ADKMOUF R]ADF} EcFbEyDipP^VfF2TZY=F/ipPR]TWi

M KQ\xT^¤TWi _ F/i

2˜=R]ADF2‘TZKUR]F/i‘TZY=F/ipPRSTZi

K(ε) =M+ε²B KM\oP| "GoB } kTZ;R]ADFcEF/E)yDiSPZVDF TZY=F/ipPR]TWib–)@BADFcžiSPZEcFbŽ“TWi]¡TZx~L<YXL"€wƒ‚#„^…MNPOQP`ML"€SR|Z{TŒžTZi\Se/P^OQP^ixFtNfP^R]KMTZVf\oebP^V y8F2P _ P^Y[RSF _ R]T

R]ADKQ\X\]KšRSNfPRSKUTWV±˜DŽ#ADFbi]FxR]AfF$FtNfPRSKUTWVKQ\#P2\]`L\aR]FbE– !“TWEyDKMVDKMVD~cR]ADFd\aFoRSFbepADVfKtNDFd\#Ž#KšRSAR]ADFožTWi]E2P^O±\]F/iSKMFb\

i]F _ NfeRSKUTWVTZxRBAdsbT~ZKM‹LKUVD~)R]AfF\kRSi]NfeR]NDiSFoT^”R]ADF| #:y8TWNDV _ PZi]` OMP‡`WF/ip\/˜LŽFxTWy[RSPZKUVR]AfFožTZOMOUTŽ#KMVD~ci]Fd\aNDOURS\hH

AZ– Š F)\]ADTŽ R]AfP^RmRSADFz# _ KQ\]YDOMPWeFbEcFbVtRx\]TZOMN[R]KMTZV™TZŒR]ADFmTWKšRSF/iXF tNfPRSKUTWV¥P _ EKURS\xP eTWEYDOUF/R]F)ENDOU

RSKM\Se/PZOUFoFg[YfPZVf\]KUTWVKMVfeOMN _ KMVD~y=TZNfV _ P^iS`OQP‡`ZFbi“RSF/iSE2\#T^Œ\]ebP^OMF

ε^1/2 Nf\]KUVf~Pc\]KMVD~ZNDOQP^i#Y=F/i]R]Nfi]yfP^R]KMTZV RSADF/TWi]`e/OUTt\aFxR]TvR|Z{T–

z[–#§‘\aKMVD~xR]ADF#iSFb\]NDOUR;KUVRBAds\T˜ZŽFR]ADFbV2\aADTŽ R]AfP^RŒR]ADFX| # _ KQ\aYfOMPWeF/EFbVWR;P _ EKšRp\;PoeTZEYDOMFRSF#ENDOUR]KQ\Se/P^OMF

F/gLY8P^Vf\]KUTWVf\Ž#KUR]A:z # y8TWNDV _ PZi]` OQP‡`ZFbiS\ T^\Se/P^OMF

ε^1/2 OMKU¡WF¥žTZiR]AfF"z# _ KM\]YDOQPZeFbEF/VtRb˜mPZV _ | # y=TZNDV _ P^iS`2OMP‡`WF/ip\T^”\Se/PZOUF

εOUKM¡ZFxžTZi#YfOMP^R]Fb\b–

|D– Š FNf\aFcRSADFb\]FFg[YfP^V8\aKMTZVf\‘RSTy=TZNfV _ RSADF _ KD=Fbi]FbVfeFy=FRkŽF/FbV RSADF2|# _ KQ\]YDOMPWeFbEcFbVtRoPZV _ z# i]F/

e/TZVf\aR]iSNfeRSF _™_ KQ\]YDOMPWeFbEcFbVtRS\xPZ\XKMV Rz^{T”TWi%RGD˜’z^|bT–x@BAfFb\]FFb\aR]KME2PRSFb\xP^iSF\aA8P^iSY¥KUV¥R]ADF\aFbVf\]F$R]AfP^R

RSADF$F/iSi]TWi¤RSF/iSE AfPW\R]ADF\SP^EFoTZi _ Fbi“RSAfP^VRSADF.>fiS\aRXVDFb~ZOMFbeRSF _ R]Fbi]E!KUVR]ADFPW\a`LEY[RST^R]KQe^–

p V R]ADFžTWOUOMTŽ#KUVf~ŽF2P^OMŽP‡`[\‘RSPZ¡ZF2PZ\

K(ε) R]ADFmTWKšRSF/ioET _ F/O– Š F2VDTŽ YDiSFb\]F/VtRoR]ADF| #P^V _ z#

YDi]TWyDOMF/E2\/˜?P^V _ ~ZKM‹ZF$RSADFE2P^KMV™RSADF/TWi]FbE\b– Š FRSADF/V¥iSFbebP^OMO±RSADFyfPW\aKQe$iSFb\]NDOURS\XTZCRA‡s\TŒP^V _ ~WKU‹WFoR]ADFcYDOQP^V

T^”R]ADF$YfPZY8Fbib–

! "#$%

@BADF‘y=TZNfV

_

P^iS`cT^±RSADF‘\]ADF/OMO

Ω^{ε_ F>fVDF _ KUV+œ AZ–BA}   AfPW\ŒR]Afi]FbFxe/TZEY8TWVDF/VtRp\Hƒl OQPRSF/ipP^O8y8TWNDV

_

PZi]`

Γ^ε₀^KUE2PZ~ZF

T^ ∂S×(−ε, ε) ^yL`¥RSADF PZYDYDOMKMebPR]KMTZV

Φ^{ε˜ƒP^V _} NDYDY=F/icP^V

_

OMTŽ“Fbio•PZe/Fb\

S− +ε

KUE2P^~WFb\$TZ

S× {−+ε}^{– Š F}

\aNDYfY8Tt\aFmR]AfP^R¤R]AfFmE2PRSF/iSKMPZO8eTWVf\aR]KUR]N[RSKUVD~R]ADFx\aAfF/OMO8KQ\;ADTWETZ~ZFbVDF/TWNf\;PZV _ KQ\aTZR]iSTZYDKQe^˜tepAfP^ipPZeR]F/iSK'&/F _ yL`cKšRp\

RkŽ“T²P^E meTLF@2e/KUFbVtRS\

λ^{P^V _} µ–Œ@BADFmOUTtP _ KMVD~ožTZipeFd\;P^YfYDOUKMF _ RST$R]AfFm\aAfF/OMO8P^iSFXiSF/YDiSFb\]F/VtRSF _ yt`P\]ETtTZR]A

‹ZFbeR]TWi}>fFbO_

f ^{_ F>fVfF _ TWV} Ω^{ε– Š} F2\]NDYDY=TW\]FRSAfPRoR]AfF\]ADFbOUOƒKQ\xe/OMPZEY8F _ PZOUTWVD~

Γ^ε₀ ^{PZV _} ŽFcKUEY=TW\]FRSADF R]ipPZeRSKUTWVži]FbFmeTWV _ KUR]KMTZV2TWV

S+ε

P^V _

S₋ε

–ƒ@BADF _ KM\]YDOQPZe/F/EF/VtR¤TZ=RSADF‘\]ADFbOUO8KM\“i]FbYDiSFb\]F/VtR]F _ yL`)R]ADF%AžTZiSE

>fF/O_

u^–pV !P^i]R]Fb\]KQP^Ve/TLTZi _ KUV8PR]Fd\

{tⁱ}RSADF$YDiSTZyDOMF/ER]ADFbVŽ#iSKUR]Fb\





−∂jA<sup>ijk`</sup>ek`(u) = f^{i KUV} Ω^ε, Tⁱ(u) = 0 ^TWV S

− +ε, u = 0 ^TZV Γ^ε₀,

œ'AZ–|  

Ž#KšRSA

A^{ijk`</sup> =λδ<sup>ij</sup>δ<sup>k`}+µ(δ^ikδ^{j`</sup>+δ<sup>i`}δ^jk)˜8Ž#ADF/iSF

∂j

KQ\BR]AfF$YfP^i]R]KQP^O _ Fbi]KM‹PRSKU‹WFxŽ#KUR]AiSFb\]Y8FdeRBR]T

t^{i P^V _} eij(u) =¹₂(∂iuj+∂jui)^Ž#KšRSA u=uidt^{i KUV} !PZiaRSFb\]KMPZVeTLTWi _ KMVfP^R]Fb\b–¤v‘VR]ADF\SP^EF$ŽBP‡`

f^{i _} F/VDTZR]FoRSADF eTZEY=TZVDFbVtRS\ƒT^=R]AfFX‹ZFdeR]TWi^>fF/O_ KUVcRSADF#yfPZ\]KQ\

∂

∂tⁱ

–”@BAfF#TZY=F/ipPR]TWi

Tⁱ(u)KQ\ƒR]AfF#VfPRSNDiSPZOLR]ipPZeR]KMTZVcTZY=F/ipPR]TWi TZVR]ADFo•PWeFb\

S− +ε

P^YDY=FbPZi]KMVD~PŸRSF/iXKMVtR]F/~WiSP^R]KMTZVyt` YfP^i]RS\BKMVR]ADFPW\]\]T[eKQPR]F _ yDKMOMKUVDFdP^iBžTWi]EH

(u,v)7→

Z

Ω^ε

A<sup>ijk`</sup>eij(u)ek`(u) dt¹dt²dt³. ^œ'AZ–G  

@BADKM\KM\cRSADF¥eOQPZ\S\aKQe/PZO#YDiSTZyDOMF/E TZ‘OMKUVfFbP^i2FbOMPW\kRSKMe/KšRk`›\aF/RKUV !P^i]R]Fd\aKQP^V¦eTLTZi _ KMVfPRSFb\cTZVP"\]ADF/OMOš\]AfP^Y=F _

_ TZE2P^KMVT^

R^{3–;‘TZiSV} KMVDFtNfPZOUKURk`{RBAb|bTKMEYDOMKUFd\BR]AfP^RBR]ADKQ\#YDiSTZyDOMF/E AfPZ\#PcNDVDKtNDF\]TZOMN[R]KMTZVKUV

H¹(Ω^ε)^3–

v‘V Ω^ε˜BŽ“F¥ebP^OMO(kVfTZiSEPZOXeTLTZi _ KMVfPRSF™\a`[\aR]F/E*)¥P\a`[\aR]FbE T^xR]AfF™žTZiSE

(xα, x3) ^{KMV _ Nfe/F _ yL` RSADF}

EPZYDYDKMVD~œAW–A

˜±Ž#ADFbi]F

xα

KQ\$Pe/TLTZi _ KUV8PR]F\]`L\aR]FbE&TZV

S ^{PZV _} x3

R]ADFcRSiSPZVf\a‹WF/ip\aFceTLTZi _ KMVfPRSFœ¢\aFbFjRBAdsbT

žTZi _ FRSPZKUOQ\

  – T^RSF™R]AfP^R RSADF _ TZE2P^KMV

Ω^{ε KM\} žTZOMKQPR]F _ yL`›RSADF+\]NDi]•PZe/Fb\

Sx3

KME2P^~ZFd\TZ

S× {x3} ^yL`

R]ADF _ KD=FbTZETZiSYDADKQ\aEœ'AZ–BA

  – pV¥R]ADFcžTZOMOMTŽ#KUVD~8˜8ŽFŽ#KMOMOƒPZOUŽBP‡`[\XK_ F/VtRSKšž`RSADFEcFdP^V \]NDi]•PZe/F

S0

Ž#KšRSA¥RSADF

P^yf\aR]ipPZeRBE2PZVDKšžTWO_

S^–

# ,+-$./ 01

v‘V2R]ADFxEFbPZV \aNfia•PWeF

S0

˜LP)z # _

KQ\]YDOMPWeFbEcFbVtR“KQ\¤iSF/YDiSFb\]F/VtRSF

_

yL`cR]ADFoeTWNDYDOMFmT^P9AžTZiSE >8F/O

_

zα

PZV _ P

žNDVfeRSKUTWV

z3

– Š F _ FbVDT^RSFXyL`

z= (zα, z3)∈Γ(T1S0)×C^∞(S0)\]NfepAPe/TZNDYfOUFW–;rmF/iSFZ˜

Γ(T1S0)^_ F/VfT^R]Fd\

R]ADF\]YfPZe/FoT^zA/¢žTWi]E >fF/O_ TZV

S0

–;l‘\#KšRXŽ#KMOMO’y8F$TZƒe/TZVf\aRSPZVWR#Nf\]FZ˜[ŽF$\aF/R

Σ(S0) := Γ(T1S0)×C^∞(S0)

R]ADF#\]YfPZe/F“TZ”œ•\]ETLT^R]A

z# _ KQ\aYDOQPZe/F/EF/VtRp\/– ¯ TZiSF“~WF/VDFbiSPZOUOM`Z˜‡ŽF _ F/VDTZR]FyL`

H^k(S0)R]ADF#\]YfPZe/F“TZZA/¢žTWi]E2\

Ž#ADTW\]F2y8TZR]A e/TZEY8TWVDF/VtRp\y=F/OMTZVD~RST™R]ADFILTZy=TZOMF/‹ \aYfPWeF

H^k(S0)^{– Š F ¡ZFbF/Y} R]AfFVDT^RpPRSKUTWV

H^k(S0) ^žTWi

žNDVfeRSKUTWVf\b–ƒ@“`LYDKQe/P^O?\]YfPZe/Fb\ŒžTWiBz# _ KM\]YDOQPZeFbEF/VtRS\P^iSF

H¹×L²(S0)^{PZV _} H¹×H²(S0)^{– Š Fx\]FR} aαβ

RSADF

EcF/R]iSKMe‘RSF/Vf\]TZi#TWV

S^{˜DPZV _} bαβ

R]AfFeNDiS‹PR]Nfi]F‘R]FbVf\aTWib–Œ@BADF $xiSF/Fb¡2KUV _ KMe/Fb\XPZi]F‘RkŽT^_ KUEFbVf\aKMTZV8P^O?‹P^iS`tKMVD~

KUV _ KMe/Fb\b–)@BADF2eTWVtR]ipPZeRSKUTWV¥yL`™R]AfFcEF/R]iSKMe)R]FbVf\]TZio`LKUFbO_ \xKQ\aTWETZiSYDADKQ\aE2\‘y8F/RkŽ“FbF/V+R]F/V8\aTWio\]YfPWeFb\xTWV

S0

–

Š F$AfP‡‹WFmžTWi#FgDP^EYDOMF

b^α_β =a^ασbσβ

–

@BADF‘TZKUR]F/iBTWY8FbiSP^R]TZiBKQ\R]ADF$TWY8FbiSP^R]TWi

K(ε) : Σ(S0)→Σ(S0)Ž#i]KURaRSF/V

K(ε) =M+ε²B.

rXF/iSFZ˜

MKM\RSADF6ii6k :#BTWY8FbiSP^R]TWi _ F>fVDF _ yt`

( M_σ = −λDe σγ_ν^ν−2µDαγ_σ^α, M₃ = −λbe ^α_αγ_ν^ν−2µb^β_αγ_β^α,

Ž#ADF/iSF

λ^{PZV _} µ P^iSFoR]AfF²PZE )eTLF@eKMF/VtRp\XT^ƒRSADFE2PRSF/iSKMPZO¢˜

λe= 2λµ(λ+ 2µ)^−1˜ Dα

KQ\#RSADFe/T‹‡PZi]KQP^VtR

_ F/iSKU‹P^R]KM‹ZFoTZV

S0

˜8P^V _

γαβ(z) =¹₂(Dαzβ+ Dβzα)−bαβz3

œ'AZ–s  

KM\BRSADFepAfPZVD~ZFxT^”EcF/R]iSKMexRSF/Vf\]TZi#TWV

S0

–

@BADFTWY8FbiSP^R]TZi

MKM\‘PZ\S\aT[e/KMP^R]F _ Ž#KUR]A™RSADFyDKMOUKMVDFdP^i#žTWi]E _ F>8VDF _ žTWi‘P^VL`

z ^{PZV _} η ^KMV H¹×L²(S0)

yt`

(z,η)7→a_M(z,η) = Z

S

M^αβσδγαβ(z)γσδ(η) dS0, ^œ'AZ–u  

Ž#ADF/iSF

M^αβσδ=λae ^αβa^σδ+µ(a^ασa^βδ+a^αδa^βσ)^–

@BADF$TZY=F/ipPRSTZi

B KM\RSADF9kln"go$TZY=F/ipPR]TWi

_

F>fVfF

_

yL`

( B_σ = −¹₃eλb^α_σDαρ^ν_ν−¹₃eλDαb^α_σρ^ν_ν−²₃µb^α_σDνρ^ν_α−²₃µDνb^α_σρ^ν_α, B₃ = ¹₃λDe ^αDαρ^ν_ν+²₃µD^αDνρ^ν_α−¹₃eλc^α_αρ^ν_ν−²₃µc^β_αρ^α_β,

Ž#ADF/iSF

c^β_α=b^ν_αb^β_ν ^{P^V _}

ραβ(z) = DαDβz3−cαβz3+b^σ_αDβzσ+ Dαb^σ_βzσ

œ'AZ–(F

KM\xRSADFepAfP^Vf~ZFcT^e/NDiS‹‡P^R]NDiSF)R]F/V8\aTWib–@BADKQ\oTZY=F/ipPR]TWi‘KQ\$PZ\S\]TLe/KMP^R]F _ Ž#KUR]A R]ADFyDKMOMKUVDFdP^iožTWi]E _ F>fVDF _ žTWi

P^VL`

z ^{PZV _} η^KUV H¹×H²(S0)^yL`

(z,η)7→a_B(z,η) = Z

S

M^αβσδραβ(z)ρσδ(η) dS. ^œAW–a

—DTWiXP)~WKU‹WF/V

g∈Σ(S0)˜DŽF$eTWVf\aK_ FbiBR]ADF\]TZOMN[R]KMTZV

z∈Σ(S0)TZRSADF$YDiSTZyDOMF/E

(K(ε)z=g ^KMV S0, z

∂S0 = 0 ^{PZV _} ∂rz3(ε)

∂S0= 0.

œ'AZ–r  

@BADF$Fg[KQ\kRSF/Vfe/FxTZ

z KM\BYfi]T‹WF

_

KMVqRzT –

.

Š F\aF/R

(r, s) ^{P e/TLTZi _} KUV8PR]F\a`[\aR]F/E KMVR]AfF ‹LKMe/KUVfKšRk`"T^

∂S0

\]NfepAR]A8PR

r KQ\R]ADF~ZFbT _ Fb\]KMe _ KM\aRSPZVfeF R]T R]ADF)y8TWNDV _ PZi]`W˜fP^V _

s KQ\#R]AfF)P^ipeOMF/Vf~^R]A™PZOUTWVD~

∂S0

– Š F _ F/VfT^R]Fyt`

bss(r, s)^˜ brs(r, s) ^{P^V _} brr(r, s)^R]AfF

eTZEY=TZVDFbVtRS\#TZRSADFeNfi]‹PRSNDiSF‘RSF/Vf\]TZimKUVR]AfKM\‘eTLTZi _ KMVfPRSFo\]`[\kRSF/E–“@BAfF$•PZeR#R]AfP^R

S0

KQ\#F/OMOUKMY[RSKMe$KMEYDOMKUFd\

R]AfP^RxŽF)ebP^V+epAfTtTt\aFR]ADFcTZiSKUFbVtRSPRSKUTWV™T^

S0

\]NfepA™RSAfPR

bss

PZV

_

brr

PZi]F)Y8Tt\aKUR]KM‹ZFcP^OMTZVD~

S0

– Š F _ F/VDTZR]F

yt` χ(r)^P C∞ ^{eN[R] T} D™žNDVfeR]KMTZVVDFbPZi

∂S0

–

@TxeTWVf\kRSi]N8eR±RSADF“F/g[YfP^Vf\]KMTZV$T^[RSADF#z # _ KM\]YDOQPZe/F/EF/VtRb˜‡ŽF“\]NDYDY=TW\]F;RSAfPRR]ADFi]KM~ZAtR]AfPZV _ \]K_ F

g=g^ε

_ F/Y=F/V _ \#TZV

ε^{PZV _ P _} EKšRp\BR]AfF$Fg[YfP^V8\aKMTZV

g^ε'X

k≥0

ε^kg^k, ^{œ AW–Ad{}  

Ž#ADF/iSFxžTZiXP^OMO

k^˜ g^k ∈Σ(S0)–;@BADKQ\#EFbPZVf\R]AfP^R#žTZiXPZVt`ILTZy=TZOMF/‹2VDTWi]E TZV

S0

P^V

_

PZVt`

N˜[Ž“FoAfP‡‹WF

kg^ε− XN k=0

ε^kg^kk ≤CNε^N,

Ž#ADF/iSF

CN

KM\#KMV _ FbY8FbV _ FbVWRXTWV

ε^–

©L­ … ©

z^ε k{<&v ($ v<&6# :IkBi œAW–r  

"<& y ob&G&#nt n

g^ε <="go

œAZ–BAb{

&

z^ε ni%" !qb =iCl m"6

%

ε^1/2 z^ε'X

k≥0

ε^k/2

ζ^k/2(xα) +χ(r)Z^k/2( r

√ε, s)

, ^{œ AW–AIA}  

& : !"

k ζ^k/2∈Σ(S0) ^{."ln #nI_x} ε ^ln Z^k/2(T, s) 2l_"(=9n :b "go?"

T

i4(=."

slnC i6h"&

R⁺×∂S0

{ :

I&^gn

s <&ll$

T 7→e^ηTZ^k/2(T, s)

2k#n nj

R⁺

!"

η < η1

& :

η1= 3µ(eλ+µ) (eλ+ 2µ)²

!1/4

pbss(0, s). ^{œ AZ–BAdz}  

@BADF0>fip\kRBRSF/iSE

ζ⁰ KQ\R]AfF\aTWOUN[RSKUTWVTZRSADF$EF/E)yDiSPZVDFxYDi]TWyDOMF/E

Mζ⁰ = g^{0 KUV} S0, zα_∂S

0 = 0,

œ AW–Ad|  

Ž#ADF/iSF

g⁰ KQ\BR]AfF0>fiS\aRBR]Fbi]E!T^RSADFPZ\]`LEY[R]TZR]KQe‘F/g[YfP^Vf\]KMTZVT^

g^ε–;@BADFo•PWeRBR]A8PR#R]AfF$EcFbEyDipP^VfFoebP^VDVfT^R

\aTWOU‹WFžTWi‘RSADF2y=TZNDV _ P^iS`¥eTZV _ KšRSKUTWVf\xTZV

z3

KQ\xR]AfFciSFbPW\aTWV¥žTWi‘RSADF2YDiSFb\]F/Vfe/F)TZ¤RSADFz # y=TZNDV _ P^iS`™OMP‡`WF/i

R]F/iSE2\/–.p V _ FbF _ ˜?RSADF)R]ADKMi_ eTWEcY=TZVfF/VtR

M₃ KQ\oP^V¥TWY8FbiSP^R]TZi‘T^;TWi _ Fbi

0^KUV z3

˜?Ž#ADKUOMF

B₃ KQ\xT^;TZi _ Fbi

4^KMV z3

–;@BADF0>fip\kR#y=TZNDV _ P^iS`OQP‡`ZF/i“R]Fbi]E2\X\SPRSKM\'>fFd\

Z_α⁰= 0 ^yDN[R Z₃⁰6= 0KMV~ZFbVDF/ipP^O –

§m\]KUVf~)RSADF$Fg[YfPZVf\]KUTWV œ'AZ–BA A  

ŽFoTZy[RpP^KMVFd\kRSKUE2PRSFb\By=FRkŽF/FbV

z^{ε P^V _} ζ⁰–;—fTZiBFgDP^EYDOMF$ŽFx~WFR

kz^ε−ζ⁰k_H¹_×L²_(S)≤Cε^{1/4 œ AW–AbG}  

Ž#ADF/iSF

C ^{KM\KMV _ FbY8FbV _ F/VtR#TZV} ε–;@BAfKM\Fb\aR]KME2PRSF‘KMEYDOUKMFb\BKMV Y8P^i]R]KQeNDOQP^i“R]ADF$e/TZVL‹ZFbi]~WF/Vfe/FXi]Fd\aNfOšRBT^xRGTPZV _ KUEYDiST‹ZFd\¤RSADF$iSFb\]NDOšR#KMVqRz^GT–

@T¦eTWVf\kRSi]N8eRRSADF F/gLY8P^Vf\]KUTWVT^oR]AfF+|# _ KQ\aYfOMPWeF/EFbVWRp\ŽF+\]NDYDY=TW\]F™R]A8PR RSADF iSKM~ZAtRaAfP^V _ \aK_ F

f =f^{ε _ F/Y=F/V _ \oTZV} ε KUV+R]ADFcžTWOUOMTŽ#KUVf~ iSF/~ZNfOMPZi‘ŽBP‡`lH}p

(xα, x3) ^KQ\$P VDTWi]E2PZOƒe/TLTZi_ KUV8PR]Fc\a`[\aR]FbE&TZV

Ω^{ε ŽF\]FR} X3 = ε⁻¹x3

˜¤P^V _›_

F>fVDFR]ADF‹WFbeRSTZi%>fF/O

_

f(ε)(xα, X3) = f^ε(xα, x3) ^TZV R]ADFE2P^VfKšžTWO

_

Ω :=S×(−1,1)^{– Š} F\aNDYfY8Tt\aFxR]A8PR

f(ε)^{P _} EKšRp\BR]ADF$F/g[YfP^Vf\]KMTZV

f(ε)'X

k≥0

ε^kf^k, ^{œ AW–A‡s}  

Ž#ADF/iSFcžTZiP^OMO

k^˜ f^{k KM\$KMV _ FbY8FbV _ FbVWRTZ} ε ^KUV Ω–@BADKQ\oAt`LY=T^RSADFb\]KM\oKQ\\SPRSKM\'>fF _ KMV+RSADFe/PW\aFŽ#ADFbi]F

f ^KQ\

KUV _ F/Y=F/V _ FbVtRxTZ

ε KMV™RSADF)YfAt`[\]KMebP^O”e/P^i]R]Fd\aKQP^V¥e/TLTZi_ KUV8PR]Fd\/–}pV™R]ADKQ\oe/PW\aF$RSADF@P‡`LOMTZiXF/gLY8P^Vf\]KUTWV™T^

f ^PR x3= 0^{PZi]TWNDV _ R]ADF$EK_ } \]NDi]•PZe/Fx`LKMF/O_ \R]AfFeTLF@eKMF/VtRp\BT^”R]ADF$F/gLY8P^Vf\]KUTWV+œ'AZ–BAds

  –