On Topological Derivative in Shape Optimization
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On topological derivative in shape optimization ˙ J. Sokołowski and A. Zochowski. N ˚ 3170 Mai 1997 ` THEME 4. ISSN 0249-6399. apport de recherche.
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(82) L o E o " E E E KL O^GX_XV ^ *TV`YPHZ\ s TVpTgfW fO h ^ s \ L Oiw#O WYPHZ5`l` ^aWY` h Z s jmo^G` h TV\eT_^G` f = L v EE EE $v E $ * L O s O L vE $ T[fE\ L OWY`YT_\ ^GWY_\ *Z s h `Y^ s PHZ5XlUOmo\e^ s ^G` -? `\ L T[fE`Y^a\ Z5\]T_^a` \aL TVUOiOc`f\ s Ovjfef\ L \eO O#` UfkO^ ms \e^T[f s s O dYL XgZGmo O v_h vE j\ E L ZOv U OvmoZ\e`^ sh \ L LOöfkW s Jv_ZavEmoO$ \ $ s E ZGmov \e$ TV ^a` ¨ f fk\Z s s ZO T_`Hh Oo\e Oc`Y`O fk^h s T[j f $$ $ \ L O^GX_X_+^ *TV`Y^ s P WYX[Z5O E L L
(83) E QL = @0}G C KL O h s fk\f L Z5d O:WY` mo\]T_^G` Z5XnWY` h O s mo^a`fkT h O s Z\eT_^G` h Ocd Oc` h f^G` \ L O h T[fkdYX[ZamoO P:Oc`q\ OcX ¨ ? ? ? C i L @@ ? C ? C i @0}q ~ C @ CL @ @ IZCL x z @ ? C -:E \Tgf L ZXgfk^s WfkOoWYX TV` \ L O# s ZP:AO *n^ s +i^57OcX[Zaf\eTgmT_\j\]^T_`q\ s ^ h W moOx\ L Oj2TVOcX h Wl` mo\eTV^a` ZX ^5 \ O^ P Z@ IZC L xttz @ ? C @ ? C' i - L x tz @ ? C @ ? C i -:E @0} C * L O s O TgfiZ5`T[fk^G\ s ^GdYTgmPHZ5\ s T_p f^a\ s ^adYT[moT_\jP:OZ5`f L O s OU¨\ L Z5\ P4Zvj O:OcpU d s OvfefkO h Zaf^aXVX_^+*xf "' L jL "' Q KL L O^aXVX_+^ *T_`l:ZGfefkWlP: dY\eTV^a` föZafefW s O\ L Z5\ +E bZ s O *nOcX_X h Oo `lO h ^ s fk^GX_WY\]T_^a`f#^5 \ O$OcX[Zaf\eT_\jbfkjlfk\]OcP . . . . . . (. . . . . . . . . . . .
(84) . . . . . . 7. . . . . . . . . . . . . . . . . .
(85) .
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(90) - " x L x z - @0}G} C KL O Z h ueh ^aTV`2? \xf\Z\eO$v OUW Z5\]T_^a`L fCL ^ s \ L Os WY` ms\eT_^G` Z5X v Z s O$T_`q\ s ^ h W moO h+ TV` ! G @-bCCfkW m \ Z\c¨2^ OcUO j G @-bC ¨ @ C. . . $. . . . $. . . . . . . . . . $. . . . . . . 0}Q T_`Z5X_XVjU¨+ ? G v @-bCNTgfE\ L OZ h ue^aTV`2\ fk\Z\eO^ s 3 $Z5` h feZ\eT[fúj^ s Z5X_Xl\eOf\EWY` mo\]T_^G` f G v @-bC \ L O^aXVX_^+*T_`YTV`2\]Oc s Z5XT h Oc`q\eT_\j { ? ? ? U x tz @ C
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(99) O h f h OcZ5`Y` ^Gh \eO! f\ TVL ` O:@0Z} `YaCXV¨nO Z5` Och \_*n^ Ocs Oc`h Tgd fdYs T_Xg` ZGmomoTVOcd P:Z5XnOc`qf\\ s OvfeOcf X h h f T s OmoZ5\]`T_^ah `fö ^T_` s @0}U C jid s TV` moTVd Z5X f\ s Ofef h T s Omo\]T_^G` f *OPbOZ5`¨©ZaftW fkWZ5X_XVjU¨U\ L Oxm^U^ s h T_` Z\eOöfkjlfk\eO P T_$`>5* ¦L¦ T[m L \ L OOc\f\ W s fOfem fZ5Xg\]mOcWY` XgfkZ^ \es OT[\ f L h O TgZh aO ^Gs TV` GZ5Z5X \e TVUOf%^t\ L OWY` mo\]T_^G` Z5X + @ IZC*T_\ LHs Ofd.Ovmo\ \ePH^ Z5\\]LO s ObT[Z5d X Z h s O Z5s P:TVGOcZ5\]\eO TVsUO#I Pb<¨ * Oc\ L L Tg^m L h, h KOc\eL O Tgs fP:XVOT_Z`lhOf#f\]\ ^L O fT_ cOb^5ö\ L O L ^aX_O= ? ¨ jAW fT_`Y\ L O @ IZC L xtz @ ? C ? - U x z @ ? C E @0} C Z5` h T_`H\ L Of]Z5P:O *Zcj ^ s \ L Of\Z\eO$OUW Z5\]T_^a` 3 U x z @ ?C
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(106) 2~.
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(118) . . . . . . . . . . ¹g×é¹gì.
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(120) !"!#$&% '. $. . . L v_$ v "' $$ $ U' _v v $ $ ". v$ $ . . KL T[f T_` \e W s `mo^ ses Ovfkd ^a` h f4\]^\ L O Tgf^a\ s ^adlTgmP4Z5\ s T_p *TV\ L L U~Z5` h L } * } X_KO L`2O\P:P4^Z5pqh T_WYPHX_^iZXUd \s Z5^G`ld.a^ Ocs \]`qT_\^a`f\ Z5s X_OvTV\fej4ftj2JZGTVOcmoX \eh ^ s m s \eT_^\]O \ s L TV^aO`.^G@ÀX_^X_^+\e*Oc`TV`YW fksO Och X[CZ5¨\]*T_^aL ` T[3 m L TgftOUWYT_GZc $ L v_$ v " $ U' v_v $ $ "?Q v$ $$ $ KL T[f T_` \eW s `mo^ ses Ovfkd ^a` h f4\]^\ L O Tgf^a\ s ^adlTgmP4Z5\ s T_p *TV\ L L U~Z5` h L ~ KL OHs fOmo^a` hh h O s s T_GZ5s \]T_UOf#^x\ L OvfkOs WY` mo\]L T_^a`Z5XgfiZ s O:f L +^ *`T_` T_ ) TV h qs h KL O Oc `Y O aKj L Oij2TVX_OcOcXUO X.m XVT_TV`Y\eOvO fT_Z^as `O T[h f:Tgffk\ TVs P:T T_WYX[Z \]O h \]^ WY`Y\ T³O^ s Pbmo^GX_P:jbdYZaX_m T[sZ5^U` femfxO4\ L WlO ` s moZ\e`YTV^aa` OiZ^5X$tmo^GWl` ` fkmoT \eTVO ^a` O f T_ ` \ P:h Os Z` fo¨a\ L Z5\ * L TVX_On\ L O h T[fk\ s T WY\eTV^a` fE^+s WY`L mo\eTV^a` s ftZ` h \ L L O h O ` fkTVs \eT_Ovf s ^\ Ls OxfkOmo^G` h OZaO m LT_GZ5^a\]\ T_L UO Os f ^%Wl` mo\eTV^a` ZXgfX_^U^ +fkTVP:T_X[Z ¨2\ OcjZ Oi`Y^a\ ¨ +^ *O UO ¨Yd ^ad ^ \eTV^a` ZX%\]^ hfkT Of#b!Z` T h XV^UZ h OcO \$h W ^f#UO\ Zs +U\ OL O\ L fOPHOcZ5X_XV^GXt`YfUOcZa\eP:O h O `2s \OvmoTV`\Z`Y\ L aOXVOUP:¨T hYp2hO hXVO^G^` n\ L ^GO\ L WYdlX_Ood.O\ös Z5f` T hh O s TV L \ \f T_fe`YZTVP:\eT[O Z5XnjqZ5T_Oc` X h?h h WYTgf` \e^mo\]s T_\e^aO `h fo¨mo^aWY`2` h aO W s s Z5ZG\efeTVfk^aWl`íP:TgdYfi\efTV^aL ` +^ *o `íL T_` TV KLp O, `2UWYZ5PbTVQ` O s*T[mcO4Z5moX ^as ` OffTWYh X_O\ sf \ Z L s OO f L ^+*`T_` T_ ) T_ . . . . . . . . . éEéº Ý5ð]òoá. . . . . . .
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(124)
(125) !"!#$&% ' . T _GhW O s O$` fkQ TV\k3 j K L O h T[fk\ s T WY\eTV^a`H^5 \ L O W O s jqT_OcX h WY` m\eT_^G`Z5` h \ L Oimo^ ses Ovfkd ^a` h T_`Y . . 3 h h h s L\ T_GOiW mo^ O ses Ofkd K^aL` O h T_`YTg fk\ s T WYh \eOcTV`^afk` T_\j^5x \ L O P4Z5pqT_PHZX\Z5`laOc`q\f\ s Ovfefj2TVOcX Wl` mo\eTV^a`?Z5` . éEéº Ý5ð]òoá.
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(129)
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(137) ¦ 7 ]¦ 7 $7¢ ¢ h Àlo ¶£íh £7o¢ L Ûy $ L s h Ofkjlcfk\\]OcW PfmoTgf^a`ZfkX_TVT a`YO O h \ *OTV\ fkL jlfk\ \eL O O$PHd f s T_@` mT_d C|ZZ5X` fk\ s O@ af]f Ch T Zs `Ovmo\eTVZa^afe` ffWY¨YP:fk^OU¨t\ L \ Z5\Z5\\v OHL mo ^U ^ OcT_``YZ5^G\e\eOO $ Z5X[fk^ C R }-< ^ E [L v_v @ C#" @ @ÀQ© C $$ C R }-< ^ [L v_v @ CaU @ @ C $$ Oc\HW f:T_`q\ s ^ h W moO\ L OAd ^GXgZ s mo^U^ s h TV` Z5\]OfjYf\eOcP @ +E C , \:OZam L d ^aTV`2\HT_`\ L O dYZ5`XgZh `YO h Oo*O`YO h h Oo j`YO \ L ZOXgfWY^ `Y\ TV\#L OUOv^ mos \e\]^ ^as Gf ^a` ZEXmo^U¨ ^ hs h T s T_Ov` moZ\e\eO O?h Z5ZpqX_^aO`lfo¨$ f\eTVZ5X_X ` hh O d `YOc^a`Y\]Od h Oc` h j TgmoWl@ XgZ E s X_Cj ¨ \e^íT_\ ¨xZ5`q\eTgmX_^q m +*L T[fkO s TVUOc`\ h L O h TgfLdYXgZGmoOcs P:Oc`q\OcX h ¨ *OPHh Zcjím^aP:dYL Wl\eO \ L Os momo^a^UPb^ s d.h ^GT_``YZ5Oc\e`qOv\ff$C 3 ^5x\ OHfk\ Z5TV`A OcX @ÀTV`\ O:^ \e^aG^a` ZXxfkjlfk\]OcP Z` W fkTV`Y \ O:d ^aX[Z . .. . . .. . . . . . . .
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(158) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.
(159)
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