Un algorithme d'identification de frontières soumises à des conditions aux limites de Signorini

Texte intégral

(1)Un algorithme d’identification de frontières soumises à des conditions aux limites de Signorini Slim Chaabane, Mohamed Jaoua. To cite this version: Slim Chaabane, Mohamed Jaoua. Un algorithme d’identification de frontières soumises à des conditions aux limites de Signorini. RR-3453, INRIA. 1998. �inria-00073237�. HAL Id: inria-00073237 https://hal.inria.fr/inria-00073237 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Un algorithme d’identification de frontières soumises à des conditions aux limites de Signorini Slim. . & Mohamed .

(3) . N˚ 3453 Juillet 1998 THÈME 4. ISSN 0249-6399. apport de recherche.

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(75) TÃ 7 4*¾ D'&À B DJ BAÀ 0ÊehUkUXWVg *%ehgTl j,bdu u)bdjaj WV`fWVu `fWjaueh a`fSVUXW~hbdu]Abdc]fehgagTWV` Vu]¤£,j,bduÊ¸ rI `ª ;ehcehgTlÊj,bdu r Wetvu8w yT·a`§ 1Qr W » rH¼ º ¸ rI W%lc!bd`fehu<l =^C ACBleh`^_ac]fehg m _ jaueh a`fSVUXW7bdu]Abdc]fehgagTWV`l_a]^bdg {c 3 Queh_aWVu~¸ cWV`^`fW7´K_TW p r »Ñ¸ cl ¸J¼ O"!$# ' 9µ» (l ¸J¼ C . o e m %l]^gTW7`fW¶p%ehg WsraW ¥ WVuUX m W º }» / ¼ m s£Tga]µj,bd{u 3. W&%o¸ º }» / ¼ &¸ 4 L WeX,¸ OzXjª jªTl_auq(¦ ' p o )*c r Àb ¥ ehuUXW a]^`^]^gTbd]ûW m s£Tga]fW4l_au º }» / ¼ j,bd{u 3 r ¼ Wed f g m WVc» rK*¼ }» 1 r,+\1QrK¼ k ¸ k ?SR p »Ñ¸ * c o ¬µb ¥ ehuUXW¶ a]^`^]^gTbd]ûW r W%lc p%ehg c]^g _TW§&p%eKWVupV]^W4l_au `fW¶leh_Tl²©W%lj,bhp%W m .W -!]^`^ WVu{c 3 / W.- 6 º }» / ¼ W[Xjª jMl_au 4L 2 )*gjauWVg,bdgK1c 0 32 }» / ¼JWVc3WVgkbdjaja`^]f´K_,bdg c p3 pR,bhpV_agTW m W%l ¥ ehgTpVc]fehgTl3cW%lc º W40:w¢¸ rI WVc W1¸ rI l 0 ´K_a]bdjaj,bduc]fWVgagTWVg c~ceh_acW%l m WV_HrMbd_ p%ehg Wsrao W ¢§aehg m m _a]^cj,bduq]^g cVu)bdc]fehgij,bdu j,bduc]fW%l ´K_TW 3 n. ßáàâßáã.

(76) m ^ ] O= m WVc» rK¼)»}1 r(+\1 rK¼ ¸ rI W[X ?HCp r k o )*c!j,bdu!l_a]^cW¶¸ I bdjaj,bduc]fWVg c`«¯WVgTlWVU a`fW 3 -¸ ¸ º }» / ¼ m ]^ = m WVc» rK¼)»}1 r + 1QrK¼ k ¸ »}/ ¼<2 |g¢j WV_ac m s£Tga]ûÀb m Vu]^%Wp%ed²«gTehuUÄbd`fW o eh]ûj,bdukWsrHWVUkja`fW=^VYBp4p%ehUkUXWÀnb ¥ ehuUXW`^]^gTbd]ûWp%ehgKc]^g _TW l_au »Þ¦¼ m WÀbkUÄbdga]fSVuW7l_a]^bdg cW 3 t3eh_au0 2 »Þ¦¼¨WVcl] 0F W%lcq_ag uWV`fSVWVUXWVg c m W 0 m bdgTl º }» / ¼Vu]¤£&bdg c 0 W[Xkl_au7»}-P/ ¦µ¼ -,¸ rI 0 W r »Ñ¸ rI 0¨ ¼ ?S? p - o )*gyjauWVg,bdg c_ag uWV`fSVWVUXWVg cp%ehg c]^gK _

(77) m W »Þ¦µ¼ m bdgTl º }» / ¼)§TehgMeh ac]fWVgKc!j eh_au _agTW7p%ehgTlc)bdgKcW rklcu]fps² cWVUXWVgKc!j el]^c]^W¶ÀbUÄbOvehu)bdc]fehgyl_a]^bdg cW 3 r ii -&¸ I 0 ii @

(78) r 0 . ii i. @. - . ii. . i. »Þ¦¼ H S &

(79) # Fh # S( #jy H \ = a?> -,¸ rI -,¸&¹I W wyk\» y ¼ - -#Z [oeh ]^Kc 0 ¶ y»Þ¦µ¼ WVc0F _ag jaueh`fehgaWVUXWVg c m W 0l_au º }» / ¼ 0 W[Xl_au7»}-P/ ¦µ¼)ªTz`fehul{3 K. &9 ;

(80) )% \k»\y ¼. ~eh_Tl!b%ehgTl 3. . . " # ! " $&%('*),+ d.-0/21.3547698;: .<"= 96 > @ ? > 0<BA % AD'FC EG dH-(/ =IA # % AD'FC E. ;. o. . ? p. JLKNMPO. m WVc» ro¼)»}1 r(+}1QrK¼ k ¸ rI W k ¸ ¹I w yQ wyVK»\y ¼ bW%p 3 Q[W m ]^ · ¸ ¹I l » ·>w + ·o¼ ¸ ¹I w ¸ Iº k k k WVc r@6 ¹ o\» y'¼ W[X m bdgTl7» »}/ ¼¼ n n |gyb m ehgTp 3 i -&¸ rI -&¸ ¹I ii Wey ii dfhgRTS 0* U4w dfhgR}o»\y'¼ 0V U ii. 0 ii l. ii ii ii -Z ii - k k i i i )*cehg m m _a]^c `«¯WsrH]flcWVgTp%W m _agTW7p%ehgTlc)bdgKcW º lcu]fpVcWVUXWVgKc!j el]^c]^WcWV`^`fW7´K_TW 3 r fhg ii -&¸ I l -&¸,¹I 0 ii " @ y 0 º º ii - i -Z ii i )*gpRTeh]fl]fl)l)bdg c0 WW

(81) y» 0¼)§ehgeh ac]fWVg c3j'eh_au_agTW p%ehgTlc)bdg cW X ]^g m Vj WVg m bdg cW m W 0WVc m W 7y ÀbqUÄbOvehu)bdc]fehg l_a]^bdg cW 3 r ii -,¸ I l -,¸ ¹I 0 ii " ii - ii @ y 0 -#Z i. ââ;ÚSHõ)ó 5°õ. i.

(82) @R. O'` P' v

(83) h. . 4¾*D'À&B DJ BAÀ! yÃ

(84)

(85) OÀ&BAÀ&Bá¾ D ! . ¦. 0ÊehUkUXWVg*%ehgTlj,bdu!Vc)bd a`^]û `fW¶`fWVUkUXW4l_a]^hbdgKc{3. @. K -,-¸ I. K¶[oeh]^c 0 r. |gyb23. ¦ Ir i»Þ¦¼ cWV`µ´ _TW¶l_ajaj» 0*¼ ¦ I r §aWVc!leh]^c0 WW

(86) i» 0¼)ª. . W X:%! `. -,¸ rI. 0 W r. - . |u{3. Ñ» ¸ rI 0 ¼cWed f R k #¸ r I t bdu!]^gKcVu)bdc]fehgMj,bdu j,bduc]fW%l§,ehgMeh ac]fWVg c{3 -,¸ rI . r. - . k. 0. »Ñ¸ rI 0¨ ¼. 0 r0U¡b%W%p 3 0 rQW 0 . r. º. W[X. 465 K¾ GK E &%&

(87) 6# Sh% FHS ju6

(88) )% }S# 4

(89) v% % %! ` ¦ ' ^6 % H!%

(90) µ¦ Ir vP c FH . % \&% SaP ; % GFH m

(91) # !Fa c# c jy H > r r@6 ¹ uT»Þ¦ I ¦Ì7¼cW[X. K ¶ »ÞH¦ L `¦ I7¼¨WVc4»Þ¦ Lr ¦ Ir ¼ r ¥ ehuUXWVg c m WV_Hr j,bduc]^c]fehgTl m Wk¦3§J]^`gTeh_Tl4l_ ,Xc m ehgTp m WkUXehgKcuWVu§p%ehUkjacWÄcWVg _ m _ `fWVUkUXW¶´ _T W r@6 ¹ *»Þ¦ I `¦ I¶c¼ W[X`~

(92) _ag m F] UJ%ehUXehujaRa]flUXW7jauS%l ehgMj'WV_ac l_ajaj elWVu ´K_TW¶¦ W X WVc ´K_TW 3 ¦ Ir W / r r º bW%p 3 X @(/ rº @ ºr @ / r @ r @ ~6~6~ @ / r @ r @ [K_ajaj elehgTl~´ _TW¶ÀbXl_a]^cW¶ ehugT%W *»Þ¦ Ir H¦ I¼ÊbX_an gTW¶hbdn `fWV_au m ®b m RTVuWVgTp%W YRHª&|g j WV_ac~bd`fehul!WVgyWsrocu)bd]ûW _agTW7l)eh_Tl l_a]^cW¶WVgTp%ehuW7]^g m Wsra%W¶j,bdû y cWV`^`fW7´K_TW 3 sl `fW%l l_a]^cW%lG/ r WVc r p%ehg WVuWVgKcWVul / WVc C - ~6~6~6~ 2 r ? p l r@6 *»Þ¦ I `¦ I¶c ¼ W zX o ¹ |gybd_au)bÄbd`fehulqj'eh_au!_ag p%WVuc)bd]^g ¹ - ~6~6~6~ 2 *» / `¦ I4¼ zX ? p o [oeh]^c 0 y» / ¼)"ª !~u ,hp%W r o r ? ps§ ehgl)bd]^c´ _ ]^` Wsro]flcW_aggTehU4 auW¡u%WVV` y ¹ XycWV`q´ _TWj eh_auceh_ac y$#zy ¹ ehgMbd]^c!l_ajaj» 0% ¼ / ª ßáàâßáã.

(93) r |u§ -&- ¸ I. kl_au / r r ª )*gibdjaja`^]f´ _,bdgKc!`fW`fWVUkUXW?k0 §Tehgyeh ac]fWVg cbdjauS%l!j,bhll)bdW7Àb`^]Ûk]^cW7WVgyJ3 -,¸ ¹I 0 W[X. W[X. -#Z. x`qWVgu%l_a`^cW ´ _TW -&-#¸ Z ¹I W X l_au / ª[K_au`«¯WVgTlWVU4 a`fW » / ¦HI¶¼)§*´K_a]qW%lc m WUXW%l_auW¡j'el]^c]^W m ®bdjauS%l o ? ps§'ehg b m ehgTp -&-¸,Z ¹I W XHªJ|up%WVc~WVgTlWVU a`fWp%ehgKc]fWVg c m W%l~eh_aWVucl§ pbdu!¦HI bk_agigTehU4 auW£Tga] m W p%ehUkj el)bdg cW%l~p%ehgagTWsraW%l§aWVc!p%W%l eh_aWVucllehg c m S%l `fehul p%ehg cWVgK_Tl m bdgTl ¦` L0p%W7´K_a]p%ehg cuW m ]^c o CASpsª.

(94). L & #" $ % &(') @ K E %&

(95) 6# h% Fh ¸ Lº 60

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(100) _ageh_aWVuc m W¦ cWV`q´ _TW

(101) ¦ L ª ¬W cRT%ehuSVUXr xW ?H§ gTeh_Tl m ehgagTWy`«¯WsrH]fr lcWVgTp%W m _ag0gTehU4 auW u%WV4` y ¹ XFcWV`~´K_TW§¨j eh_auceh_ac y # y ¹ §Êehg5bd]^c

(102) ¦ L ª,|g WVg m m _a]^c!´K_TW¸ L W[XXl_au

(103) j eh_au ceh_ac y #zy ¹ §aWVcqj,bdu l_a]^cW¶´K_TW¶¸ Lº W[XkjajMl_au

(104) yª,|u ¦HL0b_ag gTehU auW¶ 'ehugT m W¶p%ehUkj el)bdgKcW%l p%ehgagTWsrHW%l o _ag WVgTlWVU4 a`fW¶gTV`^]^Wbd a`fW7jauS%l&ps§aWVcq]^`JWVg u%l_a`^cW¶´ _TW ¸ Lº W[Xl_au H¦ L§ap%W4´ _a]µbhp)RTSVW¶Àb m VUXehgTlcu)bdc]fehgª . . 465 K¾ GK . µ¦. ». ·o¼)»Þ¦µ¼ ¦. P S

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(114) @?. O'` P' v

(115) h. %l]^gTehgTl ,j bdu º º VW c º uW%lj'W%pVc]^WVUXWVgKcq`fW%l m Vu]^%W%l j,bdu u)bdjaj ehuc bd_ m ehUÄbd]^gTW m W%l ¥ ehgTpVc]fehgagTWV`^`fW%lJb*§ ¸ ¹ º VW c ¸ ¹ º ªTz!`fehul 3 dfhg ¸ ¹I ¸ ¹L W d # '89]M k k 0ÊW¶cWVuUXW4Vc)bdg cp%ehgTlc)bdg c§Tl)b m Vu]^4j,bduqu)bdjaj'ehuc~bd_ m ehUÄbd]^gTW7W%lcqgK_a`WVcqj,bdu!l_a]^cW(b º W b I º wzb L º bW%p 3 b I º W dSfhg m ]^ · ii ¸ ¹I ii l dSfhg ¡· ¸ ¹I ¸ ¹I w dSfhg ¸ Iº ¸ ¹I ? p k k k k o ik in WVc b L º W d fhg*m ]^ J · iii k ¸ ¹L iiin l d fhg · k ¸ ¹L k ¸ ¹L w d fhg k ¸ Lº k ¸ ¹L S R p o )*gMbdjaja`^]f´K_,bdg c ? p Àb ¥ ehgTpVc]fehg cW%lc º W¸ Iº w¸ ¹I §Kja_a]fl W1¸ Iº l ¸ ¹I ´K_a]Jbdjaj,bduc]fWVgagTWVgKc!ceh_acW%l m WV_HrMbd_Hr p%ehg Wso raW §aehg m m _a]^c~´ _TW 3 n d fhg ¸ ¹I ¸ Iº W d fag · ¸ ¹I ¸ ¹I Ql d fhg m ]^ · ii ¸ ¹I ii ik in k k k k p%W7´K_a]«§Tb%W%p o ? ps§agTeh_Tl m ehgagTW 3 b I º Wdf g ¡· ¸ ¹IF ¸ ¹I l df g m ]^qj · iii k ¸ ¹I iii n HC p k k o ¬W7`fWVUkUXW¶YÄWVc o S R pÊgTeh_Tl m ehgagTWVgKc!j eh_au `fWV_au j,bdu{c 3 b L º W d fhg m ]^ · ii ¸ ¹L ii l d fag · ¸ ¹L ¸ ¹L ik in k k p%W7´K_a]µbhpRTSVW4Àb m VUXehgTlcu)bdc]fehg m _MjauWVUk]fSVuW7j,bduc]fW m _ cRT%ehuSVUXWª ¬µb m WV_Hro]fSVUXW¶j,bduc]fW m W¶p%W¶cRT%ehuSVUXW7l¯eh ac]fWVgKc!j,bdu ]^gKcVu)bdc]fehgMj,bdu j,bduc]fW*ª )*gM W UJWV{c 3 -,¸ ¹I -&¸ ¹I ¹ dfhg ¡· ¸ ¹I ¸ ¹IF W dfhg · ¸ wdfhg · ¸ ¹I. F I º -L/ - k k k k k k n p%W7´K_a]«§Hj,bdu ]^gKcWVu)bdc]fehgMj,bdu j,bduc]fW%l§TgTeh_Tl m ehgagTW 3 -,¸&¹I -&¸,¹ -,¸&¹I ¹ dSfhg · ¸ ¹I ¸ ¹I W l d fhg· º m ]^ ¸ wd· º I I -L/ k -*/ -#Z k k K Vb b I Jb L vb I Wii I ii f g Jvb L W ii L ii f g i i n i i n. x`WVg u%l_a`^cW7´K_TW 3. · m ]^ -&- ¸, ¹I k ¸ ¹I. l dSfhg. n. . ¨· -&- ¸, ¹I &- -#¸,Z ¹I. wd. n. · k ¸ ¹I k ¸ ¹I W l dfagk· º m ]^ -,-*¸ / ¹I k ¸ ¹I -&¸ ¹I -&¸ ¹I ¹ ¸ wd · ¸ ¹I I l df g · m ]^ - k -Z k n |u§,·W%lc j'ehucj,bdu!ÀbgTehuUÄbd`fW4l_auq¦3§ m ¯eMehg m m _a]^c{3 dSfhg . o. S? p. ßáàâßáã.

(116) · k ¸ ¹I k ¸ ¹I W l df g · º m ]^ -,-*¸ / ¹I k ¸ ¹I -,¸ ¹I -&¸ ¹I ¹ d f g · m ]^ ¸ wd ·&% I - k -#Z n. dSf g . |gybkbd_Tl)l] 3. l. · k iiik ¸ ¹I iii n ) wd · % iiik ¸ ¹I |u{3 m ]^ -&-L¸ / ¹I ¸ ¹I l -*- / ii ¸ ¹I ii W -&-L¸ / ¹I ¸ ¹I k ik i n WVc L -,¸&¹I , & ¸ ¹ I m ]^ k ¸ ¹I l - iii k ¸ ¹I iii n W - ¸ ¹I HCps§ o ps§ o dYvps§ o ps§TWVc o p gTeh_Tl m ehgagTWVg c§'b%W%p7`« R,bduUXehga]fpV]^c m W7¸ ¹I L 3 o -&¸ ¹I d

(117) *&· % ii ¸ ¹I ii b I º W( d¨·&% -#Z n l ik in WVc!WVgTp%ehuW 3 -,¸ ¹I -,¸ ¹I ! ' b I º Wed +· % d l -#Z n - ) n KW¶# U VUXW7ehg UXehgKcuW4´K_TW 3 -&¸,¹L -&¸,¹L w

(118) d ! ' b L º W l d!X·&% -#Z n -*) n d fhg. *m ]^ · iiik ¸ ¹I. n. . ii W l d fhg in. . p o. dYvp. ii. in o. p o. p o. ASp o. o. p.

(119) $ *:& $ > [K]ÊehgF]Ûkj elWÄ_ag 7 _Hr 9 O X l_au 4I7§µ`fWXjau]^gTpV]^j W m _FUÄbOro]Û_aU"gTeh_Tl4j WVuUXWVc m W m m _a]ûW´K_TWXÀb p%ehg m ]^c]fehgF[K]^gTehu]^ga]l_au¦ l)W4u m _a]^c7Ä_agTWp%ehg m ]^c]fehg -&¸ m ¹ Wc©noj W(;~WV_aUÄbdgagª )*gíW U WVc§'`fWUk]^ga]Û4_aU m W gTWXj'WV_ac VcuWbdccWV]^g cga]¨l_au 4I4§Jga]¨l_au¶¦ja_a]fl´K_TW -#Z I # X WVgceh_ac7j eh]^g c m WXÀb ¥ uehgKc]fSVuWÄe p%W Uk]^ga]Û_aUW%lc~bdccWV]^gKcª )*gip%ehgTl%´K_TWVgTp%W§,`fW4Uk]^ga]Û4_aU1W%lc~bdccWV]^gKc~l_au?4L~§TWVc]^`W%lc m ehgTp7g _a`«ªJ|gyWVg p%ehgTpV`^_ac´ _TWH¦ L W%lc¶K] m W ¥ bd_acW m Wk´K_Teh]*`fWUk]^ga]Û4_aU l)WVu)bd]^c4V bd`fWVUXWVg cbdccWV]^g c4l_au¶¦HL m ehgTpXl_au¶¦ª ¬¨¯WsrojauW%ll]fehg m _ u)b m ]fWVg c m W¶Àb ¥ ehgTpVc]fehgagTWV`^`fW b m WVK]fWVgK{c 3 -,¸ ¹ -,¸ ¹ b º W d!'"» L ¼ l » I ¼

(120) ¼ $y&· % -*) - ) n n p¯W%lc m ]ûW7´K_TW`«¯ehg uWVcueh_aW m W%lqu%l_a`^c)bdcl m {v m VUXehg cu%l m bdgTlq`fW7pbhlq`^]^gTbd]ûW =®T B©ª > [K]ehg bdjaja`^]f´K_TWk_ag 7 _H r 9 @ X¡l_au 4I4§ `fWjau]^gTpV]^j W m _íUÄbOrH]Û4_aU.gTeh_Tl7j'WVuUXWVc m Wkp%ehgTpV`^_au¹W m WÀb U#VUXW ¥ b*%ehgk´ _TW `fWÊUÄbOro]Û_aU ¸ " m W ¸ ¹I gTWÊj'WV_ac VcuWqbdccWV]^g cWVg_agj eh]^g c m W=4Ipbdu*ehgXb -,-#¸ Z I X WVg ceh_acqj eh]^g c m W~Àb ¥ uehg c]fSVuW7e ¡`fWUÄbOrH]Û4_aU W%lc!bdccWV]^gK c*ª 0ÊW~UÄbOro]Û_aU W%lc m ehgTpg _a`«§Hja_a]fl)´ _ ]^`W%lc bdccWV]^gKc4leh]^c7WVgí_agíj eh]^g c m W 4 L§Jl)eh]^cWVgí_agíj eh]^g c m W H¦ L~ªJx©`*WVgíu%l_a`^cWk´K_TW`¦ I W%lc¶K] m Wkpbdu7l]^gTehg ââ;SÚ Hõ)ó 5°õ .

(121) VY. O'` P' v

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