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Optimal control of Navier-Stokes equations using Lagrange-Galerkin methods

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(1)Optimal control of Navier-Stokes equations using Lagrange-Galerkin methods Gilles Fourestey, Marwan Moubachir. To cite this version: Gilles Fourestey, Marwan Moubachir. Optimal control of Navier-Stokes equations using LagrangeGalerkin methods. RR-4609, INRIA. 2002. �inria-00071976�. HAL Id: inria-00071976 https://hal.inria.fr/inria-00071976 Submitted on 23 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. Optimal control of Navier-Stokes equations using Lagrange-Galerkin methods Gilles Fourestey — Marwan Moubachir. N° 4609 Octobre 2002. ISSN 0249-6399. ISRN INRIA/RR--4609--FR+ENG. THÈME 4. apport de recherche.

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(46) _ Yª} Ç uvrŒjs{_-·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· = · = btrŒjYlmdfjsiÄjwr]r]~mnª|tdo^jwlvdfnspœns«lm[]_

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(62) {nIpBlv~mnsiTlv_Hu€l¯{Kjsuv_ ·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6`e– Œ· Ç · ` x _~mdo¨Yjwlvdf¨s_-{‹[]_K{‹µedfp]™qÆO{jsuv_‚nw«.lv[]_

(63) uvg]•#{yt{ifdopŒ™$•Šjs{‹µBlv~‹js{‹µedfp]™hr]~mnt{_ x g]~m_/·&·6`I` Œ· – Z [Œ_ x ~vdf¨s_Kp/{Kjª¨ed®l€y·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6` X Œ· –]· Ç ¸¯_ ½ p]dolvdfnspqnw«.lv[]_-r]~mns•Œio_K^ ·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6` X Œ· –]· – ‡nsp]­ x doÉ#_~m_pBlvdkjw•]dfifd®l€y,dfioifgŒuzlv~‹jYlvdfnsp,«ÁnI~§lm[]_-j x jsrtlmjwlvdf¨s_¯•Šjs{‹µBlv~‹js{‹µedfp]™

(64) uv{‹[Œ_^,_¤` X Œ· –]· = ¸¯dkuv{~v_lv_-™s~‹j x do_KpIl({nIpŒuvdfuzlv_KpŒ{y¾gŠuzdfp]™$lv[]_

(65) uvg]•#{yt{ifdfp]™,•Œjs{‹µBlv~‹js{‹µedfp]™jsio™Inw­ ~vdolv[]^ ·-·&·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6`B Œ· –]· ` „£pt´Œg]_KpŒ{_&ns«lv[Œ_

(66) uzr]ifdolzlvdfp]™rŒjs~mjs^,_lv_K~·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· ]X Ç Œ· –]· X be_K{nsp x ns~ x _~(¿.À um{‹[]_^,_&jwp x dfpBlv_™I~mjwlvdfnsp/~vgŒio_·&·-·&·&·-·&·-·&·-·&·&·‚·&· X ` Cb\ ŸY  F Ÿ Ë Ð .¡w;ž B Ï A  ]Ÿ Ï A Ð  ’Ÿ Ï žH tž B AF ¡ Ì (B A Ð Ë Ÿ ae Z · Ç 3nIpBlv~mnsi#r]~mns•]if_^ uv_lvlvdfp]™ ·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· X  Z · – jIuz_-ns«.juvdop]™Iio_-[Œjs~v^,nIp]df{‚jwpŒ™sg]ikjw~ ¨s_Kiont{d®l€yU·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· X/Z Z · –]· Ç 3nspBlvdfpeg]nsgŒunIrtlvdf^jwi{nspBlv~mnsi ·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· X/Z Z · –]· – ‡g]^,_~mdk{jwi5_|trŠ_K~vdf^,_pBlmu¯·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· X — Z · = jIuz_-ns«.uv_¨s_K~mjsi#[Œjw~m^,nsp]dk{u)·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I Ð BÈž B Ï A  S Ÿ Ï

(67) K? Ð Ë A ž B .¡I tž B Ï A Ï SS  tŸ Í Ë (ÈЛ Ï A Ð. tŸ Ì ¡ Ï A   B?Ð ( Ï  ]Ð. Ï A › ( ś Ï GÐ B Ë  eP> —Œ· Ç }§~vnI•]io_K^)uv_lvlvdfp]™Iu6·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· B– —Œ· – Z [Œ_&~v_H{”lmjsp]™sgŒifjs~{yBifdfp x _~·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· s` —Œ· = jw~m^,nsp]dk{‚rŠ_K~zlmg]~v•ŠjYlvdfnsp/nw«lv[]_&dfpt´ŒnY¦Å¨s_Kiont{dol€y ·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  X —Œ· ` btyBpBlm[]_lmdf{-ifnIj x ·-·&·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I —Œ· X ƒ¯~v•Œd®lm~mjs~vy[Œjs~v^,nIp]df{‚ifnIj x u ·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  Z Ð BÈž B Ï A  S Ÿ Ï

(68) K? Ð Ë A ž B .¡I tž B Ï A Ï SS  tŸ Í Ë (ÈЛ Ï A Ð. tŸ Ì ¡ Ï A   B?Ð ( Ï  ]Ð. Ï A

(69) Ï VPB AGF › ( Q› Ï GÐ B Ë  e ‘Œ· Ç }§~vnI•]io_K^)uv_lvlvdfp]™Iu6·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I‘ ‘Œ· – †_Kuv[/^,nY¨I_^,_pBl(jwif™sns~mdolv[]^·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z  ‘Œ· = ¸ˆdkuv{~v_lv_&™s~‹j x do_KpIl({nIpŒuvdfuzlv_KpŒ{y ·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z = ‘Œ· ` jw~m^,nsp]dk{‚rŠ_K~zlmg]~v•ŠjYlvdfnsp/nw«lv[]_&dfpt´ŒnY¦Å¨s_Kiont{dol€y ·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z'Z Ï A !¡ ( . B Ï A > $I eK\ B. . . . . . . . . . . . . . . . ÑÁÒ.ÓÄÑÕÔ.

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(78) Ç Z Ǘ Ǒ –s –Ç –I– –= –w` –X –s –Z –s— –s‘ =  = Ç = – ='= =` = X =  = Z = —. X. 

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(80) +œ(' –s¸ r]~mnw€_H{”lvdfnsp/jwifnsp]™j$™sdf¨s_Kpœ¨I_K{”lmns~ ·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· – X }§~vns€_K{”lmdonIpnIpœlm[]_&_ x ™s_-nw«Ejlv~mdfjsp]™sif_·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· –s = ¸ r]~mnw€_H{”lvdfnsp·-·&·&·-·&·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· – Z }§~vns€_K{”lmdonIpnIpœlm[]_&_ x ™s_-nw«Ejlv_lm~mjs[]_ x ~vnIp ·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· –s— “ˆr]lvdf^$df¼Kjwlvdfnsplvnensiu€lm~vgŒ{lvg]~m_ ·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· = Z }Ejs~mjsioif_iÄuzlv~‹jYlm_™sy,«ÁnI~ lv[]_&™I~mj x df_pBl {ns^,r]gtl‹jYlmdonIp·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· = — }nIdfuv_g]dfifio_ f.ionY¦ x ns^jsdop²·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6` Ç f.df~muzlns~ x _K~ {nspe¨s_K~v™I_pŒ{_¯ns«lv[Œ_ ½ p]dolv_ x doÉT_K~v_KpŒ{_&™I~mj x df_pBl3«ÁnI~ lv[]_&}nIdfuv_g]dfifio_ f.ionY¦ ` Ç “ˆr]lvdf^$df¼Kjwlvdfnsp/r]~mne{_Kumu«Áns~ lm[]_&}nsdkuz_Kg]doifif_ˆ´ŠnY¦&·¯·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6` = Z~‹js{‹µedopŒ™${nIuzl«ÁgŒpŒ{”lmdonIp x _K{~m_KjIuz_ˆ«Áns~ lm[]_&}nsdkuz_Kg]dfioif_‚´ŒnY¦&·’·-·&·&·-·&·-·&·-·&·&·‚·&·6` = f.df~muzlœjwp x uz_H{nsp x nI~ x _~¾uvg]•T{yt{ifdop]™•Šjs{‹µBlv~‹js{‹µedfp]™Iu$uv{‹[Œ_^,_<™s~‹j x df_pBlœ{‹[]_H{‹µedop]™ ¦(dolv[/lm[]_-lv[]_Kns~m_lmdf{KjwiÄuvionIrŠ_‚«Áns~ lm[]_&}nsdkuz_Kg]dfioif_‚´ŒnY¦ ·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6`I` f.df~muzlOns~ x _K~¿jw™I~mjsp]™s_­£À-jsio_K~vµedfp

(81) uv{‹[]_K^,_ jsr]r]ifdo_ x lvn-lv[Œ_ x ~vdf¨s_Kp${Kjª¨ed®l€y&´ŠnY¦­À‚~‹jY­ x do_KpIl¯{nspŒuvdku€lm_pŒ{yœ{‹[]_K{‹µedfp]™$¦(d®lm[q‡‚be„ = ·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6`B f.df~muzl§nI~ x _K~§jwp x uz_H{nIp x ns~ x _~§¿jw™I~mjsp]™s_­£À-jsio_K~vµedfpum{‹[]_^,_¯jsr]r]ifdo_ x lvn&lm[]_ x ~mdf¨s_p {Kjª¨ed®l€y´ŠnY¦³­À‚~‹j x df_pBl{‹[]_K{‹µedfp]™ ·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6` Z pŒns~m^0nw«lv[]_ x ~mdf¨s_pq{jª¨edol€yœjYluzlv_Hj x yœuzlmjwlv_ ·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·6`B— l pŒns~m^0nw«lv[]_&ifdopŒ_Kjw~mdf¼_ x/x ~mdo¨I_pq{jª¨edol€y¾jYl¯uzlv_Kj x yœuzlmjwlv_ ·-·&·&·-·&·-·&·-·&·&·‚·&·6`B‘ l btr]iodolzlmdopŒ™rŒjw~‹jw^,_lm_~ _ ¾{do_KpŒ{yœ«Áns~ lv[Œ_

(82) uz_H{nsp x nI~ x _K~(¿À uv{‹[]_K^,_‚¦(dolv[ ∆t = 0.2 X – btr]iodolzlmdopŒ™rŒjw~‹jw^,_lm_~ _ ¾{do_KpŒ{yœ«Áns~ lv[Œ_

(83) uz_H{nsp x nI~ x _K~(¿À uv{‹[]_K^,_‚¦(dolv[ ∆t = 0.05 X = 3nI^,r]gtlmjwlvdfnspŒjsi5^,_Kuv[qjs~vnIg]p x lv[]_

(84) {yBifdfp x _~7·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· X ‘ = ¸{nIpBlvnsgŒ~rŒionslnw«lm[]_

(85) {nBu€l «Ág]pŠ{”lvdfnsp ·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I S ) –s¸{nIpBlvnsgŒ~rŒionslnw«lm[]_

(86) {nBu€l «Ág]pŠ{”lvdfnsp j(ρ, ·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  Ç ¸ˆ~‹jw™,_¨Insifgtlvdfnspœ«Áns~ x doÉT_K~v_KpBl¯{nspBlm~vnIiTrŠj(ρ, jw~‹jw^,S_lm)_~‹u ·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· B– ¿d®«Õl_K¨snIiogtlmdonIpœ«ÁnI~ x doÉT_K~v_KpIl¯{nspBlv~mnsi#rŒjs~mjs^,_lv_K~mu ·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  =  nI~zlm_|uv[]_ x]x dfp]™,dop/lv[Œ_&g]pŒ{nIpBlv~mnsifio_ x {Kjsuv_œ·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  =  nI~zlm_|uv[]_ x]x dfp]™,dop/lv[Œ_&nsrtlmdo^jwi{nspBlm~vnIioif_ x {jsuv_ ·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  = “ˆr]lvdf^$df¼Kjwlvdfnsp/rŒjw~‹jw^,_lv_~‹u_¨Insifgtlvdfnsp ·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· s` “ˆr]lvdf^,jsi¸ˆ~mjs™hlmdo^,_&_¨Insifgtlvdfnsp­ = [Œjw~m^,nsp]dk{u({Kjsuv_ ·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  X †_Kjwp x ~‹jw™,_¨Insifgtlvdfnsp]­ = [Œjs~v^,nIp]df{Ku {jsuv_ ·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I 3nBu€l «Ág]pŠ{”lvdfnspq™I~mj x do_KpBl x _K{~m_KjIuz_ˆ­ = [Œjw~m^,nsp]dk{u({Kjsuv_6·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&·  Z ¿d®«Õl¯jsp xqx ~‹jw™«Áns~ lv[Œ_

(87) ‰(`,~v_H{”lmjsp]™sgŒifjs~ {yeiodfp x _K~ ·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I— }§~v_Huvuvg]~m_‚rŒ~vn ½ io_‚«ÁnI~ lv[]_

(88) ‰(`,~m_K{lmjwpŒ™sg]ikjw~ {yeifdop x _~ ·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· I‘ 3nIpBlv~mnsi#rŒjw~‹jw^,_lv_~‹u x g]~mdopŒ™$nIrtlvdf^,do¼HjYlmdonIp/uzlv_KrŒu·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z  ¸ˆ~‹jw™,[]dku€lmns~my,­§dopŒd®lmdfjsięsg]_Huvu jsp x ½ pŒjwiÄ{nspBlv~mnsi/·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· ZtÇ À‚~‹j x do_KpIl x _H{~m_KjIuz_ x g]~mdopŒ™$nIrtlvdf^,do¼HjYlmdonIp/uzlv_KrŒu<·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z – ¿d®«Õl[ŒdfuzlvnI~vy­Odfp]dolvdkjwięIg]_Kumu(jwp x ½ pŠjwi{nspBlv~mnsi·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z = 3nIpBlv~mnsi#rŒjw~‹jw^,_lv_~‹u x g]~mdopŒ™$nIrtlvdf^,do¼HjYlmdonIp/uzlv_KrŒu·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z ` 3nBu€l «Ág]pŠ{”lvdfnsp x _H{~m_KjIuz_ x g]~mdopŒ™nsrtlmdo^,df¼KjYlmdonIp/uzlv_KrŒu)·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z2X À‚~‹j x do_KpIl x _H{~m_KjIuz_ x g]~mdopŒ™$nIrtlvdf^,do¼HjYlmdonIp/uzlv_KrŒu<·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· Z  †_Kuv[/¨I_~vlvdk{_Hu3rTnIuvdolvdfnspqjYl(l  Iu Áif_«Õ l (jwp x t = 16s Á~mdo™I[I l ·&·-·&·‚·&·-·&·&·-·&· Z2Z 2 2. ÓÓ ï

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(96) ^,nY¨edfp]™œ‰(` rŒ~vn ½ io_‚­3À‚~‹j x df_pBl lv_Hu€l˜·-·&·-·&·&·‚·&·-·&·-·&·&·-·&·-·&·‚·&·-·&·&·-·&·-·&·-·&·&·‚·&· ƒ¯^,r]iodolvg x _h{nIpBlv~mnsi#rŒjw~‹jw^,_lv_~ x gŒ~vdfp]™$lv[]_&nIrtlvdf^,do¼HjYlmdonIp/uzlv_KrŒu ·-·&·-·&·-·&·&·‚·&· fŒ~v_H¶Bg]_pŒ{yœ{nIpBlv~mnsi#rŒjw~‹jw^,_lv_~ x gŒ~vdfp]™$lv[]_&nIrtlvdf^,do¼HjYlmdonIpquzlv_KrŒu ·-·&·-·&·-·&·&·‚·&· 3nBu€l «Ág]pŠ{”lvdfnspq•T_[Œjª¨edfnsg]~ x g]~vdfp]™$lv[Œ_&nsrtlmdo^,df¼KjYlmdonIp/uzlv_KrŒu·-·&·&·-·&·-·&·-·&·&·‚·&· À‚~‹j x do_KpIl(ns«lm[]_

(97) {nBu€l «Ág]pŒ{lvdfnspq•T_[Œjª¨edfnsg]~ x g]~mdfp]™$lv[]_&nIrtlvdf^,do¼HjYlvdfnspquzlv_rŠu ·&·‚·&·. Z. —. ‘ —I —Ç —B– Z. ÑÁÒ.ÓÄÑÕÔ.

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(108) nw«T{‹[Œjw~‹js{lv_~mdku€lmdf{rŒjwlv[Œu.dfphnI~ x _K~Älmn‚jsr]r]~mnª|edf^jYlm_Elv[]_^jYlm_~mdfjsislmdo^,_ x _~mdo¨Yjwlvdf¨s_s· Z [Œ_

(109) gŒuv_

(110) nw«lv[Œdfu‚jwif™snI~vdolv[] ^  ‘ .dfpBlv~mn x gŒ{_HuuznI^,_&lv_H{‹[]p]dk{jwi x d {g]iolvdf_Ku¯j x]x ~m_Kumuz_ x dop<lm[]dkurŒjsrŠ_K~K· “ˆpŠ{_¾lm[]_ x dfum{~m_lm_œifdopŒ_Kjw~mdf¼_ x uzytuzlv_^ dku_Kuzlmjw•Œiodkuz[Œ_ x ¥Odolmu$uvnsifgtlvdfnsp¬_pBlv_K~muhlv[]_q{nI^,r]gtlmjwlvdfnspns« {nIuzlO«Ág]pŒ{lvdfnsp¾™I~mj x df_pBlmuEdfpe¨snIio¨I_ x dop,lm[]_¯nIrtlvdf^,do¼HjYlmdonIpionensr¦([Œdf{‹[^,dfp]do^,df¼_Hu3j&™Ido¨I_pnI•t€_K{lvdf¨s_s· Z¦n x d®É#_~m_pBljwr]rŒiodk{jwlvdfnspŒu(js~v_-j x]x ~v_Huvuv_ x Æ. “ˆp]_,{nspŠ{_~mp]dfp]™lm[]_ x ~mjs™œ~m_ x gŒ{lvdfnsp©js~vnIg]p x jœ~vnslmjwlvdfp]™{do~‹{g]ikjw~‚{yeifdop x ~vdk{jsi•Tn x y<gŒuvdop]™ d®l‹uOjwpŒ™sg]ikjw~E¨I_ifne{d®l€yI·Z []dkuEr]~mns•Œio_K^[Œjsu•T__Kp¾jwif~v_Hj x y-lm~v_HjYlm_ x •By$uz_K¨s_K~mjsiejsgtlv[Œns~‹ u  –st¥I–w‘ >· “ˆg]~

(111) ™InIjwiEdku&^jwdfp]ify†lmn<¨ªjsiod x jwlv_nsgŒ~

(112) jwr]r]~mnIjI{‹[’•ey±jwpŒjsioytuvdop]™/dolmuhjs•]dfiodol€y©ns« x _HjwifdopŒ™/¦(dolv[ p]nsp]­?lm~vdf¨edfjsiÄ{nspBlm~vnIiTrŒ~vnI•]io_K^0•T_pŒ{‹[]^js~vµtuK·. ’_jsifuvn x _Kjsit¦(d®lm[$jspd x _pBlmd ½ {KjYlmdonIpr]~mns•Œio_K^ lv[ŒjwlE^,jªyh_pBlv_K~jsp$js_~mne_ikjsuzlvdk{§uzlmjw•Œdoifd®l€y

(113) lvnensi x _Kum{~mdf•Š_ x df

(114) p  = ` >·„ùl&{nspŒuvdku€l‹uˆdfp’lv~myBdfp]™lvn/d x _pBlvdo«Áy<«Èjw~v­ ½ _i x •Tnsg]p x jw~my†{nIp x dolvdfnspŒuˆ«Á~mns^ lv[]_hµBpŒnY¦(io_ x ™s_&ns«´ŒgŒd x ifnIj x ulvdf^,_

(115) []dku€lmns~myœnIp†j,™sdf¨s_pq•]ifgtɱ•Šn x y/dop ½ |e_ x ns~rŒ~v_Huv{~vdf•Š_ x ^,nY¨Bdfp]™&{nIp ½ ™Ig]~mjwlvdfnspÄ·Z []_ ½ pŒjsiŒjwr]rŒiodk{jwlvdfnsp,uz[]nIg]i x •T_ lv[]_js_~mne_ikjsuzlvdk{3uzlmjs•]doifdol€yhjwpŒjsioytuvdfu nw«E{df¨edoi5_Kp]™sdfp]__K~vdfp]™uvg]rŠ_K~z­£uzlv~mgŒ{”lmg]~v_Hu  = ‘t¥Š–e¥ Ç – ù·   -8/ - 

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(117) ¶BgŒjspIlmd®l€y. σ(u, p) = −p I +ν(∇u + ∇T u). ÓÓ ï

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(129) Z

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(132). Γs. J(u, p) =. Z. J(u, p) =. Z. = .

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(134) σ(u, p) · n d Γ · e1 − Fd (t)

(135)

(136) d t. Z []_&^,_Kjsp/um¶BgŒjw~m_-¨ªjsiogŒ_ˆlmnj,™sdf¨s_pœlmjs~v™I_lK¥ 0. T. Z. È` . (u − ud )2 d Ω d t. ¸ˆdf~vdk{‹[]if_l3•TnsgŒp x jw~my{nIpIlm~vnIi]«ÁnI~§lm[]_‚‡ˆjª¨edo_K~z­€bBlvnIµs_HuOuzytuzlv_K^»[Šjsu3•Š_K_pj x]x ~m_Kumuz_ x nsplm[]_ˆlv[]_Kns~m_lz­ dk{jsi5io_K¨s_Ki5dop Ç ‘t¥Š– X Ädop/lm[]_h–w¸ {Kjsuv_s·EZ []dku(dfu(j$p]nIpœlm~vdf¨edfjsi5r]~mns•]if_^q¥t_KuvrT_K{dkjwififyœ{nIpŒ{_K~vp]dfp]™lv[]_ ~m_™Ig]ikjw~md®l€yqnw«3lv[]_•TnsgŒp x jw~my†{nIpBlv~mnsi  Ç —ù·$±_¾uv[ŒjwifiEp]nwl

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(138) {nIpIlmdopeg]nIgŒu™s~‹j x df_pBl ∇J · . 0. Γout.  

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(191)     ! % . . ƒllv[Œdfu(uzlmjs™s_s¥euv_¨s_K~mjsiŠnIrtlvdf^,do¼HjYlmdonIpœu€lm~mjwlv_K™sdf_Ku{jwp•Š_&{nspŒuvd x _~m_ x ¥B¦3_‚[Œjª¨I_ˆ{‹[]nenIuv_p¾lvn$dope¨s_Hu€lmd®­ B™ jYlm_&lv[]_hgŒuz_nw«l€¦n¾d®lm_~‹jYlmdo¨I_&nsrtlmdo^,df¼KjYlmdonIp<r]~mne{_ x gŒ~v_Hu lv[Œjwlˆ^jªy/p]__ x nsp]ify/lv[]_h{ns^,r]gtl‹jYlmdonIp ns«lm[]_

(192) {nBu€l «Ág]pŒ{lvdfnspq™s~‹j x do_KpIljwl _Hjs{‹[/^,dop]df^,do¼HjYlmdonIpquzlv_KrÄ·EZ []_ ½ ~‹u€l(nIp]_&dfu ~m_«Á_K~v_ x jIulv[]_

(193) {nspt­ €gŒ™sgŒjwlv_$™s~‹j x df_pBl‚jsio™Ins~md®lm[]^ dop’lv[]_,{nIpBlv_|el-nw«§pŒnsp]ifdopŒ_Kjw~ˆ­(p]nspŠ{nspe¨I_|qnsrtlmdo^,df¼KjYlmdonIp©r]~mns•Œio_K^,u jsp x dkugŒuvgŒjwifioyuv__Kpjsuj ½ ~muzl ns~ x _~nsrtlmdo^,df¼Kjwlvdfnspœjsio™Ins~md®lm[]^q·Z [Œ_‚uv_K{nsp x nIp]_‚dfu~m_«Á_~m_ x jsu3lv[]_ ' f.À&b$^$_lv[]n x jsp x •Š_KionIp]™IuElvn&lm[]_¯uv_K{nIp x ns~ x _~3¶IgŠjsuvd®­£‡_K¦ lvnsp,nIrtlvdf^,do¼HjYlvdfnspjsio™Ins~md®lm[]^uO{ikjsumu· ±_

(194) uv[Œjwifi x _Kum{~mdf•Š_-•Tnwlv[<uzlv~‹jYlm_™sdf_Kudfp/lv[]_&p]_|el¯uz_H{”lvdfnsp/dfpqjsp/js•Œuzlv~‹js{”l(uv_lzlmdopŒ™Œ· Ï A F  tž Ë F ŸY ]GÐ B Ë A ž

(195) Ë ž E Ï Ð D

(196) D

(197) > ¿_l(gŒu{nspŒuvd x _~lv[Œ_-«ÁnsifionY¦(dfp]™jw•Œuzlv~‹js{l^,dfp]df^$df¼Kjwlvdfnspqr]~mns•]if_^q¥  ȑ u ∈ H, j(u) ≤ j(v), ∀ v ∈ H ¦([Œ_~m_ H u€l‹jwp x uO«ÁnI~3jwp doif•T_~vluvrŒjs{_¯jwp x j : H → R dfuj x doÉ#_~m_pBlvdkjw•]if_(«Ág]pŠ{”lvdfnspŠjwiTnsp H ·EZ []_ 3nIpY€g]™IgŒjYlm_À‚~mj x df_pBl$jsio™Ins~md®lm[]^jsr]r]ifdo_ x lvn’r]~mns•]if_^ ȑ

(198) {Kjwp•T_ x _Huv{~vdf•T_ x dfplm[]_¾«ÁnIioifnY¦(dopŒ™ ^jsp]p]_~H¥. bBlv_Kr 0 3[]nenIuv_ u ∈ H  bensif¨s_‚«ÁnI~ uzgŠ{‹[œlm[ŒjYl g ∈ H, < j (u ), v > , ∀ v ∈ H (g , v) = ’_

(199) uv_l . . 0. 0. 0. H. 0. H∗. 0. H. w0 = g 0. Bb lv_Kr n + 1 f]nI~ ¥t¦_-jIuvuvg]^,_‚lv[Œjwl (u , g , w ) dfujwl(nsg]~ x dkuvrŠnBuvjsi?¥]jsp x ¦_&uz_K_µ«ÁnI~ n≥0 lv[Šjwp]µtulvn,lm[]_-«ÁnsififnY¦(dop]™,uzlv_rŠu-Æ (u ,g ,w ) Ç · ¿dopŒ_

(200) uz_Hjw~‹{‹[ÄÆE3nI^,r]gtlv_ ρ uvgŒ{‹[/lv[ŠjYlK¥  benIio¨I_ˆ«ÁnI~ uvgŒ{‹[lv[Œjwl ρ ∈ R, (LS) j(u − ρ w ) ≤ j(u − ρw ), ∀ ρ ∈ R ±_

(201) uv_l n. n+1. n+1. n. n. n+1. n. n. n. n n. n. n. un+1 = un − ρn wn ,. ÑÁÒ.ÓÄÑÕÔ.

(202) ÇIÇ. 

(203)  

(204)  !#"$%&'!()*+

(205) 

(206) (,+-(!/.10 .2

(207) -.'!3

(208) 4&

(209) 5 6879;:<(. –]·(À‚~‹j x do_KpIl{ns^,r]gtl‹jYlvdfnsp±Æ  bensif¨s_ˆ«Áns~. g n+1 ∈ H, n+1 (g , v)H = H ∗ < j 0 (un+1 ), v >H ,. =. ·3nIpe¨s_~m™s_KpŒ{_¯lv_Hu€lHÆ „ù« ¥elv[]_

(210) uvnsifgtlvdfnsp/dku(™sdf¨s_Kpœ•ey ≤. uzgŒ{‹[lv[ŠjYl ∀v ∈ H. kg n+1 k kg 0 k. u = un+1. gŒp]io_HuvuK¥t¦_

(211) {ns^,r]g]lv_-nsp]_-ns«lm[]_-«ÁnsifionY¦(dfp]™,¶BgŒjwpBlvdolvdf_KuK¥ j kg k f.io_lm{‹[]_K~z­£‰(__K¨s_Hu , γ = kg k • (g ,g − g ) }nIifjI{‹µI­£‰(df•]df\~m_ γ = , kg k Z [Œ_pÄ¥]¦_&uz_l n+1 2. n. n 2. n+1. n. n+1. n. n 2. wn+1 = g n+1 + γn wn. Š` ·„ùlm_~‹jYlmdonIpifnensr±ÆE¦3_&uv_l n := n + 1 ¥]¦_-{ns^,_-•Œjs{‹µ¾lvnu€lm_r n + 1 · Z [Œ_œdfpB¨I_Kuzlvdf™Ijwlvdfnspnw«(lm[]_{nspe¨s_K~v™I_pŒ{_nw«(lm[]_<3nIpY€g]™BjYlv_œ™s~‹j x df_pBljsio™Ins~md®lm[]^ ¦ jsurT_~v«Áns~m^$_ x nIplv[Œ_/dfp ½ pŒd®lm_ x do^,_pŠuzdfnspŒjsi3if_¨I_i3«Áns~¶BgŒj x ~mjwlvdk{œ{nIpe¨s_|±«Ág]pŒ{lvdfnspŒjsifuK·±_/~m_«Á_K~hlmn

(212) 㖠Z «ÁnI~$j ~m_¨edf_¦»nw«uvgŒ{‹[²~v_Huzg]iolmuK·qZ []_Kuv_~v_HuzgŒi®l‹uh{jsp]p]nsl

(213) •Š_œjsr]r]ifdo_ x dopnIg]~h{Kjsuv_s¥uzdfpŒ{_¦_ x _HjwiO¦(dolv[j pŒnspt­£{nIpB¨I_|5¥spŒnspt­ùiodfp]_Hjw~O«ÁgŒpŒ{”lmdonIpŒjwi>·§bBlvdfifi?¥enspœlv[]_‚peg]^,_~mdf{KjwiTrŠnIdopBl nw«¨edf_¦lm[]_& À ^$_lv[]n x [Œjsu uv[]nY¦(p™I~v_HjYl§{nIpe¨s_~m™s_KpŒ{_ «Á_HjYlmg]~v_HuOdfp¾^jwpey,jwr]r]ifdk{jYlmdonIpŒuK¥Bjsp x rŒjw~vlvdk{gŒifjs~vify

(214) «ÁnI~3nIrtlvdf^jwiŠ{nspBlv~mnsi ns«´ŒgŒd x ^,n x _iku 㖠= ¥Š–s— ù·. . ] B Í U<ËBÎ ž Ï A   ( F Ï

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