Adaptation de maillage non structuré 3D pour les problèmes instationnaires.Application à la mécanique des fluides
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Adaptation de maillage non structuré 3D pour les problèmes instationnaires. Application à la mécanique des fluides Frédéric Alauzet — Pascal J. Frey — Paul Louis George — Bijan Mohammadi. N° 4981 31 octobre 2003. ISSN 0249-6399. ISRN INRIA/RR--4981--FR. THÈME 4. apport de recherche.
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(11) . Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
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(26) Æ\ÇsË{ÊDÈ3Æ Ë/ÇÊDÈÌ-ÈkɪÅÉÍÌÍÇ ÅTÌFË/Ê?È. . . zTvtu-y/r{zT|RµLß,|·l =lky¯ºLvª© u-y/rtvtr{wBy/r{zT| lvª©`w wB~-ywBy/r{zT| ljwBrtvtvxwBlv{zT © u-|·¯wBv{ku-v=rt| ywBy/r{zT|\² | wBrt lrt|y/ z u-rty¯º lkvxwBy/rt¼lkjmlk|y°mh wT³ul*h wB|-lkjmlk|y lojwBrtvtvxwBlà l r{ k ¬ky/r{wBy/r{zT|»lky lzT|-|lky/rt¼Lrty ¬¾Ä?ºu-|l zTu- l ©Âlk/ lku-.rtjH~"zT/ywB|y l ulo°*vª© rt|y lk/~"zTvxwBy/r{zT| lvxw*zTvtu-y/r{zT| l9vª©`wB|kr{lk|fjwBrtvtvxwBlo/u-v{l9|zTu-¼l¯wBuRµ zTu-} lkjm¬ r{lk¯º¾rtvTÐÞwBu-y zTrty=y/ zTu-¼lk u-||zTu-¼l¯wBu h¬kjw © rt|sy lk/~ zTvxwBy/r{zT|*³u-rszT//rtl l~- zT-v{ikjmlº( zTrtyrtvVЪwBu-y rtjHrt|ulkHv{l|zTj- l ©`w wB~-ywBy/r{zT| ljwBrtvtvxwBl·°fl =lky/ulk wA¿| lÎzT|y/" !Tv{lk&vª©Âlk/ lku- ul®°±vª© rt|sy lk/~ zTvxwBy/r{zT|Rµ¨Á w lku\\r{ikjml®zTvtu-y/r{zT|¦°±vª©`w¼wB|ywBl © ky/ l ©`wB~-~-vtr{¯wBy/r{zT| ¬k|¬kwBv{l&lkyrt| ¬k~ lk| wB|sy l lvxwjm¬ky/hz l l ¬ zTvtu-y/r{zT|cu-y/rtvtr{¬lº |zTu}wBvtv{zT| zT|ohzTr{/rt.lky/y lwB~-~- zL hlµ Ô9wB| lky/y lVwB~-~- zL hlº¯vxw¨~- zL¬ u- l ©`w wB~-ywBy/r{zT| l(jwBrtvtvxwBl zTrty ¬ u-rt l(v{l(|zTj- l y zTywBv ©`w wB~-ywBy/r{zT||¬l DwBrt l}~"zTu-9l "lky/ulk9vxw&/rtju-vxwBy/r{zT|»jwBr{9lkvtv{l zTrty9zT|lk/¼lk vxwc~- ¬kr{/r{zT| lfvxw zTvtu-y/r{zT|y zTu-yÎlk| ¬k¼LrtywB|y l±jwBrtvtv{lku-|l±y/ zT~wB| l±~ wB/y/r{l u zTjwBrt|l l,¯wBv{ku-vªµ¾ 1zTu-lkvxw\º lku\9~"zTrt|y 1kv{l?ÐÞ zTrt¼lk|y ky/ l(~-/r{ lk|zTjH~-y 0l / ~- ¬ rt l vª©Â¬k¼zTvtu-y/r{zT| l~-h¬k|zTjmik|l~-h\/r{³ul wB|v{l zTjwBrt|l&lky (ii) lk|-/r{ h-rtÃÞw (i) wB~-y lkDÄ9v{l jwBrtvtvxwBl wB|y zTu-y lv{l ¬kr{zT|$z #®l ~-h¬k|zTjmik|l¬k¼zTvtulk|y¯µ §¨lky/y lwB~-~- zshl zTrty ky/ lH¼TwBvtr ¬l&/u-u-|Ò~- zT-v{ikjmlÏÃF ¬lkvýÄ9 lk~- ¬ lk|sywBy/rÀÐ l/rtju\² vxwBy/r{zT|9rt| ywBy/r{zT|-| wBrt l¯µ ¶}| ¬/rt lmw¼zTrtu-|l¬zTjm¬ky//r{lHzTjH~-v{l?-lHlky l~-h¬k|zTjmik|l |zT|&~- ¬ r{ky/rt-v{loÅ}Æ Ë?ÌÍÇËDÌͺ©Âl/yѲë°A² rt l9³suR© rtv wBrty¯ºs~ wB¨l?-lkjH~-v{lºu-|&wB| |zTj- l © rt|\² y lkwTky/r{zT|lk|y/ lmlku\=µ= ¹wBzT| ¬³sulk|y¯º=|zTuo|zTurt|y ¬k l zT|wBu\±~- zT-v{ikjml zT|yvxw zTvtu-y/r{zT| |R©Âl/ymzT|-|ulf³ul ¯ÌÍÅu-|lÎ zTvtu-y/r{zT| |su-jm¬k/r{³ul«ÃÍlγu-rl ym¬k|¬kwBv{lkjmlk|y v{l ¯wT?Ä?µ 1zTu-*¬ky/u r{lku-|l /rtju-vxwBy/r{zT| wBu /r¨/rtjH~-v{l³sulm~"z /rt-v{lºl&³su-rV¬¯wB/y l&v{l~-h¬?² |zTjmik|ly/u-/-u-v{lk|y lkymv{lH~- zT-v{ikjml $z # vxwfÐÍ zT|y/r{ik l u zTjwBrt|l®l/yHjmzT-rtv{lº½|zTu ~- zT~"z zT|¨wBv{zT ,v{l ~- zT-v{ikjml}/u-rt¼TwB|sy½$z #vxw~- zT~ wBwBy/r{zT| ©ÂzT| l l hzL,|zT|\²ªvtrt|¬¯wBrt l wB|¹u-|l3¬zTjm¬ky//r{l¨¿--l3l/y(/rtju-v{¬lµL§¨lky/y l¨ rtju-vxwBy/r{zT|l/y¹ wT ¬l3/u-¹u-|l3¬k|¬kwBvtr{wBy/r{zT| u~- zT-v{ikjml lor{lkjwB|-|Rµ ÁRl~- zT-v{¬kjwBy/r{³ul*vtr{¬lH°®vª©`w wB~-ywBy/r{zT| ljwBrtvtvxwBl ~"zTu-v{l/rtju-vxwBy/r{zT|*rt| ywA² y/r{zT|-| wBrt l3wBrt|/r ³ul v{lV zTvtu-y/r{zT|V~- zT~ z ¬lV lk zT|y¨¬ky/u r{¬lVlky3wB| wBvtL¬l½lk| rtjmlk| r{zT| lku\Ï/u-l~- zT-v{ikjmlµ=Á,©`wB| wBvt\ l®¬kywB|sy l¯wBuzTu-~»~-vtuwTl /rt-v{l*lky9-r{lk|±~-vtu~-wA² y/r{³sulµT zTu-,lkvxw\ºzT|HzTjH~ wB lkwv{l( zTvtu-y/r{zT|,w wB~-y ¬lV°u-|l zTvtu-y/r{zT| rty l l¨ ¬?ÐÞ¬k lk|lµ . . . . . 1 /21 5'> . /2(1 . ,Á ©`w wB~-ywBy/r{zT| ljwBrtvtvxwBl|zT|»/y//uky/u- ¬~ zTu-v{l9/rtju-vxwBy/r{zT| ywBy/r{zT|-| wBrt lowHЪwBrty vª©ÂzT\/lky lm|zTj- lku\«y/w¼wBu\"µ¹ ¹wBzT|sy/ l lkvxwBy/rt¼lkjmlk|yH~"lku l&y/w¼wBu\czT|y*~- zT~"z ¬ l. zTvtu-y/r{zT|~"zTu- y/wBrty lkv{l ~- zT-v{ikjml}rt|/ywBy/r{zT|-| wBrt l9w¼lvª©`w wB~-ywBy/r{zT|Rµ zTu- vª©`w wB~-ywBy/r{zT| l}jwBrtvtvxwBlo°o|zTj- l l9 zTjHjmlky V¿-\l3lky3wB|3 h wB|-lkjmlk|y l zT|-|lky/rt¼Lrty ¬®Ãñ0Ì ýÊ EVwBuku-|l· wT ku-v{l ©`wB ky lmzTu l&ЪwTlѲëwB ky lDÄu-|lwB~-~- zshlÜ}ÁRß ÃÍlk|«wB|-vxwBr{ ëÜ}/-rty/wB/»Á wBwB|-l&ß,u-v{lk C¯Ä9~"lku-y ky/ lHu-y/rtvtr{ ¬lµß(vtv{lm~"lk/jmlky l~- lk| l lk|ÒzTjH~-y lmv{l*jmzTu-¼lkjmlk|y u»jwBrtvtvxwBllk|»rt|y/ z u-r{wB|yu-|»y lk/jml wB|vª©Â¬³su wBy/r{zT| u ~- zT-v{ikjmlº"lkvtv{l ¬k~ lk| Ï zT| uf~- zT-v{ikjml*¬ky/u r{¬µR§¨ly lk/jml lkjH~-vxwTlvª©Â¬kywB~ l © rt|y lkѲ ~"zTvxwBy/r{zT|&°9h wT³ulh wB|-lkjmlk|y l3jwBrtvtvxwBlµ ±wBr{,lky/y l jm¬ky/hz l |l3|zTu½rt|y ¬k l l~ wT ¯wB wB| |zTy/ lwB~-~- zL hl ÃFjmzTrt|/rtr lo³sul9vxw*jm¬ky/hz lÜÁRßVĨv{l9|zTj- l l zTjHjmlky . CDC ô E.FHGJI K. . .
(27) tžÊkÉ
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(29) wB|v{lÎ-u-y lfzT|y/" !Tv{lk v{l|zTj- l ©`w wB~-ywBy/r{zT|¾µRÔwB|lky/y lwB~-~- zshlº=vª©`w wB~-ywBy/r{zT| l*jwBrtvtvxwBlml y wT¬lH/u-9vxw ¬ zB² vtu-y/r{zT| © u-|¤~- zT-v{ikjml lm~"zTrt|yo¿--lHy/wB|/rty zTrt lµ }zTuw¼zT|/u-rt¼Lr½lky/y l&wB~-~- zL hl&lk| vª©`wBjm¬kvtr{zTwB|sy l9jwB|-r{ik l/rt|-rÀ¿ ¯wBy/rt¼lµ. 1. 2. . & 1 (> /2887(&'>. zTu-Vv{l½/rtju-vxwBy/r{zT| ¬k~"lk| wB|y umy lkjH~¯º|zTu,~- ¬ lk|sy zT|½u-|l.|zTu-¼lkvtv{ljm¬ky/hz l Bw u-y zTjwBy/r{³sul ©`w wB~-ywBy/r{zT| l·jwBrtvtvxwBlº,³u-r¬ky lk| v{lH ¬/u-vtywBy H~- ¬kvtrtjHrt| wBrt l l -º
(30) ¸µ§¨lky/y lwB~-~- zL hll/y ©`wB~-~-vtr{¯wBy/r{zT|άk|¬kwBv{lº-©Âl/yѲë°A² rt l9³suR©Âlkvtv{l9~ lku-y3¯©`wB~-~-vtr{³sulk.° y zTu-y l1v{l¹kvxwT l lV~- zT-v{ikjml¹rt|/ywBy/r{zT|-| wBrt lÃFrt| ¬k~ lk| wBjHjmlk|sy uy¸L~ l ©Â¬³u wBy/r{zT| v{l9jmz ¬kvtr{wB|syDÄ?µ }zTu ~- zT~"z zT| u-|ll?\y lk|/r{zT| l9vª©`wBvtzT/rty/h-jml ©`w wB~-ywBy/r{zT| l9jwBrtvtvxwBl³u-r=~"lkѲ jmlky l~- ¬ rt l}vª©Â¬k¼zTvtu-y/r{zT| lV~-h¬k|zTjmik|l¨~-h\/r{³sul¯µÁRl}~-/rt|krt~"l.l/y l ¬ zTu l.u-| ~- zT-v{ikjml l¨~"zTrt|y¹¿--l¨y/wB|/rty zTrt l3~ zTu-¹v{l3zTu-~-v{lVÐÍzT/jm¬3~ wB¹v{l¨jwBrtvtvxwBl lky¹vxw zTvtu-y/r{zT|° h wT³ul.rty ¬kwBy/r{zT| l.vxw9 zTukv{l ©`w wB~-ywBy/r{zT|Rµ %ëv l/yVzTjH~ z ¬ l lku\m zTukv{l¾ºvxw9"zTukv{l ~-/rt|krt~ wBv{l ©`w wB~-ywBy/r{zT| lkyou-|lm"zTukv{lmrt|sy lk/|l wB|vxwT³ulkvtv{lv{lH~- zT-v{ikjmly/wB| rty zTrt l l y ¬ zTvtuRµ §¨zT|-| wBr{ wB|y¹vª©Â¬k¼zTvtu-y/r{zT| lVvxw. zTvtu-y/r{zT|RºArtvsÐÞwBu-yjwBrt|y lk| wB|y½w wB~-y lk¹v{lVjwBrtvtvxwBl wB| v{l} ¬kr{zT|9$z #®v{l~-h¬k|zTjmik|l9¬k¼zTvtulk|yowA¿| l~- ¬ lk/¼lkvxw~- ¬kr{/r{zT| lvxwzTvtu-y/r{zT|Rµ ß,| ©`wBu-y/ ly lk/jml¯º|zTu¼zTu-v{zT|*zT|y/" !Tv{lk*vª©Âlk/ lku-y zTu-y*wBucv{zT|- u«¯wBv{ku-v¨©Âl yѲë°A² FG8C F î.
(31) . ÅDÆ ÉªÅÉÍÌÍÇ Ê mÅTÌ tÅ Ê
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(33). rt l}rt|y/ z u-rt l9v{l}y lkjH~ wB|Vvª©Âl y/rtjwBy lku- ©Âlk/ lku-¯µ- 1zTu- lkvxw\ºL|zTuVrt|sy/ z u-r{zT|3u-|l ~- zL¬ u- l © rt|y lk lky/r{zT| l}jm¬ky//r{³sul lk| y lkjH~ wB|3vxwzT|/y//uky/r{zT| lvxwjm¬ky//r{³sulµ Á,©`w wB~-ywBy/r{zT| ljwBrtvtvxwBlH|zT|»/y//uky/u- ¬ml yr{kr¹ wT ¬lH u-v{l*¯wBv{ku-v l9v{zT|-ulku- lfwB ky l wB|u-|l»jm¬ky//r{³sul r{ k iky l¦Ã ¬?¿|-r{l¤wBu\ zTjHjmlky u jwBrtvtvxwBl¾Ä?µ9§¨lky/y l jm¬ky//r{³sulHÃÞwB|-r{ zTy/ zT~"l¾Ä½l y¨zT|/y//u-rty l9lk| u-y/rtvtr{wB|y3u-|l/y/rtjwBy lku- ©Âlk/ lku-3¬zTjm¬ky//r{³ul ÅÆ\ÇTÈ?ɸÊ?Ë?ÌÍÇËDÌ\ wT ¬ u-,u-|l.wB~-~- z\rtjwBy/r{zT| r{ k iky l umhl/r{lk| l vxwzTvtu-y/r{zT|RµÁRl-u-yVl/y ©Â¬³su-rÀ²ª ¬k~ wB/y/rtovª©Âlk/ lku- © rt|sy lk/~ zTvxwBy/r{zT|¤ u-v{l*jwBrtvtvxwBlHlk|»zT|sy/" !TvxwB|yvxw zT|/y//uky/r{zT| l}wB ky l ujwBrtvtvxwBlµ . . .0/ 495'> . >8 & . Ô9wB|lfwB~-~"zT/y¯º3|zTu·~- ¬ lk|y zT|Îlky |zTuÎwB| wBvt\ zT|·vxwcjm¬ky/hz lÏ~- zT~"z ¬l±krÀ² l/u¯µ qLlky/r{zT|»\º|zTu}~- ¬ lk|y zT|vª©`wBvtzT/rty/h-jmlkvxwT /r{³sul ©`w wB~-ywBy/r{zT| lojwBrtvtvxwBl¾µ ,u-r{¯ºszT|&wB~-~ lkvtv{l}-/r{ik¼lkjmlk|y y zTu-y l¨v{l¨¬kywB~ l l}vxw~- zL¬ u- l ©`w wB~-ywBy/r{zT|Rµ\ÔwB|¨vxw qLlky/r{zT|«\º"v{lm h¬kjw ©`w wB~-ywBy/r{zT| l*jwBrtvtvxwBll ywB| wBvtL ¬H u-u-|Ò~- zT-v{ikjmlHjmz ikv{l lk~- ¬lk|ywBy/rÀÐ lH/rtju-vxwBy/r{zT|Hrt| ywBy/r{zT|-| wBrt lmlk| rtjmlk|/r{zT| lku\"µ zTuÒjmzT|y/ zT| v{lvtrtjHrty l l&vª©`w wB~-ywBy/r{zT| kvxwT /r{³sulº wB|*v{l ¯w l uc lkjwBrtvtvxwBlº¹~"zTu-Hlyës~"l l rtju-vxwBy/r{zT|Rµ1qLlky/r{zT|cnº=|zTuo~- ¬lk|y zT|lk| ¬kywBrtv,vxw·|zTu-¼lkvtv{l wB~-~- zshl ©`w wB~-ywBy/r{zT| ~"zTu-½v{lV/rtju-vxwBy/r{zT|¨rt|/ywBy/r{zT|-| wBrt l¾µ zTu½~- ¬kr{zT|½v{l.|zTu-¼l¯wBu h¬kjw ©`w wB~-ywBy/r{zT| lky3|zTu3rt| r{³suzT|3vxw~- zs¬ u- l © rt|sy lk lky/r{zT| ljm¬ky//r{³sul lk| y lkjH~¾µ §¨lky/y l}|zTu-¼lkvtv{l wB~-~- zL hll/y¼TwBvtr ¬lH u-v{l*j kjmlm~- zT-v{ikjmlHjmz ikv{lµ= ,u-r{qLlky/r{zT|¤\º"|zTurtvtvtu/y/ zT| vª©`wB~-~- zL hl~- zT~ z¬l/u-u-|Îl?-lkjH~-v{lozT|k lky ©`wB~-~-vtr{¯wBy/r{zT|Ïlk| rtjmlk|/r{zT|Îy/ zTr{¯µ. CDC ô E.FHGJI K.
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(37) . ÅDÆ ÉªÅÉÍÌÍÇ Ê mÅTÌ tÅ Ê
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(46) & ,7.0> * > 4 1 . 9Ô wB|lHwB~-~ zT/y¯ºR|zTuwBvtv{zT|wB~-~-vtr{³sulkv{ly lh-|-r{³ul ©`w wB~-ywBy/r{zT| l*jwBrtvtvxwBl °Hvxwjm¬¯wB|-r{³ul l ´u-r l¯ºlk|f lvtrtjHrtywB|y wB|v{l}/rtju-vxwBy/r{zT|9wBu\άzTu-v{lkjmlk|y }jmzB² ¬kvtr{¬~ wBv{l*¬³u wBy/r{zT| ©Âß,u-v{lk¯µ1gzTu-y l?ÐÍzTr{¾ºv{l*/rtju-vxwBy/r{zT|m zT|syl=lky/u¬lH/u- l ¬zTjm¬ky//r{l}zTjH~-v{l?\lÃF ¬¯wBvtr{/y lDÄ?µ Ô9wB|1v{l¨¯w l l1¬³su wBy/r{zT| ©Âß(u-v{lk¯ºBv{l¨ w ¨l/y¹/u-~-~"z ¬,~ wBÑЪwBrty¯ºBrtvLl y|zT|¼sr{³ulku\ lky¹rtvL|R© w.~ wT l Er u/r{zT|*y/hlk/jHr{³sulµ¶}|*|zTy l ρ vxw lk|/rty ¬º U~ = (u , u , u ) v{lV¼lky lku- ¼Lrty l lº T vxwHy lkjH~"¬kwBy/u- lº E = T + vª©Â¬k|lk/r{ly zTywBv{lº p = (γ − 1)ρT vxwH~- l r{zT| w¯¼l lky vª©ÂzT~"¬kwBy lku- w r{lk|y¯µ-ÁRl3¬³su wBy/r{zT| ©Âß(u-v{lk¯©Â¬k/rt¼lk|sy zTu,ÐÞzT/jml zT|lkγ/¼TwB=y/rt¼1.4 l lovxw∇jwB|-r{ik l/u-rt¼wB|y 0l / ~ k2 kU 2. ∂ρ ~) = 0, + ∇. (ρU ∂t ∂(ρU ~) ~ ⊗U ~ ) + ∇p = 0 , + ∇. (ρU ∂t ∂(ρE) ~) = 0, + ∇. ((ρE + p)U ∂t. 1. 2. 3. $z #fzT|®wH/ul /rt¼lkjmlk|sy.v{l¬³su wBy/r{zT| lzT| lk/¼TwBy/r{zT| l9vxw*jwT lº ujmzTjmlk|y9lky l vª©Â¬k|lk/r{ll?L~-/rtjm¬l} zTu3ÐÞzT/jmly lk| zT/r{lkvtv{lµ ÁRl·¬³u wBy/r{zT| ©Âß(u-v{lk zT|sy& ¬ zTvtul°¤vª©`wBr l u zTvt¼lku-7=?>A@ BAC 1 Hà ¬k¼lkv{zT~-~ ¬ °®vª© % } %ëÜ.²¸.zL³ulk|zTu-/yDÄ?µ¹§©Âl/y*u-|«zTvt¼lku-) D,zTvtu-jml 1rt|-r{~ zTu-v{l ¯wBv{ku-v ©Â¬zTu-v{l?² jmlk|sy,lk| rtjmlk|/r{zT| lku\*lky(y/ zTr{(u-y/rtvtr{wB|y l¹jwBrtvtvxwBl,|zT|\²¸/y//uky/u- ¬¯µÁRl(¬³u wBy/r{zT| ©Âß,u-v{lk zT|sy} ¬zTvtul9°mvª©`wBr l uf´u\ l}zslµ"¶}|fu-y/rtvtr{lou-|l lzT|/y//uky/r{zT| lyës~"l ÏÓq-§¨Á lky lmvtrtjHrty lku- El D½wB| Ü}vt w w±~"zTu-zT-y lk|-rtmu-|l~- ¬kr{/r{zT| ©ÂzT l lku\. CDC ô E.FHGJI K.
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(48) "Ë Ê.Ê/ÇTË ÊHÊkÉ ±Ç LÅ mÅ . Ì. ÃFvtrtjHrty ¬¾Ä,lk|&l/~ wTlµsÁ,©`wBvtzT/rty/h-jmlu-y/rtvtr{ ¬.l y½l?L~-vtr{krty llk|y lkjH~¾µ 1zTu-½v{ly/wBrty lkjmlk|sy l ~- zT-v{ikjml9rt| ywBy/r{zT|-| wBrt l¯º"u-|Ï h¬kjw lyës~"lu-|-l?² 9u-y/yw°&³u wBy/ l~ wT9l/y}u-y/rtvtr{¬ ~"zTu- vª© rt|y ¬kwBy/r{zT|±lk|Îy lkjH~¯ºwA¿| ©`w¼zTrt.u-|lo~- ¬kr{/r{zT| ©ÂzT lo/u-~"¬k/r{lku-¯µ . . . / . ,7. 5'> .0/ *. . 1$( 3,'>. 9Ô wB|·lky/y l± lky/r{zT|Rº¨|zTu wB~-~ lkvtv{zT|·-/r{ik¼lkjmlk|sy¯º3vª©Â¬kywB~ l lÏzT|/y//uky/r{zT| lÏvxw jm¬ky//r{³sulµ½Ó}|l lk/rt~-y/r{zT|¦~-vtu&~- ¬kr{ l lflky/y l®zT|/y//uky/r{zT| l/y zT|-|¬l wB| ¸µ ß,|/u-rty lº-|zTu ~- ¬kr{ zT|vxw|zT/jwBvtr{wBy/r{zT| l9vª©Âl/y/rtjwBy lku- ©Âlk/ lku-.lky vª© rt|y lk lky/r{zT| l jm¬ky//r{³sul wB|v{l¯w l l9vxw*jm¬¯wB|-r{³sul l3´u-r l¯µ ¡ Ø Í£ \Ø L¢ &;+ L¢ # × £ÒØ (R¢ ¶}|cw ЪwBrtyvª© hL~"zTy/hi lm³suR© u-| zT|sy/" !Tv{l lvª©Âlk/ lku- © rt|sy lk/~ zTvxwBy/r{zT| ~ lk/jmlky l·zT|y/" !Tv{lkHvª©Âlk/ lku- ©`wB~-~- z\rtjwBy/r{zT| Tµ zTy/ l9l/y/rtjwBy lku- ©Âlk/ lku-ÅÆ\ÇTÈ?ɸÊ?Ë?ÌÍÇËDÌ wB|-r{zTy/ zT~ l¼w zT| ky/ l wT ¬/u-¨vª©Âlk/ lku- © rt|\² y lk/~"zTvxwBy/r{zT|Rµ-ÁRl-u-y l/y l}¬k|¬k lk u-|·jwBrtvtvxwBlow wB~-y ¬/u-¨v{l³sulkv vª©Âlk/ lku- © rt|sy lk/~ zTvxwA² y/r{zT|fl/y¬³u-rÀ²ª ¬k~ wB/y/r{l wB| y zTu-y lv{l rt lky/r{zT|¯µ zTu-}y zTu-y.¬kv{¬kjmlk|y K uÎjwBrtvtvxwBl u zTjwBrt|lº zT|fwvxwHjwAÑzTwBy/r{zT|fwB|-r{zTy/ zT~ l l vª©Âlk/ lku- © rt|y lk/~"zTvxwBy/r{zT|Rº wT ¬l/u- v{l ¬k/rt¼¬l l lzT| zT l lvxw¼wB/rxwB-v{lu / ÃÑ¾Ä ku − Π uk ≤ c max max h~e , |H (x)| ~e i = ε , $z # c l yHu-|lÎzT|/ywB|y lϳsu-r ¬k~ lk| ¦ lvxw rtjmlk|/r{zT|Rº E vª©Âlk| lkj-v{l lwB ky l l vª©Â¬kv{¬kjmlk|sy K lky |H | = R|Λ|R l/y3vxw¼wBv{lku-.wB zTvtul uhl /r{lk| l9vxw¼TwB/rxwB-v{l u ºL$z # l/y}vxwmjwBy//r{l l.¼lky lku- ~- zT~- l9lky |Λ| = diag(|λ |) vxwmjwBy//r{là rxwBzT| wBv{l¾Ä l R ¼wBv{lku- .wB zTvtul l ¼wBv{lku- ~- zT~- l¯µ s£ (R¢ : Ì Ê Ê 5ÊkÉ3ÊDÈ?ÉÍÌ mÅÉ¸Ê LË : ÊkËDË Ê LËÊDÈ?É Ê9Ë Ê ÅTË Ê? Ë ÀÅÈÇ LÉÍÌÞÇ 5Ç mÊ Ê 0 +DÆ-Ê?Ë ¾È sË kAÅ 5Ê&ÊÉ Ê 5Ç ÉÍË tÊk6 Ë : 5 ÅTËkÉ 5DÊkÉÞɸÊ*È sË Å 5AÊ ,ÊkÉÞɸÊmÊ?È?ÉÍÌ HÅÉ¸Ê sËÊDÈkÉ Ç 5 Ì ëÆ-Ê Å É Æ ËÑÇ HmÊHÉÍËÑÅTÌÍÉ ÊkÉ½Ì ÊDÈkÉ½Ë Ê öÌ Å -ÊDÈÈ?ÌÞÊ Ê tÅmÈÇ LÉÍÌÞÇ 2 ¶}| u-y/rtvtr{ l9wBv{zT ¨u-|lwB~-~- z¯LrtjwBy/r{zT| r{ k iky l l}vxw @ × ¡¯Ø B¢Ú Ø F× & Û f£¤Ø (R¢ jm¬ky//r{³sulm³su-r(u-y/rtvtr{ lmzTjHjml&/u-~-~ zT/yv{lo zTjHjmlky u»jwBrtvtvxwBlµ¹¶}|Ò|zTy l h ÃF l ~Rµ Ävxw&v{zT|-ulku-jHrt|-rtjwBv{l®ÃF l ~Rµ jwA\rtjwBv{l¾Ä9wBu-y zT/r{ ¬l*~"zTu-9v{lowB ky l uÏjwBrtvtvxwBl h lky ε vª©Âlk/ lku- © rt|y lk/~"zTvxwBy/r{zT|®y zTv{¬k ¬lµÜ}v{zT lk|Î h wT³sul zTjHjmlky u·jwBrtvtvxwBlº ©`wB~- i.vxw }lkvxwBy/r{zT|cÃѾĨzT| ¬?¿|-rtyu-|®y lk| lku- l9jm¬ky//r{³sulwB|-r{ zTy/ zT~"l lvxwjwB|-r{ik l u-rt¼TwB|sy 0l / z $ # k l y 1 c |λ | 1 ˜R , ˜ = diag(λ˜ ) M = RΛ λ˜ = min max , Λ , . ε h h zTu9w¼zT|rt|y/ z u-rty wB|}vxw ¬?¿|-rty/r{zT| lvxwmjm¬ky//r{³ul l}v{zT|-ulku- ©`wB ky ljHrt|-rÀ² jwBv{llky¨jwALrtjwBv{l9wA¿| ©Â¬k¼Lrty lk3v{l½jm¬ky//r{³sul¨rt/ ¬¯wBvtr{/y l3lkyV|zTywBjHjmlk|sy ~ zTu-3zT|y/" !Tv{lk v{l9~ wT ly lkjH~ uÎ zTvt¼lku-}l?L~-vtr{krty lµ Aèªþ KDAí?ä Ñüá éÑø(éÑøÑè í ÑøÞïBäÂò¸â òªø½ô¯ïAá ñèªã ¯ïAéÑø¹þ AòªéÑô¯ïBøòªéÑôTê¾éÑôò }ìÑþDô Bèªá éëè,ìÑéñòÞòªé ù þDò ÑøÞé¨éÑô Aè¸âòªã ïBé þ?ï¾èVôAþ?ø .
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(51) Æ\ÇsË{ÊDÈ3Æ Ë/ÇÊDÈÌ-ÈkɪÅÉÍÌÍÇ ÅTÌFË/Ê?È. . ÔzT|Î~ zTu-¬³su-rt ¬k~ wB/y/rtvª©Âlk/ lku- © rt|y lk/~"zTvxwBy/r{zT| /u-v{l·jwBrtvtvxwBlº½zT| l/y&zT| u-rty Ò° jmz rÀ¿ lk v{lf~- z u-rty· ¯wBvxwBrt lϳu-r zTuѲªy lk| vxwÒ|zTy/r{zT| l r{ ywB|l®u-y/rtvtr{ ¬lf~ wB v{l ¬k|¬kwBy lku- l9jwBrtvtvxwBl¯º °vª©`wBr l l9vxwjm¬ky//r{³sul9v{zs¯wBv{l M ³su-r= lkjH~-vxwTlvxwjm¬ky//r{³ul lkukvtr r{lk|-|lÒu/ulkvtv{lµ¶}| hlk hlcwBv{zT ΰczT| y//u-rt l±u-| jwBrtvtvxwBlÒw¯¼l l¬kv{¬kjmlk|y ¬ku-vtr{lk .u-|-rty ¬ wB| vxw|zTu-¼lkvtv{ljm¬ky//r{³ulµ Ô9wB|¨vxw/u-rty lºL|zTu3~- zT~"z zT|3u-|l|zT/jwBvtr{wBy/r{zT| lvª©Âl/y/rtjwBy lku- ©Âlk/ lku- wB|3v{l ¯w l lvxwjm¬¯wB|-r{³ul l1´u-r lV³u-r-~"lk/jmlky l3jHr{lku\H¯wB~-y/u- lkVlk/ywBrt|(~-h¬k|zTjmik|l ~-hsL r{³ul¯µ-ß,|/u-rty lº|zTu ~- ¬lk|y zT|}u-|ljm¬ky/hz l³u-r=~"lk/jmlky l9~- lk| llk|fzTjH~-y l Er =¬k lk|y l.¼TwB/rxwB-v{l.~-h\/r{³sul wB|vxw ¬?¿|-rty/r{zT| l9vxw*jm¬ky//r{³ulµ ¡ Ø Í£ \Ø L¢ &;+ L¢ 6!,× ¢ *Û f£Òs#Ú (R¢ & s¡(¢D & ¡ ß(| ~-wBy/r{³sul wB|¨v{l¨/rÀ² ju-vxwBy/r{zT||su-jm¬k/r{³ul¯º zT|Îl/y zT|\ÐF zT|sy ¬wBu·ÐÞwBrty ³sul9v{l zTvtu-y/r{zT| ¼wB/r{lk|y l~-vtu/r{lku- zT l l*wB| lku- ÃF~-h¬k|zTjmik|loju-vty/rÀ²¸¬hlkvtv{lDÄ?µ1qsr(zT|Òl/ylk|»~- ¬ lk|l l~-vtu/r{lku- ~-h¬k|zTjmik|l l Er "¬k lk|sy l½wB| lku- ÃF lkrt ku-vxwBy/r{zT|¯ºL hzL¯º µtµtµåÄ?ºsrtv l/y rÀÕkrtv{l l}¯wB~\² y/u- lkv{l~-vtuoЪwBrt-v{lw¯¼lvª©`w wB~-ywBy/r{zT| l&jwBrtvtvxwBlµ¹§©Âl y ©`wBu-ywB|sy~-vtu rÀÕkrtv{l lk| jm¬¯wB|-r{³sul l¨´u-r l v{zT ³sul l hzL zT|sy ~- ¬ lk|y wB|vª©Â¬zTu-v{lkjmlk|y¯µ Áw*¯wB~-y/u- l ll}~-h¬k|zTjmik|lÐÞwBrt-v{l9l/yrtjH~ zT/ywB|sy l~ zTu-9zT-y lk|-rt9u-|l zTvtu-y/r{zT|Ï~-vtu~- ¬kr{ llk| ~- lk| wB|sy}lk|ÎzTjH~-y lv{lku- rt|y lkwTky/r{zT|9w¼lv{l.wBu-y/ l~-h¬k|zTjmik|l lvª©Â¬zTu-v{lkjmlk|y¯µ %¸v(w ¬ky ¬jmzT|sy/ ¬H³uR© u-|l*l/y/rtjwBy/r{zT| lvª©Âlk/ lku- lkvxwBy/rt¼lv{zs¯wBv{l~"lk/jmlky l ¬zTu l l}~- zT-v{ikjml 1\º kn ¸ºs©Âl/yѲë°A² rt l9zT|&|zT/jwBvtr{l}vxw.lkvxwBy/r{zT|ÒÃѾÄ,~ wBVvxwo¼TwBv{lku- wB zTvtul l vxw¼wBv{lku-v{zL¯wBv{l l9vxw*¼wB/rxwB-v{l u /. u − Π u ÃÞÄ |H (x)|. ≤ c max max h~e , ~e i , |u|. |u(x)| $z # |u| = max(|u|, kuk ) w¯¼l u-|l zT|/ywB|y l.~ lky/rty l lk¼wB|y 1 µÁRl y lk/jml kuk l y,u-|m~ wBwBjmiky/ l ly/ zT|¯wBy/u- l}~ zTu-½¬k¼Lrty lk½v{l½~- zT-v{ikjml½|su-jm¬k/r{³ul¾µ %ëv ¬?¿|-rty¨wBu/r vª© rt|sy lk/¼TwBvtv{llk|y/ lvxw~-vtu3~"lky/rty llky vxw~-vtu wB| l¬ hlkvtv{l³ulvª©ÂzT|¼lku-y¯wB~-y/u- lk.©Âl yѲ °A² rt l·³sulv{zT ³uR© u-| ~-h¬k|zTjmik|l·~-h\/r{³sul°Îu-|l wB| lku-Hrt|\ÐÞ¬k/r{lku- l·°Ïl&y lk/jmlº¹rtv |R©Âl y ~ wT¯wB~-y/u- ¬µ Ø A¡ sÚ Ø F× & £ÒØ (R¢ ¡ zTu(¼lk|-|zT| l¼zTrt,~- ¬¬ lkjHjmlk|y3vxwzT|/y//uky/r{zT| l*vxw jm¬ky//r{³sul*~ wBowB~-~"zT/y° u-|l*¼TwB/rxwB-v{l ¯wBvxwBrt lµ ÏwBr{¾º"rtv(~ lku-y ky/ l*|¬l DwBrt l l zT| r ¬k lk½~-vtu/r{lku- V¼TwB/rxwB-v{lV~"zTu-,vxwozT|/y//uky/r{zT| lvxw9jm¬ky//r{³sul*ÃF~ zTu-,v{l³ulkvtv{lVzT| ¬ rt l.zT|y/" !Tv{lk vª©Âlk/ lku- © rt|y lk/~"zTvxwBy/r{zT| Ä3wA¿| l.¯wB~-y/u- lk Er "¬k lk|sy ¨~-h¬k|zTjmik|l3~-hs² r{³ul¯µß,|&l =lky¯ºlk/ywBrt|,~-h¬k|zTjmik|l¨|l zT|sy, lk~- ¬ lk|sy ¬V³sul ~ wB½lk/ywBrt|lV¼TwB/rxwB-v{l¾µ qsr-~-vtu/r{lku- ,¼wB/rxwB-v{l½ zT|sy½zT|/r ¬k ¬l¨wBv{zT (~-vtu/r{lku- ,jm¬ky//r{³sul½zT|y,~- l k/rty l½lk| h wT³ul zTjHjmlky¯µ zTu-zT-y lk|-rt9u-|lojm¬ky//r{³sulu-|-r{³ul³u-r1v{l}~- lk| y zTu-y l9lk|±zTjH~-y lº u-|lm~- zs¬ u- l © rt|y lk lky/r{zT| lHjm¬ky//r{³sulo¼w ky/ lmu-y/rtvtr{ ¬lµqLzTr{lk|sy M lky M lku\ y lk|lku- lVjm¬ky//r{³sul zT|-|¬,wBum zTjHjmlky P µTÁRl¨y lk| lku- l¨jm¬ky//r{³ul M zT/ l/~"zT| wB|y °vª© rt|y lk lky/r{zT| l lky M zTrty ky/ l y lkvÃF~ wB ¬?¿|-rty/r{zT| Ä(³suR©ÂzT|mwBwB|y/rty3³ulvª©Âlk/ lku- © rt|sy lk/~ zTvxwBy/r{zT|«~"zTM u-o h wT³sulm¼wB/rxwB-v{lml/y"zT/|¬lH~ wBov{lH lku-rtv,~- lk/rty¯µ¶}|¤zT-y/r{lk|yvxw jm¬ky//r{³sul M lk| ÐÞwBr{wB|symvxwÏ ¬ uky/r{zT| /rtju-vtywB|¬l l*ÐÞzT/jmlm³su w wBy/r{³ulwT zskr{¬l wBu\·jm¬ky//r{³sul M lky M ÃÍ?Ðѵ FÄ?µ . . u. h. . . ∞,K. x∈K ~e∈EK. . ∞,Ω. ∞,Ω. . 1. 1. 1. CDC ô E.FHGJI K. 2. 2. 2.
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