Stability of Perfectly Matched Layers, Group Velocities and Anisotropic Waves
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Stability of Perfectly Matched Layers, Group Velocities and Anisotropic Waves Eliane BECACHE, Sandrine FAUQUEUX, Patrick JOLY. N° 4304 Novembre 2001. ISSN 0249-6399. ISRN INRIA/RR--4304--FR+ENG. THÈME 4. apport de recherche.
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(67) 4. 0.6. 0.4 2 0.2. –0.6. –0.4. 0. –0.2. 0.2. 0.4. 0.6. –4. 0. –2. 2. 4. –0.2 –2 –0.4. –0.6. –4. Ð5uvwf p4q&TÁs_x~ ° mp /owf}py| . t=2s. S. yp4§B ~_{ §À~{}kmp4u ~{} ~fu4nUp uzy². t=5s. t=20s. t=500s. Ð5uvwf p1T¹s ~nUpW};y|m}km~{ y|{ uqppo po_{/{}uvnUp §À~{}kmpLu ~{} ~fuWnUp uzy². ZO8 "B£¡98 Î 7 8 7. _UT. £ 7. £ 7 8B Î ¡¢ Î[> B8 ¥. ÊÏ. 8 . µ¼{}kfu4p`my|nGfxp ³ {}kmp!poxzy}{}uouv{¶~ p`ȬoupoO{ Wy| p9uvpo. 6 P. 7MPMH " : QPQH|D
(68) ac: H|Dy| OM QH q cq . j kmp}x~ ° mp 4owf}pWy| ° yp9§¨ ~O{ L pof pom6poO{ p uvdÐ5uvm²WUuvxvxvwm}{}6y|{ p9{}kmpGy|fu ~{} ~_~|§¹{}kmp9nUp uvwfn¾² ~{ p&{}k;y|{5{}kmp¹ po{ )pomox~ p O4{}kmp¹ x~ ° mp 5owf}p× pon¬y|uv~Op`y uv1{}kmp¹u ~{} ~fu]y p²)j kmpÁ};y|m}km~{ ~|§{}kmpW~} p}~ uvf¬_wfnUpo}uy|x)p`-po}uvnUpo_{Iy| p4uvpo¾uv·Ð5uvm²/Y ² 4. 0.6. 0.4 2. 0.2. –0.6. –0.4. 0. –0.2. 0.2. 0.4. 0.6. –2. –1. 1. 2. –0.2 –2. –0.4. –0.6. Ð)uvwf p WTÁs_x~ ° mp Iowf}pIy| Ô)Ô Þ(sutv`ø+t. S. –4. yp4§¨ ~O{ §À~ §À~{}kmpW~}{}km~{} ~fuWnUp uvwfn «¨µ ².
(69) t=2s t=10s t=20s t=500s Ð5uvwf p YuT¹s ~nUpW};y|m km~{ Iy|{ qu p´po poO{/{}uvnUp§¨~{}kmpW~}{}km~{} ~fu1nUp uvwfn «¨µ ². j kmpo¬ km~ ° {}k;y|{Á{}kmpI&© ª ÂÄ ° ~)]Uf po{}{¼ ° poxvxò&µ¶¬;y|}{}uowfxzy| ³ {}kmpoy| pI}{6y|fxp(T{}kmp/ ~xvwf{}u~ ~ p¹m~{]fx~ ° wf ³ popoy¤§B{ po/yx~f¬{}uvnUp² ZO8 "B£9¡ 8 Î 7 8 7 £ 7 £ 7 8B Î ¡¢ Î[> B8 ¥ {}uouv{¶%~ p`ȬoupoO{ 9T. ÊÏ. 8 . 696 P. j kfu{}uvnUp ³ {}kmp4n¬y|{ po}uzy|x×u k;y|6yo{ po}uvÅp O%uv{ poxzy . MPM H4 QPQ H|D
(70) ac: H|DF: M Q 4 H q. . 0.6. 4. 0.4 2. 0.2. –0.6. –0.4. –0.2. 0. 0.2. 0.4. 0.6. –4. –2. 2. 4. –0.2 –2. –0.4. –0.6. –4. Ð)uvwf p GT¹s_x~ ° mp Iowf}pIy| S. ypL§B ~_{§À~{}kmpW~}{}km~{} ~fuWnUp uvwfn «¨µ µ ². t=2s t=8s t=14s t=500s Ð5uvwf p$HT]s ~nUpW};y|m}km~{ /y|{ qu p´po poO{I{}uvnUp§À~{}kmpW~}{}km~{} ~fuWnUp uvwfn «¨µ µ ². µ¶Uºmwf p G ³ {}kmp/nUp uvwfn y|fpy| ¹{ ~1-p/n1wm6knU~ pLy|fu ~{} ~fuI{}k;y|¬{}kmpf po u~wm]~mp²µ¼U;y|}{}uowfxzy| ³ {}kmp po{ pomox~ p OU{}kmp}x~ ° mp owf}p~|§ {}kmp ° ypu¹m~!x~fpo~Op` ³ ° kfu k%uvp]}u p/{ ~1{}}uvfxvuy|{}u~m ~|§5{}kmp ° ypL§¨ ~O{² F~ ° popo ³ ~mpLy|%6pp/uv+ºmwf piH!{}k;y|{]{}kmpL&© ª nU~ pox {}uvxvx ° ~)] po} ° poxvx y| % ~_p m~{xpy { ~Uy|O%uvm}{6y|fuvxvuv{¼² ZO8 "B£¡98 Î 7 8 7 oupo_{ XT. £ 7. £ 7 8 Î ¡¢ Î[> 8 ¥. ÊÏ. 8 . 6R696 P. Ðm~{}kfup`po}uvnUpoO{ ³ ° W p km~ ~ p1ypoxzy}{}uouv{¶¾~_p`È. OMPM H " : QPQH|D
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(72) 4. 0.6. 2. 0.4. 0.2 –2. –0.6. –0.4. –0.2. 0. 0.2. 0.4. –1. 1. 2. 0.6. –0.2. –2. –0.4. –0.6. Ð)uvwf p¬T¹s_x~ ° mp Iowf}pIy| . t=2s. –4. ypL§B ~_{§À~{}kmp4~}{}km~{} ~fuWnUp uzy«¨µ µ µ ² S. t=10s. t=20s. t=160s. Ð5uvwf pU(T]s ~nUpW};y|m}km~{ /y|{ uqppo po_{{}uvnUp§À~{}kmpW~}{}km~{} ~fuWnUp uzy·«¨µ µ µ ² ª ~ ~#] uvf½y|{+{}kmp}x~ ° mp %owf}pd«¨ ppÐ5uvm² `³ {}kfu¬nUp uvwfn pponU+{ ~Ç-pxp %y|fu ~{} ~fu{}k;y| {}kmp f po u~wm~mp² FI~ ° popo ³ {}kmp+};y|m}km~{ 9~|§{}kmp¬6~xvwf{}u~m9 km~ ° {};k y|{y|Çuvm}{6y|fuvxvuv{¼u popox~fuvfuvÇ{}kmp &© ª xzypo4;y|6y|xvxpox5{ ~+{}kmp GRQ uv po{}u~»«¨ ppÐ5uvm²UGy|{L{}uvnUp [ H $C #-a-, ²9I{}kmp~O{}6y|} ³ {}kmp9¹© ª uv {}kmp G M uv po{}u~ pponU{ ~y|m ~}¾{}kmp ° ypI®_wfuv{ p ° poxvxò ZO8 "B£9¡ 8 Î 7 8 7 £ 7 £ 7 B8 Î ¡¢ Î[> B8 ¥ poxzy}{}uouv{¶¾~_p`Ȭoupo_{ XT. Ê ÏI8 . 6 P. Iwf/xzy}{nUp uvwfn u/ k;y|6yo{ po}uvÅp O%{}kmp4§À~xvx~ ° uvf. MPM H " : QPQ H|D
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(110) JLG9: [ S#H J V L < M N' H6: * ? .·µ Q : H .¾µ Q : P . q j)~U}{}w ¯}wm k¾6~xvwf{}u~muv{u wm p`§¨wfx { ~ p`§Àpo{ v ~{}kmp4uvfuvv{}uzy|x k__po}~xvuW}}{ pon «À =T < Z ] M T M Z ] Q T Q Z H|ac: {}k;y|{Iy nGuv{ /fxzy|mp ° yp1 ~xvwf{}u~mI~|§5{}kmp~ §À~}n Z JLG : [ S#. H J V L < M N' "\: ? .¯µ Q : ".·µ : P . : u §&y| ~fxv+u §#?¯y| P y| pW poxzy|{ p O%{}kmp u}po }u~¯ poxzy|{}u~ «À yJ P :?SgHbac: v v ° kmpo p «À yJ P :?Sg H
(111) OJ P ] ? M M ] ?WQ QoS: u¬ykm~nU~pomp~wm¬-~xv m~nGuzy|xuv P y| ?3~|§ po pp ²3µ¼§ ° p¾~m}u po«À yUy| p®_w;y|{}u~ uv P³ y ~m p®_wmpomp1~|§{}kmpLkO po}-~xvuouv{¶¾~|§5{}kmp4}}{ pon «À u{}k;y|{{}kmpW ~xvwf{}u~·~|§«À y| p4 py|xT P H P h J(?S: < H C-: qqq : : {}kmpWpouvpoOy|xvwmp/~|(§ !J(?S q ©~ p~po ³ {}kmp1§¨wfmo{}u~m PRh J(?Sy| p!km~nU~pomp~wmW§Bwfmo{}u~m4~|§ po pp¯Gy| qu p´po poO{}uzy|fxp ° uv{}kd p}-po{ { ~f?%y/ ~_~·y{}kmpW}}{ pon «À u}{}}uo{}xv%kO po}-~xvu² .
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(113) j kfu¹u¹y-~xv m~nGuzy|xp®_w;y|{}u~Uuv P ~|§ po pp D&²j kmpo p`§À~ p{}kmp u}po }u~U poxzy|{}u~Uuv¬{}kmp&© ª p`ºmmp D nU~ p ³ P h J(?@: S ³ <IH C-: qqq :&D ³ ~xvwf{}u~mL~|§&{}kfuLp®Ow;y|{}u~ ²Lµ¶{IuI;y|{}wf6y|x{}k;y|{L{}kmp1_wfn1poL~|§ÁnU~ p & u4{}kmp ~wffxpG~|§ {}kmpG~mpuvd{}kmpG~}uvuv;y|xf ~fxpon ³ uvmp ° pGk;yp ~wffxp {}kmpOwfn!-po9~|§Áwf ] m~ ° mW_ ;y uvfG§B ~n «À { ~·«À ² SpG ponGuv. y|;y|xv}u². km~ ° {}kmpGm~{}u~m1~|§ ° poxvx Ã~ p mp 9y| }{6y|fuvxvuv{¶ y| pGuvO{}uvn¬y|{ poxvd poxzy|{ p { ~¯{}kmpGfxzy|mp ° yp. ,Ê " " Ìn>&Î Ê Ï 7 Ê (P j kmp¯}}{ pon «À@Y u
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