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Application of the PML Absorbing Layer Model to the Linear Elastodynamic Problem in Anisotropic Heteregeneous Media

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(1)Application of the PML Absorbing Layer Model to the Linear Elastodynamic Problem in Anisotropic Heteregeneous Media Francis Collino, Chrysoula Tsogka. To cite this version: Francis Collino, Chrysoula Tsogka. Application of the PML Absorbing Layer Model to the Linear Elastodynamic Problem in Anisotropic Heteregeneous Media. [Research Report] RR-3471, INRIA. 1998. �inria-00073219�. HAL Id: inria-00073219 https://hal.inria.fr/inria-00073219 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. Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media Francis COLLINO, Chrysoula TSOGKA. N ˚ 3471 Aoˆut 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(210) ?˜. .  +   " 3 #! 5 #! "  + D  " 3 #! 5 #!   +   " 3 #! 5 #!     +    " 3 #! 5 #!   + D  " 3 #! = 5 #!    +    " 3 #! . . . . .  +   " 3 #! . 5. . #! .  + " 3 #! $5 #! "  +   +  " 3 #! 5 #!   + . . . . +. . #!   + D  " 3 ! = #!  ˜    5  # 5 " #!   ' ˜   + D  " 3 #! = 5 #!   +   " 3 #!  5 #!   + D " 3 #! = 5 #! "   " 3 #!      5 ˜  '  " $#!  + " "#! 3 #!  5  3!  # 5 !  " = #!   !   3 #!  5    + " 3 #!  5 #  + " 3 #! =  5 # "    " 3 !  5  !  ˜  #  ! ! ! " 3 #! = 5 #   + " 3 #! 5 #  + " 3 #! = 5 #   "  "       5  ˜ . . . ' " 3 #!  5 #!   +   '. . . ˜.  + D " 3 5"#! #! "  + D " 3 5"#! #!   + D " 3 5"#! #!     + D " 3 5"#! #!   + D " 3 5= #! #!    +   " 3 5$#! #!   +   " 3 5= #! #! . . . . . . . "    " 3 !  5      " 3 5 !    " 3 #! 5 #! 5 #  , . . . . 0.    " 3 5 . " 3 #! . !    " 3 #! 5 #! , 5 #    0. . . . . . .   " 3 #!  5 #!    " 3. . . #!  5 #!.  !# ˜  + " = #!  "  " #!  3 5 #!  "   3  5 !   3  5 C ˜. . .  ' ! !  + D " 3 5 #! #   + D " 3 5= #! #   +  " 3 5    ! 5# ' . ". . .  0. ­b cc ff#„ yf—„ ­ f¨Qflwvmo”o”|uf|7~ |uqs |}c suffg xqvzvsurD¦ s}f l{vz&Qxo”rD s}l{vz" x”|7 € v „±„ fUrz|ux yA~ vzx ~ |6" s}v-s  c f «ƒ ! g#v ~ fp3l{xs c f€v „ xqf „ |¼”­ c f „ f1­ f8xqff ~ r ~ rzg#y”l{xq™Hl{x §A|uqvD|}s suc fg ~ l „ v f¦ €fs}l{&Qvzo”x”rD|s}« l{vzZŠx”v | „ ­ s c l{s fc € vzg" y”oql”s±¼Qrt­sulwvmc x4l{pwf v ¦¦ v s „ c ƒf!|uvDpnoqs}pnr?l{vzDx4f „ l{|¤xŠl{|}x4l ~ s f1c sf c A f ~ ƒl !„ f€s}l{pnvzr?x4Df ­¤„ | f

(211) lnsux4r z s cf f @ " ~ l „l”f« €£sulwvmf1xHy ­ „ fUf|}fUo”xm|us¤f—lns xc f8Z3l{|±™zrDo g#„ ff j s c f1¨ rtpnoqf|6v ¦  rDx ~ ¦ v „ s c f ~ l)bXf „ fUxms ƒ! pnr?Df „ |«. .  . . .                                                                                                                                                                                                        y. support of.  

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(214) ƒ! ªÓ‚wq "

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(217)  ƒƒ!! ª´8q  " ªÓ‚ݼ  l. l. x. initial datas. l{™zo „ f j q ƒ p{r?zf „ |« Z.  .   %0 

(218) &     4!  . ¦ZAo”v xŠ„ €s}s l{c vzl{xŠ|8| |u€­ c lwfUs g#c fD|}vm¼2gs c f1fJy”¨Dr f„ pwsuvql{€€lwoqsZpn;r „ ln|€vmrzxmy”s}y lnxm„ v?o”‚QlwsZlnHgrty su„ f vm~ yŠf §m„  sulwfUyq| lwfUq €f­ l{|uf-€vmx”|}s±rDxms ¦ o”x”€s}l{vzx”|rzx ~ s c f-|us „ f|±|—s}fUx”|uv „ + m§  + D ln| €vzxmsul{xQoqvmo”|6l{xN§AvDs c ~ l „ f€sulwvmx”|(q$l{s l{|7rzy”y „ v?‚ql{grtsuf ~ §z €vmxzsul{xQoqvmo”| ¦ o”x”€s}l{vzx”|« ~ ln|u+ €Dvz xms}l{ln|xQoq€vzvzxmo”s}| lnxQlnx oqvzA o”~|l vz„ fUx”€pwsu lwvmlnxÝx « s c f @ ~ l „ fU€sulwvmx qHl{sln|#rDy”y „ v?‚ql{grDs}f ~ §z ¦ o”x”€s}l{vzx”|€vzxms}lnxQoqvzo”|lnx @rDx ~ ÊÌË!Í2ÊÏÎ :. :. . . .

(219)  ~ ln|u+ €vzxms}l{ln|xQoq€vmvzxmo”s}| lnxmlnx o”vz@ o”|~ l vz„ fxq€p{s}.l{vzlnxÝx\« s c f A ~ l „ f€s}l{vzx qNl{sln|JrDy”y „ v?‚qlng#rDs}f ~ §z  ¦ oŠx”€sulwvmx”|€vzxmsul{xQoqvmo”|#l{x A Dr x ~ £ fy „ f|ufxms6l{x Z lw™mo „ fŽ8s c fg#l ‚qf ~ °Ax”lwsuf1fp{fg#fxms7o”|uf ~ « :. . .   . h σ xx b σ xx. (vx, vy). 

(220) 

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(223). d. σyy g.  . σyy.  . σxy. l{™zo „ f8Ž'q b c f—°Ax”lwsuf1fp{fg#fxms „ , *f" ™zoq£ pnr „ ™ „ l ~ ¼X~ s c „f8¨Dc fpwvq€lwsZ „  "    ,B ¦ " ln|(€vzgy”„ oqsuf ~ rDs0s c f#yAvDlnxmsu|„   #! ~ , #! „" rzx ~ ~ s c #f c |us „ f|±|7sufx”|uv „ rts0s „ c f

(224) ˆ ; x r yA/ vD¼Šlnxmrzsux | ~  ­¤f ~ « fUxqvDf8suf€vz§mx” |ul f s c f1s lnf8xm¨D™zf f„ xq|uff(vrD¦p2(€rz|}f1v rDxrDxqln|uvDs vzy”l{€—fp{rz|}sul{€grDs}f lnrtp fU|u€ ln§Af §m4s 8f fp{rz|}sul{€lwsZgrts lw‚ Z. . .   "/ =  " HE   8  3.  3 I  83 L  33.  8  878  83. b c fxQo”g#f „ " l{€rtp |uc € c fg#f#€rDxs c fx §Af­ „ lwsus}fUx;lnx s c f ¦ vzpwp{v ­ lnxq™H­6r? Ó|uffx h y ¦ v „ ~ fsurDlwpn| " «(ZŠv „ s c f° „ |us

(225) f &Qo”rtsulwvmxv ¦ |uq|}sufg uU˜ ­ f r?¨Df    " 3 #! 5 #!     " 3 #!  5 #! "   + " "#!  + " "#!  + " $#!  + " "#!  ' ˜ *    D 3 !  !5    3  5 !     3 !  ! 5 !    3 5 ! ! . . . . . . . . .  +  " 3  5 ! #   +  " 3 5 #.   " 3 #! . .   "  5 #!  3 #!  5 #! " '. ®¦b „ fc f f~ vz|ufg €vmrDx |±~ |}vqf €&mlnrtoŠsurtf s}~ l{vzs}x v v + ¦ rD|us Q|ufs}rzfU€ g c. . .  +  " 3 !  5 # !   +  " 3 !  5#  . . #!   +  " 3 ! $5 #!!   +  " 3 5$! #!       + 4 "  ! #!   + 4 "  #!   + " !  #! !   + " !  #! 3 5 3 5 3  5 3  5 . . . ;pwvq€rDp |uq|}sufgN¼R€vmo”yqp{l{x”™ s c Hf °Š¨zf ~ f™ „ ffU|v ¦. }U„˜ "1„ f|±oqp{su|l{x\s „ c f ¦ " vDp{pwv ­ lnxq™rm ¨zf s}f‚!´ |}ff Z lw™mo f8Ž «. / m.  ! ! 3 5 #  3 5 = # "  =      =      '    "  + D  , +    , + 4 , +  , +  . ­ c f„f   ". .     +  " 3 ! "5 #!   +  " 3 5   ˜*. . Í!Í ØC e†6‡#ˆe‰. . . ˜. . EGG l. ˜. GG. F. GG   8 GG  GH  8   3. ˜. l. ˜.  8.  8. ˜. ˜. l. ˜.  8  8B8.  .  8   3. ˜.  8. l. ˜ 83.  8. ˜. ˜  878 ˜  ˜83 h. 3 KI JJ J 3 JJ  J 38 JJ J 38 L 33 .

(226) h. . rzx ~   "   8 "  3 "   "   "  ". .   "  x   ,  8 , l , l

(227) ,   y   "  x l , l ,

(228)  3 ,   ,  y  . . ­ l{s c.    " 3 #!  5 #!     " 3 #!  5 #!   " 3 #!  5 #!    " 3 #!  5 #!   " 3 #!  5 #!    " 3 #!  5 #!   " 3 #!  5 #!    " 3 #!  5 #!   " 3 #!  5 #!    " 3 #!  5 #!    " 3 #!  5 #!    " 3   " 3 #!  5 #!    " 3 #!  5 #!    " 3 #!  5 #!    " 3 .  0. .  0. 0. . 0. . 0. B0. B0. . 0. B0. . ‡xNs c f€rD|uf—v ¦ rDxNl{|uvDs „ vmyqln€0fp{rz|}sul{€—grtsuf „ l{rDpÝs c fgrts „ lw‚ Y. . . . 0. GG. ln| ™Dl{¨DfUxH§m. . $. . EGG. .  = " G $ GG GGH $ l. $. . #!. JJ JJ.  l. #!. l JJJ. .  $. B0. #!  5 0 #!  5 0. IKJJ. $Vl $ l V. $. l. B0. B0.  = . . 0. l l. L. ­ l{s c  8 "8  8  " ˜* ˜ ˜  ˜ " , $ " ˜ ,  " ˜  ˜ " , " ˜ C u• " £ f7€rzx8­ „ lwsuf xqv ­\s c f ƒ! #g v ~ fp”rz|u|uvq€lnrtsuf ~ suv1s c ln|$|±€ c fg#fD¼zo”|}lnxq™—s c f7|urzgf6suf€ c xql'&Qoqf rD| ¦ v „ s c f`0l „ lwfUoQ‚#|±€ c fg#f . . . . .   " 3 #!  5 #! "    " 3 #!  5 #!     " 3 #!  5 #!     " 3 #! 5 #!      " 3 #! 5 #!       " 3 #!.     " ! !  ! 3 #  5 # " 5 #       #! '     " 3 #!  5 #!     " 3 #!  5 #!     " 3 #!  5 #!     " 3 #!  5 #! "     ! '  #  . . . .     +   " 3 ! 5 #!   +D  " 3 5 #!   +   " 3 ! 5 #!!   +D  " 3 5 ! #!      ˜*     + D " 3 5 ! #!   + D " 3 5 #!   +  " 3 ! 5 #!!   +  " 3 ! 5 #!      ˜*. . ˜. .   " 3 #! 5 #! "    " 3 #! 5 #!    " 3 #! 5 #!       " 3 #!  5 #!     " 3 #!  5 #!      " 3 #!  5 #!     " 3 #!  5 #!. rzx ~. " 3 #! . . #!. . . . . . . . . ˜. . . '     " 3 #!  5 #!    " 3 #! 5 #! " !5 #   " 3 #! 5 #!         ! '  # . . ˜.  !   !  3 5 # "   " 3  5 #    " 3 5 #!     " 3  5= '  !   " 3 5 #    " 3 5= '. #!.   " 3 5. . . . . . . .     +  " 3 ! 5 #!   +  " 3 5 #!   +  " 3 ! 5 #!!   +  " 3 5 ! #!      ˜*     + " 3 5 ! #!   + " 3 5 #!   +  " 3 ! 5 #!!   +  " 3 ! 5#!      ˜* . . . 4. ˜. ˜. . . . #!. . . . . ". . . .   !      " 3 5 #   ! !#     " 3 5 #  . . . ˜.  . . . . . 4. .    " 3 5= #! "   " 3  5= #! ". . . . . . . . . . . . =   =  . ÊÌË!Í2ÊÏÎ.

(229) /m  3 0" #  'H3 Y‡xs c ln| |}fU€sulwvmx­ f¤­ l{pwp”|usuo ~ 1s c f y „ vzyAf „ s}l{f|!v ¦ s c f ~ ln|u€ „ fs}f ƒ! g#v ~ fp ¦ v „ fp{rz|}suv ~ QxŠrDg#l{€|2lnxs}f „ g|3v ¦ r

(230) xQo”g#f „ l{€rtp £~ ln|±f~ yŠ€f vz„c |ux”lw|uvm~l x~ f rD„ x”s rDc pwqf|u€l{|rD¼”b |ufs c c v rt¦ s(ƒ rDl{x|(r-c vzyqg#pnrDvDxq™z„ ffxq­6fr?vz¨Do”f| c rzlnx”|uvDrts p{„qvz|}y”ln| l{«€—£ fp{rzf8 |}suy l{„ €f|ug#fxmf ~s7l{„soŠc g ln|(" € rzc x”r rt„~ p{rDq€|}sulnf| „ vzlw¥x”f pw~ ~ §m¦ v N„„ s s c cc f f°XrDxqg lwsuff ªF¦f€p{vQffg#³Jf€xmlws

(231) fUxm|usu€ | c fUg#, fD„ « £rzx f1pws vQv f z¦ fUv x”„ |uyqlwsZpnrDx”* f1« ­ r?¨zf f

(232) |uvDpnoqs}l{pnvzr?xŠD|¤f v l{¦ |¤s l{c xHf s ¦ v f „ g @  '  Z lw™mo f8 rDx ­¤f

(233) €vzxŠ|}l f s f—€rz|}f0v rlnxQ°Ax”lwsuf(pnr?Df « . 1 9.  .    " 3 #! 5 #! #!    " 3 #! 5 #! #!  +D   " 3 5 "  + D  " 3 5 "  +  " 3 5 "  +  " 3 5 "  + D " 3 5 " . .  . . . . . . . 4 . . . rzx ~ ­¤f1|ufs. . . . . ! "    " !  =" # 5 #     # ! "    " !  =!  5 #     #  +      "   =!  5     3.  + D   "   !=   5     3.  + 4   "   = "  5     3.  +    "   = "  5     3.  +   "   = "  5     3. U’ ’ U’ U’ U’. ’ ’. "#! . 3 "! . # .  . , 3. . , , ,. ,.  ,

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(240) "  , . ’  ’  " #! 6 ’ " #! "  ’  " #!    ’  " #! +’    " ’ +’ D    " 6 +’ D  " " +’ D   "  +’     " + ’ " ’ + ’   " 6 +’  " " +’    "  +’    " + ’  " ’ + ’ D " C ƒ p{oq™z™Dlnxq™#s c fU|}f1f‚qy „ fU|u|ulwvmx-lnxms}vJs c f ~ l{|±€ „ fsuf|u€ c fUg#fD¼q™Dl{¨DfU|     " ' ’  #! "   A + ’ D " ! ˜ *  + ’ D " / ! +  "  +  " # ’   " ! "  8 ’   ! ’ D # ' ˜ ! *  + ’    " + "  D     ’  =  8 ˜* +’  " !  U+ ’ "    " ! "   A ' ’ # ˜ +* ’  " !  U+ ’  " / !  #    " ! " €vz| , ˜ ' ’ # * ’  ". . #!. . 4. . . "    " !   # "  +      "   " + 4  "   "  +   "  . . . . . 4. . . . . . . . . . .  "  . . . . .  "  . . . 4. . . ’ ’ ". Í!Í ØC e†6‡#ˆe‰.   +   " "     ' /  +  " "     '   +   " "    ' /  +   " "    '. "  . .  ’  ". "  9  A. #! .   ’  " #! ˜ ’ " #!   ’ ". €vz| ˜   " !   " $ A ’ # ’ ". $. . . &. . . . &. .   +   ' /  +   '. ’ ’ ’ ’. . $. . . .  0. . .  0. #!. ’. ’. #! ,. ’.   . ’.    .    "  #!   = "   8    "   8   " !   " !  $  A   #   0 # *"    8       = "   8    #! . B0. . . ’ ".  0. #!. ’ ". 50. #! ,.  .

(241) j. . rzx ~. ’ ".   + " "   A    ' 4  /  +  " " $ 4  $  '  +   " "   A      ' /  +  " "   $ $ '  , / . U’ U’ U’ U’. . #!. .   ’ ". . . #!.  0. ˜ e€vz| ˜ ’  " #!   ’  " #! ’ " #!    ’ " #!    ’  " #!   ’  " #! , ˜ e€vz| ˜  ­ c f „ f  , $ , —r „ f(™zlw¨zfxN§m!}U• " rDx ~ rzx ~ r „ f ~ f°Axqf ~ §m    " ˜  |ul{x  ' ,  " ˜  |}lnx ˜' ˜' !    / " ˜  |}lnx ˜ 'q€vm| ˜ , "  ,  # C o”x ¦ s}z f x”„ v ­7|uvzx”g#|« f;£ suf f1~ vzl{vz§”o”su|Nrtlnx€rDp{€oqpnrts}l{vzxŠ|¼ ­ f €rDx „ f­ „ lwsuf s c f y „ f¨Ql{vzo”|4|}q|us}fg v ¦ f &Qo”rtsulwvmx”|-­ l{s c ’  , ’ rD|-s c f vzxqp{  8   " "  8  ’  " #  *˜  ’  " #!   ’  " #! ! ' 8 ’ # * h8         ݀vz|   "  ’ " #   ’ " #!   ’ " #!   ’ " #! 8: ! 8: * : #! !   "   "  "   " ! !   €* vz| ˜  ’ ˜ #  8 : ’! #  ’ # ˜  8 : ’ #   ’ " #   ’ " #!  8 : #!  F˜ l "  8  " "   8   ’ " #  ˜   ’ " #!    ’ " #! ! ˜8 ' 8 ’ # *    ’ " #   ’ " #!   ’ " #!   ’ " #!  €vm| 8  8 : ! : #!  8 : : #!  ˜ *    ’  " #   ’  " #!   ’  " #!   ’  " #!   $ €vm| 8: ! 8: ˜ *   ’  " #   ’  " #! ˜  8 : #!  , ­ l{s c / ! $. 50. . 50. . &. . &.

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