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(1)Dynamique des structures et lois d’interface Franck Jourdan. To cite this version: Franck Jourdan. Dynamique des structures et lois d’interface. Mécanique [physics.med-ph]. Université Montpellier II, 2006. �tel-00583223�. HAL Id: tel-00583223 https://tel.archives-ouvertes.fr/tel-00583223 Submitted on 5 Apr 2011. 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..

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(58)  u|∂0 Ω0 = Ud. and F = I +. I4*"J. ∂u ∂X. /  

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(60)  )     '  ! ' ! I4*4%J*   ˆ ˆ ˆ.  ' (1 Γ  (1    1 S = F , P 6! : /       Fˆt = A(Pˆτ , τ ≤ t) , I4*<J. ! S e ! 1>(   '  (1 Γ  Ad * !  (,  ! ! 0.(  +   !3   I :0! 4*$ J*. Γ Sˆn K0 H0 S. Ad. e. Sn+1. Sn.  4*$ P 0.(  +.   ∆Sn+1  F' Sn+1 − Sn * ! S e  1!      !  ) e. S = S0 +. N . (Sn+1 − Sn ) = S0 +. n=0. N  n=0. ∆Sn+1.

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(64)   . Q S0  !   ,   3(,  ! ! 1>(   ' /  * ((      ! '!  !    .(0         '!,     '   !3 S˙ = F˙ , P˙ * / (>     (. '  ( S˙ˆn '   S˙ n   '  '.'. K0 *   0 !   '    * / '     (. '  ( S˙ n+1 '   S˙ˆn   '  '.'. H0 *   0 !   01   *  ' 6! ''  '  '.'.,  ! ' 6!(, ! ' 1>(  (  +(,  '  '((  ! K0   !  2 '(( '  ' H0 *   ' ' ,  0.( '.(     :0! 4*$ '  ! (.  R (:*.   '!    (.      +(!    /         '!* ! '  ! ' !,  !      !  ∀v ∗ (X, t) ∈ U. . 0. T. . Ω0.   tr (P˙n+1 − P˙n )T ∇v ∗ dXdt = 0. I4*=J. Q U = {v∗ , v∗ |∂ Ω = 0}*  6! 0   +  !  '  !  (*   !  6!  ! ' &  '.    (.  ( :  '/(* F' + (   ' !3 (.        '!*           0 . ! * 0. 0.

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(67)      )  Ω0. ρ0 u¨.v ∗ dX +.  Ω0. tr(P T ∇v ∗ )dX =.  ∂1 Ω0. Fd0 .v ∗ dS +.  Ω0. fd0 .v ∗ dX ,. I4*4%J.

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(69)        ∀t ∈ [0, T ] and ∀v∗ ∈ U0 = {v∗ , v∗ |∂0 Ω0 = 0} ,. !  ( '.    (?(*  + , ' '.  '    '   (.  '   F    '!   ' ,     A   (1    1  (* ,   ν = ρ¨ u.    +'    '. '((   1 !    '( u*   ( > 6!  ! ! 1>(   '  (1    1   (1 Γ   (. S = ((F, u), (P, ν)) ∈ Ad    1 Sˆ = (Fˆ , uˆ), (Pˆ , νˆ). :     '(/ I4*44J. Fˆt = A(Pˆτ , τ ≤ t) ,.  νˆ = ρ0. ∂2u ˆ(t) . ∂t2. I4*4$J.   I4*4$J  ' '(( !   '((*  !  > (    ( '.*  (      (?( ( >    1 νˆ , uˆ     1 Pˆ , Fˆ *  !   0.( ',  (>   /       ) ! S˙ˆn = ( F˙ˆn , P˙ˆn ), (uˆ˙n , νˆ˙n )  6!  ˆ ˆ   P˙ n − P˙n = −K0 ( F˙n − F˙ n )   ˆ ν˙ n − ν˙ n = −ka ( uˆ˙n − u˙ n ).   ˙ˆ ˙ˆ    Pn = Kτ Fn et.  2ˆ    νˆ˙n = ρ0 ∂ u˙ n ∂t2. I4*45J. Q ka  ! '  '.'. ! (?(  6! K0 *   '  , /        ) ! S˙ n+1 =.   (F˙n+1 , P˙n+1 ), (u˙ n+1 , ν˙ n+1 )  6!.  ˆ ˆ   P˙n − P˙n+1 = H0 ( F˙n − F˙n+1 )   ˆ ν˙ n − ν˙ n+1 = ha ( uˆ˙n − u˙ n+1 ). et. [Sn+1 ∈ Ad ] ,. Q ha  ! '  '.'. ! (?(  6! H0 *  !  '+  (      '.  '  '.'. ha  ka * !  (?(   6! K0 ,  ' ka  '. ' *.

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(89) u ]){U } = {Fu } + {Λ} ([M. u ] + [K

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(93)        . t 9. 10. 11. 12. 5. 6. 7. 8. ∆t. ∆t x 2. 1. 3. 4.  $*$ P 3( ! (  0 $  '/(.  [T ] 11  [T21 ]  0  .  0 0. [T12 ] [T22 ] [T32 ] . 0 0. 0 [T23 ] [T33 ] . 0 0. 0 0 [T34 ] .. [Tn/n−1 ] 0. 0 0 0 .. [Tn/n ] [Tn+1/n ]. 0 0 0 .. [Tn/n+1 ] [Tn+1/n+1 ].   {U } 0   {U1 }  .  . . . {Un }.     .  3(, !  (  0  '/(   :0! $*$,  . . . u1  u2   {U0 } =  , {U1 } =  u3  u4. . . u5 u6   et {U2 } =  u7  u8. . u9 u10  u11  u12.  !  ' &  !( ,  ( '   [T ]   !/( '   & 1 *   *

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(98) !,  '  ' 6!  '!    + !3 ' !3 (   {Λ}    +( T. {Λ} = ({Λ0 }, 0, ..., 0, {Λn}).  6! {Λ0 }       ' !    ,  6! {Λn }  '!*  '6!,  ! ! &>( I$*9J  F'!   ( > !  )  ( &>( 6! , [T11 ]{U0 } + [T12 ]{U1 } = {F0 } + {Λ0 }. I$*;J.  {U1},  &>( 6! , [Ti/i−1 ]{Ui−2 } + [Ti/i ]{Ui−1 } + [Ti/i+1 ]{Ui } = {Fi−1 } 2 ≤ i ≤ n ,. I$*"J.    '( {Ui}    &>( 6! , [Tn+1/n ]{Un−1 } + [Tn+1/n+1 ]{Un } = {Fn } + {Λn } ,. I$*<J.  {Λn}* G    !  +  6!    +>   '! ' 6!     (.  ( :, !6!  '  {U0}  ( &>( 6!  I$*;J       ( {U1}*   !3>( I$*"J 6! + !  !*  ! , '     ' *   ! +  6!  ! ! 1>( !*  ,      '(,  (1!3 ' '!    *  ',  !  6!  ( '.  !   3(  !(6!*

(99)  : 6!  ( '  ! [Ti/i+1 ]  0 (  &(6!,  6!  ( '   [T ]  &(6!* , !  0/ .(  '/!,  + ! ! ! !  &(6!*  !   A     (  (  ' '!*  , '(( 1#'+  ! ' '. !    1>(  ' ' ' +/ (  6!  !    6! !   D;E   &  !    , '  &(  F'    (  ' '!*.

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(101)  

(102)        .   " #     $%& $.  (.       0 . '  '(  > 6!  !  (  0  '/(  :* 6!        !  '  (. '(  ' 6!  & 8/8*  >  1  ! !       1  '  (*  '!  (.  ( : !  '     '  (.  & F' : !  '  (*   F'! ' '(    ! + '   , /   (0 !  ) / !  B(3  & nC, ! ( :  '/( / (   +'   &(   0 n*  3(, !  0   T! ! !  >  9 T!   (3  & 4 I:0! $*5J* / !  B(!3C, ! ( :  '/( (   +'   6!   ! ! +' ! (   ! +'   '*  3(, ! 6!  0  9 T! ! ! .3 >  = T!   (!3 I:0! $*5J*  ( :  '/( 9      1* Espace. Simplex. Multiplex. 1D t. t x. x. x. 2D y. t x. y x. t. y x. 3D z. y. Non representable. Non representable. x.  $*5 P 3( ( :  '/(  & 4. / !  (  0  '/( B0!C, ! (  0  :   (  '!    6!*  3(,  (  0 .

(103) $<. 

(104)        ! . :0! $*9  $*;   (  0  '/( 0!*  !  , ! + '  '!  ! ,    0.( / 0   R( 2*   : '(( ! )    . ¨i+1 } + [K]{Ui+1 } = 0 [M ]{U   ¨i } + δ{U ¨i+1 } {U˙ i+1 } = {U˙ i } + ∆t (1 − δ) {U       {Ui+1 } = {Ui } + ∆t{U˙ i } + ∆t2 1 − θ {U ¨i } + θ{U ¨i+1 } 2 t. t. ∆t. ∆t. ∆t. x. ∆t. x. Fig. a. Fig. b.  $*9 P   0 $  '/( 0! '(  (3 t. t. ∆t. ∆t. ∆t. x Fig. a. ∆t. x Fig. b.  $*; P   0 $  '/( 0! '(  (3. !  (  '(  * !  6!   ! '.   !   '(, !   & ( :   !'! ! (  0 6!  !,  '.(  0   F*    !   !(     1 ! !  I 1 ! $*4J ) ! !  !  ' I 1 ! $*4J,  8 !   A '(( ! 0    (. 8/8* ! @( 6!  8  ( ! 0    (. 8/8,  + !  !, ! '. 6! '.(  F' : ! (  0  !    '/ (   ! 0.( 6! *  & !, 6! !     ' (   1 1( '. 0(*  ! 6! 6!  (  0   0!,   !    '.(   & 8/8 6! *  ' 6! '' !  ( : ! '. I 0 !/ !J, ! ! (( (  ! '(   !(6!  4  '/.

(105) $=.  #   

(106)  

(107)        . '.   '(   . '.  . 4 (t).

(108)  .  .

(109)  .  .  . . $ (x, t).  .  .  .  .  .  . 5 (x, y, t). U U U U. 9 (x, y, z, t). δ = 23  θ =.    1 4. δ = 23  θ =. 2 3. δ = 23  θ =. . δ = 76  θ =. U U U U.      . 1 4.  1 4. . δ = 23  θ =. 1 4. . δ = 23  θ =. 1 4.

(110)   $*4 P (     8   (. 8/8. ( I1  ! ' ! (J !  !   6!  / 6! I:0! $*"J*   ' ' ,  '  6! ! ! (?(    (,  8   !  !(6!  4% + ! ' 6! '!3 1!    (.   ''   * −30 −50 −70 −90. 10 ln(errmax). −110 −130 −150 −170 −190 −210 STFEM interpolation lineaire STFEM interpolation quadratique Schema d’Euler Implicite Schema de l’acceleration lineaire Schema des differences finies centrales. −230 −250 −270 0.00. 0.01. 0.02. 0.03. 0.04. 0.05. 0.06. 0.07. 0.08. 0.09. 0.10. pas de temps.  $*" P ! 1! ( 3(!( ! ! . :,  '  ! 6! '  (!3,  '.( 0   ' 6! 6!   ( ! 1>( I 1 ! $*4J*  ', '     :    (3*  !   ! *  F, !  '.   '(,    ! '.( 0  ('  4  ! '.( 0  3'  $* !  '.  ,    ! '.( 0  ('  4, '(.

(111) $7.  ".  1  ! '.( 0  ('  $, '(  1*.     ' '. 0(  6! !   ( ! 1>(,  (3   0 F*  F, ! ! '!'. ( [ti , ti+1 ], !   ( ! 1>(,    ! ! !! O  '! I:0! $*<J* t. t. ti+1 Couche temporelle ti. x. x. Zone de discontinuite t. ti+1 Couche temporelle. y. ti x. SIMPLEX. MULTIPLEX.  $*< P V  '!.  ', '  (!3, ! ! '!'. (     !'! O  '! I:0! $*<J*.  ! '.   '   0 . #   '.6!  (  0  '/( 6! !  *     !,  !  (1!3 ' !  (  0* ( '  , ! 1 (1    (  0  '/ (*   !  (.   2 '!*  ,    ! ,  '.  '( *    6!  (  0 F'/ !      ( *  ! !,   !  '! ! ! ! (  0  1!   3(  !    '   ' (  0, ' 6! !  #'*  !,   !   & (!3*    ' D9E, !  ! '.6!      (  0 '(   (3,  !  '.   '(*  '.6!  '    #'*   '!   1>(  ' ' +  I D;EJ* G     .

(112) 5%.  #   

(113)  

(114)        .     '.6!  (  0, (    '!   ' D9E*  ', #   ,   '   0 ., ' 6! (   A ' ! /  *   0 ! '.6!  (  0  '( , 1  !  (  0  '/(  !'!*   6!   # '. ! ' 1>(*  ! '    !3  !0!  !1 D4<E  ! '(( '!3  (    D4=E*  !  (.   2 '!*    !3 '.,  ' '! F'! ! !  (   '/( Ω × [0, T ]* , ! ! (  Ω  ( d  ! (1   N  T! ! (  0  '/ (,  ( ! 1>(    !  d × N , ' 6!   0  6! d = 2 ! d = 3*  ! ! (!  (  !  !  ' '!   >*   '.   D4=E*   '   0 . !  ! !  '( , 6! !1/ !        (, '  B+  '/(C*    !    ! 1>(,   ( ! ! +  0 * ! :3  ,  d = 1  '  (  0  !'! :0! $*=* 8. 15. 21. 22. 23. 24. 17. 18. 19. 20. 11. 12. 13. 14. 16. pas de temps. 9. 10. t x 1. 2. 3. 4. 5. 6. 7.  $*= P 8  '/(. ! ' '!   !  T! ! (  0,  '>   +  '/ (*  '  ' '!   '(  T!      / '(  T! ! +  '/( '* ! ! ' / ( '., ! !  (  0 $  '/( :0! $*=* ! ' (  0,  ! 0!  +  '/(*  ( +  '!  T! I=,7,4%,44,4$,45,49J* !  '(  ' '!     '!3  T!    t = 0*  !3>( +  '!  ! I4;,4",4<,4=,47,$%J   >(,  T! I$4,$$,$5,$9J*   '(  T! ! + k + 1  ' '!     '!3 ! +  '/( k , #!6! 1  !   '(    : *.    0  ' '.   (!     &>(    !*  ( &>(   ( <,  '  ( ", '***  ',   ! '. ',  6!    D4<E,  + ! ! ! &>(  ( 4<* W((, ! ! 3( ! ,  0   (  ' '!    ' ((    0:' +, ( .

(115) 54.  ". !  '  ! '(6! I5  9  '/(J  0   ' 1* ! ! ' (.  ' '!, !    ! 1>(  (  0 (  !  (  0  '/(  !'! :0! $*7* 13. 8. 14. 15. 11. 12. 9. 10 7. 5. 4. ∆t/2. 6. t. ∆t x 1. 2. 3.  $*7 P   0  '/(  0!    :.  !  !(6! 1!   ' (. + , ! 3( ! 1   1 ,   '(   '!3 1! '  (  0 0! ! :, :0! $*4%* 19. 20. 21. 16. 17. 18. 13. 14. 15. 10. 11. 12. 7. 8. 9. 4. 5. 6 ∆t. 2. 3. t. ∆t/2. x 1.  $*4% P   0  '/( 0!   :. F'   ( 3(!(    !   '( !  !3 (!  I:0! $*7  $*4%J   4.10−3X I:0! $*44J*  3(   ( 6! ' '.6!  (  0 +   A      '* 3(  '    ! (  '  0   (  ' '!* 8 !  !(    ' (. + , !  !    !     0 * !+,.

(116) 5$.  #   

(117)  

(118)        . deplacement (mm). 0.00. −0.01. maillage non regulier (methode frontale) maillage regulier −0.02. 0. 25. 50. longueur de la barre (mm).  $*44 P  '(  T! !  (  0 :0! $*7  $*4%    5∆t.  '.   A > (!   '! ! '      *.  (  ) &     *.  (. ( :  '/(  !    !  (  1>( (   #!  ' !3 ( !   ' +( '  !(1* G    '   0 . ! &.>   (.   ! * !  '(  ' ' !3 ( !  ,  #!     !  +'   ', 6!  '   ( > !  . T 0. Q.  ∂c Ω. rvdsdt =. p . {Ve }T [Ce ]{Re }. I$*=J. e=1. I$*7J   '! (1   ' ! ( Ee *   '(  '/ !  ' rie 3' ! '. 6! ! i  ( Ee *   '  ! '' !3 !  '  ! !   (*.  ! [Ce ]   0    1'*  !/( ' 3 × 3, Ceii  !   ! i  1  0   hI , Q I   ( '   h     ( I!  ( (    J,   ! i  '   ! ' '* , 6!  ! (!( '  {Re } = (r1e , ..., rne e )T. u ] + [K

(119) u ]){U } − [C]{R}  ([M = {Fu } + {Λ}. I$*4%J.

(120)  . 55.    ( '  0    1'   !3 +'  ' '  {R} Q [C]   '! (1   '  ' '*  ! '. 6!   i ! (  0,   ! [Ti/i+1 ]{Ui } − [Ci ]{Ri } = {Fi−1 } − [Ti/i−1 ]{Ui−2 } − [Ti/i ]{Ui−1 } 2 ≤ i ≤ n ,. I$*44J ! (  '  ' ' I{Ri}J,  '! ' &>( 6!  '    ' '   +(* ! ! &>(    F'!  !  ! (.  & B (.  ' & ('C   ! '    (.  '/(*         ' D;E*  !  !(6! (   '.! ! 1' ! ! 1 '     '(   '!3 1!    0'  *  !   (  +'    '(   ( '. I D;EJ*  ! !     0 ! K   ! (  (  , ' !''>,  (.  (  0 '(  I D;EJ*.  + " .  ! ! (  !    ' 1  !  ! '  '  *  ' (!  ' '.,   '!    (. +   (  0  '/(,  1 ( ? !!   ! .>  ' *  & #! ' (, ! ! !  (  0  9* !6!         M  ' ((*.

(121) 59.  #   

(122)  

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(124) 

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(126) .  '>  1>(      +(    ) !  6!. u ∈ H∂1u (Ω). . F (u) =. inf 1. v∈H∂ (Ω) u. |∇v|2 dx − 2 f, v ,. I5*4J. Ω. Q F (·)   +'* Ω  ! ! 1  2 ' ! +> !@ (( 0!> [ H∂1 (Ω)   '  +'  H 1 ! !    ∂u Ω   +> ∂Ω [ f ∈ H∂1 (Ω)  ,  ·, ·   ! '    H∂1 (Ω)   ! *  1!   '! ! !  1>(  ((    '  !  +' '> : '  +' '    (' !3 ! ! + (   0!   Ω* u. u. u. 5;.

(127) 5"  #     

(128)  

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(130) 

(131) 

(132)           

(133) . '!'  ' !  +' '      :/  ! (  0 (   !   Ω, !   ' ! 0  0 *   Th  (  0 (   Ω       '! !  :0! 5*4  Tˆh  (  0 !   Th       '!* / ' h   : ! (  0*   ' '! '((    ! 0  (   0 ! (  0*  '  ! ! (  0 (    xi *   Nh  (1    ! ! (  0 ( *.  5*4 P   0 (   ! *.  '  +'   Yh  :    Yh := v ∈ L2 (Ω) : v = '. ! Tj ∈ Th , ' v = 0   Ω \ Ωh. . .. I5*$J.   v¯h  +'  Yh 0   O   2 *    6! ' +'     F 1,   '   : ! 0  0  !   1!*   D¯vh  0  !   / 1!  v¯h *   :    < D¯ vh , qˆ >= −. Ωh. v¯h ∇ˆ q dx. ∀ˆ q ∈ Xh .. I5*5J. Q Xh   '  +' @ ! Tj  '! ! Ωh I    +' &6! ! Th J*  ' 0   :  +'.

(134) 5<.  

(135)  Jh (¯ vh ) :=. . |Dh v¯h (xi )|2 |Tˆi | + ηh2. ti  . | [|¯ vh |](t) |2 ,. I5*9J. i∈Nh t=1. i∈Nh. Q Nh  (1  '  ! ! (  0  η  !  '/ ( +*  (  ti  . ηh2. I5*;J. | [|¯ vh |](t) |2 ,. i∈Nh t=1. +    A   !  '! [|¯vh |](t)  v¯h  0 ! 0( (xi , x(i) t ), (i) Q x(i)    (1 x , t = 1, ...t ,  !   x *  i i t t  '  ' (  6!    (.  ! (.   2 '!*. (>    !,   ' (   *   I! ! '! ,  D4$EJ, ( 6! !'  ' (  I5*;J (  !  Γ/'0'   +'      |∇v|2 dx . J(v) :=. Ω. ! v ∈ H∂1.    +∞. u. (Ω). I5*"J. ! v ∈ L (Ω) \  .>( (  '! ! !  3(  ! 1>(   I5*4J     ((    +' Q ·, ·h  :  .  f = f. − div f. 1.   0  . Ω. *. H∂1u (Ω). I5*<J. Fh (v) := Jh (v) − 2f, vh ,. f, vh := 0. 2.  v ∈ L2(Ω) \ Yh f 0 v + f 1 · Dh v dx.  v ∈ Yh ,. I5*=J. !      . 1#'+  ' ' ( 0  ' '!     ! (  !/ (6! !  !3  '0'   (.   '(  ' +( ' '  (.  ( : ' 6! I 2J*       ,  ! ( 6!  +' Jh (¯vh ) !  '   ( > !  Jh (¯ vh ) =. . .  1 |Tˆi | i∈Nh. . ti   1  (i)  vh |](t)nt xi − xt  [|¯ 2 t=1. 2 +. ti  t=1.  ηh2 [|¯ vh |](t) |2  ,. I5*7J.

(136) 5=  #     

(137)  

(138) 

(139) 

(140) 

(141)           

(142)  (xi , x(i) )*   vhi   '( Q nt   '! ("  ! 0( #T ! ( Ti,  Vh = vh1 , ..., vhP ,  '! (1  !   / '(* (1  ! (1,  Vad  (1  Vh  6!, ! '. 6! ( 6! ! 'K '((! '  +> 1 ∂t Ωh ,   !  '!  ! !,  ! '. 6! ( Ti, 6! ! 'K '((! '  +> 16! ∂u Ωh ,   !  '! : h. [|vhi |] = vhi.  1>(  ((  '  ) ! Vh ∈ Vad ((!(   +( 6!  6! !  Jh (Vh ) =. 1 T V AVh − F T Vh 2 h. Q A  ! ( ' &(6! :   F   '! ' ((1*   '( ( ! ∂u Ωh    '(      !  '!   +>*  ' F'  ( ' A   ' ((1 F * !  ' (. #   ! 0' ':6!    I  3(   (1  1(J* ?(   1>(  ((  '  ' 6!, #  !  (?( '  @'! '.6!     (.*  3(,  (1 0   ( ' A  +  !  (   !  !*  ' + , ! 1 ! ( ' 1 ,  + ! (  !(   (* G   ! '.6! (   (  !&  ! ' ! !  !*  !  ,  + ! '!    ! , ! '  (, '***. (.   !  6!  3(   :0! 5*$*     (,  ! 3 '  '!* !  6! >(,   '(( ! 3 ' ' 1!    (.  ( : ' ! (  0 > :* ! !     D4$E*    :0! 5*5  5*9 ,   !  !  !(6!  ! I 1  0 .(6!J '(( !  0   +'   : ! (  0* :0! 5*5  (1  !  1!    0'    5*9 '!3 1!    0'  *  '  6!   '0'  '.  4 !   ( 3(  '.  4\$ ! 3( 9*  ' 6! ''  '(   '  ,  !  6!   0 !   (?(*  &    (  F' ( #!*  ', ! ! (  0 > : I4<%%% (J,  (  ' '! 1!      5<X !!  '!   *  36!    (1  0  1 6!   !3 + !! '  (. I4<%%% ( ' =<%% !J*.

(143)    

(144)  . 57.  5*$ P  !   '(. ! ! "   #.  !  ' (  ! .6!   6! #   ' ! ' ( !  ( .> !  '0'   (.  (.  ' & ('*  ! ! H!' ! (    '!  ! '     *   ' ! !3 (, 1       I J  !  0  0!  ' 1 !* (. (  '  '   ( I  (   &>(  !J*  !      '!',  (1   !    1>( Q  !  '!, '((    '    :! * !   (  '.'.   ' '*.

(145) 9%  #     

(146)  

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(150)           

(151)  1 Example 1 Example 2 Example 3 Example 4. error. 0.1. 0.01. 0.001 0.01. 0.1. 1. 10. h.  5*5 P !. Dh uh −∇uL2 ∇uL2. +'  h I  J*. 1. error. Example 1 Example 2 Example 4. 0.1. 0.01 0.01.  5*9 P !. 0.1 h. Dh uh −∇uL2 ∇uL2. 1. +'  h I J*.

(152) 

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(155)

(156) 95 !

(157)  

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(159) 99.

(160) 

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(163)

(164)      . (

(165)  ,  .  (>     '( '(  .(> (''6!,    ' ! >( ' I1 !!J*  '  01     '(     '@' !!   +(*  (>      ! '(   ( !* G +  .&.> 6!  F .(6! ! !! ! ? 00    '  !    .>*   ϕ   +(  6!       '! X ! ( t* I9*4J   !1  I    !3  (10   D47E, D99EJ,   '(, u  '(   ( > !  u = ue + uw I9*$J Q uw    '( ]  !!  ue    '(( *.       ( (>   (?(, (  ! '  ' 1#',  '(  +  !  ! ϕ* ϕ(X, t) = ϕe (X, t) + ϕw (X, t) sur ∂Ωc I9*5J ϕw    +(  !  !!  ϕe    '((  I :0! 9*4J* x = ϕ(X, t). N. φ. X. φ. e. x. T. φ. w.  9*4 P '(    +( .  +(  !  !!     !  !! 6!      '. *       !3  '.    ! *  1 6!  !( !! w !   +( : w = Kf s I9*9J Q K   '@' 01  !!, 6!   (  !3  ' ', f   +' 6!  s   '    0(*   !. .

(166)  #

(167) . 9<. (    ! !( !! w1   ! (  ! 4  '   ! (  ! $,  !     ! . w2. w = w1 + w2. . w1 = K1 f s, w2 = K2 f s. Q, Ki,   '@' 01  !! ' ! ' i = 1, 2*    01 *      ,  ' * G    ' ! ! 0    '   '. 6!   ' '   !  *   (T, N ),  > '  ' !   ' '  !3 '* N   '! (   T  '!  0*   ϕ1   +(  !  ( ',  ϕ2 !  !3>( '*     0  :   vT = (ϕ˙ 1 − ϕ˙ 2 ).T I9*;J  r = (rT , rN )  +'   ' 3' !  ( ', 3(    > ' ,  '(  (     !!,        !  )  d w  w    dt (ϕ1 .N ) = k1 rN |vT |   d w   (ϕ .N ) = −k2w rN |vT | dt 2 !! '  ! ' i = 1, 2*. I9*"J. Q kiw   '@'.  '(   0/    +(  !!  ! !* w ϕw I9*<J 1 .T = ϕ2 .T = 0  6! '  '  !!  '  *    1 ! :  ' 1#'*    ' D$7E # ( 6!       ! '. 0( 1 !* (      !  +  6!           '. 0( 1 !*   !!  '! !3   ' '  0   +( '  !(1*  gN = (ϕ1 − ϕ2 ).N  '   !3 ' '   ! ' ',  ' !    0 ! ? '   ( > ! , rN = proj+ (rN − ρn gN ). Q proj   !  #' !   ρN        +. +. I9*=J > 0*   !(1. rT = projC(rN ) (rT − ρT vT ) I9*7J   Q C(rN ) = rT ∈ 2  6! rT ≤ µrN   ' ! 'K  !(1, µ   '@'  +(  ρT > 0*. $ .

(168) 9=.  #  

(169)

(170)      . P  >   !! I9*"J  I9*<J, '   A '(( ! +'  rN  vT * P       0     ,   !! /   ,  vT = (ϕ˙ 1 − ϕ˙ 2 ).T = (ϕ˙ e1 − ϕ˙ e2 ).T I9*4%J P  $,   !!,  !  ' ' +       (/ ' !3* 6! & !! '  !  /    (' !3*   ! !    ' (      0(*. (     .  0   '(       ! (.   0 0  '! * !  '    ,  !  (.  ( :  !  '  (, ! (. ! ('* 6! /  ! (!( '   &>( /  !  M (q + )(q˙+ − q˙− ) − hFint (q + ) − hR − hFext = 0. I9*44J. Q h      (, M (q+ )   ( '  ( , q+   '!      ! ! ( t + h, q˙+   '!      ! ! ( t + h, Fint (q+ )   '!    +' !, R   '!    +'   '  Fext   '!    +' 3! *  >   !! I9*"J, q+  ! +'     '((/   I   6! ! ((   ', !        6!J qe+ ! '!   ! '!  '  ' ' R ) I9*4$J  !  ,    '((  ! .& 6!,   +' !  !(      6!   +(  ) q + = fw (q e+ , R). I9*45J.  &>( 6!  I9*44J      ! (.  R (/ :, 6!     1 qe+ * , q˙e+  '.'. '((  (   ! (q˙ke+ )k∈N :  Fint (q + ) = Fint (q e+ ). e+ (M +h2 K)(q˙k+1 − q˙ke+ )−hRk+1 = −M (q˙k+ − q˙− )−hFint (qke+ )+hFext (k). I9*49J.  ! (.  R (: '     !  0,  ∂fw '1! ! 0  ∂q    '       (  e+ ∂M ∂q e+.  00*.

(171) 97.   .  &>( I9*49J  '! !3  !!,  ' '   +(* ! !  1>( '!, #   ! ,    !!,   (. B/ (.  ' & ('C*  6!   !    > ' !3* ! '. 6! ! α '   ! ' ',   Hα  ( '  '. 0(  1 *  '!  rα ,  '!  ' ! ! α    k + 1  veα ,     6!         k + 1*  >  :   e+ v eα = Hα (q˙k+1 − q˙0e+ ) I9*4;J Q q˙0e+    !   0 I' ! > ' J* , ! '. 6! ! α, 6!  ! (!( '  !  v eα − hW rα = vfeα I9*4"J Q vfeα    1*    6!  ! α ! ,   1'   ' rα  !   !   '  ! !     '* ( ' W  :   W = Hα (M + h2 K)−1 HαT. I9*4<J. ! '. 6! ! α    ' '  +( '  α α α rN = proj (rN − ρn g N ) I9*4=J +. . α eα α ) (r rTα = projC(rN T − ρT vT ). ' gNα        ! . I9*47J. I9*$%J Q gNα−  ' ! ( t*  &>( 6!  I9*4<J, I9*4=J, I9*47J  !   ! (.  R 0 , '    ! !  6!  I9*4=J, I9*47J  /   /F 1*  (.  6! !3   1 '  yα = (vTeα , vNeα , rTα , rNα )* >  ' '! y α ,     ! '   ! ' ' !  α+1    ! #!6! '0'* α− α eα α eα gN = gN + h(vN + k w rN |vT |). ( !  .  (>    ! ! 3( ' 6!  +( ! '& ! !   !*  !  !(6!   '(  '  !  3( !3*  ! ! ' '(      ' D$7E* G  '.    !  !(6! ''   .>  0!3*  !   ! L!  1 ! (>     '   ''  ( *.

(172) ;%.  #  

(173)

(174)       Muscle forces. Quadriceps tendon. Femur Patella ACL PCL. Patellar tendon. Tibia tray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

(175) ;4.    

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