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Stabilized Finite Element Formulations for Shells in a Bending Dominated State

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(1)Stabilized Finite Element Formulations for Shells in a Bending Dominated State Dominique Chapelle, Rolf Stenberg. To cite this version: Dominique Chapelle, Rolf Stenberg. Stabilized Finite Element Formulations for Shells in a Bending Dominated State. [Research Report] RR-2941, INRIA. 1996. �inria-00073758�. HAL Id: inria-00073758 https://hal.inria.fr/inria-00073758 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. Stabilized finite element formulations for shells in a bending dominated state Dominique Chapelle and Rolf Stenberg. N ˚ 2941 Juillet 1996 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(289)       ¬   ,ª

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(294) ¬. 

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(296)  5. 

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(298).  .  . 

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(300) .  . 

(301)   .     

(302)    .  . .  . %.  n ]^ž    *  /. . %. >.    

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(304) `­M]­]^d ]ml f†h 5 ™  ]^h}b `^`gpŸ” W`^Wd 5. 

(305)        " *     

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(314)       ¬ ]ml ” W`^` n Ws"‘}haW n Jf v b*`g`"ª

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(317) ¬ Ÿ ” W n ] v W rd•`^pfN—ad b*]^h  Nª     1  2 )

(318)  +     ¬   ª     1  2 )

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(322)  +   ¬   %  ª  ¬     ¯ 1 & ¯  ¯  ª

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(330)  

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(334) Wh;W`gWXWhAdšl ™ T U ]ml]gXia`g]^W l d U b*dZd U Wp rb*h—}W . . ‚Y.

(335) W`^]gX]^h˜b*deW n b*dd U WHW`gWXWhAdZ`^W

(336) žW`"deffN—jdšb*]gh bia_ v W`^p n ]ml{ia`mb†rWXW

(337) hQd /B—}b†lqW n hA_aXW v ]mrb ` ler U WXW Ÿrڐ ™ W œ_}b*d•]^f†hª4„ ™ ±¬ — W

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(339) `^`0d U ]mrš AhaW lel"]ml ]gh–d U W v b ha NW ²   Ÿa”?]gd U   ‘}tAW n0™ (£ Œ   (    

(340)   ªq\}‚ ¬H²     : 

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(344)    "                     )     (/)    

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(346)    Nª '      8 ) !       ¬  :  '      8  " *  ªÎY ™ ‚

(347) ¬ . . .  .  !  .  . "   *. ~‚ %. ªÎY ™ ‚¬. ¯ Wdª'      8 ¬     —}Wb v —a]gd v b v p ™ T U Wi v ff* ” W”?]^`g`“haf ” n ]gžQ] n WZ]^hAdef d Uav WWl{deWi}l ™   #"   ] v lqd

(348) Ÿab n ] v W rd?r

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(350)   %     n ]^ž    8   *    n ]^ž   ª'   ¬    *     

(351)  %     n ]^ž 8 %  ¯   *     ¯ ª'   ¬   *   %   .  . . 

(352)  . ‚°.

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(354)       n ]^ž ª    ª '   ¬e¬   *   : ª ( ª'   ) '    ¬ %   '     ¬ %. ªÎY ™ €Q‚ ¬ Ϊ Y ™ N€ €¬. 

(355) . f v r`mb v ]gdqp Ÿ´” W#”?]g`^`(Wxlqdšb —a`^]cl U ªJY ™ Q€ ‚*¬ßb*h n ªÎY ™ N€ €¬bNlb)lqWi˜b v b deW v W l{_a`^do]^h WXXHbY ™ ± —}W`^f ” ™  Wb `ml{f U bžW      ¯ ª'    ¬   *   :   ª'    ¬    ªÎY ™ €N±¬ . . b*h n  v f†X. 

(356)  . WXXHb © ™ N‚ ¯ ( ª'   ) '   ¬  ‚ € lqfd U b d?” W NW

(357) d . /.    

(358) .  ª '   ¬}¯   ¯ ª '   ¬ 143  :  ª'   ¬      ¯ ª'    ¬   *   :  ( ª'   ) '    ¬ %. ªÎY ™ €*YA¬ ªÎY ™ €N°¬. + fNX—a]^hj]^ha oW œ_˜b*de]gfNh}lZªJY ™ €Q‚*¬ Ÿ“ªÎY ™ €N€¬?b h n ªJY ™ N€ °¬ ”?]^d U ªÎY ™ N€ ²¬šŸ}” WZf†—adšb ]^h  ª'      8 ) '        8 ¬ ªe‚ . . % %. . %. /. . :  ª'  ¬}¯ ¯  ª' . 5 : 5. . . ª•‚ . . 5 . . . . ¬. . /. 

(359)   %      

(360)  .  n ]^ž. 8 %. ) '   ¬  :   '     ' ' 143 ªÎY ™ €N„¬    ª  ¬   ª   ¬ / %  ª•‚  8   8 143   ¬  n . ¬ ( ª'  /. 1432%.   ¯   ¯ .  8  * .  . ¬. : . %. . . 143.   

(361)    ¯   *  %. ]gž  .  8. \]^h˜rW   . Q² Ÿ   Y  °QŸ}d U WW`g`^]giade]cr]^dBp v W l{_a`^d?i v f žW n ]^h  „AŸ0‚

(362) “ N]^žWxl ª•‚   5 : / 5   :  :  ¬ ( ª'   ) ' /   ¬ %     ª '  ¬}¯ ¯  ª'  ¬ 143 %     ª'    ¬   ª'   ¬ 143  ” U Wh)r U ffl{]^ha 5

(363)  ªB€!:  ¬ *   ~‚  €     :  % b*h n Jf v )lqXHb `^`0W

(364) hafN_a U ™ ’WtAdŸ W

(365) XXHb ©™ ‚Z]gXia`g]^W l /   ¯   ¯  143  :     %  _ v d U W v ŸA]^d U f†` n l / % >}¬  8   % 8   8 143  €Qªe‚.  * . :.  '. .  5 . ªÎY ™ €³†¬ ªÎY ™ €

(366) ¬. . . . ‚„. ªÎY ™ €N¬ ªÎY ™ ±N²¬.

(367) \]^h˜rW. ‚      ‚      brfNX—a]gh}b*d•]^fNhkf*"d U Wob*—}f žW v W`m b de]gfNh}l †]^žW l  Nª '      8 ) '        8 ¬ . ' 5     . : %. . 5 .      8   * . 

(368)  %      

(369) . %.  n ]gž. . 8.   . %.   

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(371) . . 

(372) . ªÎY ™ ±N€¬.   ]^hGªJY ™ €N„¬ ™. n W n ah W l•l f*  Ÿ. 

(373)  .  .  8   *  %. Wh}rsWŸ—QpHl{Wded•]^ha r`^f lqWWhaf†_a U deffNhaWŸa” WZfN—jdšb*]gh   ª '      8 ) '        8 ¬  :    '    5 %     8   %    8 

(374)     "’WtAdŸ” W_}l{Wd U W v Wxlq_a`gd?d U b dd U W v W¡WtQ]clqd lZª

(375)   ² ¬   lq_}r U d U b d /  ª   > 

(376) %  8 > 

(377)   1!8  :    8  . ªÎY ™ ±N±¬. b*h n. ªÎY ™ ±*YA¬ ªÎY ™ ±N°¬. 

(378)     . 

(379)   5   .  %. Ϊ Y ™ ±N„¬ T U Wzi v ff (f Md U ]cl v Wxlq_a`gd5]cl v *b d U W v deW r U ha]crb*`}b h n U W

(380) h}rWz” WZ”?]g`^`ai}fl{dei}fNhjW]gd •d f WXXÜb Y ™ Y— W

(381) `^f ” ™  ]^d U d U ]ml

(382) # ” W‘ v lqdz” v ]^d•W  Nª '       8 )

(383)   ²  ²  ² ¬   % &  ªÎY ™ ±³†¬ ”?]^d U /    ( ª'   )

(384)   ² ¬ %    ª'  ¬˜¯ ¯  ª

(385) Q ¬ 143 ªÎY ™ ±

(386) ¬  / % ª '    ¬  ª

(387)   ² ¬ 143   5  n ]gž ª '   ¬  8  n ]^ž ª

(388)   ² ¬           n ]^ž 1 ª '     ¬   8  n ]gž 1 ª

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