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Multilevel Parametrization for Aerodynamical Optimization of 3D Shapes

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(1)Multilevel Parametrization for Aerodynamical Optimization of 3D Shapes Nathalie Marco, Alain Dervieux. To cite this version: Nathalie Marco, Alain Dervieux. Multilevel Parametrization for Aerodynamical Optimization of 3D Shapes. RR-2949, INRIA. 1996. �inria-00073750�. HAL Id: inria-00073750 https://hal.inria.fr/inria-00073750 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. Multilevel Parametrization for Aerodynamical Optimization of 3D Shapes Nathalie Marco and Alain Dervieux. N˚ 2949 Juillet 1996 THE`ME 4. ISSN 0249-6399. apport de recherche.

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(196) š;X'xh{œmcfxqc{WYXeoX'xgp ¤(Xi6h”¤ p€eqxoXpopReqTceD­_cfb‰eqT6hjaVŽjT"X'c{pg^‰b‰sYhjšVec^di6X r š†sŸeqTVXlWaVb‰eo^‰bdX¨jXb·WYXeqTVh r ­†eoT6X hjmVeo^`WcfbMt'h{iFeoxoh{b;t'c{i pgT6h”¤™pgm6a6x€^dhjaVpyTV^dŽ{T\œ½xoX'¬Fa6Xi6t^`X'p¦ ˜ X(eoTV^`i6´,eoTcfe eoT6X hjx€^dŽf^di\hfœ;c{iŸh rVr ¯¹X¨jX'i r X't'h{a6mVb‰^di6Žb‰^dXp ^`iÈeqT6X\œ=cfte}eoTcfei6h{xoWcfb ¨{X'teohjxopcei6h r X'pc{xgX þc{x€^`eoT6WYXeg^dt «WX'c{i6ph{œ i6h{xoWcfb9¨{X'teqhjxgpeoh œ=c{tX'p hfœHeqT6X}pgTc{m;X{¦ SxohjW‚eoTV^dp(mh{^`i†eh{œH¨F^`X¤­VeoT6XªyaVb`X'x £h”¤Ç^dp(xqcfeoT6X'x(^`i6pgX'i6p€^`eg^`¨{X,eoh\TV^`ŽjT œ½xgX'¬Fa6X'iVts¢hjpot^`b‰büceo^`hji6p

(197) hfœ eqT6X"šh{a6i r c{x€sj¦kS TV^`p r Xœ½X'te\^dpc{WYmVb‰^`²6X r š†sÊeqT6Xeqxqcfi6pomV^`xqceo^`hji œ½hjxgW

(198) aVbdcfeg^dhjiŸ^`iŸ¤ TV^dtT"TV^`ŽjTŸœ½xoX'¬Fa6Xi6t^`X'p(c.PX'te i6h{xoWcfb`pyš6aVe i6h{eRtXb‰b¨jh{b`a6WYX'p : º a6x c{i6p€¤(Xx eoh¢eqTn^dp$mVxohjšVb`X'W ^`p%eqh c”¨{h{^ r eqha6pgX¶eqTVX¶²i6Xpge$b`X¨{Xb ½¤ TV^dtT¾^`i†¨jh{b‰¨jXpLeqT6X¶¤ TVh{bdX¶poXe%h{œ Ù¥ÚÛÙ½Ü.

(199)  

(200)   !"$#&%(') *,+-./0132546 /. Š”Œ. š;hja6i r c{xgs%i6h r X,thnh{x r ^`icfeoX'p »¦ S T6XWYhjp€e ^`iFeoX'xgicfb³b`hnh{m$^dp(eoT6XŸŠ”¸ b`hnt'cfb³xgX'pgXDc{xgtT©h{œHeoT6XpgeoX'XmXpge r X'pgt'XiFelmcfxqc{WYXeoX'x   C ¦ 3D coordinates. 3D mesh. tetraedra. (airfoil, aircraft). OPTIMIZATION METHOD Transpiration conditions applied on the skin mesh 3D coordinates triangles Solve state equation Ψ( W,γ ) = 0 (Euler) adjoint-state ( Π). External loop. optimality conditions ( j’(γ ) = 0) Choice of a level. About 10. Find the steepest descent ρopt automatically n+1 n γ = γ − ρ opt L P P* L* j’ ( γ n ). iterations. 3D remeshing. S^`Žja6xgX‹Y º xgŽ†c{iV^`¡Dceo^`hjih{œeoT6XhjmVeg^dWŸ^`¡Dcfeg^dhjib`hnhjm·¦ ).  ! . ! #.  ! . S T6XŸœ½h{b‰b`h8¤^`i6ŽXvVc{WYmVb`X'p,xoXb`sÈhjiÊpoX¨{X'xocfby¨jXxgsÊpg^`WYmVb‰^`²X r m6xghjšVb`X'WYp'¦ ¹iÊm6xghjm;hjpg^`i6ŽLeqT6XW­ ¤¼X,cf^dW eqhYpoT6h”¤§TVh8¤eqT6X,c{mVmVb`^`tDceo^`hji©hfœeoT6XWXeqT6h r eqhcŸi6X¤ ŽjX'hjWYXeqxgsL^`p XDc{p€sj¦ ¹WYm6xoh¯ ¨F^`i6Ž\eqT6X«X t^`X'i6tsLh{œHeoT6X«£h”¤poh{b‰¨jXxTc{p i6hfe š;X'X'ithji6pg^ r XxoX r X»vVtX'mVe š†s%pohjWYX«eoa6iV^`i6ŽŸh{œ Æ hjaVxqc{i†eliFa6W

(201) š;X'xgp«c{i r iFa6WšXxlhfœ9b‰^di6X'c{xpg¤¼X'Xm6p r aVxg^`i6ŽŸeqT6X^`Wmnb`^`t^‰eueg^dWYXpgeoX'm6mn^di6ŽV¦ G  @  9 ) B D B BX •n’ B=Ä D ÄF< ’&= À ‘"= Ä – Å“• L Ä  D     ) ] à B D L ‘ B D L +   G DÄÅÀ‘ S T6Xpo´†^`i¶WYX'pgT¶h{œeqTVX¤^`i6ŽY^dp thjWYmhjpgX r šFs  +

(202) i6h r X'p¦ S T6Xthjpge œ½a6i6teo^`hjicfbH¤¼X,TcD¨jXa6poX r ^`p h{œeoT6Xœ½hjxoW Y  ,     .  (   3  (    &  &    &     ,+ c{i r     , G ¦  . .   .   3   &. &32. ¤^‰eqT   , +)G  ,+ S T6XŸ^`iV^‰eo^dcfbRt'hji r ^`eg^dh{& i6p,c{xoX r X²i6X r š†sAc$œ=c{xg²Xb r ®¶c{tTAiFa6W

(203) š;X'x,h{œ G 4   c{i r c{i¢c{i6Ž{b`XŸh{œ cfegec{t´%h{œ "4 G  r XŽjxoXX'p­M¤ Tn^dtT©Ž{^`¨{X'p$Y   . , + 4  +09 c{i r   . ,       , G 4 +  G $ ÛÛ öba'Þ»ïàï.

(204) 5C 0 + ' % 

(205)  . Š. S^`Žja6xgX,ŒY ®¶‹¯þ¤^di6ŽhjmVeg^dWŸ^`¡Dcfeg^dh{i ^`iV^‰eo^dcfbpoT6c{mXf¦. S ^`Žja6xoX  Y ®¶‹¯±¤^`i6ŽhjmVeg^dWŸ^`¡Dcfeg^dhji ²i6cfbpoTcfmX,cfœ¥eoX'x +)G hjmVeg^dWŸ^`¡Dcfeg^dhjiL^‰eqXxqcfeg^dhjiVp'¦ ˜ XRTcD¨jXRa6poX r cpqcD¤eohnh{eoT,°l¯¹tsntbdXpp€eqxocfeqXŽ{sš;Xe¹¤¼X'X'ieqT6Xyœ½hja6x€eqTb`X¨jXbþ­¤^`eoT

(206) m6c{xqc{WYXeqX'xgp c{i r eoT6X²i6XpgelbdX¨jXb?¦ \uœ¥eqXx +)G hjmVeo^`WŸ^d¡'cfeo^`hjiL^‰eqX'xocfeo^`hji6p­eqT6XhjšVeqcf^`i6X rr xoc{Ž^`p    . , G 4    . c{i r    . , G 4

(207)       S T6X Æ wR˧eo^`WX^`plc{šh{aVe \T6hja6xgplh{i¶cY¸}ª Æ p€ecfeg^dhji·¦ S^`Žja6xgX'p,ŒLc{i r  m6xgX'pgX'i†eeoT6XŸ^din^`eg^ücfb poTc{m;XYc{i r eoT6XŸ²icfbypgTc{m;X{¦ º i S^`Žja6xgXY V­³¤(XYpgX'Xc xgX r a6teo^`hjikh{œeoT6XpoT6h_t´LhjiLeqTVX¤^`i6Ž6¦. G  @  @  ; ’ B B X •V’ B½Ä D Ä< • D¾• B “–j“8• < ’  J • =* – Ä D à B ’&=. +)G +  . DÄÅÀ‘ Ù¥ÚÛÙ½Ü.

(208)  

(209)   !"$#&%(') *,+-./0132546 /. ŠD . ~ X r a6teg^dhji©h{œeqT6XV¯¹poTVhnt´M¦ S^`Žja6xgX Y ~lX r a6teo^`hji©hfœeoT6X^diV^‰eo^dcfbV¯¹pgT6hnt´$cfœ¥eqXx +)G hjmVeo^`WŸ^d¡'cfeo^`hji^`eoX'xocfeo^`hji6p¦ S T6Xpg´F^`i¶WYXpoT¶hfœeoT6X,cf^dxgt'xocfœ¥e^dp thjWYmhjpgX r h{œ

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