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Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization

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(1)Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization Régis Duvigneau. To cite this version: Régis Duvigneau. Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization. [Research Report] RR-5949, INRIA. 2006, pp.40. �inria-00085058v2�. HAL Id: inria-00085058 https://hal.inria.fr/inria-00085058v2 Submitted on 18 Jul 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. Adaptive Parameterization using Free-Form Deformation for Aerodynamic Shape Optimization Régis Duvigneau. N° 5949 Juillet 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5949--FR+ENG. Thème NUM.

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(75)   "   K. ² g‰b)ypˆ‰ˆŠ[k¨F±¶nkfyŸ‘|g‰°|Qab§P1y|{d{dg‰bd‘ M y|b)~«y cdfhfhQb>^]_N)y|{}Q S ¨F^`NdQ²²%´ ~\Q±¶nkf`P1y/^hgŠnkb«ª&Ndg‰€N7»d^€] S gŠb¡y ˆ‰Qayk]q^`£¤]hi>c)ypfhQa]]_Qab)]`QŒ±¶nkf&^`N)QºP1y|{d{dg‰bd‘ M gj]{dfhn/°>gj~\Qœ~*©K[*]`n|ˆ‰°KgŠbd‘t^`N)Q{df`Qa°KgŠnkc)]ˆŠg‰bdQœypf&]_[\]_^`QaP®±¶n|f ∆P ¥ (kopt ) Ml. l+1. (kopt ) Ml (0) Ml+1. l+1. . . +)+)%& + <

(76)   /. MNdQRP1yp{){dgŠb)‘{dfhn\~\c)gŠbd‘^`N)QR²%²%´®n>nkfh~dgŠb)yp^`Qœ] ±¶fhn|Pµ^hNdQRˆ‰yp^_^hg‰Qn>nkfh~dgŠb)yp^`Qœ] gj]c)]`Qa~ ^hnŸPnkbdgŠ^`n|f ^hNdQŒyk~dyp{\^hgŠnkb1{dfhnKQa]h]¥%MNdQaf`Q±¶n|fhQ|¨p^hNdgj] P1yp{){dgŠb)‘N)(s,y|]mt,^`nu)©LQU{)f`Qœgj]_QaˆŠ[R~\Qa]hfhg‰©}Qœ~1ypb)~R^hNdQŒ(ξ,yk~dypη,{d^`ζ)g‰n|b1°/ypfhg‰y|©dˆ‰Qa]m~\Q»)bdQœ~ ¥ M n$]_g‰PR{dˆ‰g¢±¶[*^`NdQ{dfhn|©)ˆŠQaP¨K^hNdQ±¶n|ˆ‰ˆ‰n/ª&gŠbd‘Ry|]h]`cdPR{\^`g‰n|b)]&y|f`QºP1y|~\Q. · Gœk¹ s = φ(ξ) t = ψ(η) u = θ(ζ). MNdgj]€Ndn|gjQUn|fhf`Qœ]_{Ln|b}~d]%^`ntypb1cdb)n|cd{)ˆŠQœ~1f`Qaˆ‰yp^`g‰n|b)]`Ndg‰{1©}Q^qª(QaQbR^hNdQŒˆjy/^`^`gjQUnKn|f€~\gŠb}y/^`Qœ]mypb}~R^`NdQº²%²´ n>nkfh~dgŠb)yp^`Qœ]¥ MNdQœ]_Qyk]`]`cdPR{\^hgŠnkb)]&ypfhQf`Qœy|]`n|bdb)y|©dˆ‰Qºyk]ˆŠnkbd‘1y|]^`N)QPRy|gŠb ~\g‰fhQa

(77) ^hgŠnkb)]np±^`N)Qˆ‰yp^_^`gjQy|f`QªQˆ‰ˆF]`cdg¢^hQa~^`n^hNdQ]`N)yp{LQº^`n nk{\^`g‰PRgŠšaQ|¥ MNdQab¾¨}±¶cdb)

(78) ^hgŠnkb)] ¨ y|b)~ ypfhQ]_c){d{}n>]_Qœ~^`n$©LQ{Ln|ˆ‰[Kbdn|PRgjypˆj]znp±m~\Qa‘|fhQQœ] n ¨ n ypb)~ n ¨Ly|b)~ ypfhQQ¼\{dfhQa]h]_Qœ~gŠb ^hNdQ &Qfhb)]q^hQg‰b{Ln|ˆ‰[Kbdn|φPRψgjypˆj]©)y|]`θgj]. 0 i. n0. 0. φ(ξ) =. ni X. n0i. Bi (ξ)φi. ψ(η) =. n0 Bj j (η)ψj. j j=0,...,n0j. i i=0,...,n0i. · G Gœ¹ ypfhQºn|b)]`gj~\QfhQa~y|]&yk~dyp{\^hgŠnkb°/y|f`gjyp©dˆ‰Qa]a¥ ypfhQ]`Q^U^hn*ˆ‰gŠb)Qaypfº°/ypˆ‰cdQa]a¨LgŠb§n|f€~\Qf. θ(ζ) =. j=0. i i=0,...,n0i. 0 k. 0. j X. ² g‰b)ypˆ‰ˆŠ[k¨dªQg‰‘|N>^`g‰bd‘RnKQ$g‰Qb>^€] (φ ) ¨ (ψ ) py b}~ (θ ) ³bdgŠ^`gjypˆ‰ˆ‰[|¨@ª(QagŠ‘kN>^`g‰bd‘*nKQ 1gŠQab>^h] ¨ (ψ ) py b)~ (θ ) ^hn$~\Q»)b)Q^`NdQgj~\Qb>^hg¢^q[$P1y|{d{dg‰bd‘ (φ ) i=0. 0 j. n0. Bk k (ζ)θk .. k=0. k k=0,...,n0k. j j=0,...,n0j. nk X. k k=0,...,n0k. i i = 0, n0i , n0i j ψj = 0 j = 0, n0j , nj φi =. k n0k. θk =. · G½‹|¹. k = 0, n0k .. MNdQab¾¨^hNdQ›y|~)yp{\^hgŠnkb {)f`n\Qœ~\cdfhQ¡nkb)]_gj]_^h] g‰bPRnK~dg¢±¶[Kg‰bd‘ ^`N)Q›y|~dy|{\^`g‰n|b °/ypfhgjyp©dˆ‰Qa] (φ ) ¨ (ψ ) ypb)~ ` ^ 1 n R P ‰ g d b Š g R P ‰ g  š  Q h ^ d N  Q | y d ~ | y \ { ` ^ ‰ g | n Ÿ b   > n q ]  ^ ¶ ± d c } b

(79)  ` ^ ‰ g | n F b ¥ (θ ) ³bn|f€~\Qf^hn~dQ»)bdQ&yºn|b)Q£G^`np£¤n|b)Q P1yp{d{)gŠbd‘}¨/±¶cdb)^`g‰n|b)] ¨ ypb)~ ]`Ndn|c)ˆ‰~t©}QNdnkPRQn|PRnkf`{dN)g‰]`P1]F±¶fhn|P ^`n ¥ YKc 1gŠQab>^+n|b)~dg¢^hgŠnkb)]^`ny|€Ndg‰Q°kQ ^hNdgj]+fhg¢^hQfhgŠnkby|bt©LQ&~\QφfhgŠ°kψQa~±¶f`nkP θ ^`NdQ^`Qaf`PRg‰b)y/^hgŠnkb{dfhn|{LQf`^`g‰Qa]%y|b)~t~dQ[0,fhgŠ°/y/1]^hgŠnkb[0,fhcdˆ‰1]Qa] n|± &Qfhb)]_^`QagŠb{}nkˆŠ[KbdnkPRg‰y|ˆ‰]  K ¤¥ YKc $g‰Qb>^Unkb)~\gŠ^`g‰n|b)]&aypb©}QQ¼\{df`Qœ]`]`Qa~y|]. i i=0,...,n0i. j j=0,...,n0j. k k=0,...,n0k. φ0 = ψ0 = θ0 = 0,. ·G3 ¹. φn0i = ψn0j = θn0k = 1, φi < φi+1. i = 0, n0i − 1,. ψj < ψj+1. j = 0, n0j − 1,. θk < θk+1. k = 0, n0k − 1.. MNdQœ]_Q n|b)~\gŠ^`g‰n|b}]1ypfhQ^€yHkQb gŠb>^`n¬yknkcdb>^Ryk]RˆŠg‰bdQœypf$nkb)]_^`f€ypg‰bk^€]R~\cdfhgŠbd‘7^hNdQŸPg‰bdg‰PRgŠšœy/^`g‰n|bžn|±U^`N)Q­y|~dy|{\^`g‰n|bžn>]q^ ±¶cdb}

(80) ^`g‰n|bF¥ MNdQ&~\Qa‘|fhQQœ] ¨ ypb)~ f`cdˆ‰Q^`NdQzn|PR{dˆ‰Q¼\gŠ^q[np±}^`NdQ&fhQa]`cdˆŠ^`g‰bd‘PRy|{d{dg‰bd‘)¥ ¤^ ]_QaQP1]%c)]`Qˆ‰Qa]h]^`n~\Q»)bdQzyºP1yp{d{dg‰bd‘ ª&Ndn>]_Qn|PR{dˆ‰Q¼\gŠ^qn[*gj]&nNdg‰‘|NdQaf&n^`N)y|bŸ^`N)yp^Unp±^`N)Q~\Q±¶nkf`P1yp^`g‰n|b¾¥MNdQb¾¨)^`NdQt~\Q‘kf`QaQa]np±%^hNdQP1yp{d{dg‰bd‘ n ¨ n ypb}~ n y|f`Q ]`gŠPR{dˆ‰[]`Q^zQœi>c)ypˆ@^hnt^hNdnk]`Qnp±^`NdQ~dQ±¶n|fhP1y/^hgŠnkb n ¨ n y|b)~ n ¥ (ŒbdQ&aypbb)np^`gjQ^hN)y/^%^`N)Q&PRy|{d{dg‰bd‘~\Q»)b)Qa~t©>[· Gak¹%ypb)~· G Ga¹n|PR{Lnk]`Qa~tª&g¢^hN^hNdQ&²%²´ ~\Q±¶nkf`P1y/^hgŠnkb· Gœ¹fhQa]`cdˆ¢^€]gŠb   2 )

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(95) +^hgŠ1)nkb × (n + 1) × (n + 1). XXX · GœŽk¹ 1. J = ∇δP . i i=0,...,ni ni. 1. 0. i. i. i−1. 0. i=1. i i=0,...,ni. 1. 0. 1. 0. 1. 0. 3. i. 3. j. k. ni. AD. ni nj nk. nj. nk. ijk. i=1 j=1 k=1. g‰]yPRyp^`fhg¢¼ bdnkf`Py|b)~ y|b†Qœ]q^hgŠP1y/^hQtn|± (ξ, η, ζ) n/°|QfŒ^`N)QtQaˆŠQaPQab>^hypfh[Ÿ°|nkˆŠcdPRQnp± g‰b)~\gjQœ] ijk ¥MNdQ c)]`Qnp±^`N)Q²dfhn|©LQbdg‰c)]bdnkf`P0∇δP ypb)~Ÿyt»)bdgŠ^`Q£¤~dg !@QfhQb)Q∇δP yp{d{)f`n½¼\g‰PRyp^`g‰n|bnp±^`NdQ‘kfhyk~\g‰Qb>^n/°kQf^hNdQQˆ‰QPRQb>^€ypfh[$°|nkˆŠcdPRQnp± g‰b)~\gjQœ] ijk [KgŠQaˆ‰~d]. k.k. ijk.  3 X. n2i. l l l l (∆Pijk − ∆Pi−1jk )2 + (∆Pij−1k − ∆Pi−1j−1k )2 ∇δPijk = 4 l=1. l l l l + (∆Pijk−1 − ∆Pi−1jk−1 )2 + (∆Pij−1k−1 − ∆Pi−1j−1k−1 )2.  n2j l l l l (∆Pijk − ∆Pij−1k )2 + (∆Pi−1jk − ∆Pi−1j−1k )2 + 4 l l l l + (∆Pijk−1 − ∆Pij−1k−1 )2 + (∆Pi−1jk−1 − ∆Pi−1j−1k−1 )2  n2k l l l l + (∆Pijk − ∆Pijk−1 )2 + (∆Pij−1k − ∆Pij−1k−1 )2 4. +. l (∆Pi−1jk. −. l ∆Pi−1jk−1 )2. +. l (∆Pi−1j−1k. −. l ∆Pi−1j−1k−1 )2. . · GJp¹. .  21. ,. ª&NdQaf`Q ∆P gj]^`N)Qn|PR{Ln|bdQab>^ l n|± ^hNdQ~\gj]_{dˆjy|QPRQb>^&np±^`N)Qn|b>^hf`nkˆL{Ln|g‰b>^ ∆P ¥ ²g‰b)ypˆ‰ˆŠ[k¨>^`NdQy|~dy|{\^`g‰n|b*{df`n\Qa~\cdfhQUn|b)]`g‰]_^h](g‰b*PRgŠb)gŠPRg‰šg‰bd‘^hNdQºyk~dyp{d^`g‰n|b*nk]_^ ±¶cdb)^`g‰n|b Q°/y|ˆŠc)yp^`Qœ~1c)]`gŠbd‘Ÿ· GœŽ|¹ y|b)~«· GJp¹

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