Aerodynamic Shape Optimization with Uncertain Operating Conditions using Metamodels
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Aerodynamic Shape Optimization with Uncertain Operating Conditions using Metamodels Régis Duvigneau. N° 6143 Mars 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6143--FR+ENG. Thème NUM.
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(124) `bP_z IY Z o}. µ = J (x, W (x, a)) ρ (a) da, . 2 J. J. A. . 2 A. A. . 2 J. J. J. A. Ω(A). σJ2 =. Z. o_!. (J (x, W (x, a)) − µJ )2 ρA (a) da.. K LbP}[]Pe`<\^P_yd%wnh[7| wn`N`bln\{JPwn`wnYi\fe|_wnY P ¥wnaNw\fP_zª[^e`N| P`bl:wn`wnYi\fe|_wn>¢¯lid^Orabwn\^ely` ln¢ e[ w³¥wnewybPikK2lP}[o\feOwn\^P \fLbP Oªkli`bP [^Lblyabhz«P_e \fLbP drxbd^l¥IehzZP wy`©wn`NwyY<\fe|_wn±wyxbxbdfl³²IeOw\feli`ly¢7\^LbP |l<[o\r¢¯ab`N|°\feli` JlydzZeh[^| d^P \^e P\fLbP e`<\^P_yd%wnh[ KLbPhw\]\fP dª[]liab\^ely` eh[ giaNe \fP[^eOewyd \fl©\fLbPOab\^exlie`<\¨wnxbxNd^l<wy|%L®{[^e`|P+\^LbPe`<\fP yd%wnh[:JP_| lyOP "P eyL<\fP_z [^abO[ ly¢| li[]\¢¯ab`N|°\feli`N[:P_¥wnaNw\fP_z ¢¯lyd [^lyOP¨xbdfP_[f|dfeP}z©lixP_dfwn\^e`b«|ly`zZe \feli`N[_ KLbP_`®"\^LbP¨¢¯lid^OP_d:wnxbxbdfliwi|%L a[]e`b«wn` wyxbxbdfl³²IeOw\feli` J˜ e[{aN[]awnY:xNd^P ¢¯P dfd^P}z Z oR J. µ ˜ = J˜(x, a) ρ (a) da, Ω(A). J. A. Ω(A). σ ˜J2 =. Z. Ω(A). o}. (J˜(x, a) − µJ )2 ρA (a) da.. ½ RQ b_G@ XG ¹<¼ P ½ G F K lP_[]\^eOw\fP\^LbP OP}wn` wy`Nz¤¥wndfewy`N|Ply¢"\fLbP d%wn`zZlyOV¥³wyd^ehwnNP J Dly`bP |_wn` [^eOxbYaN[]P [o\%w\^eh[]\^eh| wn±P_[]\^eOwn\^lidf[e`©w | wi[^[^e|_wn T¨li`i\fP¬]µ"wndflwnxbxNd^l<wy|%L® ¡[^wyOxNP#ln¢DlyxJP d%w\fe`b|li`NzZe\^ely`N[ ly¢D[^e_P N eh[yP `NP d%w\^P}z wi| |lidfzbe`b \fl\^LbPj Y ρ kKLbP_`®bab`Ibehwy[^P_z P_[]\^eOw\flyd%[ ly¢2\fLbPOP_wn`¨wy`Nz ¥wyd^ehwn`N(a| P)wyd^P o. 1 X M = J (x, W (x, a )), M C M. N. i i=1,...,N. A. N. J. N. i. i=1. o#L. N. SJ2 =. 1 X (J (x, W (x, ai )) − MJ )2 . N − 1 i=1. KLbeh[wnxNxbd^l<wy|%L zbl<P}[#`bln\)[^a >P d#¢¯dflyO ]xlie`<\^¬ElixZ\^eOe}w\^ely`+P >P_|\ [^e`N|P\^LNP[fwnOxbP e[iP `bP_dfwn\^P_z dfwy`Z¬ Zz liOY©wi| | lyd%zZe`N \fl+\fLbP¨j Y ρ 5{l"P ¥yP_d_±e \:eh[ P_7§I`bl7`©\^LNwn\\^Lbeh[[]\^lZ|%LNwy[]\^(aeh|ª)wnxbxNd^l<wy|%L dfP_g<abed^P}[w+hwndfyP [fwnOxbP{\flrxbdfl¥IezbPSwn` wi| | abdfwn\^P{P}[o\feOwn\^ely`:ly¢ \^LbP#[o\%w\fe[]\^eh| [_'Yblyd7µ Y wyxbxbe|_w\feli`N[ IwzbedfP_|\T¨li`<\^P¬]µ"wndflOP \^LblZz eh[)`bly\|li`N|P_e¥wnNPxbdfP_[^P `<\fYi Q{P_¥yP_d]\fLbP P_[f[ w¨|%LbP_wyxP_dwnxbxNd^l³²ZeOwn\^ely` J˜ ln¢\^LbP |li[]\¢¯ab`N|\^ely` |_wn`¤Pa[]P}z¤e` w T¨li`<\^P¬]µ"wndflwyxbxbdfliwy|%L:\^lP_[]\^eOw\fP \fLbPOP_wy`ªwy`Nz \^LbP¥wndfehwn`N| PyZwy[7[^abyiP_[]\^P}z e`$ i X o# V. ˜ = 1 M J˜(x, a ), i i=1,...,N. A. N. J. N N. S˜J2 =. \ \^] _`#acbd. i. i=1. 1 X ˜ ˜ J )2 . (J (x, ai ) − M N − 1 i=1. @y.
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(126) 9. , -,. !' 2 8 !' 7 !$ 7 6!'. Sxbxbdfl³²IeOw\fP±\fLbP{|l<[o\¢¯ab`N|\^ely`e[Owy`Nzbw\flydfY#¢¯lyddflybaN[]\kzZP_[^ey`®ywi[DP²Zxbhwne`bP_zwnJl¥yPi³\^l#e`<\fP yd%w\fP±\fLbP7|l<[o\¢¯ab`N|\^ely` l¥iP d\^LNP{dfwy`byPly¢JlixP_dfwn\^e`b|ly`zZe \feli`N[lid\^l)P}[o\feOw\fP\^LbP [o\%w\^eh[]\^eh| wn|%LNwnd%wy|\^P dfeh[o\fe|_[ a[]e`bwT¨ly`<\fP¬]µ"wndfl#OP\fLblZz K lxJli[f[]ebee\^eP_[SwndfP P ²ZwyOe`bP}z e` \^LbP`NP²I\S[]P}|°\feli`N[ KDw³YIlid7WZP dfeP}[ÇK{W7 "lid{v{wyzZehwnXWwi[]eh[ Ybab`N|\^ely`N[ Çv W YN[ ° M M C º % ½ ¹ R G ¹ r P+PD ½ G Z# G ½ F KLbP:| li[]\¢¯ab`N|\^ely` | wy`+JP wyxbxbdfl³²IeOw\fP_zIY+e`bP_wyd)K{W J˜ lyd)g<aNwyzbdfwn\^eh|K{W J˜ e`\^LbP ¥<eh|e`be\oYln¢"\^LNP:P²ZxJP_|°\fP_z lixP_dfwn\^e`b|ly`zZe \feli`N[ µ
(127) X ∂
(128) @b . J˜ (x, a) = J (x, W (x, µ )) + J (x, W (x, A))
(129)
(130) (a − µ ), . . L. Q. A. M. m. L. A. J˜Q (x, a) = J (x, W (x, µA ))+. ∂Am m=1. m A. µA.
(131)
(132) ∂
(133) (am − µm J (x, W (x, A)) A)
(134) m ∂A µA m=1
(135) M L
(136) ∂2 1 XX l l
(137) (am − µm + J (x, W (x, A)) A )(a − µA ).
(138) m l 2 m=1 ∂A ∂A µA M X. @i. K LbPa[]P#ln¢D\^LNPe`bP_wyd7wnxbxbdfl³²ZeOwn\^ely`$EZ "ly¢ \fLbP|l<[o\7¢¯ab`|°\^ely` \flP}[o\feOwn\^P\^LbPe`<\^P_yd%wn oRJ "YIeP_zN[7wYDedf[]\]¬ d%zZP d Y kwyxbxbdfl³²ZeOw\feli`ln¢>\^LbP[o\%w\fe[]\^eh| wyOP_wy`®KLbe["wnxNxbd^l<wy|%Leh[k§I`bl7`:wy[k\^LbP YDed%[o\^¬ d%zZP_d YDed%[o\^¬TlyOP `<\( Y YT3. OP\fLblZz @y. (˜ µ ) = J (x, W (x, µ )). KLbPaN[^PSly¢ \^LbPe`bP}wnd wyxbxbdfl³²ZeOw\feli` @Z ly¢®\^LbP|l<[o\ ¢¯ab`N|°\feli`:\flP_[]\^eOw\fP{\^LNP e`<\^P idfwy ]_ ±xbdfl¥<ehzZP}[kw
(139) Nd%[]\]¬lyd%zZP d wyxbxbdfl³²IeOw\feli` ln¢ \^LbP¥wyd^ehwn`N| PyZYIeP hzZe`Nr\fLbP YDed%[]\]¬ dfzZP_d{WIP_| ly`NzI¬¶T¨liOP_`< \ Y WZ3T "OP \^LblZz
(140) ! X
(141) @n!. ∂ (˜ σ ) = J (x, W (x, A))
(142)
(143) (σ ) . ∂A S \fP df`Nw\fe¥iP Yyk\fLbPªaN[^P ly¢S\^LNPªg<aNwizZd%w\^eh| wnxbxbdfl³²ZeOwn\^ely` Ey rln¢ \^LbP¨| li[]\r¢¯ab`|°\^ely` YIeP_zN[wWIP}|li`NzI¬ dfzZP_d EW . wyxbxbdfl³²IeOw\feli` ln¢ \^LbPOP}wn`®kKLbPWZP_|li`NzI¬ dfzbP d Y ed%[o\^¬T¨liOP `<\ @W Y3T "OP\fLblZz d^P}wyzb [
(144)
(145) @ J. 1 X ∂ (˜ µ ) = J (x, W (x, µ )) + J (x, W (x, A))
(146)
(147) (σ ) . 2 ∂A ∂A ¢\^LbPg<aNwyzZd%w\fe|±wnxbxbdfl³²ZeOwn\^ely` @i ln¢Z\fLbP7|li[]\2¢¯aN`N|°\feli`eh[ P_OxblYiP_z#\flSP}[o\feOw\fP±\fLbP"¥³wyd^ehwn`|Pyw{¢ÇwndOlydfP |liOxbP ² ¢¯lid^Orabwreh[lyZ\%wne`bP_z <\fLNw\7dfP_g<abed^P}[±\fLbP\^Lbed%z wy`Nz¢¯lid]\fL OlyOP_`i\%["ln¢ A kKLbP `2I\^LbP)WIP}|ly`zI¬ d%zZP_dWIP}|li`NzI¬¶T¨lyOP `<\ EW WZ3 T DOP\fLblZzaN[^aNwnYd^P_eP_[li`\fLbP7e`<\fP yd%w\feli`ly¢ ]_ DaN[^e`b\^LbP7e`NP_wndkwyxbxbdfl³²ZeOw\feli` EZ Dly¢\fLbP{|l<[o\¢¯ab`N|\^ely` wy`Nz \^LbP)[^P_| ly`NzI¬lyd%zZP_d P}[o\feOw\feli` ly¢ \fLbPOP_wy4` E@ J b7Lbeh|%L YIeP hzb [ l=1. J. FO. A. 2. M. 2 FO J. m 2 A. m. m=1. µA. M. J. 2. SO. A. m. m=1. (˜ σJ2 )SO. m 2 A. m. µA.
(148) !2
(149) ∂ m 2 = (σA ) + J (x, W (x, A))
(150)
(151) m ∂A µA m=1
(152)
(153) M L
(154)
(155) 1 XX ∂2 ∂2 m 2 l 2
(156) J (x, W (x, A))
(157) J (x, W (x, A))
(158)
(159) (σA ) (σA ) . m m l l 4 m=1 ∂A ∂A µ ∂A ∂A µ M X. @y. _P ¥yP d}be¢kly`bPr[]abxNxl<[]P}[7\^Lw\ e[Sd^aNP}zª<Yªw10#wyaN[^[^ehwn`¨j Y"N\fLbPrg<aNwizZdfwn\^eh|)wnxNxbd^l³²ZeOw\^ely` @i {|_wn`¨JPP}wy[^eY P_OxNlYiP_z e`¨|li`moab`N|\^ely` 7e \fL \^ALbP)[^P_| ly`NzI¬lyd%zZP_d P}[o\feOw\feli`ªln¢2\fLbPOP_wy` E@J kKLbP `2Z\^LbPW WZT OP \^LblZz d^P}wyzb[ 5Sl. l=1. (˜ σJ2 )SO. A. A.
(160) !2
(161) ∂ m 2 = (σA ) + J (x, W (x, A))
(162)
(163) m ∂A µA m=1
(164) !2 M L
(165) ∂2 1 XX m 2 l 2 (σA ) (σA ) . J (x, W (x, A))
(166)
(167) m l 2 m=1 ∂A ∂A µA M X. l=1. @<. 829 \,8+:.
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(177) ?#@-AB8. [:| wy` JP[]P_P `® ¢¯lyd\fLbP Y YTXOP\fLblZz En "\^LbP¨OP_wy` ly¢ \fLbP|l<[o\¢¯ab`|°\^ely` eh[:[^eOxbY«\^LNP|li[]\¢¯ab`N|\^ely` _P ¥³wyaw\^P}z wn\\fLbPOP_wn`+¥wnabPrly¢ lixP_dfwn\^e`b | ly`NzZe\^ely`[ KLNe[wi[^[^abOxZ\^ely`+e[li`bYª\fd^aNP¢¯lide`bP_wyd#[^YZ[o\fP O[_ ` \fLbP W YT OP \^LblZz$@@J °Zwr| lydfd^P}|°\feli`r\fP dfOe["wyzNzZP_z\fLNw\aN[^P_[±\^LbP#[^P_|li`Nz:zZP_d^e¥w\fe¥iP_[kly¢®\^LNP| li[]\±¢¯ab`N|\^ely`®QSln\^P \^Lw\ \fLbP
(178) df[]\#zbP dfe¥w\fe¥iP_[ LNw³¥yP`bl e`#NaNP `N| Prli`¨\^LbPOP_wy` ¥wnabPly¢k\^LNP| li[]\S¢¯ab`N|\^ely`®)KLNPrP}[o\feOw\feli` ly¢\^LbP¥wndfewy`N|P dfP_g<abed^P}[D\^Lb&P
(179) df[]\±zZP dfe¥³wn\^e¥yP}[e`\^LNP Y WZT¦OP\fLblZz @! wn`Nzr\fLbP{[]P}|li`NzzZP dfe¥wn\^e¥yP_[e`\^LbPSW WZT¦OP \^LblZz En³¬oi/ xJlyYI`blyOehwn±wyxbxbdfl³²IeOw\feli` ln¢"\^LNP |l<[o\#¢¯ab`N|\^ely` |_wn`«wnh[]lªJP:lyb\fwne`bP}z aN[^e`bw ®P}wy[]\]¬oWZg<aNwndfP_%[ D7W
(180) b\^\^e`b wyxbxbdfliwy|%L2ie`N[]\^P_wizln¢ aN[]e`bK{W¢¯dflyO \^LNP `bliOe`Nwy>lyxJP d%w\^e`b|ly`zZe \feli`N[_ ` \^LNwn\7|_wy[^Py<`blzZP dfe¥wn\^e¥yP_[ wndfPSdfP_g<abed^P}z bab\ []P_¥yP d%wn{µ Y (wn`wnYI[^P_[rOa[o\:JPªxP_d]¢¯lid^OP_z® T¨lid^P_l¥yP_d_\fLbP¨|li[]\¢¯ab`N|\^ely` ¥³wyaNPªwn\\^LbP¨`bliOe`NwylyxJP d%w\^e`b | ly`NzZe\^ely`[lyZ\%wne`bP_z IY µ Y ~eh[7aN[^aNwnY zZe >P dfP `<\¢¯dflyO\fLbPly`bPxNd^P}zZe|\^P}z I Y DW
(181) N\]\^e`b KLb=P Y WbT OP\fLblZz wy[wnxbxNeP_z \^l ab`N| P d^\fwne`<\oYªxbd^lixNwn<w\feli`¨e`¤wnP_d^lZzZYI`NwyOeh| [Se` _Z <I i µ lyOxNwyd^eh[^ly`N[ JP\o"P P_` Y WZTVwy`NzªW WZT OP\^LNlIzN[7wndfP xbdfl¥IezbP_z:e ` < ±WIlyOP#d^libaN[]\"lixZ\^eOe}w\feli` xbdflyNP_O[e`ªwnP dflZzZYI`NwnOeh| [ LNw³¥iP JP P_` wizbzZdfP_[f[]P}z:aN[^e`b\fLbP Y WZTVlyd{W WbT OP\fLblZzb["\^l[^ly¥yP \^LbP#xbdflybP O ]y (@T | d^e\^P_d^ely` n ®lid \fLbP \o"ln¬lyZmoP}|°\fe¥iP xbdflyNP_O o³ I F y G ½ F M M M
(182) b+¾ GÇ %b G@ v{wizZehwn7 Wwy[^e[ YNab`N|°\feli`N [ Çv W Y[ #wyd^P`bli`Z¬ExJlyYI`blyOehwnDe`<\fP dfxliwn\^ely` OP \^LblZzb[#¢¯lid[^|_w\]\fP dfP_z+zbw\%$w y/ EKLbP Y LNw³¥yP JP P_`¨¢¯lyab`z \^l:JP¥iP dfY wy|_|abd%w\fP#¢¯lyd{LNeiLbY `bli`Z¬Ee`bP_wydSzbw\%we`¨LbeiLzZeOP_`N[^eli` E{v W Y[S[^P P § wn`wyxbxbdfl³²IeOw\feli` ly¢ \fLbP)|l<[o\¢¯ab`N|\^ely`ªln¢D\^LNP ¢¯lid^O X @@ L. J˜ (x, a) = ω (x) φ(ka − a k), 7LbP_d^P (a ) wndfP{| wyP_zv W YN[ |P_`i\fP d%[ WIP_¥yP d%wnZd%wyzbewyZ¢¯ab`N|\^ely`N[ φ |_wn`JPS|li`N[]ehzZP_d^P}z YNlydk\fLbP{xbdfP_[^P `<\±[]\^aNzbY w 0#wyaN[^[^ehwn`:¢¯aN`N|°\feli` eh[7P_OxNlYiP_z @@ V. φ(r) = e , 7LbP_d^P eh[7wxwnd%wnOP\fP d| wnP_z w\]\fP `IaNwn\^ely`:¢Çwi|°\flyd}WIaNxbxl<[]P\fLNw\\^LNP#| li[]\ ¢¯aN`N|°\feli` ¥wyabPeh[§<`Nl7`:¢¯lyd7w[]P \7ln¢ N lixP_dfwn\^es`b | ly`NzZe\^ely`[{\fLNw\#|lydfdfP_[^xli`Nz \^l \^LbPv W Y |P_`i\fP d%[ (a ) KLbP `®>\fLbP"P eyL<\%[ (ω (x)) wyd^P zZP \^P_d^Oe`bP_z ¢¯dflyO\fLbPe`<\fP dfxliwn\^ely`ª|ly`zZe \feli`N[ X Çi. J (x, W (x, a )) = ω (x) φ(ka − a k) i = 1, . . . , N . KLIaN[_ (ω (x)) eh["\^LbP)[^lyaZ\^ely` ln¢D\^LNP#¢¯lyl7e`be`bP_wyd{[]YZ[]\^P_O . . . Nc. RBF. j. j. j=1. j j=1,...,Nc. 2. − rs2. c. i i=1,...,Nc. j. j=1,...,Nc. Nc. i. j. i. j. c. j=1. j. j=1,...,Nc. . φ(0) φ(ka1 − a2 k) . . . φ(ka1 − aNc k) φ(ka2 − a1 k) φ(0) . . . φ(ka2 − aNc k) ... ... ... ... φ(kaNc − a1 k) φ(kaNc − a2 k) . . . φ(0). ω1 (x) J (x, W (x, a1 )) J (x, W (x, a2 )) ω2 (x) . = ... ... J (x, W (x, aNc )) ωNc (x). ÇN . KLbP7Own\^dfe ²ly¢\fLbP{[^YI[]\^P_O/eh[lyI¥IeliaN[^Y)[^YIOOP \^dfe|y \keh[kwnh[^lxJli[^e\^e¥yP¬¶zZP
(183) N`be\^Pe ¢J\^LNP{v W YN[k| P `<\^P_df[ wyd^P Zz eh[]\^e`N|°\} `bP|_wn` `bln\fe| PS\fLNw\e \7eh[w)¢¯ab Ow\fd^e² 5Sl P_¥yP_d_ie \%["zbeOP `[]ely` N eh[ a[]aNwyYOlZzZP d%w\fPS(ae` )xbd%wy|\^eh|Pwn`Nz e\f[e`<¥iP d%[]ely`¨eh[ `bln\\^eOP| ly`N[^abOe`NN#KLbPrwn\]\^P_`IaNw\feli`¨¢Çwy|\^lyd s | wy`Plyxb\^eOe P}zlid zZP \^P dfOe`NP_zP ²ZxP_d^eOP `<\fwyY IY:\fLbPaN[^P d}'Yblyd\fLbPxbdfP_[^P `<\{[o\faNzZYiI\^LbPP_OxNedfe|_wn>¢¯lid^Orabweh[7wyzZlixZ\^P}z3 i/ Ç<. s=d (M N ) , 7LbP_d^P M eh[ \^LbP`IabOJP dln¢ lyxJP d%w\^e`b|ly`zZe \feli`N["ab`NzZP_d7|ly`[]ehzZP d%w\feli` a = (a ) wn`z d \fLbP Own²IeOaNO zZeh[]\fwn`|PJP\o"P P `ªv W YN[S|P `<\fP d%[ K ld^P_xbdfP_[^P `<\ \fLbP| li[]\"¢¯aN`N|°\feli`¨¥³wyd^ehwnNee \oY aN[]e`b\^LNe[{\^P_|%LN`beg<abPiN[^lyOP|li[]\7¢¯ab`N|\^ely`ªP ¥wyaNwn\^ely`N[7wyd^P#dfP_g<abed^P}z \fLNw\ | lydfd^P}[]xJly`z\^l#\fLbP{v W YN[±|P `<\fP d%[ (a ) KLbP_[^PP ¥wnaNwn\^ely`N[[^LblyaNzJP{lZ|_w\^P}ze`w#d^P_yely`r\^LNwn\±zZP xJP `zb[ li` \fLbP)j Y ρ "KLbeh[7xlie`<\S7e®PrzZe[f|a[^[^P_z hw\^P_d7¢¯lyd{wxNdfwi|°\^eh| wy®wnxbxNeh| wn\^ely`®"KLbP `®N\^LbP)wyxbxbdfl³²ZeOw\feli` ly¢ \fLbP | li[]\"¢¯aN`N|°\feli`¨e[baNe\{IY [^ly¥<e`br\^LbP)[^YZ[o\fP O @b `N| P\^LNPwyxbxbdfl³²IeOw\feli` eh[|lyOxbab\^P_z®N\fLbP| li[]\{¢¯ab`N|\^ely`[o\%w\^eh[]\^eh| [S|_wn`JPP}[o\feOwn\^P_z®JP_e \fLbP daN[^e`N:`IabOP dfe|_wn e`<\^P idfwn\^ely`+ln¢ e`<\^P_yd%wnh-[ ]# J wy`N*z ]_! Slid#aN[]e`b¨T¨li`i\fP¬]µ"wndfl P_[]\^eOw\^P} [ o# V wn`Nz @y °KLbP|liOxbaZ\fwn\^ely`Nwyk|l<[o\ dfP hw\fP_z \^l\^LNP_[^PP}[o\feOwn\^ely`N[_Z7Lbeh|%L dfP_g<abedfP li`bY v W YN[7P_¥³wyaw\^ely`[ Zeh[`bP ieyebPy j j=1,...,Nc. c. max. c. 1 −M. m. j j=1,...,Nc. A. \ \^] _`#acbd. m=1,...,M. max.
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