Dec 15, 2004 - I INTR O DU С TI O N. Multidisciplinary modelling is an important ingredient for the solution of an engineering problems of utmost importance in ...
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Optimization loops for shape and error control: extended lecture notes A. Dervieux, F. Courty, T. Roy, M. Vázquez, B. Koobus
N° 5413 December 15, 2004
ISSN 0249-6399
ISRN INRIA/RR--5413--FR+ENG
Thème NUM
apport de recherche
"!# $%!& ' )(*! +, !- $ !&$. !&'/0 ! !& 132546879 :;68=5?A@CBD2CEDFG)wysutv{M|§wAic)
Qt|¬hxAh¹©¸{vhxA¦§Mtv|§)w£&|¬tve,))|§wtc)wAiª sveA)t
Qtv|§h,|¬¯ctv|§)w3M¦§){v|¬tveAhsMwy"h xy¦¨t|¨ª«¦¬iI¤)iI¦
Qtv|§h,|¬¯ct|¬w¥
A{iI)wyQ|¬tv|§)wy|¬wAX² dfeAi,M
A
y¦¬|~Mtv|§)w®t¶sveyM
Gi AiIsv|¬w
A{)A¦§ihº{i¦~tiItDsv)wy|§lX\h»hD|§wA|§h,|¬¯ctv|§)w |§s {i
G){vtvic¼²5´µw Dsuic )wX±
y){utc\M©½tviI{fsv)h,il{viIh,){v¾Qs¿)w h,icsue¥AM
Att|¬wZ`£¿ii ÀAMh,|§wAiteAiD
A{vA¦§ihÁM©©¸{vhxA¦§Mtv|§wAh,icsue¥AM
Att|¬w®|§w®tviI{vhslM©¿)w®)
Atv|§hD|§¯IMtv|§)w
y{vA¦¬iIh¶8
y{vicsuiIwt|¬wyDsv)h,iM
A
A¦§|~t|¬wys¿tv,|§wti{
X¦§Mtv|§)wMwy±tv5M{vtv|~M¦¼Â|¬Ã`i{iw8tv|~M¦ZÄC}8xyMtv|§)wysI²dfeAi|¬h,
A¦§|¬ª It|¬wys-)©teAi
y{v
X8suic¶hDitveAQOGiI)h,i¦¬icM{c°¿\r¶ h A|§wA|§wAtv)itveAiI{ÂGMtve¥|§Qic)sl sutliÃGicKt|¬¤ih,i tvey\ys IMwOGiQiI¤)iI¦¬
Xic¶£&eAi{itveAi)
Qt|¬h,|§¯IMtv|§)wO¦§8
ys-|§w\¤)¦¬¤iX)tve¥sveyM
Gi
Qtv|§h,|¬¯ctv|§)w>yt±|¬h,
A{¤)i)w®MiI{vQQr8ª wXMh,|§Qicsu|§)w>>ÅÆGÇh,icsueO)yM
Qt|¬¤\|¬tr)\tv±)
Gi£&|¨te teAiQ|¬È± xA¦¬tv|§iIs&)©>teAiÉ Ê5
A{)A¦§ih |¬w\¤)¦¬¤iI¼² ËCÌ Ç)ÍcÎÏQÐ Ì ÆyÑGÒXÓÔKÕ Ì ÆyÑ ÌÂÖ Å × Ì ÓÕØ;ÍMÆ®Ç)ÔÑ«ÅMØ Ö½Ö Ô Ì Ç Ì ÕÂÆGÍMÑ Ì ÕlÇ Ì ÎÍÏQÓÕlÅMÏOÙ§ÚÛÜ ÝßÞà)áÂâ3ãXä\ÍMÓKÑ Ë ÍMÏQÓÕ Ì ÍMÆ ÝOÏ Ö Ñ³Ø;ÇØ~ÕKå ÎØ Ò Ö Ø½ÆGÅÓKæ±Ý®ÍcÇ ÌÖ½Ö Ø½Æ\ç±ãGؽÐÏ Ö ÅѳØ;ÍÆ"ÅMÆGÇèàXÅ Ö Ø;Ç)ÅMÑéØ;ÍMƥؽÆá Ì ÓvÍÆ`ÅÏQѳØ;ÎKÕê~ë/ì ÏQØÕAí Ì ÕÑ¿Ñ Ì Æy϶î ïlÅÓvÎ ÌÖ ÍÆ Ì±ð;ñ ò ë Ö¬Ì Õó8ôMåµó8õ ö ÏQؽÆ"ó8÷÷ øA² ùOú8û`üý þ ÿ)¢ 5M{vtv|~M¦>l|¬ÃGiI{viIwt|§)¦ ÄC}8xyMtv|§)wysI`Ê |§wA|¬tviDĦ¬iIh,iw8tm¶itveAQ¼¼m¶icsue &A)
Qtv|§¤\|¨tr`mOxA¦¨t|¬¦§i¤i¦ l
Qª t|¬h,|§¯IMtv|§)wZXÉ¿hD
yxQtMtv|§)wy)¦ZÊ ¦§xA|~¶Âr\wy)hD|~sI l
Qt|¬h)¦>p\eyM
GiliIsv|¬w
B "!$#%&(')*) "!(,+.-0/12,3154678 ':9 ')7?*)')!( @,A 9= &" C " "!$!$#%#%&(&(')')*)*) " "!(!(,,+.+.-0-0//1122,,33115546467788 ':':9 ')')7?7?*)*)')')!(!( @,@,A A 99== &"&" DFE V E == '' G>G> " "AHAH!I!I'' I W99° $ !% & é dfeyi¤){v|~My¦¬ics Mwy Qicsv{v|§Gi,tveAi
A{)y¦¬iIh £&|¬tveA|§w iI{utM|§wèsv
y iIs)© £ ){¾`² ZitxyssrOtveXt ){viDteAi )w8tv{)¦Q¤){v|~MA¦§iIs)wy M{iftveyiÂstti&¤){v|~My¦¬ics²dfeAir,)wtM¾i-Q|¨Ã`i{iw8tC{)¦§iIsC){Q|¬wyÂttveAiÂ
A{vA¦§ih £ i ){vil|¬w\¤)¦§¤)iI,|§wZ(° y£ ·yiI¦§¤M{|§)A¦¬ics/{¿)
A
A{À\|§ht|¬wi{{v{C©¸){ Mwy±svey)
Xi
yM{Mh,itvi{|§¯It|¬w,){f h,i t{v|~ Ai ·ywA|§wA,D
yM{vtv|~ xA¦~M{ ¦§\IM¦`h,iIsve Qicsv{v|§
Qt|¬w±©¸){ ²CdfeAi|§QicD|§s¿tv,h,|§wA|¬h,|§¯itveyi Í ö Ì ÎÑéØ½× Ì ÏQÆGÎÑéØ;ÍMÆ ¦§\)¾\|§wA ©¸{ $O£&eA|~e¶st|§su·yics tveAi Ì ìÏyÅ Ö Ø½Ñ³æÎÍÆAÕKѳÓvŠؽÆXÑ $ !'" ² Zi t&xysÂ)ssuxAh,itveXt&tvey)i () )y|§)w * !,+ ;% + |~s&M¦§£fr\s |§w\¤)i{vtv|§A¦§i)² p8tM{vtv|§wAD©¸{vhteAi $.- !% - - A£ iQi ·XwAiMw|¨ti{t|¬¤i
Qtv|§h,|¬¯ct|¬w
A{Q icQxA{i\rtM¾\|§wA
!
* -
!
$ - $ - & + . +
´µw¶){QiI{ftv,i ȱ |§iw8t¦¬r )h,
AxAtvitveyi{vic}xy|¬{iIsurQsutviIh )xAtv
AxQtÂAi{|¬¤t|¬¤iIsf£&|¬tve¶{iIsv
XicKtftv,teAi|¬wy
AxQtsIy£ i Xx suitveyi )|¬w8t&h,i teAQ¼A£&eA|~e¶
A{v¤\|~QiIs iwAiI{)¦¬|§¯ic M){Mwy)iÂhxA¦¬tv|§
A¦¬|§i{s&|§wtveAi©¸{vh)©5,sv
yt|§)¦`·yi¦~M© teAisMh,i A|¬h,iwXsu|§)wO)©teAi {v|§)|§wy)¦Z
A{)A¦§ih xAwA¾£&wys²-Êy){-ic)eO|¬tviI{Mtv|§)w0/`XtveAi¶ÎÍå«ÕKÑ«ÅMÑ Ì ){,Å)Ç ö ÍMؽÆyÑ&ÕKѵÅÑ Ì 1-|§s&,sv)¦§xQtv|§)wM© 1- ! ¸ 2 + $.- 4365.7 + $.- 8& '3iQiwyMtviteAi ÏQÆGÎѳØ;ÍÆGÅ Ö ç)ÓvÅ)ÇMØ Ì ÆXÑZ\: r 9;- ! + 9 $! + $.- @? A 1=- + $.- CB é ´µwteAi)svi)wwAMteAi{-)wysutv{M|§w8tI8tveyisttv|§+)wXM{|¨tr)© teAi©¸xAwyKt|¬+ wyM¦>£&{v|¬tvics° 9 $ - 1 - >!D"E
F%
) wytveAisvi t-)© teA{viIiiI}8xyt|¬wysfyGF)wytveAi)wysutv{M|§w8t&Ai ·ywA|¬tv|§)w¶|§wè|~s tveyi)HM{xysveQªIHxyeAwQªµd>xy¾i{ HJHd svi tM©&)
Qt|¬h)¦¬|¬tr")wyQ|¬tv|§)wXs² ÂssttiIè|§wètveAi |¬w8tv{QQxy tv|§)wZ£¿i£&|§¦§¦ sviieA£ºhxyeèX)tve
Gi{¤)sv|¬¤i±Mwy xXsui©¸xA¦Z|~s tveAi)wyi
Qt&M©tveyi)|¬w8t&
A{)A¦§ih¶² ,L .MONQP » K PRP Á,LTS ´µwtvey|§s5sviIKt|¬wZM£ i AiIs {|¬Gi/teA|~s©¸){h,)¦¬|~svhM
A
A¦§|§iI tvb l
Qtv|§hM¦yp\ey)
XifliIsv|¬w ©¸{5l
y){ut|§xA¦§){)i{QQr\wyMh,|~s
y{vA¦¬iIh¶² UWVYX
Z@[8\%]^_^=`Ya bt }
Iwyp?{w= | I}~>tn.:
=
}
4
ãXä\ÅKÒ Ì ÅÆ`Ç Ì ÓKÓvÍÓ ÎÍMÆyѳÓvÍ Ö
£&eyi{i tu t } )wy t M{i wystMw8ts&tveyMtl
A{iIs {|¬Gitveyi £ i|§)e8ts&)©5tveA{ii svxAQªµ {|¬tvi{|~AA{i¦~tiItv¦¬|¬©½tI¼Q{M )wy¥sv)wy|§X\h ih,|§ssv|¬wZ² dfeAiIr¥M{i iI¤)¦¬xXtvic®\r¥sv)¦§¤\|¬wAtveAiÄCxA¦¬iI{iI}8xyMtv|§)wysM{)xAwXOtveyiDihDitv{r sv
GiI |¬·yicè8r"tveyisueXM
Gi±
yM{Mh,itvi{c² C{Ktv|~)¦¬¦§r) tveyi±)ysvi{¤t|¬w3GÀ q r |§sO
yM{vtM©fteAi h,
AxQtMtv|§)wy)¦ A)hM|§w¶
A¦§ ic Gi¦§£ tveAiM|§{v
A¦~Mwyi)²C´«ts-xA
A
Gi{lXxAwyA){vr |§sÂ
A¦~MwAi¦¬8suiGi¦§£ teAi )|¬{ {©½tc² © )xy{svi)A|§w
y{vicsuiIwy iM©¿sveAQ¾QsXtveAiD¤M¦§xAiDM©CteAi,pGo Ä tvi{h |~sÂ|§wQ·ywA|¬tvi,|§w®teAiD)w8tv|§w\xA)xXslI)svi)`AxQt·ywA|¬tvi ;)wy®h,iIsve Ai
GiwyQiIw8t |§w teAiQ|~sv{vitvisui²dfeAi wt{v¦`
yM{Mh,i ti{ £&|§¦¬¦ZGi{i¦~tvic±tDtveAisveyM
Gi M© tveAi)|¬{ {©½tc²
Sonic Boom Downwards Emission Control Box (near field)
Tail Wave
Bow Wave
Sonic Boom N−Wave (far field) p
Ê|§)xA{i®°,dfeyi sv)wA|~G8h¶²3p\¾i te )©-wAiI){)wyè©;M{ ·yi¦~ sveAQ¾"£f¤)i
Xtuti{wys)©Â¥svxA
Gi{suwA|§M|§{{M©½tI² p o Ä s- w8tv{)¦`XÀ q8rf² G ¸c¡M ú ú ÿ > þ ú
dfeAi¿Q|§s {i ti¿É Ê5 h,QQi¦xysuicsZMwxy
8£&|§wyÄxA¦§i{su¦¬¤i{Z)
A
A¦§r8|§wAftÂMwxAwysutv{xy tvxA{iIltvicsvsvi¦~t|¬w )h,
GsviI )©¼tvitv{MeAicQ{y²dfeyisveyM
GiÂ|~sfeyMwy)iI,\r,h,¤\|¬wA tveAiwAQQics/wtveAiG)xAwyyM{rD)©¼tveAih,iIsve )¦¬wAwA{vhM¦~s/tv teytXxAwyA){vr²ÂdfeAiI|¬{Q|§sv
A¦~)ih,iw8tl|~sÂtM¾iwO|§w8tv )xAw8tÂ\r¶tv{MwXsu
A|§{tv|§)wO)wyQ|¬tv|§)w¥|¬w®{Ai{&t ¤)|§ sutv¦§r{viIh,iIsveA|¬wysIF² *{Q|§iw8t )©>teAiQ|~sv{vitvi©¸xAwy tv|§)wy)¦¼|§s&)h,
AxQtiI£&|¨tetveAieAiI¦¬
M©5MwO))|§wt-svrQstih¶ yxA|¬¦¬t-xysv|¬wyteAbi -xAtv)hMtv|~Â|¬ÃGiI{viIw8tv|~tv|§)w tv\ ¦
`. * ³²
UWVU
>e_\n[8dCczdC^_ca i a h]cGa dfe
Om \Ai{w ·XwA|¨ti ªµQ|§hDiIwysv|¬wyM¦)
Qt|¬h,|§¯It|¬w¹h,i tvey\ys®{i¦§r8|§wA w)|¬w8ts"M{i |§ssvxAiI ©¸{)h tveAi p\iI}8xAiIwt|§)¦ xX)Q{t|§C{v){Mh,h,|¬wAh,itveAQQ¦¬)r)²
G)
Axy¦§){l
A{MtMtr\
Gi|§sÂtveyio r\{Qª"lh,M¾\xAwOM¦§)){|¬tveAh¶Xsvii 8* «² dfey|§sM¦§)){|¬tveAh#|¬w |¬ts y)sv|§D©¸{vh#©¸){)ssuxAh,icstveyMt teAi±{iIsv)¦§xQtv|§)wèM©fteAiQ|¬ÃGiI{viIw8t ¦¬|§wAicM{|§st|¬wysM©-sutMtvi svrQstihs ;jÂi£ftvw |¬tviI{Mtv|§)w M©sutMtviMwysv)¦§xQtv|§)w M©)|§w8t ¿M{ilwyMt&i ÀQ
y)wysu|§¤)i² £¿iI¤)iI{ItveA|~s&)ssvxAh,
Qtv|§)w |~s,wA)t,¤)¦¬|~ |§w l
Qtv|§hM¦-p\ey)
Xi®liIsv|¬wZ² dfey|§sD©;Ktey)s,¦§iI sv)h,iOMxQteA){s tv tvt)¾èteAi¶
A{)A¦§ih xXsu|§wA þ ú)ü þ ª³{C
A{)){iIssv|¬¤i)M{/su|§h xy¦¨tMwAiI)xysuª M¦§){v|¬tveyh,s ;svii-©¸{/iÀQ)h,
A¦¬i 8 ³ § K\£&eA|~e±M{i&y)sviI,|§wteAi ©¸¦¬¦§£&|§wA t£ ,
X|¬w8ts° i HJHdÁsvrQstih t tveAièsv)h,iOtv|§h,i"|§w iIe )
Atv|§hD|§¯IMtv|§)w " Ê|¬{stclQ¤MwyiOtveAi"tvey{viIi¥iI}8xyMtv|§)wysM©teA@ |¨ti{t|¬wZ²
Ì ÓK×cØ Ì Ï Ë ÍÏQÓKѳæ ÛlÍæ àIìÏ Ì ÍIÍ KÏ\Õ F
j-iÀ8tcM|§wystiIM©Xxysu|§wA¦¬|§wAicM{)iwyi{|§¿su¦¬¤i{scxysvi Q|~s |§
A¦¬|§wAiª«sv
Xic |¬·X¿|¨ti{tv|§¤)ihr8Gi/wy)wA¦§|¬wyiIM{5sv)¦§¤)i{s ½©¸{-i ÀAMh,
A¦§i)Q
XsuiIxyQ,xAwysutvic)Qrsv)¦§¤)iI{s¿©¸{-Ê ¦§xA|~Om¶iIey)wA|~s K² ú)ü þ ü >I ü ú þ ß þ ½
dfeyisttviic}8xyt|¬w¶sv)¦§¤)iI{ftvGi|¬w8ti){tiI|§w teAi
A{)
GsviI )
Atv|§hD|§¯iI{-|~s-)ssuxyhDic tvGi Mw¶|¬tvi{t|¬¤i·AÀQiI
G)|§w8t&)wAi²´«t-{viI¦¬|§iIsfwÒyÕ Ì ÏyÇ)ÍMå Ì Ñ«ÍMƱsti
¶)w¤M{|§)A¦¬i ° "
? $
