Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle

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1 NASA/CR ICASE Report No Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle Timothy W. Simpson Georgia Institute of Technology Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia Prepared for Langley Research Center under Contract NAS February 1998

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4 VR RYBbt $t Y 8$IIj BP 3yp d R}BtR RZ Ps{ 8BI j$t k}${sjjk 8}jBkR j sr RJZs R RR$Bt B P$ s }BjktB8$sj 8BI j B Y Rs8}j I Iss, $$t 8BI jr s {YBR t B $t }Bjs Y Iss sti s P$ ZR$t 8s-$8Z8 j$, j$ybbi R$8s$Bt wr >1(HD zsj$is$bt BP i7 8BI jr $R AsR I BtU wsd R$t Rs$R${sj Yk}BY R$R wt RR sti 3TRs$R${RD I $% I P B8 B R$8s R BP Y %s $sa$j$k $t Y Iss wad }jb$t sti {Y {,$t Y R$IZsjR sti w{d {B8}Z$t i 1 Y s$b BP Y 8BI j RZ8 BP RJZs R B Y Bsj RZ8 BP RJZs R wr >1MHD 7s{,R sj >nmh sti G j{y sj >;H ABY Rs Ys Rs$R${sj R$t $R $ts}} B} $s by t $ {B8 R B I 8$t$R${ {B8}Z -} $8 tr by${y js{, stib8 B { BRRT%sj$Is$Bt sti $t s 8 st RJZs B {st A Z$j$9 I B srr RR Y s{{z s{k BP s, $$t 8BI j *V]e sti, $$t 8BI jr Ys% PBZtI j$8$ I s}}j${s$bt $t t$t $t I R$t } Ys}R A {szr BP Y js{, BP si$jk s%s$jsaj RBPbs B P$, $$t 8BI jr Y sii I {B8}j -$k BP P$$t s, $$t 8BI j B Y sii$$btsj PPB JZ$ I B ZR s, $$t 8BI j qb {js $Pk Y$R jsr }B$t } I${$Bt b$y s, $$t 8BI j JZ$ R Y $t% R$Bt sti 8Zj$}j${s$Bt BP R % sj 8s ${ R sti Y, $$t 8BI j IB R tb -$R sr s X{jBR ITPB 8" }BjktB8$sj JZs$Bt i7 8BI j } I${$Bt Btjk JZ$ R {B8}Zs$Bt BP R$8}j }BjktB8$sj JZs$Bt Bt{ s 8BI j YsR A t P$ qy Bsj $t Y$R }s} $R B -s8$t Y sii I {B8}Zs$Btsj -} tr JZ$ I B } PB 8, $$t sti {B8}s Y } I${$% {s}sa$j$k BP, $$t sti i7 8BI jr yt 7 {$Bt 1 st B% %$ b BP Y Rs$R${sj sti 8sY 8s${sj PBZtIs$BtR BP R}BtR RZ Ps{ 8BI j$t sti, $$t $R $% t yt 7 {$Bt n Y 8Zj$I$R{$}j$ts k I R$t BP st s BR}$, tb99j $R $t BIZ{ I $ R % R sr s R } BAj 8 B {B8}s i7 sti, $$t 8BI jr PB s}} B-$8s$Bt yt 7 {$Bt ; Y i7 sti, $$t 8BI jr s {BtR Z{ I PB Y s BR}$, tb99j -s8}j yt 7 {$Bt x PBZ B}$8$9s$Bt } BAj 8R s PB 8Zjs I sti RBj% I ZR$t Y i7 sti, $$t 8BI jr 7 {$Bt E {Bts$tR s I$R{ZRR$Bt BP BtB$t bb, 1 7s$R${sj V}} B-$8s$BtR PB ]B8}Z e-} $8 tr Z$jI$t s}} B-$8s$BtR BP {B8}Z stsjkr R k}${sjjk $t%bj% RU wsd {YBBR$t st -} $8 tsj I R$t B Rs8}j Y {B8}Z stsjkr$r {BI wad {YBBR$t s 8BI j B } R t Y Iss sti w{d P$$t Y 8BI j B Y BAR % I Iss qy s s %s $ k BP B}$BtR PB s{y BP Y R R }R sr RYBbt $t 3yp 1 sti RB8 BP Y 8B } %sj t Bt R Ys% A t Y$Yj$Y I n

5 EXPERIMENTAL DESIGN (Fractional) Factorial Central Composite Box-Behnken D-Optimal!*eN ]<!y]e Polynomial (linear, quadratic) Splines (linear, cubic) i sj$9s$bt BP s!*en 3yqqyWp Least Squares Regression Weighted Least Squares Regression SAMPLE APPROXIMATION TECHNIQUES Response Surface Methodology Orthogonal Array Plackett-Burman Hexagon Hybrid Latin Hypercube Select By Hand Random Selection 7B{YsR${ _ B{ RR Kernel Smoothing Radial Basis Functions Network of Neurons Rulebase or Decision Tree R N$t s =ta$sr I _ I${B Best Linear Predictor Backpropagation Log-Likelihood Entropy (info.-theoretic) Kriging Neural Networks Inductive Learning 3yp 1 x_z7r`xwl K`7qQ7z/y L?7Ey07LzX>;dH 7$8}RBt sj >;dh I$R{ZRR R % sj BP Y si%sts R sti I$RsI%sts R BP Y Y$Yj$Y I s}} Bs{Y R j$r I $t 3yp 1 ts8 jk R}BtR RZ Ps{ 8 YBIBjBk t Z sj t bb,r $tiz{$% j s t$t sti, $$t yt Y$R bb, Y 8BI j {YB${ sti 8BI j P$$t }B $BtR BP AZ$jI$t s}} B-$8s$BtR s Y } $8s k {Bt{ t R}BtR RZ Ps{ 8BI jr s I$R{ZRR I $t 7 {$Bt 1d sti, $$t 8BI jr $t 7 {$Bt 11 y $R srrz8 I Ys Y si YsR RB8,tBbj I BP *!e sti i7 8BI j$t sti j$j B tb,tbbj I BP, $$t 1d!% %$ b BP i R}BtR 7Z Ps{ BI j$t i R}BtR RZ Ps{ 8BI j$t {Yt$JZ R b B $$tsjjk I % jb} I B stsjk9 Y RZjR BP }YkR${sj -} $8 tr sti { s 8}$ ${sjjktasr I 8BI jr BP Y BAR % I R}BtR %sjz R i R}BtR RZ Ps{ 8BI j$t }BRZjs R s 8BI j BP Y PB 8U w dd kw-d )Pw-D Lε by kw-d $R Y Zt,tBbt PZt{$Bt BP $t R Pw-D $R s,tbbt }BjktB8$sj PZt{$Bt BP - sti ε $R stib8 B by${y $R srrz8 I B A tb 8sjjk I$R $AZ I b$y 8 st 9 B sti %s $st{ σ 1 qy $ti$%$izsj B R ε $ s s{y BAR %s$bt s sjrb srrz8 I B A $ti } ti t sti $I t${sjjk I$R $AZ I qy }BjktB8$sj PZt{$Bt Pw-D ZR I B s}} B-$8s kw-d $R k}${sjjk s jbb B I }BjktB8$sj by${y $R srrz8 I B A $Y j$t s sr $% t Ak e+w w1du w 1D B JZsI s${ sr $% t Ak e+w wndu ˆ y )β 0 + k i =1 β i x i w nd ˆ y )β 0 + k 2 β i x i + β i x ii + β ij x i x j i =1 k ii =1 i< j ;

6 qy }s s8 R β ( β $ β $$ sti β $` BP Y }BjktB8$sjR $t e+w7 w1d sti wnd s I 8$t I Y BZY j sr RJZs R RR$Bt by${y 8$t$8$9 R Y RZ8 BP Y RJZs R BP Y I %$s$btr BP Y } I${ I %sjz R y ˆ w-d P B8 Y s{zsj %sjz R kw-d yt B I B P$ Y 8BI j B Y BAR % I Iss ZR$t j sr RJZs R RR$Bt Y {B PP${$ tr BP e+w7 w1d sti wnd {st A R$8s I ZR$t e+w w;d w ;D β)> : H Td :k yt e+w w;d $R Y I R$t 8s $- BP Rs8}j I }B$tR sti k $R s {BjZ8t % {B {Bts$t$t Y {B R}BtI$t %sjz R BP Y R}BtR 3B 8B I s$jr Bt j sr RJZs R RR$Bt B }BjktB8$sj i7 8BI j$t R >1MH 11!% %$ b BP $$t 11d sy 8s${R BP $$t $$t B *V]e 8BI j$t }BRZjs R s {B8A$ts$Bt BP s }BjktB8$sj 8BI j }jzr I }s Z R BP Y PB 8 $% t Ak e+w wxdu wxd kw-d )Pw-D Lw-D by kw-d $R Y Zt,tBbt PZt{$Bt BP $t R Pw-D $R s,tbbt }BjktB8$sj PZt{$Bt BP - sti w-d $R Y sj$9s$bt BP s tb 8sjjk I$R $AZ I pszrr$st stib8 } B{ RR b$y 8 st 9 B %s $st{ σ 1 sti tbtt9 B {B%s $st{ qy Pw-D 8 $t e+w wxd $R R$8$js B Y }BjktB8$sj 8BI j $t s R}BtR RZ Ps{ sti } B%$I R s XjBAsj" 8BI j BP Y I R$t R}s{ $t 8stk {sr R Pw-D $R R$8}jk s, t B A s {BtRst 8 β wr >1(H >nmh sti >;HD GY$j Pw-D XjBAsjjk" s}} B-$8s R Y I R$t R}s{ w-d { s R XjB{sj$9 I" I %$s$btr RB Ys Y, $$t 8BI j $t }Bjs R Y t R Rs8}j I Iss }B$tR qy {B%s $st{ 8s $- BP w-d $R $% t Ak e+w wed w ED ]B%>w- $ Dw-`DH ) σ 1 iw>iw- $ -`DHD yt e+w wed i $R Y {B js$bt 8s $- sti iw- $ -`D $R Y R}s$sj {B js$bt PZt{$Bt A b t stk bb BP Y t R Rs8}j }B$tR - $ sti -` i $Rst R -t R Rk88 ${ }BR$$% I P$t$ 8s $- b$y Bt R sjbt Y I$sBtsj qy {B js$bt PZt{$Bt iw- $ -`D $R R} {$P$ I Ak Y ZR 7s{,R sj >nmh sti B Yj sti!b t >1(H I$R{ZRR R % sj {B js$bt PZt{$BtR qy pszrr$st {B js$bt PZt{$Bt $t e+w wd $R 8}jBk I $t Y$R bb, n dv k =1 w D iw- $ -`D ) -}w θ k x k i x k j 2 D qy θ, $t e+w wd s Y Zt,tBbt }s s8 R ZR I B P$ Y 8BI j t I% $R Y tz8a BP I R$t $ Y %s $saj R sti -, sti -,` s Y, {B8}Bt tr BP Rs8}j }B$tR - $ sti -` yt RB8 {sr R ZR$t s R$tj {B js$bt }s s8 $% R RZPP${$ tjk BBI RZjR wr >;H >n(h sti >nmhd _ I${ I R$8s R y ˆ w-d BP Y R}BtR kw-d s Zt $ I %sjz R BP - s $% t Ak e+w wmdu w MD y ˆ ) β ˆ L q w-di Td wkt3 β ˆ D x

7 by k $R Y {BjZ8t % {B BP j ty t R by${y {Bts$tR Y %sjz R BP Y R}BtR s s{y Rs8}j }B$t sti 3 $R s {BjZ8t % {B BP j ty t R by${y $R P$jj I b$y Bt R by t Pw-D $R s, t sr s {BtRst yt e+w wmd w-d $R Y {B js$bt % {B BP j ty t R A b t st Zt $ I - sti Y Rs8}j I Iss }B$tR >- d - t R H sti $R $% t Ak e+w wd - 1 w D q w-d )>iw-- d D iw-- 1 D iw-- t R DH q yt e+w wmd ˆ β $R R$8s I ZR$t e+w wd(d w d(d ˆ β )w3 q i Td 3D3 q i Td k qy R$8s BP Y %s $st{ ˆ σ 2 BP Y Rs8}j }B$tR P B8 Y ZtI jk$t jbasj 8BI j wtb Y %s $st{ $t Y BAR % I Iss $R jpd $R $% t Ak e+w wdddu w ddd σ ˆ 2 ) (y -F β ˆ ) T R -1 (y -F β ˆ ) n s by Pw-D $R srrz8 I B A Y {BtRst ˆ β qy 8s-$8Z8 j$, j$ybbi R$8s R w$ XA R Z RR R"D PB Y θ, $t e+w wd ZR I B P$ Y 8BI j s PBZtI Ak 8s-$8$9$t e+w wd1d B% θ, k ( wr >;HD w d1d max θ k > 0 - [n s ln( σ ˆ 2 ) + ln(detr)] 2 BY σ ˆ 2 sti I i s PZt{$BtR BP θ, GY$j stk %sjz R PB θ, { s st $t }Bjs$% 8BI j Y "A R", $$t 8BI j $R PBZtI Ak RBj%$t Y,TI$8 tr$btsj Zt{BtR s$t I tbttj$t s B}$8$9s$Bt } BAj 8 $% t Ak e+w wd1d 111 V}}j${s$BtR BP *V]e sti $$t *V]e sti, $$t 8BI jr Ys% PBZtI j$8$ I ZR $t t$t $t I R$t s}}j${s$btr R$t{ $R $t BIZ{$Bt $tb Y j$ sz Ak 7s{,R sj >nmh p$zts >ddh YsR } PB 8 I s } j$8$ts k $t% R$s$Bt $tb Y ZR BP *V]e 8BI j$t PB Y 8Zj$I$R{$}j$ts k I R$t B}$8$9s$Bt BP s <$Y 7} I ]$%$j q str}b s$ { sp < -}jb R s x sti s d( %s $saj I R$t } BAj 8 BAR %$t Ys Y *V]e sti R}BtR RZ Ps{ 8BI j$t s}} Bs{Y R k$ ji R$8$js RZjR IZ B Y JZsI s${ ti BP Y R}BtR R!R$B sti V8Bt >n(h Ys% I % jb} I st - tr$bt BP *V]e 8BI j$t PB tz8 ${sj B}$8$9s$Bt by${y ZR R s 8Zj$Rs R s k PB P$t$t Y s{{z s{k BP Y 8BI j Y k Ys% s}}j$ I Y $ s}} Bs{Y B Y Y 8sj I R$t BP st 8A II I j { Bt${ }s{,s by${y YsR x I R$t %s $saj R G j{y sj >;EH I R{ $A s, $$ttasr I s}} B-$8s$Bt 8 YBIBjBk by${y Y k ZR B $I t$pk $8}B st %s $saj R I { {Z %sz sti $t s{$btr sti } BIZ{ s ZR PZj s}} B-$8s$Bt 8BI j PB bb 1( %s $saj } BAj 8R ZR$t Btjk n(tx( ZtR BP Y {B8}Z {BI Y k {js$8 Y $ 8 YBI {st {B} b$y Z} B n(t;( %s $saj R } B%$I I Ps{B R}s R$k {st A -}jb$ I BB, sj >nh RBj% s nd %s $saj Y j${b} BB R Z{Z sj I R$t } BAj 8 ZR$t st s}} B-$8s$Bt 8 YBI AsR I Bt, $$t BB, >1H - tir Y Y j${b} BB I R$t } BAj 8 B $t{jzi xe R Z{Z sj %s $saj R B -s8$t Y s B jsr${ sti Ikts8${ R}BtR BP Y BB q BRR sti qb {9Bt >;;H Ys% I % jb} I s tz8 ${sj B}$8$9s$Bt R s k by${y $t{b }B s R *V]e 8BI j$t sti }s t R s {Y 8 YBIR PB jbasj B}$8$9s$Bt ]B- sti FBYt >H Ys% I % jb} I Y 7 JZ t$sj * R$t PB!}$8$9s$Bt 8 YBI by${y E

8 ZR R jbb {BtP$I t{ ABZtIR Bt } I${ I %sjz R BP Y R}BtR PB Y R JZ t$sj R j {$Bt BP %sjzs$bt }B$tR IZ $t B}$8$9s$Bt BY s}} Bs{Y R Ys% RYBbt $8} B% 8 tr B% si$$btsj B}$8$9s$Bt s}} Bs{Y R by t s}}j$ I B s %s $ k BP RstIs I 8sY 8s${sj R } BAj 8R 1n!t zs $saj e-s8}j BP i R}BtR 7Z Ps{ sti $$t BI jr V R$8}j Bt %s $saj -s8}j A RR $jjzr s R Y I$PP t{ A b t Y s}} B-$8s$Bt {s}sa$j$$ R BP s R {BtI B I i7 8BI j sti s, $$t 8BI j 7Z sti i tszi >;1H PB 8Zjs I Y$R -s8}j B I 8BtR s RB8 BP Y j$8$s$btr BP ZR$t R {BtI B I i7 8BI jr R 3yp n qy k P$ s R {BtI B I R}BtR RZ Ps{ ZR$t j sr RJZs R RR$Bt B P$% Rs8}j }B$tR P B8 s PsA ${s I $YY B I PZt{$Bt b$y$t Y $Bt BP Y B}$8Z8 w- ) n1d V, $$t 8BI j ZR$t s {BtRst PB Y jbasj 8BI j sti Y pszrr$st {B js$bt PZt{$Bt BP e+w wd $R P$ B Y Rs8 P$% }B$tR Y B $$tsj PZt{$Bt Y P$% Rs8}j }B$tR sti Y i7 sti, $$t 8BI jr s RYBbt $t 3yp n y88 I$s jk %$I t P B8 3yp n $R Ps{ Ys Y, $$t 8BI j $t }Bjs R Y Iss }B$tR s}} B-$8s$t Y B $$tsj PZt{$Bt A Yst Y R}BtR RZ Ps{ 8BI j sti } I${$t st B}$8Z8 by${y $R 8Z{Y {jbr B Y s{zsj B}$8Z8 y $R $8}B st B tb${ Ys BZR$I BP Y I R$t R}s{ I P$t I Ak Y Rs8}j }B$tR w1( c - c ;xd t $Y 8BI j } I${R b jj sr -} { I Y, $$t 8BI j Z tr B Y ZtI jk$t jbasj 8BI j by${y $R s {BtRst qy$r $R k}${sj A Ys%$B PB s, $$t 8BI j Ps P B8 Y Rs8}j }B$tR Y, $$t 8BI j Z tr B Y ZtI jk$t jbasj 8BI j R$t{ Y $tpjz t{ BP Y Rs8}j }B$tR YsR -}Bt t$sjjk I {sk I sbsk BZR$I BP Y I R$t R}s{ ( E( Su's & Renaud's Fcn [42] Sample Points 2nd Order RS Model x( Kriging Model y ;( n( 1( d( dx 1x nx ;x xx Ex - 3yp n 5z8y 7yKq?yE qlw X LzXX` WyyzQ> 7/7z/ELQqX 7$- t % tjk R}s{ I }B$tR wtb $t{jzi$t Y Rs8}j }B$tRD s s, t P B8 b$y$t Y st w1( c - c ;xd B {Y {, Y s{{z s{k BP Y bb s}} B-$8s$Bt 8BI jr qy 8s-$8Z8 sarbjz B wi$pp t{ A b t Y s{zsj sti } I${ I %sjz RD Y s% s sarbjz B sti Y BB 8 st RJZs B w7ed by${y $RU w dnd BB 7e ) n error i =1 (y i -y ˆ i ) n error 2

9 by t B $R Y tz8a BP }B$tR w) ded s j$r I $t qv Ne d sr I Bt Y B stsjkr$r $t qv Ne d $ {st A {Bt{jZI I Ys Y, $$t 8BI j s}} B-$8s R Y B $$tsj PZt{$Bt A R$t{ $ YsR s jbb BB 7e s% s B sti 8s-$8Z8 B V 8B $t%bj% I 8Zj$I$R{$}j$ts k I R$t -s8}j $R I R{ $A I $t Y t - R {$Bt qv Ne d o L yzyqrx7xwl Lz8y 7yKq?yE q 1tI! I i7 BI j $$t BI j s- V 7w B D ndn; 1x( V% V 7w B D ddd (E BB 7e 1dxx d((; n V BR}$, WB99j * R$t e-s8}j qy 8Zj$I$R{$}j$ts k I R$t BP st s BR}$, tb99j YsR A t R j { I sr Y R } BAj 8 PB {B8}s $t Y } I${$% {s}sa$j$k BP i7 sti, $$t 8BI jr qy j$t s s BR}$, B{, t$t $R Y } B}ZjR$Bt RkR 8 } B}BR I PB Y z tz 7s >;nh ZRsAj jszt{y % Y${j $jjzr s I $t 3yp ; 3yp ; ~z0` C0y `XyKqqy`z_8_7qj70_q7zy y LX 7> L `qx7lzxrx0e>1dh qy s BR}$, B{, t$t {BtR$RR BP s B{, Y ZR {Bbj s BR}$, tb99j sti }jz AsR $BtR sr RYBbt $t 3yp x qy s BR}$, tb99j $R s Zt{s I R}$, B }jz tb99j Ys si`zrr B Y s8a$ t } RRZ sti $t s R b jj b$y jszt{y % Y${j R >n;h qy PjBb P$ ji R Z{Z {Yst R I s8s${sjjk P B8 jbb sj$zi B Y$Y sj$zi Bt Y R}$, RZ Ps{ sti $t Y AsR PjBb $Bt w{p >dxh >1H sti >nehd VII$$Btsj PjBb $R $t` { I $t Y AsR $Bt B { s st s BIkts8${ R}$, >deh by${y $% R Y s BR}$, tb99j $R ts8 sti $t{ sr R Y AsR } RRZ sti {Bt $AZ$Bt BP Y AsR $Bt B Y s BR}$, Y ZR ie]yi]=nvqy!w ]!GN 7<!] q<i=7qei w_iyvio 3N!GD e _VW7y!W 3VW Feq!=W*ViO Vei!7_y e W!Ne ]!Wq!=i 7<eVi NVOei ie]!_ie77y!w 7<!] V7e Nee* _N=p V7e ie]yi]=nvqy!w iepy!w M

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