Multi-Objective Design Exploration for Aerodynamic Configurations

Transcription

1 Mult-Objectve Desgn Exploraton for Aerodynamc Confguratons Shgeru Obayash *, Tohoku Unversty, Senda, , Japan Shnkyu Jeong Tohoku Unversty, Senda, , Japan and Kazuhsa Chba Japan Aerospace Exploraton Agency, Tokyo, 8-005, Japan A new approach, Mult-Objectve Desgn Exploraton (MODE), s presented to address Multdscplnary Desgn Optmzaton problems. MODE reveals the structure of the desgn space from the trade-off nformaton and vsualzes t as a panorama for Decson Maker. The present form of MODE conssts of Krgng Model, Adaptve Range Mult Objectve Genetc Algorthms, Analyss of Varance and Self-Organzng Map. The man emphass of ths approach s vsual data mnng. Two data mnng examples usng hgh fdelty smulaton codes are presented: four-objectve aerodynamc optmzaton for the fly-back booster and Multdscplnary Desgn Optmzaton problem for a regonal-jet wng. The frst example confrms that two dfferent data mnng technques produce consstent results. The second example llustrates the mportance of the present approach because desgn knowledge can produce a better desgn even from the bref exploraton of the desgn space. I. Introducton Ths paper dscusses a new approach for Multdscplnary Desgn Optmzaton (MDO). MDO has been a rapdly growng area of research. -3 Thanks to these poneerng works, researchers n Computatonal Flud Dynamcs (CFD) are gettng nterested n MDO research as well. MDO research s stll expandng because hgh fdelty CFD codes are becomng avalable wth the ad of ncreasng computer power. A typcal MDO problem nvolves competng objectves, for example n the arcraft desgn, mnmzaton of aerodynamc drag, mnmzaton of structural weght, etc. Whle sngle objectve problems may have a unque optmal soluton, mult-objectve problems (MOPs) have a set of compromsed solutons, largely known as the tradeoff surface, Pareto-optmal solutons or non-domnated solutons. 4 These solutons are optmal n the sense that no other solutons n the search space are superor to them when all objectves are consdered (Fg. ). Tradtonal optmzaton methods such as the gradent-based methods 5,6 are sngle objectve optmzaton methods that optmze only one objectve. These methods usually start wth a sngle baselne desgn and use local gradent nformaton of the objectve functon wth respect to changes n the desgn varables to calculate a search drecton. When these methods are appled to a MOP, the problem s transformed nto a sngle objectve optmzaton problem by combnng multple objectves nto a sngle objectve typcally usng a weghted sum method. For example, to mnmze competng functons f and f 2, these objectve functons are combned nto a scalar functon F as F = w f + w () 2 f 2 * Professor, Insttute of Flud Scence, 2-- Katahra, Senda, Japan, , Assocate Fellow AIAA Research Assocate, Insttute of Flud Scence, 2-- Katahra, Senda, Japan, , Member AIAA Research Assocate, Jndaj-Hgash, Chofu, Japan, , Member AIAA Amercan Insttute of Aeronautcs and Astronautcs

2 Ths approach, however, can fnd only one of the Pareto-optmal solutons correspondng to each set of the weghts w and w 2. Therefore, one must run many optmzatons by tral and error adjustng the weghts to get Pareto-optmal solutons unformly over the potental Pareto-front. Ths s consderably tme consumng n terms of human tme. What s more, there s no guarantee that unform Pareto-optmal solutons can be obtaned. For example, when ths approach s appled to a MOP that has concave trade-off surface, t converges to two extreme optmums wthout showng any trade-off nformaton between the objectves (Fg. 2). To overcome these dffcultes, Normal- Boundary Intersecton Method 7 and Aspraton Level Method 8 were developed. An alternatve approach to solve MOP s to fnd as many Pareto-optmal solutons as possble to reveal trade-off nformaton among dfferent objectves. Once such solutons are obtaned, Decson Maker (DM) wll be able to choose a fnal desgn wth further consderatons. Evolutonary Algorthms (EAs, for example, see Refs. 9 and 0) are partcularly suted for ths purpose. Evolutonary Algorthm s a generc name for populaton-based optmzaton methods, such as Genetc Algorthms (GAs), Evolutonary Strateges (ESs), Genetc Programmng (GP), etc. EAs smulate the mechansm of natural evoluton, where a bologcal populaton evolves over generatons to adapt to an envronment. Ftness, the ndvdual, and genes n the evolutonary theory correspond to the objectve functon, desgn canddate, and desgn varables n desgn optmzaton problems, respectvely. EAs have been extended successfully to solve MO problems. 2 EAs use a populaton to seek optmal solutons n parallel. Ths feature can be extended to seek Pareto solutons n parallel wthout specfyng weghts between the objectve functons. Because of ths characterstc, EAs can fnd Pareto solutons for varous problems havng convex, concave and dscontnuous Pareto front. The resultant Pareto solutons represent global trade-offs. In addton, EAs have other advantages such as robustness and sutablty for parallel computng. Due to these advantages, EAs have been appled to MOPs very actvely (EMO proceedngs). EAs have been also appled to sngle objectve and mult-objectve aerospace desgn optmzaton problems. 3-9 objectve functon f 2 Feasble regon G F A D B E Pareto-front C objectve functon f 2 Feasble regon A C Pareto-front B objectve functon f Fgure The concept of Pareto-optmalty objectve functon f Fgure 2 Weghted-sum method appled to a MOP havng a convex Pareto-front Ths approach of fndng many Pareto solutons works fne as t s, however, only when the number of objectves remans small (usually two, three at most, as shown n Fg. 3). To reveal trade-off nformaton from the resultant Pareto front for real-world problems wth many objectves, vsualzaton of the Pareto front becomes an ssue. Several technques have been consdered, such as parallel coordnates, 20 box plot, 2 and Self-Organzng Map (SOM). 22 The mportance of vsualzaton of desgn space s also dscussed n Ref. 23. Because such vsualzaton s a tool for data mnng, data mnng s found very mportant n ths approach. To support data mnng actvtes, response surfaces are found versatle. Once the surface s constructed, t can be used for statstcal analyss, for example, analyss of varance (ANOVA). 24 ANOVA shows the effect of each desgn varables on objectve functons quanttatvely whle SOM shows the nformaton qualtatvely. When the response surface method (RSM) s ntroduced for data mnng as post-process of optmzaton, t can be appled to pre-process of optmzaton as a surrogate model, too. Pre-process has been an mportant aspect of ntroducton of surrogate models because t would reduce the computatonal expense greatly, whle t would produce rch non-domnated 2 Amercan Insttute of Aeronautcs and Astronautcs

3 solutons effcently. In ths paper, surrogate models are ntroduced for both pre- and post-processes. However, t should be noted that RSM s needed for post-process prmarly. EAs may be appled from the begnnng n parallel to buldng the surrogate model. If functon evaluatons are very cheap, EAs may also be appled drectly. As a result, the new approach for MDO named as Mult-Objectve Desgn Exploraton (MODE) can be summarzed as a flowchart shown n Fg. 4. MODE s not ntended to gve an optmal soluton. MODE reveals the structure of the desgn space from the trade-off nformaton and vsualzes t as a panorama for DM. DM wll know the reason for trade-offs from non-domnated desgns, nstead of recevng an optmal desgn wthout trade-off nformaton. The rest of the paper wll explan the components of MODE used n our group, although the concept of MODE can be coupled wth other RSM and optmzaton algorthms. Examples of data mnng 7,24,28 wll be gven brefly. 2 objectves 3 objectves Mnmzaton problems 4 objectves? Projecton Fgure 3 Vsualzaton of Pareto front Defne desgn space Parameterzaton: PARSEC, B-Splne, etc. Choose sample ponts Desgn of Experment: Latn Hypercube Construct surrogate model Response Surface Method: Krgng Model Fnd non-domnated front Check the model and front Extract desgn knowledge Optmzaton: Adaptve Range Mult Objectve Genetc Algorthms Uncertanty Analyss: Expected Improvement based on Krgng Model, statstcs of desgn varables, etc. Data Mnng: Analyss of Varance, Self-Organzng Map, etc. Fgure 4 Flowchart of Mult-Objectve Desgn Exploraton (MODE) wth component algorthms 3 Amercan Insttute of Aeronautcs and Astronautcs

4 II. Surrogate Model A. Krgng Model The present Krgng model expresses the unknown functon y(x) as y ( x) = µ + Z( x) (2) where x s an m-dmensonal vector (m desgn varables), µ s a constant global model, and Z(x) represents a local devaton from the global model. In the model, the local devaton at an unknown pont s expressed usng stochastc processes. The sample ponts are nterpolated wth the Gaussan correlaton functon to estmate the dstrbuton of the functon value at the unknown pont. The correlaton between Z(x ) and Z(x j ) s strongly related to the dstance between the two correspondng ponts, x and x j. In the Krgng model, a specal weghted dstance s used nstead of the Eucldean dstance because the Eucldean dstance weghs all desgn varables equally. The dstance functon between the pont at x and x j s expressed as j m d( x, x ) = θ k xk x (3) k= where θ k (0 θ k ) s the k th element of the parameter θ. The correlaton between the pont x and x j s defned as The Krgng predctor 27,29 s Corr 2 j k j j [ Z ( x ), Z ( x )] exp [ d ( x, x )] ˆ y( x) = ˆ µ + r R ( y µ ˆ) = (4) Where µˆ s the estmated value of µ, R denotes the n n matrx whose (, j) entry s Corr[Z(x), Z(xj)]. r s vector whose th element s r ( x) Corr[ Z( x), Z( x )] (6) and y=[y(x ),,y(x n )]. The unknown parameter, θ, for the Krgng model can be estmated by maxmzng the followng lkelhood functon: 2 n n 2 Ln( ˆ, µ ˆ σ, θ) = ln(2π ) ln( ˆ σ ) ln( R ) (7) ( y ˆ µ ) R ( y ˆ µ ) 2 2 ˆ σ where denotes an m-dmensonal unt vector. Maxmzaton of the lkelhood functon s an m-dmensonal unconstraned non-lnear optmzaton problem. In ths paper, the alternatve method 30 s adopted to solve ths problem. 2 For a gven θ, µˆ and σˆ can be defned as R y µˆ = (8) R ( y µ ) 2 R ( y µ ) ˆ σ = (9) n The accuracy of the estmated value on the Krgng model largely depends on the dstance from the sample ponts. Intutvely speakng, the closer pont x s to the sample ponts, the more accurate ŷ( x) s. Ths ntuton s expressed n the mean squared error of the predctor ( R r) s ( x) = ˆ σ r R r + (0) R s 2 (x) s the mean squared error at pont x, ndcatng the uncertanty of the estmated value. (5) 4 Amercan Insttute of Aeronautcs and Astronautcs

5 B. Exploraton of Global Optmum and Treatment of Constrants on the Krgng model Once the approxmaton model s constructed, the optmum pont can be explored usng an arbtrary optmzer. However, there s a possblty of mssng the global optmum because the estmated value ncludes uncertanty. Fgure 5 The objectve functon and the approxmaton model In Fg. 5, the sold lne s for the real shape of the objectve functon and the dotted lne s for the approxmaton model. The mnmum pont on the approxmaton model s located near x=9, whereas, the real global mnmum of the objectve functon s stuated near x=4. Exploraton of global mnmum usng the approxmaton model s apt to result n the local mnmum. For a robust search of the global optmum on the approxmaton model, the uncertanty nformaton s very useful. Fgure 6 The estmated value and the standard error of the Krgng model Fgure 6 shows the estmated value and the standard error (uncertanty) of the Krgng model. Around x=9.5, the standard error of the Krgng model s very small because there are many sample ponts around ths pont. Thus, the confdence nterval s very short as shown n Fg. 6. On the other hand, the standard error around x=3.5 s very large due to the lack of sample ponts around there. Thus, the confdence nterval at ths pont s very wde. The lower bound of ths nterval s smaller than current mnmum on the Krgng model. Thus, t can be sad that ths pont has some probablty of beng the global mnmum. The probablty of beng the global optmum concept can be expressed by the crteron of expected mprovement (EI) 3. In case of a mnmzaton problem, the EI s express as follows: [ f y( x) ] mn f y( x) < fmn = I( x) = 0 otherwse E fmn ( I) = ( f mn y) φ( y) dy max( f mn - y,0) where s the probablty densty functon representng uncertanty about y. By selectng the maxmum EI pont as addtonal sample ponts of the Krgng model teratvely, the robust exploraton of the global optmum s possble. Then, f there are constrant as follows, () (2) 5 Amercan Insttute of Aeronautcs and Astronautcs

6 a c ( x) b =, L, k (3) EI subject to constrants s expressed as follows: E ( I ),, y c c2,... c k = E y c (max( f mn y,0) b bk f c ),..., c y d c d ck a a k k φ ( y, c,, ck In order to evaluate Eq. (4), the multvarate normal dstrbuton ), whch s very complcated, should be specfed. Thus, n ths paper, we assume that y, c, c 2,,c k are statstcally ndependent n order to smplfy Eq. (4). The modfed Eq. (4) s as follows E y, c, L, ck ( I c ) = Ey( I) y =, L, k = E ( I) P( a ( Φ ( b ) Φ ( a )) c c ( x) b ) L P( a c k c ( x) b ) In order to calculate ths value, the Krgng model s constructed for the objectve functon and all constrant functons separately. On the Krgng model of objectve functon, EI s calculated, and on the Krgng models of constrants, the probablty to satsfyng each constrant s calculated. Based on these values, the next addtonal pont for balanced local and global search s selected, whle satsfyng the constrants. k k (4) (5) III. Adaptve Range Mult-Objectve Genetc Algorthms Pareto solutons and Pareto front are exact solutons by defnton. Because t s dffcult to show that numercal solutons are exact, numercal solutons and the correspondng front are usually called as non-domnated solutons and non-domnated front, respectvely. They are non-domnated among the solutons generated by the computaton (Fg. 7). Except for the ntroducton of range adaptaton operator, the present ARMOGAs operators 5 are the same as the MOEAs. 9 Therefore, each genetc operator of the MOEAs adopted here s frstly explaned. Then the unque procedure of ARMOGAs s descrbed n ths chapter. Pareto front (exact) Approxmate Pareto front Pareto solutons (exact) Extreme Pareto solutons (exact) Non-domnated solutons (numercal) Non-domnated front (numercal) Utopa Fgure 7 Defnton of Pareto solutons and non-domnated solutons 6 Amercan Insttute of Aeronautcs and Astronautcs

7 A. Algorthm of Mult-Objectve Evolutonary Algorthms. Bnary and Floatng-Pont Representaton As GAs orgnally smulated natural evoluton, bnary numbers were often used to represent desgn parameter values. However, for real functon optmzatons, such as aerodynamc optmzaton problems, t s more straghtforward to use real numbers. Thus, the floatng-pont representaton s adopted here. 2. Codng and Decodng GAs requre both phenotype and genotype desgn varables. The phenotype desgn varable represents the actual desgn varables, such as length, angle, shape, etc. On the other hand, the genotype desgn varable s a bnary number (Bnary GAs) or a real number n [0,] (Real-coded GAs). The operators of many GAs requre genotype representaton of desgn parameters. Therefore, actual desgn varables (phenotype representaton) must be converted to the genotype representaton. The converson from phenotype to genotype s called codng, and conversely, the converson from genotype to phenotype s encodng. For real-parameter desgn problems, such as aerodynamc optmzatons, t s not favorable to use bnary representaton. One reason for ths s that phenotype desgn space s not contnuous by bnary representaton. For the present floatng-pont representaton, -th desgn parameter p s coded to genotype value r, whch s normalzed n [0,]: p p, mn r = (6) p p, max, mn 3. Intal Populaton The results of GAs can be affected by the ntal populaton f the number of ndvduals per generaton s small. It would be better to generate ntal ndvduals n a wde range of desgn spaces. Here, the ntal populaton s generated randomly. 4. Evaluaton As GAs use only objectve-functon values for optmzaton, no modfcaton of evaluaton tools s requred. In addton, t s easy to apply Master-Slave type parallelzaton systems to conserve computatonal resources because GAs do not have to compute desgn canddates sequentally, unlke gradent-based method. 5. Selecton GAs choose superor ndvduals as parents to generate new desgn canddates. Therefore, selecton has a large nfluence on search performance of GAs. For sngle-objectve optmzatons, as the am s to obtan the best soluton, selecton s based on the ftness value gven by the objectve-functon value. However, Pareto-optmal solutons must be obtaned for MO optmzaton. To obtan Pareto solutons effectvely, each ndvdual s assgned a rank based on the Pareto rankng method and ftness sharng. In the present MOEAs, Flemng and Fonseca s Paretorankng method 2 s adopted. Each ndvdual s assgned a rank accordng to the number of ndvduals domnatng t, as shown n Fg. 8. The ftness value (F ) of ndvdual s assgned based on the followng equaton: R F = N µ ( k) 0.5( µ ( R ) ) (7) k = where N s the number of solutons, and µ(r ) s the number of solutons n rank R. Thereafter, the standard sharng approach s adopted to prevent choosng smlar solutons as parents and to mantan dversty of the populaton. The assgned ftness values are dvded by the nche count: F F = nc Here, nche count nc s calculated by summng the sharng functon values: sh( d j N nc = j = sh( d j ) d ) = σ 0 j share α d j < σ others share (8) (9) (20) 7 Amercan Insttute of Aeronautcs and Astronautcs

8 M j f k f k d = j k = uk lk (2) where u k s the maxmum objectve-functon value of k at the present generaton, l k s the mnmum objectvefuncton value of k at the present generaton, and α s the sharng functon parameter. If the dstance between ndvduals and j s lower than σ share, then nche count ncreases to reduce the ftness of the soluton. The normalzed nchng parameter σ share s proposed as follows: M ( ) ( ) + σ share = N σ share (22) where N s the sze of the populaton and M s the number of objectve functons. After shared ftness values are determned for all ndvduals, the stochastc unversal selecton (SUS) s appled to select better solutons for producng a new generaton. Unlke roulette wheel selecton method, only one random number s chosen for the whole selecton process for SUS. As many dfferent solutons should be chosen to mantan the dversty, a set of N equ-spaced numbers s created. f2 M f Fgure 8 Pareto rankng method (Rank means non-domnated solutons) 6. Crossover Crossover s an operator that nterchanges the genotype parameters of selected parents and produces two dfferent desgn canddates. Probablty of crossovers and crossover method markedly affect the search performance of GAs. For the bnary representaton, crossover nterchanges the bt strngs of selected parents. However, many crossover methods have been proposed for real-parameter GAs. Smulated bnary crossover (SBX) operator 9 creates offsprng based on the dstance between the parents. If the two parents are closely related to each other, SBX s lkely to generate new offsprng near the parents. On the other hand, f the two parents are more dstantly related, t s possble for solutons to be created away from the parents. Ths operator s descrbed as follows: Chld = 0.5 [(+β q ) Parent + ( β q ) Parent2 ] (23a) Chld2 = 0.5 [ ( β q ) Parent + (+β q ) Parent2 ] (23b) ( ηc + ) ( 2 ran) = ( ηc + ) β q 2 ( ran) (23c) 7. Mutaton Mutaton mantans dversty and expands the search space by changng the desgn parameters. If the mutaton rate s hgh, a GA search s close to a random search and results n slow convergence. Therefore, an adequate value s requred for the mutaton rate. For bnary representaton, mutaton s performed to reverse the bt strngs. It s not as smple for real-coded GAs as for bnary GAs. Ths s realsed by addng dsturbances to the desgn parameters. 8 Amercan Insttute of Aeronautcs and Astronautcs

9 Polynomal mutaton, whch s smlar to the SBX operator descrbed n prevous secton, has been proposed 9 : Chld mutaton = Chld crossover + (x max x mn ) δ (24) where δ s calculated from the polynomal probablty dstrbuton: ( ηm + ) ( 2 ran2) δ = ( + ) [ 2 ( 2) ] ηm ran (25) where ran2 s a unform random number n [0,]. A value of η m determnes the perturbaton sze of mutaton. 8. Archvng To obtan Pareto solutons effcently, t would be better to nclude past excellent solutons as current solutons. In the present MOEAs, two archvng technques are combned. The frst s the Best-xN technque, whch keeps the latest better solutons and parent generaton of (x-)n sze and uses these solutons for the selecton process. The second s the standard archvng technque, whch s comprsed of all prevous solutons to prevent the loss of prevous excellent solutons. These two methods are combned n the present MOEAs as shown n Fg. 9. The procedure s as follows:. Ftness values based on the ftness assgnment operators are assgned to the present populaton and the Best-xN group. Here, x s set to Accordng to the ftness value, the top N ndvduals are chosen for the next step. In addton, the top (x-)n ndvduals are preserved as the Best-xN group. 3. Ftness values are assgned to chosen N ndvduals. 4. SUS s used to select the parents. Then, crossover and mutaton are appled to generate new ndvduals. 5. Several ndvduals from the Best-xN group are replaced by the same number of ndvduals from the archves. Best-xN [(x-)n ndvduals] Present populaton [N ndvduals] Better ndvduals Ftness assgnment [xn ndvduals] Archvng (all solutons) Selecton for top N populaton Ftness assgnment [N ndvduals] Selecton for matng pool Crossover and mutaton Fgure 9 Archvng procedure used n the present MOEAs 9. Constrant-Handlng Technque In many real-world problems, t s common to have several constrants. Many constrant-handlng technques have been proposed, however, t s not easy for GAs to solve constraned-problems compared to gradent-based methods. A popular and easy constrant-handlng strategy s the penalty functon approach n whch a penalty value s added to the objectve-functon value f the desgn volates the constrant. Although several penalty functons have been proposed, t s dffcult to choose approprate penalty values a pror. In the present MOEAs, an extended Pareto rankng method based on constrant-domnance s used. Constrantdomnance s defned as follows 9 : A soluton x s sad to constran-domnate a soluton x j, f any of the followng condtons are true:. x s feasble and x j s not. 9 Amercan Insttute of Aeronautcs and Astronautcs

10 2. x and x j are both nfeasble, but x has a smaller constrant volaton. 3. x and x j are feasble and x domnates x j n the usual sense. Fgure 0 shows the example of a Pareto rankng method based on constrant-domnance for the two-objectve mnmzaton problem wth one constrant. Based on ths approach, t would be easy to generate new offsprng that satsfy the constrants because feasble solutons are lkely to be chosen as the parents. However, t s possble for good solutons to le close to the edge of the feasble and nfeasble regon n many ndustral problems. Therefore, an adequate tolerance of the constrant (c tol ) should be ntroduced to the constrant volaton: G c tol 0 (26) where G s an orgnal constrant less than zero. As the tolerance c tol s ntroduced, solutons havng smaller volaton than c tol are assumed to be feasble for constrant-domnance. Ths enables EAs to search for solutons near the boundary between feasble and nfeasble solutons. To consder the aerodynamc optmzaton usng tme-consumng CFD, t s unfavorable to generate many volated canddates. If t s possble to determne that the soluton volates constrants before CFD computaton, such as geometrcal constrants (length, angle, etc.), t s possble to prevent generatng such solutons, as t would be a waste of computaton tme n CFD. In the case of aerodynamc optmzaton, t would be better to take account of the above problem. f2 Constrant Infeasble 2 4 Feasble Pareto front f Fgure 0 Example of constran-domnance. B. Algorthm of Adaptve Range Mult-Objectve Genetc Algorthms To reduce the large computatonal burden, the reducton of the total number of evaluatons s needed. On the other hand, a large strng length s necessary for real parameter problems. Oyama developed real-coded ARGAs and appled them to the transonc wng optmzaton. 4 Accordng to the encodng system based on normal dstrbuton (Fg. ) bult by populaton statstcs consstng of better desgns computed before, ARGAs can fnd a good optmal desgn effcently. The bass of ARMOGAs s the same as ARGAs, but a straghtforward extenson may cause a problem n the dversty of the populaton. Therefore, ARMOGAs have been developed based on ARGAs to deal wth multple Pareto solutons for the mult-objectve optmzaton. In addton, archvng and constrant-handlng technques are consdered to select better solutons to decde new search range. Ths secton descrbes genetc operators of ARMOGAs. ARMOGAs dffer from MOEAs descrbed above wth regard to the applcaton of range adaptaton. Therefore, before startng range adaptaton, the MOEAs and ARMOGAs n the present study are dentcal. A flowchart of ARMOGAs s shown n Fg. 2. The range adaptaton starts at M sa generaton and s carred out every M ra generatons. The new decson space s determned based on the statstcs of selected better solutons, and then the new populaton s generated n the new decson space. Thereafter, all the genetc operators are appled to the new desgn space. ARMOGAs are able to fnd Pareto solutons more effcently than conventonal MOEAs because of the concentrated search of the promsng desgn space out of the large, ntal desgn space. ARMOGAs can adapt ther 0 Amercan Insttute of Aeronautcs and Astronautcs

11 search regon as shown n Fg. 3. In contrast, the search regon of conventonal EAs remans unchanged. The encodng system s based on the normal dstrbuton wth the plateau regon as shown n Fg. 4. The selected desgns locate n the plateau regon, and the normal dstrbuton regon s determned based on the populaton statstcs to better preserve the dversty of soluton canddates. Re-ntalzaton helps to mantan the populaton dversty. Probablty r 0 p n, + x p =µ p Fgure Normal dstrbuton for encodng n real-coded ARGAs Intal populaton Evaluaton Termnaton crtera Selecton Crossover Mutaton Archve Samplng Range adaptaton Re-ntalsaton Fgure 2 Flowchart of ARMOGAs Search regon Probablty Superor soluton Inferor soluton Probablty Search regon x x x L x U x L x U (a) Conventonal MOEAs (b) ARMOGAs Fgure 3 Sketch of search regon Amercan Insttute of Aeronautcs and Astronautcs

12 Probablty α r r 0 r α r I II α r < r < α r III p,mn µ fl µ µ +fl p p,max Fgure 4 Sketch of probablty dstrbuton of phenotype desgn varable p n ARMOGAs. Samplng for Range Adaptaton Range adaptaton needs to select superor solutons to determne the new desgn space based on the statstcs. The solutons, whch have hgher ftness values based on Pareto rankng method, are selected to determne the reasonable search range. It would be better to select many solutons to prevent the creaton of new search regons that do not nclude the global optmum. On the other hand, many solutons for range adaptaton generally nterfere wth the decrease n sze of the search space. The solutons are selected at random accordng to ther ftness gven by the followng soluton sets:. PR non % non-domnated solutons from all solutons. (PR non =00) 2. PR arc % solutons from the archve. (PR arc =0) 3. PR prs % solutons from the latest generaton. (PR prs =0) 4. PR vo % solutons that volate the constrant. (PR vo =, at least one desgn) Soluton set 4 s ntroduced to search near the boundary between feasble and nfeasble solutons, as the globaloptmum for constrant problems s often located there. Accordng to the amount of volaton, volated desgns are sampled. The probabltes n bracket are used n ths optmzaton. In ths case, only non-domnated solutons wth several nfeasble desgns are selected to determne new desgn range. 2. Range Adaptaton In the ARMOGAs, the search regon s changed accordng to the populaton statstcs of the average and the standard devaton. The range adaptaton adopts the Normal dstrbuton to search global solutons effcently. Fgure shows the Normal dstrbuton used for encodng n the real-coded ARGAs. The real value of the -th desgn varable p s encoded to a real number r defned n (0,) such that r s equal to the ntegratons of the normal dstrbuton from - to p n, : r = p n, N(0,)( z) dz (27a) p µ pn, = (27b) σ where µ s the average of the -th desgn varable, and σ s the standard devaton of the -th desgn varable. The basc dea of encodng system n ARMOGAs s the same as for real-coded ARGAs, but a straghtforward extenson s not sutable n dversty of the populaton. To better preserve the dversty of soluton canddates, the Normal dstrbuton for encodng has to be changed. Fgure 4 shows the search range wth the dstrbuton of probablty. The search regon s parttoned nto three parts, I, II, and III. Regons I and III make use of the same encodng method as ARGAs. The real value of the -th desgn varable P s encoded to a real number r defned n (0,). In contrast, regon II adopts the conventonal realnumber encodng method. The plateau regon (regon II) s defned by the upper and lower desgn varables of chosen solutons. Then, the normal dstrbuton s consdered at both sdes of the plateau determned by the average (µ ) and the standard devaton (σ ). Ths encodng system s controlled by the parameters α r and fl, where α r (<0.5) 2 Amercan Insttute of Aeronautcs and Astronautcs

13 s the populaton rato at regon I and fl s half the length of the plateau at regon II. The encodng s conducted at each regon descrbed below: Regon I (p µ fl, 0 r α r ): r = α r (28a) r = p n, p n, r N(0,)( z) dz (28b) p ( µ fl ) = (28c) 2σ Regon II (µ fl < p < µ + fl, α r < r < α r ): r = ( 2α ) r + α (28d) Regon III (µ + fl p, α r r ): r r ' = r = p r p r ( µ fl ) 2 fl = α r + n, p n, r ( r α ) N(0,)( z) dz (28e) (28f) (28g) p ( µ + fl ) = (28h) 2σ IV. Data Mnng A. ANOVA ANOVA s one of the data mnng technques showng the effect of each desgn varable to the objectve and the constrant functons n a quanttatve way. ANOVA uses the varance of the model due to the desgn varables on the approxmaton functon. By decomposng the total varance of model nto the varance due to each desgn varable, the nfluence of each desgn varable on the objectve functon can be calculated. The decomposton s accomplshed by ntegratng out the varables of model ŷ. The total mean ˆ µ ) and varance σ ) of model ŷ are as follows: ( total The man effect of varable x s ˆµ total ( ˆtotal 2 yˆ( x,..., xn ) dx dx n (29) 2 [ yˆ( x,..., xn) ] dx dx 2 total = ˆ n ˆ σ µ (30) ˆ µ ( x ) yˆ( x,, xn) dx dx dx + dxn ˆ µ The varance due to the desgn varable x s [ ˆ( x )] 2 dx (32) µ The proporton of the varance due to desgn varable x to the total varance of the model can be expressed by dvdng Eq. (32) by Eq. (30). [ ˆ µ ( x )] dx [ yˆ( x x ],..., n) ˆ µ 2 2 dx dx Ths value ndcates how much effect desgn varable x gves to the objectve functon ŷ. n (3) (33) 3 Amercan Insttute of Aeronautcs and Astronautcs

14 B. Self-Organzng Map (SOM). General SOM algorthm SOM s an unsupervsed learnng, nonlnear projecton algorthm 32,33 from hgh to low-dmensonal space. Ths projecton s based on self-organzaton of a low-dmensonal array of neurons. In the projecton algorthm, the weghts between the nput vector and the array of neurons are adjusted to represent features of the hgh dmensonal data on the low-dmensonal map. The closer two patterns are n the orgnal space, the closer s the response of two neghborng neurons n the low-dmensonal space. Thus, SOM reduces the dmenson of nput data whle preservng ther features. A neuron used n SOM s assocated wth weght vector m = [m, m 2,,m n ] (=,.,M) where n s equal to the dmenson of nput vector and M s number of neuron. Each neuron s connected to adjacent neurons by a neghborhood relaton and usually forms two-dmensonal rectangular or hexagonal topology as shown n Fg. 5. (a)rectangular (b) Hexagonal Fgure 5 Topology used n SOMs The learnng algorthm of SOM s started wth fndng the best-matchng unt (m c ) whch s closest to the nput vector x as follow: x m = mn x m ( k =, LL, M ) (34) c k Once the best-matchng unt s determned, the weght adjustments are performed not only for the best-matchng unt but also for ts neghbors. The adjustment depends on the dstance (smlarty) between the nput vector and the neuron. Based on the dstance, the best-matchng unt and ts neghborng become closer to the nput vector as shown n Fg. 6. The weght vectors are stuated n the cross of the sold lnes. The best-matchng unt s the weght vector who s closest to the nput vector x. The best-matchng unt and ts neghbors are adjusted to be closer to the nput vector x. The adjusted topology s represented wth dashed lnes. Repeatng ths learnng algorthm, the weght vectors become smooth not only locally but also globally. Thus, the sequence of close vectors n the orgnal space results n a sequence of the correspondng neghborng neurons n the two-dmensonal map. Fgure 6 Adjustment of the best-matchng unt and ts neghbors 4 Amercan Insttute of Aeronautcs and Astronautcs

15 2. Kohonen s Batch-SOM In ths nvestgaton, SOMs are generated by usng commercal software Vscovery R SOMne plus produced by Eudaptcs GmbH. Although SOMne s based on the general SOM concept and algorthm, t employs an advanced varant of unsupervsed neural networks,.e. Kohonen s Batch SOM. The algorthm conssts of two steps that are terated untl no more sgnfcant changes occur: search of the best-matchng unt c for all nput data {x } and adjustment of weght vector {m j } near the best-matchng unt. The Batch-SOM algorthm can be formulated as follows: where * m j c = mn x arg m j (35) j m * j = (36) h h jc jc x s the adjusted weght vector. The neghborhood relatonshp between two neurons j and k s defned by the followng Gaussan-lke functon: 2 d jk h = jk exp (37) 2 rt where d jk denotes the Eucldean dstance between the neuron k and the neuron j on the map, and r t denotes the neghborhood radus whch s decreased wth the teraton steps t. The standard Kohonen algorthm adjusts the weght vector after all each record s read and matched. On the contrary, the Batch-SOM takes a batch of data (typcally all records), and performs a collected adjustment of the weght vectors after all records have been matched. Ths s much lke epoch learnng n supervsed neural networks. The Batch-SOM s a more robust approach, snce t medated over a large number of learnng steps. In the SOMne, the unqueness of the map s ensured by the adopton of the Batch-SOM and the lnear ntalzaton for nput data. Much lke some other SOMs, SOMne creates a map n a two-dmensonal hexagonal grd. Startng from numercal, multvarate data, the nodes on the grd gradually adapt to the ntrnsc shape of the data dstrbuton. Snce the order on the grd reflects the neghborhood wthn the data, features of the data dstrbuton can be read off from the emergng map on the grd. The traned SOM s systematcally converted nto vsual nformaton. 3. Cluster Analyss Once the hgh-dmensonal data projected on the two-dmensonal regular grd, the map can be used for vsualzaton and the data mnng. It s effcent to group all neurons by the smlarty to facltate SOM for the qualtatve analyss, because number of neurons on the SOM s large as a whole. Ths process of groupng s called clusterng Herarchcal agglomeratve algorthm s used for the clusterng here. Frst, each node tself forms a sngle cluster and two clusters, whch are adjacent n the map, are merged n each step. The dstance between two clusters s calculated by usng the SOM-ward dstance. 30 The number of clusters s determned by the herarchcal sequence of clusterng. A relatvely small number of clusters are used for vsualzaton, whle a large number are used for the generaton of weght vectors for respectve desgn varables. V. Data Mnng Results A. Fly-back Booster of Reusable Launch Vehcle (RLV) Desgn 24 The frst example consders the four-objectve aerodynamc optmzaton for the fly-back booster usng hgh fdelty CFD code. The resultng non-domnated front reveals trade-offs n the desgn space. Two dfferent data mnng technques were appled to the resultng non-domnated front to examne whether they wll produce consstent results. Although t s dffcult to valdate data mnng results n general, ths example gves a verfcaton of the present data mnng approach. Geometry of the fly-back booster used n the present optmzaton s shown n Fg. 7(a). The desgn varables used to defne wng shape are related to planform, arfol, wng twst and relatve wng poston to fuselage. A wng planform s determned by fve desgn varables as shown n Fg. 7(b). Arfol shapes are defned at wng root, knk 5 Amercan Insttute of Aeronautcs and Astronautcs

16 and tp, respectvely, by usng thckness and camber dstrbutons. Both dstrbutons are parameterzed by usng Bezer curves and lnearly nterpolated n the spanwse drecton. Wng twst n refned by usng a B-splne curve wth sx control ponts. Relatve poston of the wng root to the fuselage s parameterzed by x and z coordnates of the leadng edge, angle of attack and dhedral angle. Total 7 desgn varables are used to wng geometry defnton. (a) Overvew (b) Defnton of wng planform Fgure 7 Geometry of fly-back booster Fgure 8 Typcal flght sequence for TSTO fly-back booster Accordng to the trajectory analyss, the separaton of the booster and orbter takes places around Mach 3 and the booster turns over, slows down, cruse at transonc speed and lands at subsonc speed as shown n Fg. 8. In order to mantan good aerodynamc performances n wde flght range, the followng 4 objectve functons are consdered n ths desgn.. Mnmzaton of the dfference between supersonc ptchng moment and transonc ptchng moment. SUPERSONIC M p TRANSONIC M p F = C C (38) 2. Mnmzaton of the ptchng moment at the transonc flght condtons TRANSONIC C M p F 2 = (39) 3. Mnmzaton of the drag at the transonc flght condtons TRANSONIC F 3 = C D (40) 4. Maxmzaton of the lft at the subsonc flght condtons SUBSONIC F 4 = C L (4) As the optmzer, ARMOGA s used wthout a surrogate model. The populaton sze of the present ARMOGA s 8 and 40 generatons are performed. Fgure 9 shows 02 non-domnated solutons obtaned by ARMOGA. 6 Amercan Insttute of Aeronautcs and Astronautcs

17 However, t s dffcult to understand the feature of desgn space from the Fg. 9. For the better understand of the desgn space, ANOVA and SOM are performed wth 02 non-domnated solutons. Fgure 9 Non-domnated solutons projected onto three-dmensonal objectve functon space. ANOVA ANOVA s performed for four objectve functons to analyze the non-domnated front. Varance of desgn varables and ther nteractons whose proporton to the total varance s over than.0% are shown n Fg. 20. Fgure 20 ANOVA results 7 Amercan Insttute of Aeronautcs and Astronautcs

18 Accordng to the results, dv7 (x coordnate of relatve wng poston to fuselage) gves the largest effect on the objectve functons F and F 2. About F 3 and F 4, dv8 (rearward camber heght at knk) gves the largest effect. dv7 and dv8 are llustrated n the Fg. 2. These fndngs correspond to the aerodynamc knowledge. (a) x coordnate of relatve wng poston to fuselage (dv7) (b) rearward camber heght at knk (dv8) Fgure 2 Illustratons of dv7 and dv8 2. SOM SOM s also appled to the non-domnated front. Fgure 22 show the resultng SOM colored by respectve objectve functons. The plots for F and F 2 show smlar color patterns. Roughly speakng, F and F 2 can be mnmzed smultaneously, and thus they are not n the trade-off relaton. On the other hand, although the plots for F 3 and F 4 show the smlar color dstrbuton, F 3 and F 4 are n a severe trade-off relaton because F 3 should be mnmzed but F 4 should be maxmzed. (a) F (b) F 2 (c) F 3 (d) F 4 Fgure 22 SOM colored by respectve objectve functons Fgure 23 shows SOM colored by three desgn varables (dv7, dv8 and dv5). In Fg. 23(a), colored by dv7, large dv7 values can be found at the lower left corner. Ths area corresponds to large F and F 2 values as shown n Fgs. 22(a) and 22(b). Ths means that large dv7 values lead to poor performances of F and F 2. In Fg 23(b), colored by dv8, large dv8 values can be found n the left-hand sde. Ths color pattern s very smlar to those for F 3 and F 4 as shown n Fgs. 22(c) and 22(d). Ths means that large dv8 values lead to large F 3 and F 4 values. These results suggest that dv7 has a large effect on the objectve functons F and F 2 and that dv8 has a large effect on the 8 Amercan Insttute of Aeronautcs and Astronautcs

19 objectve functons F 3 and F 4. In Fg. 23(c), colored by dv5 (x coordnate of forward at knk), there s no notceable trend of color dstrbuton. Ths means that dv5 has lttle nfluence on the objectve functons. These results are concdent wth the results of ANOVA. The results ndcate that ANOVA shows the effect of each desgn varables on objectve functons quanttatvely whle SOM shows the nformaton qualtatvely. (a) dv7 b) dv8 (c) dv5 Fgure 23 SOM colored by three desgn varables B. MDO for Regonal-Jet Wng 7 Data mnng for a large-scale, real-world MDO problem s shown here. Because hgh fdelty CFD solvers are desred for transonc wng desgn, the computatonal cost for MDO wll be enormous. In ths example, nstead of searchng for the optmal soluton, we have appled ARMOGA to explore the desgn space brefly. The optmzaton process was stopped when mprovements were observed n all objectves. Then, SOM was appled to vsualze the desgn space by usng all the solutons computed so far. Based on the observaton, a new wng desgn has been suggested and the resultng wng has been confrmed to outperform the other computed solutons. Ths llustrates the mportance of the present approach because desgn knowledge can produce a better desgn even from the bref exploraton of the desgn space.. Multdscplnary Wng Desgn Objectve Functons In ths optmzaton, mnmzaton of the block fuel at a requred target range derved from aerodynamcs and structures s consdered as the prmary objectve functon. In addton, two more objectve functons are consdered: mnmzaton of the maxmum takeoff weght and mnmzaton of the drag dvergence between transonc and subsonc condtons. Geometry Defnton The desgn varables descrbe arfol, twst, and wng dhedral. The arfol was defned at three spanwse crosssectons usng the modfed PARSEC wth nne desgn varables (x up, z up, z xxup, x lo, z lo, z xxlo, α TE, β TE, and r LElo /r LEup ) for each cross-secton as shown n Fg. 24. The twsts were defned at sx spanwse locatons, and then wng dhedrals are defned at knk and tp locatons. The entre wng shape was thus defned usng 35 desgn varables. Fgure 24 Illustraton of the modfed PARSEC arfol defnton 9 Amercan Insttute of Aeronautcs and Astronautcs

20 Evaluaton Method The present ARMOGA generates eght ndvduals per generaton, and evaluates aerodynamc and structural propertes of each desgn canddate as follows:. Structural optmzaton s performed to jg shape to realze mnmum wng weght wth constrants of strength and flutter requrements usng NASTRAN. And then, weghts of wng box and carred fuel are calculated. 2. Statc aeroelastc analyss s performed at thee flght condtons to determne the aeroelastc deformed shapes (G shape) usng Euler solver and NASTRAN. 3. Aerodynamc evaluatons are performed for the G shapes usng a N-S solver. 4. Flght envelope analyss s performed usng the propertes obtaned as above to evaluate the objectve functons. Usng the objectve functons, the optmzer generates new ndvduals for the next generaton va genetc operatons, such as selecton, crossover, and mutaton. 2. Optmzaton Results The populaton sze was set to eght, and then roughly 70 Euler and 90 N-S computatons were performed n one generaton. It took roughly one and nne hours of CPU tme on NEC SX-5 and SX-7 per PE for sngle Euler and N-S computatons, respectvely. The populaton was re-ntalzed every fve generatons for the range adaptaton. A total evolutonary computaton of 9 generatons was carred out. The evoluton dd not converge yet. However, the results are satsfactory because several non-domnated solutons have acheved sgnfcant mprovements over the ntal desgn. Furthermore, a suffcent number of solutons are searched so that the senstvty of the desgn space around the ntal desgn can be analyzed. Fgure 25 shows all solutons projected on a two-dmensonal plane between two objectves, the block fuel and the drag dvergence. The non-domnated front s formed, ndcatng the trade-off between the block fuel and the drag dvergence. All solutons projected on two-dmensonal planes between other combnatons are shown n Fgs. 26, and 27. As the non-domnated solutons dd not comprse a front, these fgures ndcate that there are no global trade-offs between these combnatons of the objectve functons. The comparson between ntal and optmzed geometres s nvestgated. Although the wng box weght tends to ncrease as compared wth that of the ntal geometry, the block fuel can be reduced. Thus, the aerodynamc performance can redeem the penalty due to the structural weght. An ndvdual on the non-domnated front shown n Fg. 25 s selected, ndcated as optmzed, and then the optmzed geometry s compared wth the ntal geometry. Although the drag mnmzaton s not consdered here, C D s reduced. By comparson of the polar curves at constant C L for the crusng condton, C D of the optmzed geometry s found to be reduced by 5.5 counts. Due to the mprovement of the drag, the block fuel of the optmzed geometry s decreased by over one percent even wth ts structural weght penalty. Fgure 25 All solutons on two-dmensonal plane between block fuel and drag dvergence 20 Amercan Insttute of Aeronautcs and Astronautcs

21 Fgure 26 All solutons on two-dmensonal plane between block fuel and maxmum takeoff weght Fgure 27 All solutons on two-dmensonal plane between maxmum takeoff weght and drag dvergence 3. Data Mnng by SOM Detaled flow vsualzaton for the optmzed geometry ndcates that the man drag reducton s acheved at the knk locaton. However, the optmzed geometry has nverted gull at the knk. Fgure 28(a) shows the SOM colored by the angle between nboard and outboard on the upper wng surface for the gull-wng at the knk locaton. Angles greater and less than 80 deg correspond to gull and nverted gull-wng, respectvely. Hgher values of ths angle as shown n Fg. 28(a) correspond to hgher C D at the transonc crusng flght condton as shown n Fg. 28(b). However, at angles less than 80 deg, there s lttle correlaton between Fg. 28(a) and 28(b). The nverted gull dd not affect aerodynamc performance very much. Furthermore, SOM also shows that hgher angles shown n Fg. 28(a) correspond to hgher maxmum takeoff weghts as shown n Fg. 28(c). The nverted gull-wng s known to have a structural weght ncrease, whch s also observed n the present results. From the vsualzaton of the desgn space by SOM, t s suggested that non-gull wngs should be desgned even though the optmzed geometry has nverted gull. 2 Amercan Insttute of Aeronautcs and Astronautcs

22 (a) (b) (c) Fgure 28 SOM; (a) colored by the angle on upper surface expressng the gull-wng at the knk locaton, (b) colored by C D under transonc crusng flght condton, (c) colored by the maxmum takeoff weght. 4. Evaluaton of the Non-Gull Geometry The optmzed wng shape has been modfed to examne the non-gull wng shape (called as optmzed_mod ) can acheve better performance and to verfy the desgn knowledge obtaned by the prevous data mnng. The result s shown n Fgs. 29 to 3. These fgures show that optmzed_mod mproves both block fuel and maxmum takeoff weght. Moreover, by comparson of the polar curves at constant C L for crusng condton shown n Fg. 32, C D of optmzed_mod s found to be reduced by 0.6 counts over the ntal geometry. Due to the mprovement of drag, the block fuel of optmzed_mod s reduced by 3.6 percent. The present optmzaton s probably ncomplete because only the small number of the generatons has been performed. In addton, the automatc mesh generator may clp the desgn space severely. In the present MDO system, surface splne functon of the geometry devaton s used for the modfcaton of the wng surface mesh, and then the volume mesh s modfed accordngly by the unstructured dynamc mesh method. However, ths process made the surface mesh dstorted around the leadng edge. Ths mesh generaton mght be the prmary reason for the dffculty n fndng the non-gull geometry. However, the present result demonstrates that data mnng can produce a good desgn even from the results of the ncomplete optmzaton. Fgure 29 Fgure 30 Comparson of optmzed_mod and all solutons on twodmensonal plane between block fuel and C D dvergence Comparson of optmzed_mod and all solutons on twodmensonal plane between block fuel and maxmum takeoff weght 22 Amercan Insttute of Aeronautcs and Astronautcs

23 Fgure 3 Fgure 32 Comparson of optmzed_mod and all solutons on twodmensonal plane between maxmum takeoff weght and C D dvergence Comparson of the C L -C D curves among three geometres as ntal, optmzed, and optmzed_mod under transonc flght condton VI. Concludng Remarks A new approach, MODE, has been presented to address MDO problems. MODE s not ntended to gve an optmal soluton. MODE reveals the structure of the desgn space from the trade-off nformaton and vsualzes t as a panorama for DM. DM wll know the reason for trade-offs from non-domnated desgns, nstead of recevng an optmal desgn wthout trade-off nformaton. The present components of MODE are descrbed, although the concept of MODE can be coupled wth other RSM and optmzaton algorthms. The man emphass of ths approach s vsual data mnng. Two data mnng examples are presented. The frst example consders the four-objectve aerodynamc optmzaton for the fly-back booster usng hgh fdelty CFD code. The resultng non-domnated front reveals trade-offs n the desgn space. Two dfferent data mnng technques were confrmed to produce consstent results. Although t s dffcult to valdate data mnng results n general, ths example gves a verfcaton of the present approach. The results ndcate that ANOVA shows the effect of each desgn varables on objectve functons quanttatvely whle SOM shows the nformaton qualtatvely. The second example consders the hgh fdelty MDO problem for a regonal-jet wng. It optmzes aerodynamc performance and structural weght under aeroelastc constrants. Because the desgn space was large and hgh fdelty smulaton codes were tme-consumng, ARMOGA was used to explore the desgn space brefly. The optmzaton was stopped after mprovements were obtaned. Then, SOM was appled to vsualze the desgn space. Based on the observaton, a new, better wng desgn has been suggested. Ths llustrates the mportance of MODE because desgn knowledge can produce a better desgn even from the bref exploraton of the desgn space. Although t s not dscussed n ths paper, the flowchart of MODE shown n Fg. 4 has feedback loops. The desgn space can be redefned by analyzng the surrogate model. 35 Moreover, from data mnng, competng objectves and actve constrants can be dentfed. Ths wll lead to the re-defnton of the MDO problem tself (Fg. 33). MDO often uses conceptual performance equatons as desgn objectves. However, senstvtes of those equatons to hgh fdelty smulaton codes are not well understood. As more and more hgh fdelty smulaton codes become avalable to MDO, selecton of objectve functons wll become more crucal. The outermost feedback loop n Fg. 33 wll be essental to address ths ssue. 23 Amercan Insttute of Aeronautcs and Astronautcs