SIMULTANEOUS AERODYNAMIC ANALYSIS AND DESIGN OPTIMIZATION (SAADO) FOR A 3-D RIGID WING

Transcription

1 AIAA SIMULTANEOUS AERODYNAMIC ANALYSIS AND DESIGN OPTIMIZATION (SAADO) FOR A -D RIGID WING Clyde R. Gumbert Research Scentst Multdscplnary Optmzaton Branch NASA Langley Research Center Hampton, VA Gene J.-W. Hou Professor Department of Mechancal Engneerng Old Domnon Unversty, Norfolk, VA Member AIAA Perry A. Newman Senor Research Scentst Multdscplnary Optmzaton Branch NASA Langley Research Center Hampton, VA Abstract The formulaton and mplementaton of an optmzaton method called Smultaneous Aerodynamc Analyss and Desgn Optmzaton (SAADO) s presented and appled to a smple D wng problem. The method ams to reduce the computatonal expense ncurred n performng shape optmzaton usng state-of-the-art CFD flow analyss and senstvty analyss tools. Results for ths small problem show that the method reaches the same local optmum as conventonal optmzaton methods and does so n about half the computatonal tme. b C D C l C L C M C p Nomenclature wng semspan drag coeffcent rollng moment coeffcent lft coeffcent ptchng moment coeffcent pressure coeffcent Copyrght 999 by the Amercan Insttute of Aeronautcs and Astronautcs, Inc. No copyrght s asserted n the Unted States under Ttle 7 U. S. Code. The U. S. Government has a royaltyfree lcense to exercse all rghts under the copyrght clamed heren for Governmental Purposes. All other rghts are reserved by the copyrght holder c r wng root chord wng tp chord F objectve functon g constrants M free-stream Mach number Q flow feld varables (state varables) at each CFD mesh pont Q change n flow solver feld varables due to better convergence Q change n flow solver feld varables due to desgn changes R resdual feld of the state equatons at each CFD mesh pont R/R norm of the resdual normalzed by ts ntal value S semspan wng planform area T twst angle at wng tp, postve for leadng edge up X CFD volume mesh X LE vector locaton of wng root leadng edge x/c chordwse locaton normalzed by local wng secton chord longtudnal locaton of wng tp tralng edge z r root secton camber α free-stream angle-of-attack β desgn varables

2 operator whch ndcates a change n a varable δ sze of fcttous sde bound on desgn varables γ lne search parameter superscrpts: * desgnates updated value gradent wth respect to desgn varables Introducton Smultaneous Aerodynamc Analyss and Desgn Optmzaton (SAADO) s a procedure whch ncorporates desgn mprovement wthn the teratvely solved (nonlnear) aerodynamc analyss so as to acheve fully converged flow solutons only near an optmal desgn. Overall computatonal effcency s acheved because the many expensve teratve (nonlnear) solutons for nonoptmal desgn parameters are not converged (.e., obtaned) at each optmzaton step. One obtans the desgn n the equvalent of a few (rather than many) multples of the computatonal tme for a sngle, fully converged flow analyss. SAADO and smlar procedures for smultaneous analyss and desgn (SAND) developed by others are noted and dscussed by Newman et al. These SAND procedures appear to be best suted for applcatons where the dscplne analyses nvolved n the desgn are nonlnear and solved teratvely. Generally, convergence of these dscplne analyses (.e., state equatons) s vewed as an equalty constrant n an optmzaton problem. From ths latter pont of vew, the SAADO method proceeds through nfeasble regons of the (β, Q) desgn space. A further advantage of SAADO s the effcent utlzaton of exstng dscplne analyss codes (wthout nternal changes), augmented wth senstvty or gradent nformaton, and yet effectvely coupled more tghtly than s done n conventonal gradent-based optmzaton procedures, referred to as nested analyss and desgn (NAND) procedures. A recent overvew of aerodynamc shape optmzaton dscusses both NAND and SAND procedures n the context of current steady aerodynamc optmzaton research. For sngle-dscplne desgn problems, the dstncton between NAND and SAND procedures s farly clear and readly seen. Wth respect to dscplne feasblty (.e., convergence of the teratvely (generally nonlnear) solved state equatons), these procedures can be vewed as accomplshng desgn by usng only very well converged dscplne solutons (NAND) or as convergng a sequence of dscplne solutons from poorly to well as the desgn progresses (SAND). However, the problem formulaton and soluton algorthms may dffer consderably. About twenty SAND references are quoted by Newman et al. and Newman et al. ; these references dscuss a varety of formulatons, algorthms, and results for sngledscplne problems (mostly CFD applcatons) n the sense of SAND defned above. For multdscplnary desgn optmzaton problems, the dstncton between NAND and SAND s somewhat blurred because there are feasblty consderatons wth respect to all the ndvdual dscplne state equatons as well as wth respect to the multdscplnary system compatblty and constrants. A number of the papers n Ref. dscuss MDO formulatons and algorthms that are called SAND-lke. However, not all of these latter MDO procedures appear to agree wth the sense of SAND defned above and used heren; one that does s Ref.4. The computatonal feasblty of SAADO for quas -D nozzle shape desgn based on the Euler equaton CFD approxmaton was demonstrated by Hou et al. 5 and Man. 6 Applcaton of SAADO for turbulent transonc arfol shape desgn based on a -D thn-layer Naver- Stokes CFD approxmaton was demonstrated and reported n a later paper by Hou et al. 7 Both of these applcaton results are summarzed and brefly dscussed n Ref.. These SAADO procedures utlzed quasanalytcal senstvty dervatves whch were obtaned from hand-dfferentated code for the ntal quas -D applcaton 5, 6 and from automatcally dfferentated code for both the -D arfol applcaton 7 and the present -D wng applcaton. Dfferent optmzaton technques have also been used n these SAADO procedures. Our ntal -D wng results from SAADO, obtaned wth several optmzaton technques, are gven n ths paper. The problem was a smple wng planform desgn where changes n desgn varables were sought to produce mprovement n the lft-to-drag rato subject to lft, ptchng moment, and rollng moment (.e., soluton dependent) constrants, as well as geometrc constrants. Our ultmate goal s smultaneous aerodynamc and structural wng desgn. Problem Descrpton To evaluate the effcacy of ths SAADO procedure n -D, t s appled heren to a smple rgd wng alone. The wng conssted of a trapezodal planform wth a rounded tp. It was parameterzed by ffteen varables; fve descrbed the planform, and fve each descrbed the root and tp secton shapes. A schematc of the wng and ts assocated planform parameters s shown n Fg.. The baselne wng secton vared lnearly from an NACA at the root to an NACA 8 at the tp. The specfc parameters selected as desgn

3 varables n the sample optmzaton problems are dentfed n the secton enttled Results. The objectve functon to be mnmzed was the negatve of the lft-to-drag rato, L/D. In leu of addtonal dscplne nteractons, both aerodynamc and geometrc constrants were mposed. The aerodynamc constrants were: lower lmt on lft, C L * S, n leu of a mnmum payload requrement upper lmt on rollng moment, C l, n leu of a maxmum bendng moment upper lmt on ptchng moment, C M, n leu of a trm constrant The purely geometrc constrants were: mnmum leadng edge radus, n leu of a manufacturng requrement mnmum wng secton thckness, n leu of a structural nteracton sde constrants (bounds) on the actve desgn varables SAADO Procedure The SAADO approach formulates the desgnoptmzaton problem as follows: mn F(Q, X ( β), β) () β, Q subject to g ( Q, X( ), ) ; =,,..., m () and RQX (, ( β), β ) =. () Recall that Q, R, and X are very large vectors. Ths formulaton treats the state varables, Q, as part of the set of ndependent desgn varables, and consders the state equatons to be constrants. Because satsfacton of the equalty constrants of Eq. () s requred only at the fnal optmum soluton, the steady-state aerodynamc feld equatons are not converged at every desgn-optmzaton teraton. The easng of that restrcton can sgnfcantly reduce the excessvely large computatonal cost ncurred n the conventonal approach. However, ths advantage would lkely be offset by the very large ncrease n the number of desgn varables and equalty constrant functons, unless some remedal procedure s adopted. The SAADO method begns wth a lnearzed desgnoptmzaton problem whch s solved for the most favorable change n the desgn varables, β, as well as for the changes n the state varables, Q; that s, subject to and mn F(Q, X, β) β, Q Q X β X β g g (Q, X, β) Q g g X β X β ; R(Q, X, β) Q X X β = β =,,..., m (4) (5) (6) where Eqs. (4), (5) and (6) are lnearzed approxmatons of Eqs. (), () and (), respectvely. In ths formulaton, the resdual of the non-lnear aerodynamc feld equatons, R(Q,X,β), s not requred to be zero (reach target) untl the fnal optmum desgn s acheved. The lnearzed problem of Eqs. (4), (5), and (6) s dffcult to solve drectly because of the number of desgn varables and equalty constrant equatons. Ths dffculty s overcome for the drect dfferentaton method by usng drect substtuton to remove Q and Eq. (6) altogether from ths lnearzed problem. The dscrete adjont method descrbed n the Appendx s another way to overcome the dffculty. However, the remander of ths paper descrbes only the use of the drect dfferentaton method for formulatng the SAADO procedure and ts applcaton to optmzaton problems. Equaton (6) provdes a lnear relaton to represent Q n terms of β as Q = Q Q β (7) where vector Q and matrx Q are solutons of the followng equatons Q = R (8) Q = X. (9) X β Note that the number of columns of matrx Q s equal to the number of desgn varables, β. A new

4 lnearzed problem wth β as the only desgn varables can be obtaned by substtutng Eq (7) nto Eqs. (4) and (5) for Q: subject to mn F(Q, X, β) Q β Q X β X β g g (Q, X, β) Q g g g Q X ; Q X β β =,,..., m () () Once establshed, ths lnearzed problem can be solved usng any mathematcal programmng technque for desgn changes, β. The assocated changes n Q can be calculated usng Eq (6) or equvalently, Eqs. (7), (8), and (9). A onedmensonal search on the step sze parameter γ s then performed n order to fnd the updated values, β * =βγ β, the new mesh, X * =X(β ), and Q * =Q Q * where the search procedure must solve a nonlnear optmzaton problem of the form subject to and * * * mn F(Q, X, β ) () γ * * * g ( Q, X, β ) ; =,,..., m () * * * RQ (, X, β ) = (4) The step sze γ s the only desgn varable. Agan t s noted for emphass that the equalty constrants, Eq. (4), are not requred to be zero (reach target) untl the fnal optmum desgn; volatons of these equalty constrants should smply be progressvely reduced durng the SAADO procedure. Therefore, the updated Q * n ths study s defned as Q * =Q Q *, whch satsfes the frst order approxmatons Eq. (4) as R(Q, X, β) Q X X β * ( γ β) = (5) Equaton (5) may be solved usng Eqs (8) and (9), whch gve the updated Q * as Q * =Q Q γ Q β. Therefore, the update of Q ncludes two parts: Q and Q. The former s due to better convergence of the flow solver, whereas the latter s due to changes n the desgn varables. In fact, Q approaches the flow feld senstvtes, Q, as the soluton becomes better converged. The appearance of Q n the formulaton makes the SAADO approach dfferent from the conventonal NAND aerodynamc optmzaton method. The Q not only consttutes a change n Q but also plays an mportant role n defnng the constrant volaton of Eq. (). Snce Q, as shown n Eq. (8), represents a sngle Newton s teraton on the flow equatons, t s possble to approxmate t as the change n Q as a result of several Newton s teratons to mprove the qualty of the flow soluton. A schematc of the present SAADO procedure s shown n Fg.. Computatonal Tools and Models The aerodynamc flow analyss code used for ths study s a verson of the CFLD code 8. Only Euler analyses are performed for ths work, although the code s capable of solvng the Naver-Stokes equatons wth any of several turbulence models. The dervatve verson of ths code, whch was used for aerodynamc senstvty analyss, was generated by an unconventonal applcaton 9 of the automatc dfferentaton code ADIFOR, to produce a relatvely effcent, drect mode, gradent analyss code, CFLD.ADII. It should be ponted out that the ADIFOR process produces a dscretzed dervatve code that s consstent wth the dscretzed functon analyss code. The addton of a stoppng crteron based on the norm of the resdual of the feld equatons was the only modfcaton of the CFLD.ADII code made to accommodate the SAADO procedure. The surface geometry was generated based on the parameters descrbed n the prevous secton by a code utlzng the Rapd Arcraft Parameterzaton Input Desgn (RAPID) technque developed by Smth, et al. Ths code was preprocessed wth ADIFOR to generate a code capable of producng senstvty dervatves as well. The CFD volume mesh needed by the flow analyss code was generated usng a verson of the CSCMDO 4 grd generaton code. The assocated grd senstvty dervatves needed by the flow senstvty analyss were generated wth an automatcally dfferentated 4

5 verson of CSCMDO. 5 In addton to the parameterzed surface mesh and accompanyng gradents, CSCMDO requres a baselne volume mesh of smlar shape and dentcal topology. The 4, grd pont baselne volume mesh of C-O topology used n the present examples was obtaned wth the WTCO code. 6 Ths mesh s admttedly partcularly coarse by current CFD analyss standards. Several optmzaton technques were used n ths study. Both conventonal and SAADO procedures were mplemented, run, and compared. The Sequental Quadratc Programmng method and the Method of Feasble Drectons as mplemented n the DOT 7 optmzaton software were used and the results were desgnated SQP and MFD, respectvely, when run n the conventonal (NAND) mode. The same optmzaton methods were also used n a SAADO or SAND mode (that s, partally converged CFD analyss and senstvty results at each teraton) and these results were desgnated SSQP and SMFD. In addton, a rather neffcent, unsophstcated optmzaton technque was devsed for use n the SAADO mode n order to provde dynamc drect access to and control of parameters typcally nternally controlled by packaged optmzers. Ths technque s a sequental lnear method consstng of formulaton of a mert functon as weghted sum of objectve functon and actve constrants determnaton of the step drecton by steepest descent of the mert functon determnaton of the step sze usng a onedmensonal lne search Results are desgnated SDMF when obtaned n the conventonal mode and when obtaned n the SAADO mode. All procedures were executed wth an SGI OCTANE workstaton wth a 5Mhz R processor. The 785MB RAM was suffcent for the largest code, CFLD.ADII, to ft n core for the 4,-pont mesh and number of desgn varables consdered heren. Results Most of the results shown n ths work are for a desgn problem nvolvng only two desgn varables out of the ffteen avalable wng parameters. Prelmnary results for a fve-desgn-varable problem are brefly dscussed n the subsequent subsecton. In the last subsecton, these present SAADO results are dscussed n the context of other SAND approaches. Two-Desgn-Varable Problems Many of the benefts and many of the problems assocated wth SAADO are demonstrated wth just two desgn varables. A beneft of studyng just two desgn varables s the faclty to vsualze the optmzaton process. In the course of another approxmaton and optmzaton study 8, databases of computatonal results were created for each of several grds for the present objectve functon as a functon of and wth the same CFD code, wng, and flow condtons as used heren. One of those grds was dentcal to that used n ths study. The results n ths database were converged to R/R = -6. Thus, contour plots of the present objectve functon are avalable; on such plots, we can show the paths taken through desgn space by the several optmzaton technques. Subsonc Problem-The flow condtons for the subsonc flow case were M =.5 and α =. Fgure a shows the objectve functon contour plot for ths case; the two SAADO mode results are those labeled and SSQP whereas the conventonal mode result s that labeled SQP. The rollng moment constrant boundary wth ts correspondng nfeasble regon appears on ths plot; however, the other constrants are feasble throughout the doman shown. Ths ( - ) doman conssts of that regon bounded by the desgn varable sde constrants. All three optmzaton results le very close to the mnmum. Fgure b shows the dfferent paths taken by these three optmzaton technques. Because none of these optmzaton paths approach the constrant boundary, none of the constrants are actve for these subsonc case results. All of these paths start at (,) n the (, ) desgn space. The doman has been truncated for these plots. The paths taken by the SQP and SSQP methods appear to be much more effcent n terms of the total number of calls to the analyss code. Even wth analyss and gradents not as well converged, the step drecton and sze determned by the SSQP are qute smlar to those of the SQP method. The method takes smaller steps on a dstnctly more crcutous path. Ths crcutous path s partally due to less effectve use of past teraton results and also due to a somewhat smplstc method of determnng the step sze and drecton. Defntve computatonal tme comparsons for these subsonc cases are not avalable due to omssons n the code nstrumentaton. The results are presented to show that for two optmzaton technques, the SAADO (SAND) method fnds the same optmum as the 5

6 conventonal (NAND) method. Further, these results suggest that a more sophstcated optmzaton procedure could substantally mprove the result, whle greater user control of the step sze, partcularly early n the procedure, could further decrease the computatonal cost for the SSQP result. Fgure 4 shows the wng planform and the surface pressure coeffcent results for the subsonc cases. The three optmzed planform shapes are very smlar and can be seen to result n essentally the same chordwse pressure dstrbutons. Furthermore, all of these dstrbutons dffer lttle from the orgnal, even where the local chords dffer, away from the wng root. Supercrtcal Problem- Most of the expermentaton wth SAADO was conducted for a problem whch s more nterestng both from the pont of vew of the aerodynamc analyss and of the optmzaton problem. The flow condtons for the supercrtcal flow case were M =.8 and α =. A shock was present over a substantal spanwse porton of the wng and there were actve constrants. Fgure 5a shows the objectve functon contour plot for the two desgn varable problem. The shaded regon denotes the nfeasble regon of the lft constrant. The other constrants were not actve. The fnal terates for eght optmzaton technque results are shown to le near the optmal pont of the feasble regon. These eght technques consst of three conventonal NAND optmzatons (SQP, MFD and SDMF) and ther correspondng SAND modes (SSQP, SMFD and ) as dscussed n a prevous secton, as well as two modfcatons to the SSQP, denoted as SSQP, δ=. and SSQP, δ=.5. The parameter δ smply provdes a means to lmt the change n desgn varables at each optmzaton step by dynamcally adjustng the sde constrants. Further dscusson concernng results from these latter two technques appears n the subsequent subsecton. Fgure 5b shows the dfferent paths taken by these eght optmzaton technques. All of these optmzaton paths approach and cross the lft constrant boundary. Thus, ths constrant was actve and, at tmes, volated n all of the technques for ths supercrtcal problem. As n the subsonc problem, all of these paths start at (,) n the (, ) desgn space, and the doman has been truncated for these plots. Here too, the less-converged analyss and gradent data used by the SSQP method do not seem to mpar ts ablty to determne the step drecton and sze. Agan, the results are smlar to those of the conventonal SQP method. The smplstc SDMF and methods stll take smaller steps on less drect paths, and once more, these crcutous paths are due to neffectve use of nformaton from past optmzaton teraton results and a smplstc means of determnng the step sze and drecton. The step sze for the SDMF decreased substantally when the constrant was encountered. The SDMF requred a few more optmzaton steps than the to reach the mnmum. As expected, the MFD spends less tme n the nfeasble regon than the other optmzaton technques. However, the SMFD does not appear to track the MFD optmzaton path very well. Fgure 6 shows the wng planform and the surface pressure coeffcent results for the supercrtcal cases. Snce all of the optmzed planform shapes are essentally the same and result n the same chordwse pressure dstrbuton, results from only three optmzaton technques are shown. The shock wave on the orgnal wng has been weakened substantally n the optmzed cases, as would be expected. Computaton Cost Comparsons- In vew of the consstency of the NAND and SAND optmzaton results, the measure of success or falure of the SAADO procedure s then ts relatve computatonal expense. Fgure 7 shows the relatve cost, based on accumulated CPU tme, for the procedures usng the varous methods. The cost of the geometry generator, mesh generator and optmzaton drver are not ncluded because ther contrbutons are mnmal relatve to the cost of the flow solver and flow senstvty solver. The cost shown s for flow analyses and gradent analyses and s normalzed by the cost of one full analyss to the target resdual. The SAADO method prmarly reduces the cost of the flow analyss. In ths regard, the SAADO method does show mprovement over ts conventonal counterpart for all methods. The SAADO method appears to make the most overall mprovement for the Method of Feasble Drectons (MFD vs. SMFD) and the Steepest Descent Method wth the Mert Functon (SDMF vs. ). The least relatve reducton s shown for the Sequental Quadratc Programmng method (SQP vs. SSQP). Note that for the conventonal methods, the MFD s domnated by flow analyss whereas the SDMF s domnated by senstvty analyss; the SQP s relatvely balanced. On the computer platform used n ths study, the ADIFOR-generated gradent analyss s relatvely neffcent compared to the flow analyss. 6

7 For the SAND methods, the smaller steps taken n the path relatve to those n the SSQP path, result n smaller perturbatons n the flow and gradent felds. In addton, for each step n the, the convergence level for the subsequent step s not as strngent as that requred n the SSQP. Compared to the number of flow solver teratons requred at each desgn step n SSQP and conventonal SQP, relatvely few are requred n the. However, ths savngs s traded off wth the relatvely large number of desgn steps requred for. Some combnaton of the two methods, such that small steps are taken along a more drect path, may lead to mproved effcency. That mproved effcency was the dea behnd the two SSQP methods labeled δ=. and δ=.5. The sde constrants were adjusted at each call to DOT so as to lmt the step sze. However, an overall detrmental effect on the method resulted, snce each tme ths fcttous sde constrant was encountered, DOT called for recalculaton of the gradents. The most computatonal cost for any of these methods, other than MFD and SMFD, s spent computng gradents, even though none of the gradent resdual ratos were converged below three orders of magntude. Early n the respectve processes, the gradents for n partcular and also the DOT methods were not well converged. As the number of desgn varables s ncreased, ths proporton wll grow nearly lnearly. The need for faster gradent calculatons s apparent. Hou et al. estmated a consderable speed-up attrbuted to usng hand-dfferentated adjont code for the -D Euler equatons. Fortunately, automatc code generaton tools for more easly and effcently computng gradents usng adjont methods 9 are on the horzon. Fve-Desgn-Varable Problem Lmted prelmnary results are gven and brefly dscussed n ths secton for the same wng shape optmzaton problem but wth fve desgn varables. These fve desgn varables are root camber (z r ), tp chord ( ), tp setback ( ), wng semspan (b), and tp twst (T). Table presents a comparson of the conventonal SQP (NAND) and SSQP (SAND) results. It can be seen from the objectve functon and fnal desgn varable values that these two procedures dd not arrve at exactly the same place n desgn space. However, both had the same two actve constrants (rollng moment and ptchng moment) and actve sde constrants (wng semspan and tp setback). The reducton n relatve optmzaton cost, based on CPU tme, for SAADO s comparable to that obtaned n the two desgn varable cases and the total optmzaton cost scales wth the number of desgn varables as t should for gradents obtaned va drect dfferentaton. Further Dscusson The relatve cost, based on CPU tmng ratos, for the SAADO (SAND) versus conventonal (NAND) procedures appled to these present small -D aerodynamc shape desgn optmzaton problems range from about one-half to seven-tenths. Ths range s very smlar to that reported for -D nonlnear aerodynamc shape desgn optmzaton n Refs. and 4, even though many of the computatonal detals dffer. The results gven n Ref. were for a turbulent transonc flow wth shock waves computed usng a Naver-Stokes code; a drect dfferentaton approach (usng ADIFOR) was used for the senstvty analyss. The results reported n Ref. 4 were for a compressble flow wthout shock waves computed usng a nonlnear potental flow code; an adjont approach was used for the senstvty analyss. Snce these two optmzaton problems were also not the same, then, no tmng comparson between these adjont and drect dfferentaton soluton approaches would be meanngful. As ndcated earler, an expected speed-up for usng an adjont, nstead of the drect dfferentaton, approach was estmated n Ref.. Ghattas and Bark recently reported -D and -D results for optmal control of steady ncompressble Naver-Stokes flow whch demonstrate an order-ofmagntude reducton of CPU tme for a SAND approach versus a NAND approach. These results were obtaned usng reduced Hessan SQP methods that avod convergng the flow equatons at each optmzaton teraton. The relatonshp of these methods wth respect to other optmzaton technques s also dscussed n Ref.. Several other SAND-lke methods for smultaneous analyss and desgn are summarzed and dscussed by Ta asan. These methods are called "One-Shot" and "Pseudo-Tme" and have been appled to aerodynamc shape desgn problems at several fdeltes of CFD approxmaton, as noted n Ref.. These technques have obtaned an aerodynamc desgn n the equvalent of several analyss CPU tmes for some sample problems. 7

8 Concludng Remarks Ths study has ntroduced an mplementaton of the SAADO technque for a smple -D wng. Intal results ndcate that SAADO. fnds the same local mnmum as a conventonal technque for the two-desgn-varable cases and a smlar local mnmum for a prelmnary fvedesgn-varable case. s computatonally more effcent than a conventonal gradent-based optmzaton technque. requres few modfcatons to the analyss and senstvty analyss codes nvolved. Intal results have llustrated the need for an adjont method for effcently computng the senstvty dervatves. Fortunately, such automated adjont code generaton tools are on the brnk of beng readly avalable. The SAADO method of closely couplng the analyss and optmzer teraton loops requres optmzaton software that does not usurp all control. That s, the SAADO technque needs to have some control over certan parameters durng the optmzaton process; these parameters are typcally consdered optmzer parameters and are only set by the user ntally. Acknowledgement The second author, G. J.-W. H., was supported n ths work by NASA through several Tasks under contract NAS-9858 wth the ODU Research Foundaton. References. Newman, P. A., Hou, G. J.-W., and Taylor III, A. C., Observatons Regardng Use of Advanced CFD Analyss, Senstvty Analyss, and Desgn Codes n MDO, n Ref., pp. 6-79; also ICASE Report 96-6, NASA CR 989, (avalable electroncally at Newman, III, J. C., Taylor, III, A. C., Barnwell, R. W., Newman, P. A., and Hou, G. J.-W., Overvew of Senstvty Analyss and Shape Optmzaton for Complex Aerodynamc Confguratons, Journal of Arcraft, Vol. 6, No., 999, pp Alexandrov, N. M., and Hussan, M. Y., Eds., Multdscplnary Desgn Optmzaton: State of the Art, SIAM Proceedngs Seres, SIAM, Phladelpha, Ghattas, O. and Orozco, C. E., A Parallel Reduced Hessan SQP Method for Shape Optmzaton, n Ref., pp Hou, G. J.-W., Taylor, III, A. C., Man, S. V., and Newman, P. A., Smultaneous Aerodynamc Analyss and Desgn Optmzaton, Abstracts from nd U.S. Natonal Congress on Computatonal Mechancs, Washngton, DC, Aug., 99, pp.. 6. Man, S. V., Smultaneous Aerodynamc Analyss and Desgn Optmzaton, M. S. Thess, Old Domnon Unversty, Norfolk, VA, Dec Hou, G. J.-W., Korv, V. M., Taylor, III, A. C., Maroju, V., and Newman, P. A., Smultaneous Aerodynamc Analyss and Desgn Optmzaton (SAADO) of a Turbulent Transonc Arfol Usng a Naver-Stokes Code Wth Automatc Dfferentaton (ADIFOR), Computatonal Aeroscences Workshop 95, edted by W. J. Feeresen, and A. K. Lacer, NASA CD CP-, Jan. 996, pp Rumsey, C., Bedron, R., and Thomas, J., CFLD: Its Hstory and Some Recent Applcatons, NASA TM-86, May Sherman, L., Taylor, III, A., Green, L., Newman, P., Hou, G., and Korv, M., Frst- and Second- Order Aerodynamc Senstvty Dervatves va Automatc Dfferentaton wth Incremental Iteratve Methods, Journal of Computatonal Physcs, Vol.9, No., 996, pp Bschof, C. H., Carle, A., Corlss, G. F., Grewank, A., and Hovland, P., ADIFOR: Generatng Dervatve Codes from Fortran Programs, Scentfc Programmng, Vol., No., 99, pp Bschof, C., and Grewank, A., ADIFOR: A Fortran System for Portable Automatc Dfferentaton, Proceedngs, Fourth AIAA/USAF/NASA/OAI Symposum on Multdscplnary Analyss and Optmzaton, Cleveland, Sept. 99, pp. 4-44; also AIAA Paper CP.. Taylor, III, A. C., Oloso, A., and Newman, III, J.C., CFLD.ADII (Verson.): An Effcent, Accurate, General-Purpose Code for Flow Shape-Senstvty Analyss, AIAA Paper 97-4, June Smth, R. E., Bloor, M. I. G., Wlson, M. J., and Thomas, A. T., Rapd Arplane Parametrc Input Desgn (RAPID), Proceedngs, th AIAA Computatonal Flud Dynamcs Conference, San Dego, June 995, pp ; also AIAA Paper

9 4. Jones, W. T., and Samareh-Abolhassan, J., A Grd Generaton System for Multdscplnary Desgn Optmzaton, Proceedngs, th AIAA Computatonal Flud Dynamcs Conference, San Dego, June 995, pp ; also AIAA Paper Bschof, C., Jones, W. T., Samareh-Abolhassan, J., and Mauer, A., Experences wth the Applcaton of the ADIC Automatc Dfferentaton Tool to the CSCMDO -D Volume Grd Generaton Code, AIAA Paper 96-76, Jan Vatsa, V. N., and Wedan, B. W., Effect of Sdewall Boundary Layer on a Wng n a Wnd Tunnel, Journal of Arcraft, Vol. 6, No., 989, pp. 57-6; also AIAA Paper 88-, Jan Anon., DOT Users Manual: Verson 4., Vanderplaats Research & Development, Inc., Colorado Sprngs, May Prvate communcaton Alexandrov, N., and presentaton, Multlevel Optmzaton wth Surrogates In Wng Desgn, Sxth SIAM Conference on Optmzaton, Atlanta, May, Carle, A., Fagan, M., and Green, L. L., Prelmnary Results from the Applcaton of Automated Adjont Code Generaton to CFLD, Proceedngs, th AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, St. Lous, Sept. 998, pp ; also AIAA Paper Ghattas, O., and Bark, J.-H., Optmal Control of Two- and Three-Dmensonal Naver-Stokes Flows, Journal of Computatonal Physcs, Vol. 6, No., 997, pp Ta asan, S., Trends n Aerodynamc Desgn and Optmzaton: A Mathematcal Vewpont, Proceedngs, th AIAA Computatonal Flud Dynamcs Conference, San Dego, June 995, pp ; also, AIAA Paper Appendx - SAADO Procedure Usng a Dscrete Adjont Approach The SAADO approach formulates the desgnoptmzaton problem as follows: mn F(Q, X ( β), β) β, Q subject to g ( Q, X( β ), β ) ; =,,..., m (A) (A) and RQX (, ( β), β ) =. (A) Recall that Q, R, and X are very large vectors. Ths formulaton treats the state varables, Q, as part of the set of ndependent desgn varables, and consders the state equatons to be constrants. Because satsfacton of the equalty constrants of Eq. (A) s requred only at the fnal optmum soluton, the steady-state aerodynamc feld equatons are not converged at every desgn-optmzaton teraton. The easng of that restrcton can sgnfcantly reduce the excessvely large computatonal cost ncurred n the conventonal approach. However, ths advantage would lkely be offset by the very large ncrease n the number of desgn varables and equalty constrant functons, unless some remedal procedure s adopted. The SAADO method begns wth a lnearzed desgnoptmzaton problem whch s solved for the most favorable change n the desgn varables, β, as well as for the changes n the state varables, Q; that s, subject to and mn F(Q, X, β) Q β, Q (A4) X β X β g g (Q, X, β) Q g g X β X β ; R(Q, X, β) Q X X β = β =,,..., m (A5) (A6) where Eqs. (A4), (A5) and (A6) are lnearzed approxmatons of Eqs. (A), (A) and (A), respectvely. In ths formulaton, the resdual of the non-lnear aerodynamc feld equatons, R(Q,X,β), s not requred to be zero (reach target) untl the fnal optmum desgn s acheved. The lnearzed problem of Eqs. (A4), (A5), and (A6) s dffcult to solve drectly because of the number of desgn varables and equalty constrant equatons. Ths dffculty s overcome for the adjont method by the ntroducton 9

10 of the adjont vectors, λ and λ whch are solutons of the followng adjont equatons T λ T λ = (A7) g = ; =,,...m. (A8) The lnearzed problem of Eqs. (A4) to (A6) can then be formulated as mn F(Q, X, β) λ β λ X subject to T g (Q, X, β) λ T T R X λ X β g T g λ X λ X X β =,,..., m R T T (A9) β β β ; (A) β Once establshed, ths lnearzed problem can be solved usng any mathematcal programmng technque for desgn changes, β. A one-dmensonal search s then performed for the step sze parameter γ n order to fnd the updated values, β * =βγ β, X * =X(β ), and Q * where the search procedure must solve a nonlnear optmzaton problem of the form subject to and * * * mn F(Q, X, β ) (A) γ * * * g ( Q, X, β ) ; =,,..., m * * * (A) RQ (, X, β ) =. (A) The step sze, γ, s the only desgn varable. Agan t s noted for emphass that the equalty constrants, Eq. (A), are not requred to be zero (reach target) untl the fnal optmum desgn; volatons of these equalty constrants should smply be progressvely reduced durng the SAADO procedure. Therefore, the updated Q * n ths study s defned as Q * =Q Q *, whch satsfes the frst order approxmatons Eq. (A) as R(Q, X, β) Q X X β * ( γ β) = (A4) The update for Q * need not be determned from Eq. (A4), but rather can be taken from the results of solvng Eq. (A) n the process of determnng γ. The SAADO procedure presented here s smlar to that procedure whch Ghattas and Bark call Quas- Newton Sequental Quadratc Programmng. However, that procedure as presented dd not provde for state varable dependent constrants and t was drectly embedded n a partcular mathematcal programmng technque, SQP. Table. Comparson of Prelmnary SQP and SSQP Results for the Fve Desgn Varable Problem Objectve functon Number of actve constrants Number of actve bounds z r, %chord b T, deg. Relatve functon analyss cost Relatve gradent analyss cost Relatve optmzaton cost Intal value SQP value SSQP value ** 4.** ** 4.98**..*.*** * CPU tme for an analyss soluton converged to R/R 6 s 9 sec. ** Desgn varable sde bound s actve *** Gradent solutons converged to R/R

11 X LE =(,,) Planform vew Profle vew c r = T xt z r XLE =(,,) xt b Fg. Dagram of the parametrc wng model. Input Specfy: Desgn problem Target convergence Intalze: Geometry Flow condtons Convergence level Tghten convergence level Geometry generaton wth gradents Desgn update Optmzer Volume mesh generaton Flow analyss Mesh senstvty generaton Flow senstvty analyss Partally converged desgn Partally converged aerodynamc analyss Partally converged aerodynamc senstvty analyss Fg. Schematc dagram of the SAADO procedure.

12 SQP SSQP contours of objectve nfeasble regon startng pont Fg. a Map of well converged objectve functon and constrant boundary wth optmzaton endponts, M =.5, α =..8 SQP SQP SSQP.. Contours of objectve SSQP Fg. b. Map of well converged objectve functon and constrant boundary wth complete optmzaton paths, M =.5, α =.

13 B Upper surface pressure contours Orgnal result wng A B. B C p.5 A B.5 Secton A A.5 Orgnal SQP SSQP C p.5 Secton B B Pressure coeffcents Planform shapes Fg. 4. Comparson of orgnal wng and optmzaton results at M =.5, α = : upper surface pressure contours, pressure coeffcents at two span statons, and planform shapes.

14 start here MFD SMFD SQP SSQP SSQP, δ=. SSQP, δ=.5 SDMF Fg. 5a Map of well converged objectve functon and constrant boundary wth optmzaton endponts, M =.8, α=. MFD SMFD Fg. 5b Map of well converged objectve functon and constrant boundary wth optmzaton paths, M =.8, α=. 4

15 8 SQP SSQP SSQP δ =. SSQP δ= SDMF Fg. 5b Map of well converged objectve functon and constrant boundary wth optmzaton paths, M =.8, α=

16 B Upper surface pressure contours result A Orgnal wng B..5 B C p B.5. Secton A A Orgnal SQP SSQP A.5 C p.5 Secton B B Pressure coeffcents Fg. 6. Comparson of orgnal wng and optmzaton results at M =.8, α = : upper surface pressure contours, pressure coeffcents at two span statons, and planform shapes. Planform shapes

17 Flow Analyss Gradent Analyss Relatve cost MFD SMFD SQP SSQP δ=.5 SSQP δ=. SSQP SDMF Fgure 7. Computatonal cost comparsons 7