THE ROLE OF CFD IN PRELIMINARY AEROSPACE DESIGN

Transcription

1 Prceedings f FEDSM 03 4TH ASME JSME Jint Fluids Engineering Cnference Hnlulu, Hawaii USA, July 6-11, 2003 FEDSM THE ROLE OF CFD IN PRELIMINARY AEROSPACE DESIGN Antny Jamesn Department f Aernautics and Astrnautics Stanfrd University jamesn@babn.stanfrd.edu Abstract This paper discusses the rle that cmputatinal fluid dynamics plays in the design f aircraft. An verview f the design prcess is prvided, cvering sme f the typical decisins that a design team addresses within a multi-disciplinary envirnment. On a very regular basis trade-ffs between disciplines have t be made where a set f cnflicting requirements exists. Within an aircraft develpment prject, we fcus n the aerdynamic design prblem and review hw this prcess has been advanced, first with the imprving capabilities f traditinal cmputatinal fluid dynamics analyses, and then with aerdynamic ptimizatins based n these increasingly accurate methds. 1 Backgrund The past 25 years have seen a revlutin in the entire engineering design prcess as cmputatinal simulatin has cme t play an increasingly dminant rle. Mst ntably, cmputer aided design (CAD) methds have essentially replaced the drawing bard as the basic tl fr the definitin and cntrl f the cnfiguratin. Cmputer visualizatin techniques enable the designer t verify that n interferences exist between different parts in the layut, and greatly facilitate decisins n the ruting f electrical wiring and hydraulic piping. Similarly, structural analysis is nw almst entirely carried ut by cmputatinal methds, typically finite element methds. Cmmercially available sftware systems have been prgressively develped and augmented with new features, and can treat the full range f requirements fr aernautical structures, including the analysis f stressed skin int the nnlinear range. Als, they are very carefully validated against a cmprehensive suite f test cases befre each new release. Hence, engineers place cmplete cnfidence in their results. Accrdingly, the structural design is rutinely cmmitted n the basis f cmputatinal analysis, while structural testing is limited t the rle f verificatin that the design truly meets its specified requirements f ultimate strength and fatigue life. Cmputatinal simulatin f fluid flw has nt yet reached the same level f maturity. While cmmercial sftware fr the simulatin f fluid flw is ffered by numerus vendrs, aircraft cmpanies cntinue t make substantial investments in the in-huse develpment f their wn methds. At the same time there are majr nging effrts t develp the science f cmputatinal fluid dynamics (CFD) in gvernment research agencies and Universities. This reflects the fact that fluid flw is generally mre cmplex and harder t predict than the behavir f structures. The cmplexity and range f phenmena that characterize fluid flw is well illustrated in Van Dyke s Album f Fluid Mtin [1]. The cncept f a numerical wind tunnel, which might eventually allw cmputers t supplant wind tunnels in the aerdynamic design and testing prcess, was already a tpic f discussin in the In their celebrated paper f 1975, Chapman, Mark and Pirtle [2] listed three main bjective f cmputatinal aerdynamics: 1. T prvide flw simulatins that are either impractical r impssible t btain in wind tunnels r ther grund based 1 yright c 2003 by A. Jamesn

2 experimental test facilities. 2. T lwer the time and cst required t btain aerdynamic flw simulatins necessary fr the design f new aerspace vehicles. 3. Eventually, t prvide mre accurate simulatins f flight aerdynamics than wind tunnels can. There have been majr advances twards these gals. Despite these, CFD is still nt being explited as effectively as ne wuld like in the design prcess. This is partially due t the lng set-up times and high csts, bth human and cmputatinal, assciated with cmplex flw simulatins. This paper examines ways t explit cmputatinal simulatin mre effectively in the verall design prcess, with the primary fcus n aerdynamic design, while recgnizing that this shuld be part f an integrated multi-disciplinary prcess. The emphasis is n the secnd f the riginal gals fr a numerical wind tunnel, but als n taking cmputatinal methds a step further, t use them nt nly t predict the aerdynamic prperties, but als t find superir designs. The key idea is t embed bth analysis and ptimizatin methds in the design prcess, drawing frm bth CFD and cntrl thery. Design trade-ffs and the design prcess itself are surveyed in the next sectins. Sectin 4 discusses the state f the art fr CFD in applied aerdynamics, in particular the accuracy f drag predictin. Sectins 5-8 examine the way in which ptimizatin techniques can be integrated with CFD. The paper cncludes with case studies which apply aerdynamic shape ptimizatin methds within the design prcess. 2 Aerdynamic Design Tradeffs Fcusing n the design f lng range transprt aircraft, a gd first estimate f perfrmance is prvided by the Breguet range equatin: Range = VL D 1 SFC lgw 0 W f. (1) W 0 Here V is the speed, L/D is the lift t drag rati, SFC is the specific fuel cnsumptin f the engines, W 0 is the lading weight(empty weight paylad fuel resurced), and W f is the weight f fuel burnt. Equatin (1) already displays the multi-disciplinary nature f design. A light weight structure is needed t reduce W 0. The specific fuel cnsumptin is mainly the prvince f the engine manufacturers, and in fact the largest advances in the last 30 years have been in engine efficiency. The aerdynamic designer shuld try t maximize VL/D, but must cnsider the impact f shape mdificatins n structure weight. The drag cefficient can be split int an apprximately fixed cmpnent C D0, and the induced drag due t lift as C D = C D0 C2 L πεar where AR is the aspect rati, and ε is an efficiency factr clse t unity. C D0 includes cntributins such as frictin and frm drag. It can be seen frm this equatin that L/D is maximized by flying at a lift cefficient such that the tw terms are equal, s that the induced drag is half the ttal drag. Mrever, the actual drag due t lift D v = 2L2 περv 2 b 2 varies inversely with the square f the span b. Thus there is a direct cnflict between reducing the drag by increasing the span and reducing the structure weight by decreasing it. It als fllws frm equatin (1) that ne shuld try t maximize V L/D. This means that the cruising speed V shuld be increased until it appraches the speed f sund C, at which pint the frmatin f shck waves causes the nset f dragrise. Typically the lift t drag rati will drp frm arund 19 at a Mach number V/C in the neighbrhd f 0.85, t the rder f 4 at Mach 1. Thus the ptimum cruise speed will be in the transnic regime, when shck waves are beginning t frm, but remain weak enugh nly t incur a small drag penalty. The designer can reduce shck drag and delay the nset f drag-rise by increasing the sweep back f the wing r reducing its thickness. Increasing the sweepback increases the structure weight, and may incur prblems with stability and cntrl. Decreasing the thickness bth reduces the fuel vlume (since the wing is used as the main fuel tank), and increases the structure weight, because fr a given stress level in the skin and a given skin thickness, the bending mment that can be supprted is directly prprtinal t the depth f the wing. In the absence f winglets, the ptimum span lad distributin is elliptic, giving an efficiency factr ε = 1. When, hwever, the structure weight is taken int accunt, it is better t shift the lad distributin inbard in rder t reduce the rt bending mment. It may als be necessary t limit the sectin lift cefficient in the utbard part f the wing in rder t delay the nset f buffet, which may ccur when the lift cefficient is increased t make a turn at a high Mach number. 3 Design Prcess The design prcess can generally be divided int three phases: cnceptual design, preliminary design, and final detailed design, as illustrated in Figure 1. (2) 2

3 The cnceptual design stage defines the missin in the light f anticipated market requirements, and determines a general preliminary cnfiguratin capable f perfrming this missin, tgether with first estimates f size, weight and perfrmance. A cnceptual design requires a staff f peple. Over a perid f 1-2 years, the initial business case is develped. The csts f this phase are the range f 6-12 millin dllars, and there is minimal airline invlvement In the preliminary design stage the aerdynamic shape and structural skeletn prgress t the pint where detailed perfrmance estimates can be made and guaranteed t ptential custmers, wh can then, in turn, frmally sign binding cntracts fr the purchase f a certain number f aircraft. At this stage the develpment csts are still fairly mderate. A staff f peple is generally emplyed fr up t 2 years, at a cst f millin dllars. Initial aerdynamic perfrmance is explred thrugh wind tunnel tests. In the final design stage the structure must be defined in cmplete detail, tgether with cmplete systems, including the flight deck, cntrl systems (invlving majr sftware develpment fr fly-by-wire systems), avinics, electrical and hydraulic systems, landing gear, weapn systems fr military aircraft, and cabin layut fr cmmercial aircraft. Majr csts are incurred at this stage, during which it is als necessary t prepare a detailed manufacturing plan, tgether with apprpriate facilities and tling. A staff f thusands f peple define every part f the aircraft. Wind Tunnel validatin f the final design is carried ut. Significant develpment csts are incurred ver a 3 year perid, plus an additinal year f Flight Testing and Structural Qualificatin Testing fr Certificatin. Ttal csts are in the range f 3-12 billin dllars. Thus, the final design wuld nrmally be carried ut nly if sufficient rders have been received t indicate a reasnably high prbability f recvering a significant fractin f the investment. Fr a cmmercial aircraft there are extensive discussins with airlines. In the develpment f cmmercial aircraft, aerdynamic design plays a leading rle during the preliminary design stage, during which the definitin f the external aerdynamic shape is typically finalized. The aerdynamic lines f the Being 777 were frzen, fr example, when initial rders were accepted befre the initiatin f the detailed design f the structure. Figure 2 illustrates the way in which the aerdynamic design prcess is embedded in the verall preliminary design. The starting pint is an initial CAD definitin resulting frm the cnceptual design. The inner lp f aerdynamic analysis is cntained in an uter multi-disciplinary lp, which is in turn cntained in a majr design cycle invlving wind tunnel testing. In recent Being practice, three majr design cycles, each requiring abut 4-6 mnths, have been used t finalize the wing design. Imprvements in CFD which wuld allw the eliminatin f a majr cycle wuld significantly shrten the verall design prcess and therefre reduce csts. Majr Design Cycle Outer Lp Inner Lp Cnceptual Design Preliminary Design Final Design Figure 1. Cnceptual Design CAD Definitin Mesh Generatin CFD Analysis Visualizatin Perfrmance Evaluatin Multi Disciplinary Evaluatin Wind Tunnel Testing Figure 2. Defines Missin Preliminary sizing Weight, perfrmance The Overall Design Prcess Central Database { Prpulsin Nise Stability Cntrl Lads Structures Fabricatin Mdel Fabricatin Detailed Final Design Release t Manufacturing The Aerdynamic Design Prcess The inner aerdynamic design lp is used t evaluate numerus variatins in the wing definitin. In each iteratin it is necessary t generate a mesh fr the new cnfiguratin prir t perfrming the CFD analysis. Cmputer graphics sftware is then used t visualize the results, and the perfrmance is evaluated. The first studies may be cnfined t partial cnfiguratins 3

4 such as wing-bdy r wing-bdy-nacelle cmbinatins. At this stage the fcus is n the design f the clean wing. Key pints f the flight envelpe include the nminal cruise pint, cruise at high lift and lw lift t allw fr the weight variatin between the initial and final cruise as the fuel is burned ff, and a lng range cruise pint at lwer Mach number, where it is imprtant t ensure there is n significant drag creep. Other defining pints are the climb cnditin, which requires a gd lift t drag rati at lw Mach number and high lift cefficient fr the clean wing, and buffet cnditins. The buffet requirement is typically taken as the high lift cruise pint increased t a lad f 1.3 g t allw fr maneuvering and gust lads. The aerdynamic analysis interacts with the ther disciplines in the next uter lp. These disciplines have their wn inner lps, nt shwn in Figure 2. Fr an efficient design prcess the fully updated aer-design database must be accessible t ther disciplines withut lss f infrmatin. Fr example, the thrust requirements fr the pwer plant design will depend n the drag estimates fr take-ff, climb and cruise. In rder t meet airprt nise cnstraints a rapid climb may be required while the thrust may als be limited. Initial estimates f the lift and mments allw preliminary sizing f the hrizntal and vertical tail. This interacts with the design f the cntrl system, where the use f a fly-by-wire system may allw relaxed static stability, hence tail surfaces f reduced size. First estimates f the aerdynamic lads allw the design f an initial structural skeletn, which in turn prvides a weight estimate f the structure. One f the main trade-ffs is between aerdynamic perfrmance and wing structure weight. The requirement fr fuel vlume may als be an imprtant cnsideratin. Manufacturing cnstraints must als be cnsidered in the final definitin f the aerdynamic shape. Fr example, the curvature in the spanwise directin shuld be limited. This avids the need fr sht peening which might therwise be required t prduce curvature in bth the spanwise and chrdwise directins. Frm the freging cnsideratins, it is apparent that in rder t carry ut the inner lp f the aerdynamic design prcess the main requirements fr effective CFD sftware are: 1. Sufficient and knwn level f accuracy 2. Acceptable cmputatinal and manpwer csts 3. Fast turn arund time 4 Perfrmance Predictin The discussin in sectin 2 illustrates hw perfrmance estimatin at the cruise cnditin is crucial t the design f lngrange transprt aircraft. The errr shuld be n the rder f 1 percent r less. The drag cefficient f a typical transprt aircraft, such as the Being 747, is abut 275 (275 cunts) fr a typical lift cefficient f apprximately. The drag cefficient f prpsed supersnic transprt designs is in the range f 120 t 150 at much lwer lift cefficients in the range f 0.1 t Thus ne shuld aim t predict drag with an accuracy f 1 t 2 cunts. Manufacturers have t guarantee perfrmance, and errrs can be very expensive due t the csts f redesign, penalty payments and lst rders. In rder t achieve this level f accuracy, it is ultimately essential t use the Reynlds-Averaged Navier-Stkes (RANS) equatins. Hwever, t accelerate their initial effrts, designers typically use CFD methds based n less sphisticated flw mdels, such as the full-ptential r Euler equatins cupled with a bundary-layer methd. In rder t allw the cmpletin f the majr design cycle in 4-6 mnths, the cycle time fr the multidisciplinary lp shuld nt be greater than abut 2 weeks. Cnsidering the need t examine the perfrmance f design variatins at all the key pints f the flight envelpe, this implies the need t turn arund aerdynamic analysis in a few hurs. The cmputatinal csts are als imprtant because the cumulative csts f large numbers f calculatins can becme a limiting factr. The state-f-the-art in CFD drag predictin was recently assessed by an internatinal wrkshp n the subject. A summary f the results f the wrkshp is presented in Reference [3]. Figure 4 prvides the 28 drag plars resulting frm this drag predictin wrkshp (DPW). Als included in this figure are the available test data [4] fr the DLR-F4 wing/bdy cnfiguratin, which was the subject f the study. With the exceptin f a few ut-layers, the cmputed plars fall within a band f abut ±7% the abslute level. The slpes, dc2 L dc D, are nearly identical. When cmparing the CFD results with the test data, we nte that the CFD slutins were all run assuming fully-turbulent flw, while the test data were cllected with laminar runs n the wing up t transitin strips n bth upper and lwer surfaces. T quantify the shift in drag assciated with this difference, several independent calculatins were perfrmed yielding cunts higher drag levels fr the fully-turbulent flws. Accunting fr this adjustment, the center f the CFD drag plars band cincides with the mean f the test plars. While this indicates that the industry as a whle is clsing in n the ability t cmpute accurate abslute drag levels, in general, the errrs are nt t the level desired by aircraft design teams. Hwever, a few f the results submitted t the DPW fall within the uncertainty band f the experimental data. Fr example, the crrespnding results f Reference [5] are shwn in Figure 5. Achieving this level f accuracy is dminated by the quality f the underlying grid, but als depends n the turbulence mdel, the level f cnvergence, discretizatin scheme, etc. It is imperative that each f these areas be studied independently f each ther, therwise accurate results might be btained as a cnsequence f cancellatin f errrs. Unfrtunately, an ptimizatin based n an analysis methd cntaining such a cancellatin f errrs will mst likely emphasize its weakness and prbably yield a new design with a false perfrmance imprvement. 4

5 5 Aerdynamic Shape Optimizatin Traditinally the prcess f selecting design variatins has been carried ut by trial and errr, relying n the intuitin and experience f the designer. It is als evident that the number f pssible design variatins is t large t permit their exhaustive evaluatin, and thus it is very unlikely that a truly ptimum slutin can be fund withut the assistance f autmatic ptimizatin prcedures. In rder t take full advantage f the pssibility f examining a large design space, the numerical simulatins need t be cmbined with autmatic search and ptimizatin prcedures. This can lead t autmatic design methds which will fully realize the ptential imprvements in aerdynamic efficiency. Ultimately there is a need fr multi-disciplinary ptimizatin (MDO), but this can nly be effective if it is based n sufficiently high fidelity mdeling f the separate disciplines. As a step in this directin there culd be significant pay-ffs frm the applicatin f ptimizatin techniques within the disciplines, where the interactins with ther disciplines are taken int accunt thrugh the intrductin f cnstraints. Fr example the wing drag can be minimized at a given Mach number and lift cefficient with a fixed planfrm, and cnstraints n minimum thickness t meet requirements fr fuel vlume and structure weight. An apprach which has becme increasingly ppular is t carry ut a search ver a large number f variatins via a genetic algrithm. This may allw the discvery f (smetimes unexpected) ptimum design chices in very cmplex multi-bjective prblems, but it becmes extremely expensive when each evaluatin f the cst functin requires intensive cmputatin, as is the case in aerdynamic prblems. In rder t find ptimum aerdynamic shapes with reasnable cmputatinal csts, it pays t regard the wing as a device which cntrls the flw in rder t prduce lift with minimum drag. As a result, ne can draw n cncepts which have been develped in the mathematical thery f cntrl f systems gverned by partial differential equatins. In particular, an acceptable aerdynamic design must have characteristics that smthly vary with small changes in shape and flw cnditins. Cnsequently, gradient-based prcedures are apprpriate fr aerdynamic shape ptimizatin. Tw main issues affect the efficiency f gradient-based prcedures; the first is the actual calculatin f the gradient, and the secnd is the cnstructin f an efficient search prcedure which utilizes the gradient. 5.1 Gradient Calculatin Fr the class f aerdynamic ptimizatin prblems under cnsideratin, the design space is essentially infinitely dimensinal. Suppse that the perfrmance f a system design can be measured by a cst functin I which depends n a functin F (x) that describes the shape,where under a variatin f the design δf (x), the variatin f the cst is δi. Nw suppse that δi can be expressed t first rder as δi = G(x)δF (x)dx where G(x) is the gradient. Then by setting ne btains an imprvement δf (x) = λg(x) δi = λ G 2 (x)dx unless G(x) = 0. Thus the vanishing f the gradient is a necessary cnditin fr a lcal minimum. Cmputing the gradient f a cst functin fr a cmplex system can be a numerically intensive task, especially if the number f design parameters is large and the cst functin is an expensive evaluatin. The simplest apprach t ptimizatin is t define the gemetry thrugh a set f design parameters, which may, fr example, be the weights α i applied t a set f shape functins B i (x) s that the shape is represented as F (x) = α i B i (x). Then a cst functin I is selected which might be the drag cefficient r the lift t drag rati; I is regarded as a functin f the parameters α i. The sensitivities I α i may nw be estimated by making a small variatin δα i in each design parameter in turn and recalculating the flw t btain the change in I. Then I I(α i δα i ) I(α i ). α i δα i The main disadvantage f this finite-difference apprach is that the number f flw calculatins needed t estimate the gradient is prprtinal t the number f design variables [6]. Similarly, if ne resrts t direct cde differentiatin (ADIFOR [7,8]), r cmplex-variable perturbatins [9], the cst f determining the gradient is als directly prprtinal t the number f variables used t define the design. A mre cst effective technique is t cmpute the gradient thrugh the slutin f an adjint prblem, such as that develped in references [10 12]. The essential idea may be summarized as fllws. Fr flw abut an arbitrary bdy, the aerdynamic prperties that define the cst functin are functins f 5

6 the flwfield variables (w) and the physical shape f the bdy, which may be represented by the functin F. Then I = I(w,F ) where [ ] G = IT R F ψt. F and a change in F results in a change f the cst functin δi = IT IT δw w F δf. Using a technique drawn frm cntrl thery, the gverning equatins f the flwfield are intrduced as a cnstraint in such a way that the final expressin fr the gradient des nt require reevaluatin f the flwfield. In rder t achieve this, δw must be eliminated frm the abve equatin. Suppse that the gverning equatin R, which expresses the dependence f w and F within the flwfield dmain D, can be written as R(w,F ) = 0. (3) The advantage is that the variatin in cst functin is independent f δw, with the result that the gradient f I with respect t any number f design variables can be determined withut the need fr additinal flw-field evaluatins. In the case that (3) is a partial differential equatin, the adjint equatin (5) is als a partial differential equatin and apprpriate bundary cnditins must be determined. It turns ut that the apprpriate bundary cnditins depend n the chice f the cst functin, and may easily be derived fr cst functins that invlve surface-pressure integratins. Cst functins invlving field integrals lead t the appearance f a surce term in the adjint equatin. The cst f slving the adjint equatin is cmparable t that f slving the flw equatin. Hence, the cst f btaining the gradient is cmparable t the cst f tw functin evaluatins, regardless f the dimensin f the design space. Then δw is determined frm the equatin δr = [ ] [ ] R R δw δf = 0. w F Next, intrducing a Lagrange multiplier ψ, we have δi = IT IT δw δf ψt w F ([ ] [ ] ) R R δw δf. (4) w F 6 Design using the Euler Equatins The applicatin f cntrl thery t aerdynamic design prblems is illustrated in this sectin fr the case f threedimensinal wing design using the cmpressible Euler equatins as the mathematical mdel. The extensin f the methd t treat the Navier-Stkes equatins is presented in references [13 15]. It prves cnvenient t dente the Cartesian crdinates and velcity cmpnents by x 1, x 2, x 3 and u 1, u 2, u 3, and t use the cnventin that summatin ver i = 1 t 3 is implied by a repeated index i. Then, the three-dimensinal Euler equatins may be written as With sme rearrangement ( I T δi = w ψt [ ]) ( R I T [ ]) R δw w F ψt δf. F Chsing ψ t satisfy the adjint equatin [ ] R T ψ = IT w w (5) where w t f i x i = 0 in D, (6) ρ ρu 1 w = ρu 2, f i = ρu 3 ρe ρu i ρu i u 1 pδ i1 ρu i u 2 pδ i2 ρu i u 3 pδ i3 ρu i H (7) the term multiplying δw can be eliminated in the variatin f the cst functin, and we find that δi = GδF, and δ i j is the Krnecker delta functin. Als, { p = (γ 1)ρ E 1 ( ) } u 2 2 i, (8) 6

7 and where ρh = ρe p (9) where γ is the rati f the specific heats. In rder t simplify the derivatin f the adjint equatins, we map the slutin t a fixed cmputatinal dmain with crdinates ξ 1, ξ 2, ξ 3 where and [ ] xi K i j =, J = det(k), Ki 1 j = ξ j S = JK 1. [ ξi The elements f S are the cfactrs f K, and in a finite vlume discretizatin they are just the face areas f the cmputatinal cells prjected in the x 1, x 2, and x 3 directins. Using the permutatin tensr ε i jk we can express the elements f S as Then x j ], S i j = 1 2 ε jpqε irs x p ξ r x q ξ s. (10) S i j = 1 ( 2 2 ε x p x q jpqε irs x p 2 ) x q ξ r ξ s ξ r ξ s = 0. (11) Als in the subsequent analysis f the effect f a shape variatin it is useful t nte that S 1 j = ε jpq x p ξ 2 x q ξ 3, S 2 j = ε jpq x p ξ 3 x q ξ 1, S 3 j = ε jpq x p ξ 1 x q ξ 2. (12) Nw, multiplying equatin(6) by J and applying the chain rule, J w t R(w) = 0 (13) R(w) = S i j f j = (S i j f j ), (14) using (11). We can write the transfrmed fluxes in terms f the scaled cntravariant velcity cmpnents as U i = S i j u j ρu i ρu i u 1 S i1 p F i = S i j f j = ρu i u 2 S i2 p ρu i u 3 S i3 p. ρu i H Fr cnvenience, the crdinates ξ i describing the fixed cmputatinal dmain are chsen s that each bundary cnfrms t a cnstant value f ne f these crdinates. Variatins in the shape then result in crrespnding variatins in the mapping derivatives defined by K i j. Suppse that the perfrmance is measured by a cst functin I = M (w,s)db ξ B P (w,s)dd ξ, D cntaining bth bundary and field cntributins where db ξ and dd ξ are the surface and vlume elements in the cmputatinal dmain. In general, M and P will depend n bth the flw variables w and the metrics S defining the cmputatinal space. The design prblem is nw treated as a cntrl prblem where the bundary shape represents the cntrl functin, which is chsen t minimize I subject t the cnstraints defined by the flw equatins (13). A shape change prduces a variatin in the flw slutin δw and the metrics δs which in turn prduce a variatin in the cst functin This can be split as with δi = δm (w,s)db ξ δp (w,s)dd ξ. (15) B D δi = δi I δi II, (16) δm = [M w ] I δw δm II, δp = [P w ] I δw δp II, (17) 7

8 where we cntinue t use the subscripts I and II t distinguish between the cntributins assciated with the variatin f the flw slutin δw and thse assciated with the metric variatins δs. Thus [M w ] I and [P w ] I represent M w and P w with the metrics fixed, while δm II and δp II represent the cntributin f the metric variatins δs t δm and δp. In the steady state, the cnstraint equatin (13) specifies the variatin f the state vectr δw by δr = δf i = 0. (18) Here als, δr and δf i can be split int cntributins assciated with δw and δs using the ntatin where δr = δr I δr II δf i = [F iw ] I δw δf iii. (19) [F iw ] I = S i j f i w. Multiplying by a c-state vectr ψ, which will play an analgus rle t the Lagrange multiplier intrduced in equatin (4), and integrating ver the dmain prduces D ψt δf i dd ξ = 0. (20) Assuming that ψ is differentiable, the terms with subscript I may be integrated by parts t give B n iψ T δf ii db ξ D ψ T δf ii dd ξ ξ i D ψt δr II dd ξ = 0. (21) This equatin results directly frm taking the variatin f the weak frm f the flw equatins, where ψ is taken t be an arbitrary differentiable test functin. Since the left hand expressin equals zer, it may be subtracted frm the variatin in the cst functin (15) t give δi = δi II D D ψt δr II dd ξ [δp I ψt δf ii [ δmi n i ψ T ] δf ii dbξ B ] dd ξ. (22) Nw, since ψ is an arbitrary differentiable functin, it may be chsen in such a way that δi n lnger depends explicitly n the variatin f the state vectr δw. The gradient f the cst functin can then be evaluated directly frm the metric variatins withut having t recmpute the variatin δw resulting frm the perturbatin f each design variable. Cmparing equatins (17) and (19), the variatin δw may be eliminated frm (22) by equating all field terms with subscript I t prduce a differential adjint system gverning ψ ψ T [F iw ] I [P w ] I = 0 in D. (23) Taking the transpse f equatin (23), in the case that there is n field integral in the cst functin, the inviscid adjint equatin may be written as C T i ψ = 0 in D, (24) where the inviscid Jacbian matrices in the transfrmed space are given by C i = S i j f j w. The crrespnding adjint bundary cnditin is prduced by equating the subscript I bundary terms in equatin (22) t prduce n i ψ T [F iw ] I = [M w ] I n B. (25) The remaining terms frm equatin (22) then yield a simplified expressin fr the variatin f the cst functin which defines the gradient δi = δi II D ψt δr II dd ξ, (26) which cnsists purely f the terms cntaining variatins in the metrics, with the flw slutin fixed. Hence an explicit frmula fr the gradient can be derived nce the relatinship between mesh perturbatins and shape variatins is defined. The details f the frmula fr the gradient depend n the way in which the bundary shape is parameterized as a functin f the design variables, and the way in which the mesh is defrmed as the bundary is mdified. Using the relatinship between the mesh defrmatin and the surface mdificatin, the field integral is reduced t a surface integral by integrating alng the crdinate lines emanating frm the surface. Thus the expressin fr 8

9 δi is finally reduced t the frm δi = GδF db ξ B where F represents the design variables, and G is the gradient, which is a functin defined ver the bundary surface. The bundary cnditins satisfied by the flw equatins restrict the frm f the left hand side f the adjint bundary cnditin (25). Cnsequently, the bundary cntributin t the cst functin M cannt be specified arbitrarily. Instead, it must be chsen frm the class f functins which allw cancellatin f all terms cntaining δw in the bundary integral f equatin (22). On the ther hand, there is n such restrictin n the specificatin f the field cntributin t the cst functin P, since these terms may always be absrbed int the adjint field equatin (23) as surce terms. Fr simplicity, it will be assumed that the prtin f the bundary that underges shape mdificatins is restricted t the crdinate surface ξ 2 = 0. Then equatins (22) and (25) may be simplified by incrprating the cnditins n 1 = n 3 = 0, n 2 = 1, db ξ = dξ 1 dξ 3, s that nly the variatin δf 2 needs t be cnsidered at the wall bundary. The cnditin that there is n flw thrugh the wall bundary at ξ 2 = 0 is equivalent t s that U 2 = 0, δu 2 = 0 when the bundary shape is mdified. Cnsequently the variatin f the inviscid flux at the bundary reduces t δf 2 = δp 0 S 21 S 22 S 23 0 p 0 δs 21 δs 22 δs (27) Since δf 2 depends nly n the pressure, it is nw clear that the perfrmance measure n the bundary M (w,s) may nly be a functin f the pressure and metric terms. Otherwise, cmplete cancellatin f the terms cntaining δw in the bundary integral wuld be impssible. One may, fr example, include arbitrary measures f the frces and mments in the cst functin, since these are functins f the surface pressure. In rder t design a shape which will lead t a desired pressure distributin, a natural chice is t set I = 1 2 (p p d) 2 ds B where p d is the desired surface pressure, and the integral is evaluated ver the actual surface area. In the cmputatinal dmain this is transfrmed t I = 1 2 where the quantity B w (p p d ) 2 S 2 dξ 1 dξ 3, S 2 = S 2 j S 2 j dentes the face area crrespnding t a unit element f face area in the cmputatinal dmain. Nw, t cancel the dependence f the bundary integral n δp, the adjint bundary cnditin reduces t ψ j n j = p p d (28) where n j are the cmpnents f the surface nrmal n j = S 2 j S 2. This amunts t a transpiratin bundary cnditin n the cstate variables crrespnding t the mmentum cmpnents. Nte that it impses n restrictin n the tangential cmpnent f ψ at the bundary. We find finally that δi = ψ T δs i j f j dd D (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 ) pdξ 1 dξ 3. (29) B W Here the expressin fr the cst variatin depends n the mesh variatins thrughut the dmain which appear in the field integral. Hwever, the true gradient fr a shape variatin shuld nt 9

10 depend n the way in which the mesh is defrmed, but nly n the true flw slutin. In the next sectin we shw hw the field integral can be eliminated t prduce a reduced gradient frmula which depends nly n the bundary mvement. 7 The Reduced Gradient Frmulatin Cnsider the case f a mesh variatin with a fixed bundary. Then, δi = 0 but there is a variatin in the transfrmed flux, δf i = C i δw δs i j f j. Here the true slutin is unchanged. Thus, the variatin δw is due t the mesh mvement δx at each mesh pint. Therefre and since δw = w δx = w x j δx j (= δw ) δf i = 0, Thus, by chsing φ t satisfy the adjint equatin (24) and the adjint bundary cnditin (25), we reduce the cst variatin t a bundary integral which depends nly n the surface displacement: δi = ψ T (δs 2 j f j C 2 δw )dξ 1 dξ 3 B W (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 ) pdξ 1 dξ 3. (33) B W Fr cmpleteness the general derivatin f equatin(30) is presented here. Using the frmula(10), and the prperty (11) (δs i j f j ) = 1 2 { ( δxp ε jpq ε irs ξ r x q x q x p δx q ξ s ξ r ξ s ) f j = 1 ( 2 ε δxp jpqε irs x p δx q ξ r ξ s ξ r ξ s = 1 { ( )} 2 ε x q f j jpqε irs δx p ξ r ξ s 1 { ( )} 2 ε x p f j jpqε irs δx q ξ s ξ r = ( x q f j δx p ε pq j ε rsi ξ r ξ s ) } f j ). (34) Nw express δx p in terms f a shift in the riginal cmputatinal crdinates it fllws that (δs i j f j ) = (C i δw ). (30) Then we btain δx p = x p ξ k δξ k. It is verified belw that this relatin hlds in the general case with bundary mvement. Nw D φt δrdd = D φt C i (δw δw )dd = B φt C i (δw δw )db φ T C i (δw δw )dd. (31) D Here n the wall bundary C 2 δw = δf 2 δs 2 j f j. (32) (δs i j f j ) = The term in ξ 1 is ξ r ( ) x p x q f j ε pq j ε rsi δξ k. (35) ξ k ξ s ( x p xq f j ε 123 ε pq j x ) q f j δξ k. ξ k ξ 2 ξ 3 ξ 3 ξ 2 Here the term multiplying δξ 1 is ( xp x q f j ε jpq x ) p x q f j. ξ 1 ξ 2 ξ 3 ξ 1 ξ 3 ξ 2 10

11 Accrding t the frmulas(12) this may be recgnized as Suppse that we wish t find the path y(x) which minimizes S 2 j f 1 ξ 2 S 3 j f 1 ξ 3 r, using the quasi-linear frm(14) f the equatin fr steady flw, as S 1 j f 1 ξ 1. The terms multiplying δξ 2 and δξ 3 are ( xp x q f j ε jpq x ) p x q f j f 1 = S 1 j ξ 2 ξ 2 ξ 3 ξ 2 ξ 3 ξ 2 ξ 2 b I = F(y,y )dx a with fixed end pints y(a) and y(b). Under a variatin δy(x), δi = = b a b Thus defining the gradient as a ( ) F F δy δy dx y y ( F y d F dx y ) δydx and ( xp x q f j ε jpq x ) p x q f j f 1 = S 1 j. ξ 3 ξ 2 ξ 3 ξ 3 ξ 3 ξ 2 ξ 3 g = F y d F dx y and the inner prduct as Thus the term in ξ 1 is reduced t ( ) f 1 S 1 j δξ k. ξ 1 ξ k we find that b (u,v) = uvdx a Finally, with similar reductins f the terms in btain and, we ξ 2 ξ 3 (δs i j f j ) = ( ) f j S i j δξ k = (C i δw ) ξ k If we nw set δi = (g,δy). δy = λg, λ > 0 as was t be prved. we btain a imprvement 8 Optimizatin Prcedure 8.1 The Need fr a Sblev Inner Prduct in the Definitin f the Gradient Anther key issue fr successful implementatin f the cntinuus adjint methd is the chice f an apprpriate inner prduct fr the definitin f the gradient. It turns ut that there is an enrmus benefit frm the use f a mdified Sblev gradient, which enables the generatin f a sequence f smth shapes. This can be illustrated by cnsidering the simplest case f a prblem in the calculus f variatins. δi = λ(g,g) 0 unless g = 0, the necessary cnditin fr a minimum. Nte that g is a functin f y,y,y, g = g(y,y,y ) In the well knwn case f the Brachistrne prblem, fr example, which calls fr the determinatin f the path f quickest descent 11

12 between tw laterally separated pints when a particle falls under gravity, and F(y,y ) = It can be seen that each step 1 y 2 g = 1 y 2 2yy 2 ( y(1 y 2 ) ) 3/2 y y n1 = y n λ n g n reduces the smthness f y by tw classes. Thus the cmputed trajectry becmes less and less smth, leading t instability. In rder t prevent this we can intrduce a weighted Sblev inner prduct [16] u,v = (uv εu v )dx gives an imprvement δi = λ n g n,g n but y n1 has the same smthness as y n, resulting in a stable prcess. 8.2 Sblev Gradient fr Shape Optimizatin In applying cntrl thery t aerdynamic shape ptimizatin, the use f a Sblev gradient is equally imprtant fr the preservatin f the smthness class f the redesigned surface. Accrdingly, using the weighted Sblev inner prduct defined abve, we define a mdified gradient Ḡ such that δi =< Ḡ,δF >. In the ne dimensinal case Ḡ is btained by slving the smthing equatin Ḡ ξ 1 ε ξ 1 Ḡ = G. (36) In the multi-dimensinal case the smthing is applied in prduct frm. Finally we set where ε is a parameter that cntrls the weight f the derivatives. We nw define a gradient g such that δi = g,δy with the result that δf = λḡ (37) δi = λ < Ḡ,Ḡ > < 0, Then we have where δi = (gδy εg δy )dx = (g x ε g x )δydx = (g,δy) unless Ḡ = 0, and crrespndingly G = 0. When secnd-rder central differencing is applied t (36), the equatin at a given nde, i, can be expressed as Ḡ i ε ( Ḡ i1 2Ḡ i Ḡ i 1 ) = Gi, 1 i n, where G i and Ḡ i are the pint gradients at nde i befre and after the smthing respectively, and n is the number f design variables equal t the number f mesh pints in this case. Then, Ḡ = AG, g x ε g x = g and g = 0 at the end pints. Thus g can be btained frm g by a smthing equatin. Nw the step y n1 = y n λ n g n where A is the n n tri-diagnal matrix such that 1 2ε ε 0. 0 A 1 ε.. = ε. 0 ε 1 2ε 12

13 Using the steepest descent methd in each design iteratin, a step, δf, is taken such that δf = λag. (38) As can be seen frm the frm f this expressin, implicit smthing may be regarded as a precnditiner which allws the use f much larger steps fr the search prcedure and leads t a large reductin in the number f design iteratins needed fr cnvergence. 8.3 Outline f the Design Prcedure The design prcedure can finally be summarized as fllws: 1. Slve the flw equatins fr ρ, u 1, u 2, u 3, p. 2. Slve the adjint equatins fr ψ subject t apprpriate bundary cnditins. 3. Evaluate G and calculate the crrespnding Sblev gradient Ḡ. 4. Prject Ḡ int an allwable subspace that satisfies any gemetric cnstraints. 5. Update the shape based n the directin f steepest descent. 6. Return t 1 until cnvergence is reached. Flw Slutin Adjint Slutin Gradient Calculatin Repeat the Design Cycle until Cnvergence lift. Therefre the angle f attack is adjusted during each flw slutin t frce a specified lift cefficient t be attained, and the influence f variatins f the angle f attack is included in the calculatin f the gradient. The vrtex drag als depends n the span lading, which may be cnstrained by ther cnsideratins such as structural lading r buffet nset. Cnsequently, the ptin is prvided t frce the span lading by adjusting the twist distributin as well as the angle f attack during the flw slutin. 8.4 Cmputatinal Csts In rder t address the issues f the search csts, the authrs investigated a variety f techniques in Reference [20] using a trajectry ptimizatin prblem (the brachistchrne) as a representative mdel. The study verified that the search cst (i.e., number f steps) f a simple steepest descent methd applied t this prblem scales as N 2, where N is the number f design variables, while the cst f quasi-newtn methds scaled linearly with N as expected. On the ther hand, with an apprpriate amunt f smthing, the smthed descent methd cnverged in a fixed number f steps, independent f N. Cnsidering that the evaluatin f the gradient by a finite difference methd requires N 1 flw calculatins, while the cst f its evaluatin by the adjint methd is rughly that f tw flw calculatins, ne arrives at the estimates f ttal cmputatinal cst given in Tables 1-2. Table 1. Steepest Descent Quasi-Newtn Smthed Gradient Cst f Search Algrithm. (Nte: K is independent f N) O(N 2 ) steps O(N) steps O(K) steps Sblev Gradient Shape & Grid Mdificatin Figure 3. Design cycle Practical implementatin f the design methd relies heavily upn fast and accurate slvers fr bth the state (w) and c-state (ψ) systems. The result btained in Sectin 9 have been btained using well-validated sftware fr the slutin f the Euler and Navier-Stkes equatins develped ver the curse f many years [17 19]. Fr inverse design the lift is fixed by the target pressure. In drag minimizatin it is als apprpriate t fix the lift cefficient, because the induced drag is a majr fractin f the ttal drag, and this culd be reduced simply by reducing the Table 2. Ttal Cmputatinal Cst f Design. Finite Difference Gradients Steepest Descent O(N 3 ) Finite Difference Gradients Quasi-Newtn Search O(N 2 ) Adjint Gradients Quasi-Newtn Search O(N) Adjint Gradients Smthed Gradient Search O(K) (Nte: K is independent f N) 13

14 9 Case Studies Several design effrts which have utilized these methds include: Raythen s and Gulfstream business jets, NASA s High- Speed Civil Transprt, reginal jet designs, as well as several Being prjects such as the MDXX and the Blended-Wing- Bdy [21, 22]. Sme representatin examples f design calculatins are presented in this sectin t illustrate the present capability. 9.1 Redesign f the Being 747 wing Over the last decade the adjint methd has been successfully used t refine a variety f designs fr flight at bth transnic and supersnic cruising speeds. In the case f transnic flight, it is ften pssible t prduce a shck free flw which eliminates the shck drag by making very small changes, typically n larger than the bundary layer displacement thickness. Cnsequently viscus effects need t be cnsidered in rder t realize the full benefits f the ptimizatin. Here the ptimizatin f the wing f the Being is presented t illustrate the kind f benefits that can be btained. In these calculatins the flw was mdeled by the Reynlds Averaged Navier-Stkes equatins. A Baldwin Lmax turbulence mdel was cnsidered sufficient, since the ptimizatin is fr the cruise cnditin with attached flw. The calculatin were perfrmed t minimize the drag cefficient at a fixed lift cefficient, subject t the additinal cnstraints that the span lading shuld nt be altered and the thickness shuld nt be reduced. It might be pssible t reduce the induced drag by mdifying the span lading t an elliptic distributin, but this wuld increase the rt bending mment, and cnsequently require an increase in the skin thickness and structure weight. A reductin in wing thickness wuld nt nly reduce the fuel vlume, but it wuld als require an increase in skin thickness t supprt the bending mment. Thus these cnstraints assure that there will be n penalty in either structure weight r fuel vlume. Figure 6 displays the result f an ptimizatin at a Mach number f 0.86, which is rughly the maximum cruising Mach number attainable by the existing design befre the nset f significant drag rise. The lift cefficient f 0.42 is the cntributin f the expsed wing. Allwing fr the fuselage t ttal lift cefficient is abut It can be seen that the redesigned wing is essentially shck free, and the drag cefficient is reduced frm 1269 (127 cunts) t 1136 (114 cunts). The ttal drag cefficient f the aircraft at this lift cefficient is arund 270 cunts, s this wuld represent a drag reductin f the rder f 5 percent. Figure 7 displays the result f an ptimizatin at Mach In this case the shck waves are nt eliminated, but their strength is significantly weakened, while the drag cefficient is reduced frm 1819 (182 cunts) t 1293 (129 cunts). Thus the redesigned wing has essentially the same drag at Mach 0.9 as the riginal wing at Mach The Being 747 wing culd apparently be mdified t allw such an increase in the cruising Mach number because it has a higher sweep-back than later designs, and a rather thin wing sectin with a thickness t chrd rati f 8 percent. Figures 8 and 9 verify that the span lading and thickness were nt changed by the redesign, while figures 10 and 11 indicate the required sectin changes at 42 percent and 68 percent span statins. 9.2 Planfrm and Aer-structural Optimizatin The shape changes in the sectin needed t imprve the transnic wing design are quite small. Hwever, in rder t btain a true ptimum design larger scale changes such as changes in the wing planfrm (sweepback, span, chrd, and taper) shuld be cnsidered. Because these directly affect the structure weight, a meaningful result can nly be btained by cnsidering a cst functin that takes accunt f bth the aerdynamic characteristics and the weight. In references [23 25] the cst functin is defined as 1 I = α 1 C D α 2 2 (p p d) 2 ds α 3 C W B where C W is a dimensinless measure f the wing weight, which can be estimated either frm statistical frmulas, r frm a simple analysis f a representative structure, allwing fr failure mdes such as panel buckling. The cefficient α 2 is intrduced t prvide the designer sme cntrl ver the pressure distributin, while the relative imprtance f drag and weight are represented by the cefficients α 1 and α 3. By varying these it is pssible t calculate the Paret frnt f designs which have the least weight fr a given drag cefficient, r the least drag cefficient fr a given weight. The relative imprtance f these can be estimated frm the Breguet range equatin (1). Figure 28 shws the Paret frnt btained frm a study f the Being 747 wing [25], in which bth the wing planfrm and sectin were varied simultaneusly, with the planfrm defined by five parameters; sweepback, span, and the chrd at three span statins. In this case the planfrm variatins were limited t changes in sweepback, and the sectin changes were cnstrained t maintain the same thickness t chrd rati. When α 3 α 1 is relatively large, the ptimum wing has a reduced sweepback, leading t a weight reductin, as illustrated in figure 29. The accmpanying sectin shape changes keep the shck drag lw. Figure 28 als shws the pint n the frnt that is estimated t prduce the maximum range accrding t the Breguet equatin. 9.3 Wing inverse design using an unstructured mesh A majr bstacle t the treatment f arbitrarily cmplex cnfiguratins is the difficulty and cst f mesh generatin. This can 14

15 be mitigated by the use f unstructured meshes. Thus it appears that the extensin f the adjint methd t unstructured meshes may prvide the mst prmising rute t the ptimum shape design f key elements f cmplex cnfiguratins, such as wingpyln-nacelle cmbinatins. Sme results are presented belw. These have been btained with new sftware t implement the adjint methd fr unstructured meshes which is currently under develpment [26]. Figures 12 and 13 shws the result f an inverse design calculatin, where the initial gemetry was a wing made up f NACA 0012 sectins and the target pressure distributin was the pressure distributin ver the Onera M6 wing. Figures 14, 15, 16, 17, 18, 19, shw the target and cmputed pressure distributin at six span-wise sectins. It can be seen frm these plts the target pressure distributin is well recvered in 50 design cycles, verifying that the design prcess is capable f recvering pressure distributins that are significantly different frm the initial distributin. This is a particularly challenging test, because it calls fr the recvery f a smth symmetric prfile frm an asymmetric pressure distributin cntaining a triangular pattern f shck waves. which are t subtle t allw their determinatin by trial and errr methds. The ptential ecnmic benefits are substantial, cnsidering the fuel csts f the entire airline fleet. Mrever, if ne were t take full advantage f the increase in the lift t drag rati during the design prcess, a smaller aircraft culd be designed t perfrm the same task, with cnsequent further cst reductins. It seems inevitable that sme methd f this type will prvide a basis fr aerdynamic designs f the future. Acknwledgment This wrk has benefited greatly frm the supprt f the Air Frce Office f Science Research under grant N. AF F I have drawn extensively frm the lecture ntes prepared by Luigi Martinelli and myself fr a CIME summer curse in 1999 [14] and frm a paper prepared fr the 23 rd Internatinal Cngress f Aernautical Sciences, Trnt, September 2002 [27]. I am als indebted t my research assistant Kasidit Leviriyakit fr his assistance in preparing the Latex files fr this text. 9.4 Shape ptimizatin fr a Transnic Business Jet The unstructured design methd has als been applied t cmplete aircraft cnfiguratins. The results fr a business jet are shwn belw. As shwn in figures 20, 21, 22, 23, the utbard sectins f the wing have a strng shck while flying at cruise cnditins (M = 0.80, α = 2 ). The results f a drag minimizatin that aims t remve the shcks n the wing are shwn in figures 24, 25, 26, 27. The drag has been reduced frm 235 cunts t 215 cunts in abut 8 design cycles. The lift was cnstrained at 0.4 by perturbing the angle f attack. Further, the riginal thickness f the wing was maintained during the design prcess ensuring that fuel vlume and structural integrity will be maintained by the redesigned shape. Thickness cnstraints n the wing were impsed n cutting planes alng the span f the wing and by transferring the cnstrained shape mvement back t the ndes f the surface triangulatin. The vlume mesh was defrmed t cnfrm t the shape changes induced using the spring methd. The entire design prcess typically takes abut 4 hurs n a 1.7 Ghz Athln prcessr with 1 Gb f memry. Parallel implementatin f the design prcedure has als been develped that further reduces the cmputatinal cst f this design prcess. 10 Cnclusin The accumulated experience f the last decade suggests that mst existing aircraft which cruise at transnic speeds are amenable t a drag reductin f the rder f 3 t 5 percent, r an increase in the drag rise Mach number f at least.02. These imprvements can be achieved by very small shape mdificatins, REFERENCES [1] M. Van Dyke. An Album f Fluid Mtin. The Parablic Press, Stanfrd, [2] D.R. Chapman, H. Mark, and M.W. Pirtle. Cmputers vs. wind tunnels in aerdynamic flw simulatins. Astrnautics and Aernautics, 13(4):22 30, 35, [3] Levy D.W., Zickuhr T., Vassberg J., and al. Summary f data frm the first aiaa cfd drag predictin wrkshp. AIAA paper , AIAA 40rd Aerspace Sciences Meeting, Ren, Nevada, January [4] G. Redeker. DLR-F4 wing-bdy cnfiguratin. In A Selectin f Experimental Test Cases fr the Validatin f CFD Cdes, number AR-303, pages B4.1 B4.21. AGARD, August [5] J. Vassberg, P.G. Buning, and C.L. Rumsey. Drag predictin fr the dlr-f4 wing/bdy using verflw and cfl3d n an verset mesh. AIAA paper , AIAA 40rd Aerspace Sciences Meeting, Ren, Nevada, January [6] R. M. Hicks and P. A. Henne. Wing design by numerical ptimizatin. Jurnal f Aircraft, 15: , [7] C. Bischf, A. Carle, G. Crliss, A. Griewank, and P. Hvland. Generating derivative cdes frm Frtran prgrams. Internal reprt MCS-P , Cmputer Science Divisin, Argnne Natinal Labratry and Center f Research n Parallel Cmputatin, Rice University, [8] L. L. Green, P. A. Newman, and K. J. Haigler. Sensitivity derivatives fr advanced CFD algrithm and viscus mdeling parameters via autmatic differentiatin. AIAA paper , 11th AIAA Cmputatinal Fluid Dynamics Cnference, Orland, Flrida,