Optimum Aerodynamic Design using the Navier Stokes Equations

Transcription

1 Optimum Aerdynamic esign using the Navier Stkes Equatins A. Jamesn, N.A. Pierce and L. Martinelli, epartment f Mechanical and Aerspace Engineering Princetn University Princetn, New Jersey USA and Oxfrd University Cmputing Labratry Numerical Analysis Grup Oxfrd OX1 3Q UK ASTRACT This paper describes the frmulatin f ptimizatin techniques based n cntrl thery fr aerdynamic shape design in viscus cmpressible flw, mdelled by the Navier-Stkes equatins. It extends previus wrk n ptimizatin fr inviscid flw. The thery is applied t a system defined by the partial differential equatins f the flw, with the bundary shape acting as the cntrl. The Frechet derivative f the cst functin is determined via the slutin f an adjint partial differential equatin, and the bundary shape is then mdified in a directin f descent. This prcess is repeated until an ptimum slutin is apprached. Each design cycle requires the numerical slutin f bth the flw and the adjint equatins, leading t a cmputatinal cst rughly equal t the cst f tw flw slutins. The cst is kept lw by using multigrid techniques, in cnjunctin with precnditining t accelerate the cnvergence f the slutins. The pwer f the methd is illustrated by designs f wings and wing-bdy cmbinatins fr lng range transprt aircraft. Satisfactry designs are usually btained with design cycles. 1 INTROUCTION This paper, which is dedicated t Sir James Lighthill, is fcused n the prblem f aerdynamic design. Here, as in s many ther branches f fluid mechanics and applied mathematics, Lighthill has made a seminal cntributin thrugh his early demnstratin f a slutin f the inverse prblem fr airfil design in ptential flw 1. The evlutin f cmputatinal fluid dynamics during the last three decades has made pssible the rapid evaluatin f alternative designs by cmputatinal simulatin, eliminating the need t build numerus mdels fr wind tunnel testing, which is used primarily t cnfirm the perfrmance f the final design, and t prvide a cmplete database fr the full flight envelpe. The designer still needs sme guiding principle t distinguish a gd design ut f an infinite number f pssible variatins, since it is nt at all likely that a truly ptimum design can be fund by a trial and errr prcess. This mtivates the use f numerical ptimizatin prcedures in cnjunctin with cmputatinal flw simulatins. Early investigatins int aerdynamic ptimizatin relied n direct evaluatin f the influence f each design variable. This dependence was estimated by separately varying each design parameter and recalculating the flw. The cmputatinal cst f this methd is prprtinal t the number f design variables and cnsequently becmes prhibitive as the number f design parameters is increased. An alternative apprach t design relies n the fact that experienced designers generally have an intuitive feel fr the kind f pressure distributin that will prvide the desired aerdynamic perfrmance. This mtivates the intrductin f inverse prblems in which the shape crrespnding t a specified pressure distributin is t be determined. A cmplete analysis f the inverse prblem 1

2 fr airfils in tw dimensinal ptential flw was given by Lighthill 1, wh btained a slutin by cnfrmally mapping the prfile t a unit circle. The speed ver the prfile is q = 1 h φ where φ is the ptential, which is knwn fr the circle, while h is the mdulus f the mapping functin. The surface value f h can be btained by setting q = q d, where q d is the desired speed, and since the mapping functin is analytic, it is uniquely determined by the value f h n the bundary. Lighthill s analysis highlights the fact that a physically realizable shape may nt exist unless the prescribed pressure distributin satisfies certain cnstraints. In fact a slutin exists fr a given speed q at infinity nly if 1 q d dθ = q 2π where θ is the plar angle arund the circle, and there are additinal cnstraints n q d if the prfile is t be clsed. In the mre general case f three-dimensinal viscus cmpressible flw, the cnstraints which must be satisfied by a realizable target pressure distributin are nt knwn, and attempts t enfrce an unrealizable pressure distributin as a bundary cnditin can lead t an ill-psed prblem. The prblems f ptimal and inverse design can bth be systematically treated within the mathematical thery fr the cntrl f systems gverned by partial differential equatins 2 by regarding the design prblem as a cntrl prblem in which the cntrl is the shape f the bundary. The inverse prblem then becmes a special case f the ptimal design prblem in which the shape changes are driven by the discrepancy between the current and target pressure distributins. The cntrl thery apprach t ptimal aerdynamic design, in which shape changes are based n gradient infrmatin btained by slutin f an adjint prblem, was first applied t transnic flw by Jamesn 3,4. He frmulated the methd fr inviscid cmpressible flws with shcks gverned by bth the ptential equatin and the Euler equatins 3,5,6. With this apprach, the cst f a design cycle is independent f the number f design variables. When applied t the design f the airfils in cmpressible ptential flw using cnfrmal mapping t transfrm the cmputatinal dmain t a unit disk, it leads t a natural generalizatin f Lighthill s methd. The effects f cmpressibility are represented by an additinal term in the mdificatin f the mapping functin which tends t zer as the Mach number tends t zer 3,5. Mre recently, the methd has been emplyed fr wing design in the cntext f cmplex aircraft cnfiguratins 7,8, using a grid perturbatin technique t accmmdate the gemetry mdificatins. Pirnneau had earlier initiated studies f the use f cntrl thery fr ptimum shape design f systems gverned by elliptic equatins 9,10. Ta asan, Kuruvila and Salas have prpsed a ne sht apprach in which the cnstraint represented by the flw equatins need nly be satisfied by the final cnverged design slutin 11. Adjint methds have als been used by aysal and Eleshaky 12, by Cabuk and Mdi 13,14, and by esai and It 15. The bjective f the present wrk is the extensin f adjint methds fr ptimal aerdynamic design t flws gverned by the cmpressible Navier Stkes equatins. While inviscid frmulatins have prven useful fr the design f transnic wings at cruise cnditins, the inclusin f bundary layer displacement effects with viscus design prvides increased realism and alleviates shcks that wuld therwise frm in the viscus slutin ver the final inviscid design. Accurate reslutin f viscus effects such as separatin and shck/bundary layer interactin is als essential fr ptimal design encmpassing ff-design cnditins and high-lift cnfiguratins. The cmputatinal csts f viscus design are at least an rder f magnitude greater than fr design using the Euler equatins fr several reasns. First, the number f mesh pints must be increased by a factr f tw r mre t reslve the bundary layer. Secnd, there is the additinal cst f cmputing the viscus terms and a turbulence mdel. Finally, Navier Stkes calculatins generally cnverge much mre slwly than Euler slutins due t discrete stiffness and directinal decupling arising frm the highly stretched bundary layer cells. The cmputatinal feasibility f viscus design therefre hinges n the develpment f a rapidly cnvergent Navier Stkes flw slver. Pierce and Giles have develped a precnditined multigrid methd that dramatically imprves 2

3 cnvergence f viscus calculatins by ensuring that all errr mdes inside the stretched bundary layer cells are damped efficiently 16,17. The same acceleratin techniques are applicable t the adjint calculatin, s that the ptential payffs tward reducing the cst f the design prcess are substantial. The ultimate success f an aircraft design depends n the reslutin f cmplex multi-disciplinary trade-ffs between factrs such as aerdynamic efficiency, structural weight, stability and cntrl, and the vlume required t cntain fuel and paylad. A design is finalized nly after numerus iteratins, cycling between the disciplines. The develpment f accurate and efficient methds fr aerdynamic shape ptimizatin represents a wrthwhile intermediate step twards the eventual gal f full multi-disciplinary ptimal design. 2 GENERAL FORMULATION OF THE AJOINT APPROACH TO OPTIMAL ESIGN efre embarking n a detailed derivatin f the adjint frmulatin fr ptimal design using the Navier Stkes equatins, it is helpful t summarize the general abstract descriptin f the adjint apprach which has been thrughly dcumented in references 3,4. The prgress f the design prcedure is measured in terms f a cst functin I, which culd be, fr example the drag cefficient r the lift t drag rati. Fr flw abut an airfil r wing, the aerdynamic prperties which define the cst functin are functins f the flw-field variables (w and the physical lcatin f the bundary, which may be represented by the functin F, say. Then and a change in F results in a change [ ] I T δi = w I = I (w, F, I [ ] I T δw δf, (1 F II in the cst functin. Here, the subscripts I and II are used t distinguish the cntributins due t the variatin δw in the flw slutin frm the change assciated directly with the mdificatin δf in the shape. This ntatin is intrduced t assist in gruping the numerus terms that arise during the derivatin f the full Navier Stkes adjint peratr, s that it remains feasible t recgnize the basic structure f the apprach as it is sketched in the present sectin. Using cntrl thery, the gverning equatins f the flw field are intrduced as a cnstraint in such a way that the final expressin fr the gradient des nt require multiple flw slutins. This crrespnds t eliminating δw frm (1. Suppse that the gverning equatin R which expresses the dependence f w and F within the flw-field dmain can be written as R (w, F = 0. (2 Then δw is determined frm the equatin [ ] R δr = δw w Next, intrducing a Lagrange Multiplier ψ, we have I [ ] R δf = 0. (3 F II ] δw ([ δi = IT IT R δw w F δf ψt w { I T [ ] } { R I T = w ψt δw w Chsing ψ t satisfy the adjint equatin I [ ] R δf F ] } F ψt [ R F II δf. (4 [ ] T R ψ = I w w (5 3

4 the first term is eliminated, and we find that δi = GδF, (6 where G = IT F ψt [ ] R. F The advantage is that (6 is independent f δw, with the result that the gradient f I with respect t an arbitrary number f design variables can be determined withut the need fr additinal flw-field evaluatins. In the case that (2 is a partial differential equatin, the adjint equatin (5 is als a partial differential equatin and determinatin f the apprpriate bundary cnditins requires careful mathematical treatment. The cmputatinal cst f a single design cycle is rughly equivalent t the cst f tw flw slutins since the the adjint prblem has similar cmplexity. When the number f design variables becmes large, the cmputatinal efficiency f the cntrl thery apprach ver traditinal apprach, which requires direct evaluatin f the gradients by individually varying each design variable and recmputing the flw field, becmes cmpelling. Once equatin (3 is established, an imprvement can be made with a shape change δf = λg where λ is psitive, and small enugh that the first variatin is an accurate estimate f δi. The variatin in the cst functin then becmes δi = λg T G < 0. After making such a mdificatin, the gradient can be recalculated and the prcess repeated t fllw a path f steepest descent until a minimum is reached. In rder t avid vilating cnstraints, such as a minimum acceptable wing thickness, the gradient may be prjected int an allwable subspace within which the cnstraints are satisfied. In this way, prcedures can be devised which must necessarily cnverge at least t a lcal minimum. 3 THE NAVIER-STOKES EQUATIONS Fr the derivatins that fllw, it is cnvenient t use Cartesian crdinates (x 1,x 2,x 3 and t adpt the cnventin f indicial ntatin where a repeated index i implies summatin ver i = 1 t 3. The three-dimensinal Navier-Stkes equatins then take the frm w t f i x i = f vi x i in, (7 where the state vectr w, inviscid flux vectr f and viscus flux vectr f v are described respectively by ρ ρu 1 w = ρu 2, (8 ρu 3 ρe f i = ρu i ρu i u 1 pδ i1 ρu i u 2 pδ i2 ρu i u 3 pδ i3 ρu i H 4, (9

5 f vi = 0 σ ij δ j1 σ ij δ j2 σ ij δ j3 u j σ ij k T x i. (10 In these definitins, ρ is the density, u 1, u 2, u 3 are the Cartesian velcity cmpnents, E is the ttal energy and δ ij is the Krnecker delta functin. The pressure is determined by the equatin f state { p = (γ 1 ρ E 1 } 2 (u iu i, and the stagnatin enthalpy is given by H = E p ρ, where γ is the rati f the specific heats. The viscus stresses may be written as ( ui σ ij = µ u j u k λδ ij, (11 x j x i x k where µ and λ are the first and secnd cefficients f viscsity. The cefficient f thermal cnductivity and the temperature are cmputed as k = c pµ P r, T = p Rρ, (12 where P r is the Prandtl number, c p is the specific heat at cnstant pressure, and R is the gas cnstant. Fr discussin f real applicatins using a discretizatin n a bdy cnfrming structured mesh, it is als useful t cnsider a transfrmatin t the cmputatinal crdinates (ξ 1,ξ 2,ξ 3 defined by the metrics [ ] [ xi K ij =, J = det (K, K 1 ξi ij = ξ j x j The Navier-Stkes equatins can then be written in cmputatinal space as (Jw t (F i F vi = 0 in, (13 where the inviscid and viscus flux cntributins are nw defined with respect t the cmputatinal cell faces by F i = S ij f j and F vi = S ij f vj, and the quantity S ij = JK 1 ij is used t represent the prjectin f the ξ i cell face alng the x j axis. In btaining equatin (13 we have made use f the prperty that S ij = 0 (14 which represents the fact that the sum f the face areas ver a clsed vlume is zer, as can be readily verified by a direct examinatin f the metric terms. 4 GENERAL FORMULATION OF THE OPTIMAL ESIGN PROLEM FOR THE NAVIER-STOKES EQUATIONS Aerdynamic ptimizatin is based n the determinatin f the effect f shape mdificatins n sme perfrmance measure which depends n the flw. 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 ij. 5 ].

6 Suppse that the perfrmance is measured by a cst functin I = M (w, S d ξ P (w, S d ξ, cntaining bth bundary and field cntributins where d ξ and d 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 δi = δm(w, S d ξ δp(w, S d ξ, (15 with δm = [M w ] I δw δm II, δp = [P w ] I δw δp II, (16 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 P w and 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 δ (F i F vi = 0. (17 Here δf i and δf vi can als be split int cntributins assciated with δw and δs using the ntatin The inviscid cntributins are easily evaluated as δf i = [F iw ] I δw δf iii δf vi = [F viw ] I δw δf viii. (18 [F iw ] I = S ij f j w, δf viii = δs ij f j. The details f the viscus cntributins are cmplicated by the additinal level f derivatives in the stress and heat flux terms and will be derived in Sectin 6. 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 ψ T δ (F i F vi = 0. (19 If ψ is differentiable this may be integrated by parts t give n i ψ T ψ T δ (F i F vi d ξ δ (F i F vi d ξ = 0. (20 Since the left hand expressin equals zer, it may be subtracted frm the variatin in the cst functin (15 t give [ δi = δm ni ψ T δ (F i F vi ] d ξ ] [δp ψt δ (F i F vi d ξ. (21 6

7 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 (16 and (18, the variatin δw may be eliminated frm (21 by equating all field terms with subscript I t prduce a differential adjint system gverning ψ ψ T [F iw F viw ] I P w = 0 in. (22 The crrespnding adjint bundary cnditin is prduced by equating the subscript I bundary terms in equatin (21 t prduce n i ψ T [F iw F viw ] I = M w n. (23 The remaining terms frm equatin (21 then yield a simplified expressin fr the variatin f the cst functin which defines the gradient { δi = δmii n i ψ T } [δf i δf vi ] II dξ } {δp II ψt [δf i δf vi ] ξ II d ξ. (24 i 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 δi is finally reduced t the frm f equatin (6 δi = GδF d ξ 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 (23. 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 (21. 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 (22 as surce terms. It is cnvenient t develp the inviscid and viscus cntributins t the adjint equatins separately. Als, 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 (21 and (23 may be simplified by incrprating the cnditins n 1 = n 3 = 0, n 2 = 1, d ξ = dξ 1 dξ 3, s that nly the variatins δf 2 and δf v2 need t be cnsidered at the wall bundary. 5 ERIVATION OF THE INVISCI AJOINT TERMS The inviscid cntributins have been previusly derived in 5,18 but are included here fr cmpleteness. Taking the transpse f equatin (22, the inviscid adjint equatin may be written as C T i ψ = 0 in, (25 7

8 where the inviscid Jacbian matrices in the transfrmed space are given by C i = S ij f j w. The transfrmed velcity cmpnents have the frm U i = S ij u j, and 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 0 0 δf 2 = δp S 21 S 22 p δs 21 δs 22. (26 S 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 (p p d 2 ds 2 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 (p p d 2 S 2 dξ 1 dξ 3, 2 w where the quantity S 2 = S 2j S 2j 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 (27 where n j are the cmpnents f the surface nrmal n j = S 2j S 2. This amunts t a transpiratin bundary cnditin n the c-state variables crrespnding t the mmentum cmpnents. Nte that it impses n restrictin n the tangential cmpnent f ψ at the bundary. In the presence f shck waves, neither p nr p d are necessarily cntinuus at the surface. The bundary cnditin is then in cnflict with the assumptin that ψ is differentiable. This difficulty can be circumvented by the use f a smthed bundary cnditin δs 23

9 6 ERIVATION OF THE VISCOUS AJOINT TERMS In cmputatinal crdinates, the viscus terms in the Navier Stkes equatins have the frm F vi = ( Sij f v j. Cmputing the variatin δw resulting frm a shape mdificatin f the bundary, intrducing a c-state vectr ψ and integrating by parts fllwing the steps utlined by equatins (17 t (20 prduces ψ T ( δs 2j f v j S 2jδf v j dξ ψ T ( δsij f v j S ijδf v j dξ, where the shape mdificatin is restricted t the crdinate surface ξ 2 = 0 s that n 1 = n 3 = 0, and n 2 = 1. Furthermre, it is assumed that the bundary cntributins at the far field may either be neglected r else eliminated by a prper chice f bundary cnditins as previusly shwn fr the inviscid case 5,18. The viscus terms will be derived under the assumptin that the viscsity and heat cnductin cefficients µ and k are essentially independent f the flw, and that their variatins may be neglected. In the case f turbulent flw, if the flw variatins are fund t result in significant changes in the turbulent viscsity, it may eventually be necessary t include its variatin in the calculatins. Transfrmatin t Primitive Variables The derivatin f the viscus adjint terms is simplified by transfrming t the primitive variables w T = (ρ, u 1, u 2, u 3, p T, because the viscus stresses depend n the velcity derivatives ui x j, while the heat flux can be expressed as where κ = k R = γµ P r(γ 1. defined by the expressins κ x i ( p. ρ The relatinship between the cnservative and primitive variatins is M 1T = δw = Mδ w, δ w = M 1 δw which make use f the transfrmatin matrices M = w w and M 1 = w prvided in transpsed frm fr future cnvenience u 1 u 1 u 2 u iu i ρ 0 0 ρu 1 M T = 0 0 ρ 0 ρu ρ ρu γ 1 1 u1 ρ u2 ρ u3 (γ 1u iu i ρ ρ 0 0 (γ 1u ρ 0 (γ 1u ρ (γ 1u γ 1. w. These matrices are The cnservative and primitive adjint peratrs L and L crrespnding t the variatins δw and δ w are then related by δw T Lψ d ξ = δ w T Lψ dξ, 9

10 with L = M T L, s that after determining the primitive adjint peratr by direct evaluatin f the viscus prtin f (22, the cnservative peratr may be btained by the transfrmatin L = M 1T L. Since the cntinuity equatin cntains n viscus terms, it makes n cntributin t the viscus adjint system. Therefre, the derivatin prceeds by first examining the adjint peratrs arising frm the mmentum equatins. Cntributins frm the Mmentum Equatins In rder t make use f the summatin cnventin, it is cnvenient t set ψ j1 = φ j fr j = 1, 2, 3. Then the cntributin frm the mmentum equatins is φ k (δs 2j σ kj S 2j δσ kj d ξ The velcity derivatives in the viscus stresses can be expressed as with crrespnding variatins δ u i x j = φ k (δs ij σ kj S ij δσ kj d ξ. (28 u i = u i ξ l = S lj u i x j ξ l x j J ξ l [ Slj J ] I ξ l δu i [ ] ui The variatins in the stresses are then { [ Slj δσ kj = µ J ξ l δu k S lk J [ S λ δ lm jk J { [ µ δ ( Slj J uk ξ l [ λ ξ l II δ ( Slj ] ξ l δu j δ ( S lk J δ jk δ ( S lm J J. ξ l δu m ]}I ] uj ξ l um ξ l ]}II. As befre, nly thse terms with subscript I, which cntain variatins f the flw variables, need be cnsidered further in deriving the adjint peratr. The field cntributins that cntain δu i in equatin (28 appear as { ( φ k Slj S ij µ δu k S lk δu j J ξ l J ξ l λδ jk S lm J This may be integrated by parts t yield ( µ δu k S lj S ij ξ l J ( µ δu j S lk S ij ξ l J δu m ξ l ξ l δu m φ k φ k ( S lm S ij λδ jk J 10 φ k } d ξ. d ξ d ξ d ξ,

11 where the bundary integral has been eliminated by nting that δu i = 0 n the slid bundary. y exchanging indices, the field integrals may be cmbined t prduce { ( Sij φ k δu k S lj µ S ik φ j ξ l J J } S im φ m λδ jk d ξ, J which is further simplified by transfrming the inner derivatives back t Cartesian crdinates { ( φk δu k S lj µ φ } j φ m λδ jk d ξ. (29 ξ l x j x k x m The bundary cntributins that cntain δu i in equatin (28 may be simplified using the fact that ξ l δu i = 0 if l = 1, 3 n the bundary s that they becme φ k S 2j {µ ( S2j J δu k S 2k ξ 2 J λδ jk S 2m J ξ 2 δu m δu j ξ 2 } ds x. (30 Tgether, (29 and (30 cmprise the field and bundary cntributins f the mmentum equatins t the viscus adjint peratr in primitive variables. Cntributins frm the Energy Equatin In rder t derive the cntributin f the energy equatin t the viscus adjint terms it is cnvenient t set ψ 5 = θ, Q j = u i σ ij κ ( p, x j ρ where the temperature has been written in terms f pressure and density using (12. The cntributin frm the energy equatin can then be written as θ (δs 2j Q j S 2j δq j d ξ θ (δs ij Q j S ij δq j d ξ. (31 The field cntributins that cntain δu i,δp, and δρ in equatin (31 appear as θ S ij δq j d ξ = { θ S ij δu k σ kj u k δσ kj κ S ( lj δp J ξ l ρ p } δρ d ξ. (32 ρ ρ The term invlving δσ kj may be integrated by parts t prduce { ( θ θ δu k S lj µ u k u j ξ l x j x k } θ λδ jk u m d ξ, (33 x m 11

12 where the cnditins u i = δu i = 0 are used t eliminate the bundary integral n. Ntice that the ther term in (32 that invlves δu k need nt be integrated by parts and is merely carried n as θ δu k σ kj S ij d ξ (34 The terms in expressin (32 that invlve δp and δρ may als be integrated by parts t prduce bth a field and a bundary integral. The field integral becmes ( δp ρ p ( δρ κ θ S lj S ij d ξ ρ ρ ξ l J which may be simplified by transfrming the inner derivative t Cartesian crdinates ( δp ρ p ( δρ S lj κ θ d ξ. (35 ρ ρ ξ l x j The bundary integral becmes ( δp κ ρ p ρ δρ S2j S ij ρ J θ d ξ. (36 This can be simplified by transfrming the inner derivative t Cartesian crdinates ( δp κ ρ p δρ S2j θ d ξ, (37 ρ ρ J x j and identifying the nrmal derivative at the wall and the variatin in temperature t prduce the bundary cntributin δt = 1 R n = S 2j, (38 x j ( δp ρ p ρ δρ, ρ kδt θ n d ξ. (39 This term vanishes if T is cnstant n the wall but persists if the wall is adiabatic. There is als a bundary cntributin left ver frm the first integratin by parts (31 which has the frm θδ (S 2j Q j d ξ, (40 where Q j = k T x j, since u i = 0. Ntice that fr future cnvenience in discussing the adjint bundary cnditins resulting frm the energy equatin, bth the δw and δs terms crrespnding t subscript classes I and II are cnsidered simultaneusly. If the wall is adiabatic s that using (38, T n = 0, δ (S 2j Q j = 0, and bth the δw and δs bundary cntributins vanish. 12

13 On the ther hand, if T is cnstant T ξ l Q j = k T ( Sl j T = k x j J ξ l Thus, the bundary integral (40 becmes { 2 S2j kθ δt δ J ξ 2 = 0 fr l = 1, 3, s that ( S2j 2 J = k T ( S2j ξ 2 J T. ξ 2 } d ξ. (41 Therefre, fr cnstant T, the first term crrespnding t variatins in the flw field cntributes t the adjint bundary peratr and the secnd set f terms crrespnding t metric variatins cntribute t the cst functin gradient. All tgether, the cntributins frm the energy equatin t the viscus adjint peratr are the three field terms (33, (34 and (35, and either f tw bundary cntributins ( 39 r ( 41, depending n whether the wall is adiabatic r has cnstant temperature. 7 THE VISCOUS AJOINT FIEL OPERATOR Cllecting tgether the cntributins frm the mmentum and energy equatins, the viscus adjint peratr in primitive variables can be expressed as ( Lψ 1 = p ( ρ 2 S lj κ θ ξ l x j ( Lψ i1 = [ ( φi {S lj µ φ ]} j φ k λδ ij i = 1, 2, 3 ξ l x j x i x k [ ( ]} θ θ θ {S lj µ u i u j λδ ij u k ξ l x j x i x k θ σ ij S lj ( Lψ 5 = ρ ξ l ξ ( l S lj κ θ x j. The cnservative viscus adjint peratr may nw be btained by the transfrmatin L = M 1T L. 8 VISCOUS AJOINT OUNARY CONITIONS It was recgnized in Sectin 4 that the bundary cnditins satisfied by the flw equatins restrict the frm f the perfrmance measure that may be chsen fr the cst functin. There must be a direct crrespndence between the flw variables fr which variatins appear in the variatin f the cst functin, and thse variables fr which variatins appear in the bundary terms arising during the derivatin f the adjint field equatins. Otherwise it wuld be impssible t eliminate the dependence f δi n δw thrugh prper specificatin f the adjint bundary cnditin. As in the derivatin f the field equatins, it prves cnvenient t cnsider the cntributins frm the mmentum equatins and the energy equatin separately. undary Cnditins Arising frm the Mmentum Equatins The bundary term that arises frm the mmentum equatins including bth the δw and δs cmpnents (28 takes the frm φ k δ (S 2j σ kj d ξ. 13

14 Replacing the metric term with the crrespnding lcal face area S 2 and unit nrmal n j defined by then leads t efining the cmpnents f the surface stress as and the physical surface element S 2 = S 2j S 2j, n j = S 2j S 2 φ k δ ( S 2 n j σ kj d ξ. τ k = n j σ kj ds = S 2 d ξ, the integral may then be split int tw cmpnents φ k τ k δs 2 d ξ φ k S 2 δτ k ds, (42 where nly the secnd term cntains variatins in the flw variables and must cnsequently cancel the δw terms arising in the cst functin. The first term will appear in the expressin fr the gradient. A general expressin fr the cst functin that allws cancellatin with terms cntaining δτ k has the frm I = N (τds, (43 crrespnding t a variatin δi = N τ k δτ k ds, fr which cancellatin is achieved by the adjint bundary cnditin φ k = N τ k. Natural chices fr N arise frm frce ptimizatin and as measures f the deviatin f the surface stresses frm desired target values. Fr viscus frce ptimizatin, the cst functin shuld measure frictin drag. The frictin frce in the x i directin is C fi = σ ij ds j = S 2j σ ij d ξ s that the frce in a directin with csines n i has the frm C nf = n i S 2j σ ij d ξ. Expressed in terms f the surface stress τ i, this crrespnds t C nf = n i τ i ds, s that basing the cst functin (43 n this quantity gives N = n i τ i. Cancellatin with the flw variatin terms in equatin (42 therefre mandates the adjint bundary cnditin φ k = n k. 14

15 Nte that this chice f bundary cnditin als eliminates the first term in equatin (42 s that it need nt be included in the gradient calculatin. In the inverse design case, where the cst functin is intended t measure the deviatin f the surface stresses frm sme desired target values, a suitable definitin is N (τ = 1 2 a lk (τ l τ dl (τ k τ dk, where τ d is the desired surface stress, including the cntributin f the pressure, and the cefficients a lk define a weighting matrix. Fr cancellatin This is satisfied by the bundary cnditin φ k δτ k = a lk (τ l τ dl δτ k. φ k = a lk (τ l τ dl. (44 Assuming arbitrary variatins in δτ k, this cnditin is als necessary. In rder t cntrl the surface pressure and nrmal stress ne can measure the difference n j {σ kj δ kj (p p d }, where p d is the desired pressure. The nrmal cmpnent is then s that the measure becmes N (τ = 1 2 τ 2 n τ n = n k n j σ kj p p d, = 1 2 n ln m n k n j {σ lm δ lm (p p d } {σ kj δ kj (p p d }. This crrespnds t setting a lk = n l n k in equatin (44. efining the viscus nrmal stress as the measure can be expanded as τ vn = n k n j σ kj, N (τ = 1 2 n ln m n k n j σ lm σ kj 1 2 (n kn j σ kj n l n m σ lm (p p d 1 2 (p p d 2 = 1 2 τ 2 vn τ vn (p p d 1 2 (p p d 2. Fr cancellatin f the bundary terms φ k (n j δσ kj n k δp = { n l n m σ lm n 2 l (p p d } n k (n j δσ kj n k δp leading t the bundary cnditin φ k = n k (τ vn p p d. In the case f high Reynlds number, this is well apprximated by the equatins φ k = n k (p p d, (45 which shuld be cmpared with the single scalar equatin derived fr the inviscid bundary cnditin (27. In the case f an inviscid flw, chsing requires N (τ = 1 2 (p p d 2 φ k n k δp = (p p d n 2 kδp = (p p d δp which is satisfied by equatin (45, but which represents an verspecificatin f the bundary cnditin since nly the single cnditin (27 need be specified t ensure cancellatin. 15

16 undary Cnditins Arising frm the Energy Equatin The frm f the bundary terms arising frm the energy equatin depends n the chice f temperature bundary cnditin at the wall. Fr the adiabatic case, the bundary cntributin is (39 kδt θ n d ξ, while fr the cnstant temperature case the bundary term is (41. One pssibility is t intrduce a cntributin int the cst functin which is dependent T r T n s that the apprpriate cancellatin wuld ccur. Since there is little physical intuitin t guide the chice f such a cst functin fr aerdynamic design, a mre natural slutin is t set θ = 0 in the cnstant temperature case r θ n = 0 in the adiabatic case. Nte that in the cnstant temperature case, this chice f θ n the bundary wuld als eliminate the bundary metric variatin terms in (40. 9 IMPLEMENTATION OF NAVIER-STOKES ESIGN The design prcedures can be summarized as fllws: 1. Slve the flw equatins fr ρ, u 1, u 2,u 3, p. 2. Smth the cst functin, if necessary. 3. Slve the adjint equatins fr ψ subject t apprpriate bundary cnditins. 4. Evaluate G. 5. Prject G int an allwable subspace that satisfies any gemetric cnstraints. 6. Update the shape based n the directin f steepest descent. 7. Return t 1. Practical implementatin f the viscus design methd relies heavily upn fast and accurate slvers fr bth the state (w and c-state (ψ systems. This wrk emplys a well-validated Navier Stkes slver develped by tw f the authrs 19. iscretizatin th the flw and the adjint equatins are discretized using a semi-discrete cell-centered finite vlume scheme. The cnvective fluxes acrss cell interfaces are represented by simple arithmetic averages f the fluxes cmputed using values frm the cells n either side f the face, augmented by artificial diffusive terms t prevent numerical scillatins in the vicinity f shck waves. Cntinuing t use the summatin cnventin fr repeated indices, the numerical cnvective flux acrss the interface between cells A and in a three dimensinal mesh has the frm h A = 1 2 S A j ( faj f j da, where S Aj is the cmpnent f the face area in the j th Cartesian crdinate directin, ( ( f Aj and fj dente the flux fj as defined by equatin (12 and d A is the diffusive term. Variatins f the cmputer prgram prvide ptins fr alternate cnstructins f the diffusive flux. 16

17 The simplest ptin implements the Jamesn-Schmidt-Turkel scheme 20,21, using scalar diffusive terms f the frm d A = ɛ (2 w ɛ (4 ( w 2 w w, where w = w w A and w and w are the same differences acrss the adjacent cell interfaces behind cell A and beynd cell in the A directin. y making the cefficient ɛ (2 depend n a switch prprtinal t the undivided secnd difference f a flw quantity such as the pressure r entrpy, the diffusive flux becmes a third rder quantity, prprtinal t the cube f the mesh width in regins where the slutin is smth. Oscillatins are suppressed near a shck wave because ɛ (2 becmes f rder unity, while ɛ (4 is reduced t zer by the same switch. Fr a scalar cnservatin law, it is shwn in reference 21 that ɛ (2 and ɛ (4 can be cnstructed t make the scheme satisfy the lcal extremum diminishing (LE principle that lcal maxima cannt increase while lcal minima cannt decrease. The secnd ptin applies the same cnstructin t lcal characteristic variables. There are derived frm the eigenvectrs f the Jacbian matrix A A which exactly satisfies the relatin A A (w w A = S Aj ( fj f Aj. This crrespnds t the definitin f Re 22. The resulting scheme is LE in the characteristic variables. The third ptin implements the H-CUSP scheme prpsed by Jamesn 23 which cmbines differences f f A and w w A in a manner such that statinary shck waves can be captured with a single interir pint in the discrete slutin. This scheme minimizes the numerical diffusin as the velcity appraches zer in the bundary layer, and has therefre been preferred fr viscus calculatins in this wrk. Similar artificial diffusive terms are intrduced in the discretizatin f the adjint equatin, but with the ppsite sign because the wave directins are reversed in the adjint equatin. Satisfactry results have been btained using scalar diffusin in the adjint equatin while characteristic r H-CUSP cnstructins are used in the flw slutin. dual cell σ i j Figure 1: Cell-centered scheme. σ ij evaluated at vertices f the primary mesh The discretizatin f the viscus terms f the Navier Stkes equatins requires the evaluatin f the velcity derivatives u i xj in rder t calculate the viscus stress tensr σ ij defined in equatin (11. These are mst cnveniently evaluated at the cell vertices f the primary mesh by intrducing a 17

18 dual mesh which cnnects the cell centers f the primary mesh, as depicted in Figure (1. Accrding t the Gauss frmula fr a cntrl vlume V with bundary S v i dv = u i n j ds x j V where n j is the utward nrmal. Applied t the dual cells this yields the estimate v i = 1 x j vl S ū i n j S faces where S is the area f a face, and ū i is an estimate f the average f u i ver that face. In rder t determine the viscus flux balance f each primary cell, the viscus flux acrss each f its faces is then calculated frm the average f the viscus stress tensr at the fur vertices cnnected by that face. This leads t a cmpact scheme with a stencil cnnecting each cell t its 26 nearest neighbrs. The semi-discrete schemes fr bth the flw and the adjint equatins are bth advanced t steady state by a multi-stage time stepping scheme. This is a generalized Runge-Kutta scheme in which the cnvective and diffusive terms are treated differently t enlarge the stability regin 21,24. Cnvergence t a steady state is accelerated by residual averaging and a multi-grid prcedure 25. Cnvergence is further accelerated by the use f lcally varying time steps (which may be regarded as a scalar precnditiner r the matrix precnditiner methd develped by Pierce and Giles 16,17. Optimizatin 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 lift. Therefre the angle f attack is adjusted during the flw slutin t frce a specified lift cefficient t be attained. The search prcedure used in this wrk is a simple descent methd in which small steps are taken in the negative gradient directin. Let F represent the design variable, and G the gradient. Then the iteratin δf = λg can be regarded as simulating the time dependent prcess df dt = G where λ is the time step t. Let A be the Hessian matrix with element A ij = G i = 2 I. F j F i F j Suppse that a lcally minimum value f the cst functin I = I(F is attained when F = F. Then the gradient G = G(F must be zer, while the Hessian matrix A = A(F must be psitive definite. Since G is zer, the cst functin can be expanded as a Taylr series in the neighbrhd f F with the frm Crrespndingly, I(F = I 1 2 (F F A (F F... G(F = A (F F... As F appraches F, the leading terms becme dminant. Then, setting ˆF = (F F, the search prcess apprximates d ˆF dt = A ˆF. 18

19 Als, since A is psitive definite it can be expanded as A = RMR T, where M is a diagnal matrix cntaining the eigenvalues f A, and Setting the search prcess can be represented as RR T = R T R = I. v = R T ˆF, dv dt = Mv. The stability regin fr the simple frward Euler stepping scheme is a unit circle centered at 1 n the negative real axis. Thus fr stability we must chse µ max t = µ max λ < 2, while the asympttic decay rate, given by the smallest eigenvalue, is prprtinal t e µmint. In rder t imprve the rate f cnvergence, ne can set δf = λp G, where P is a precnditiner fr the search. An ideal chice is P = A 1, s that the crrespnding time dependent prcess reduces t d ˆF dt = ˆF, fr which all the eigenvalues are equal t unity, and ˆF is reduced t zer in ne time step by the chice t = 1. Quasi-Newtn methds estimate A frm the change in the gradient during the search prcess. This requires accurate estimates f the gradient at each time step. In rder t btain these, bth the flw slutin and the adjint equatin must be fully cnverged. Mst quasi- Newtn methds als require a line search in each search directin, fr which the flw equatins and cst functin must be accurately evaluated several times. They have prven quite rbust fr aerdynamic ptimizatin 7. An alternative apprach which has als prved successful in ur previus wrk 18, and is used here, is t smth the gradient and t replace G by its smthed value Ḡ in the descent prcess. This bth acts as a precnditiner, and ensures that each new shape in the ptimizatin sequence remains smth. T apply smthing in the ξ 1 directin, fr example, the smthed gradient Ḡ ma be calculated frm a discrete apprximatin t Ḡ ξ 1 ɛ ξ 1 Ḡ = G where ɛ is the smthing parameter. If ne sets δf = λḡ, then, assuming the mdificatin is applied n the surface ξ 2 = cnstant, the first rder change in the cst functin is δi = GδF dξ 1 dξ 3 ( = λ Ḡ ɛ Ḡ Ḡ dξ 1 dξ 3 ξ 1 ξ 1 ( ( 2 = λ Ḡ 2 Ḡ ɛ dξ 1 dξ 3 ξ 1 < 0, 19

20 assuring an imprvement if λ is sufficiently small and psitive, unless the prcess has already reached a statinary pint at which G = 0. It turns ut that this apprach is tlerant t the use f apprximate values f the gradient, s that neither the flw slutin nr the adjint slutin need be fully cnverged befre making a shape change. This results in very large savings in the cmputatinal cst. Fr inviscid ptimizatin it is necessary t use nly 15 multigrid cycles fr the flw slutin and the adjint slutin in each design iteratin. Fr viscus ptimizatin, abut 100 multigrid cycles are needed. This is partly because cnvergence f the lift cefficient is much slwer, s abut 20 iteratins must be made befre each adjustment f the angle f attack t frce the target lift cefficient. The new precnditiner fr the flw and adjint calculatins allws the number f iteratins t be substantially reduced in bth the flw and the adjint simulatin. The numerical tests s far have fcused n the viscus design f wings fr ptimum cruise, fr which the flw remains attached, and the main viscus effect is due t the displacement thickness f the bundary layer. While sme tests have been made with the viscus adjint terms included, it has been fund that the ptimizatin prcess cnverges when the viscus terms are mitted frm the adjint system. This may reflect the tlerance f the search prcess t inexact gradients. 10 RESULTS Precnditined Inverse esign The first demnstratin is an applicatin f the precnditining technique fr inverse design with the Euler equatins. The ONERA M6 (Figure 2b wing is recvered fr a lifting case starting frm a wing with a NACA0012 sectin (Figure 2a and using the ONERA M6 pressure distributins cmputed at α = 3.0 and M = 4 as the target (Fig. 3. Thus, a symmetric wing sectin is t be recvered frm an asymmetric pressure distributin. The calculatins were perfrmed n a mesh with 294,912 cells. The mesh had a C-H tplgy with the C-lines wrapping arund the wing leading edge. Each design cycle required 3 multigrid cycles fr the flw slver using characteristic-based matrix dissipatin with a matrix precnditiner and 12 multigrid cycles fr the adjint slver using scalar dissipatin and a variable lcal time step (scalar precnditiner. Cmpared t a test in which the 3 multigrid cycles using the matrix precnditiner were replaced by 15 multigrid cycles using a standard scalar precnditiner, and 15 cycles were used in the adjint slutin, each design cycle required abut 3/8 as much cmputer time, while the number f design cycles required t reach the same level f errr als fell frm 100 t abut 50. Use f the matrix precnditiner therefre reduced the ttal CPU time n an IM 590 wrkstatin frm 97,683 sec ( 27 hurs t 18,222 sec ( 5 hurs fr rughly equivalent accuracy. Viscus esign ue t the high cmputatinal cst f viscus design, a tw-stage design strategy is adpted. In the first stage, a design calculatin is perfrmed with the Euler equatins t minimize the drag at a given lift cefficient by mdifying the wing sectins with a fixed planfrm. In the secnd stage, the pressure distributin f the Euler slutin is used as the target pressure distributin fr inverse design with the Navier-Stkes equatins. Cmparatively small mdificatins are required in the secnd stage, s that it can be accmplished with a small number f design cycles. In rder t test this strategy it was used fr the re-design f a wing representative f wide-bdy transprt aircraft. The results are shwn in Figures 4 and 5. The design pint was taken as a lift cefficient f.55 at a Mach number f.83. Figure 4 illustrates the Euler re-design, which was perfrmed n a mesh with cells, displaying bth the gemetry and the upper surface pressure distributin, with negative upwards. The initial wing shws a mderately strng shck wave acrss mst f the tp surface, as can be seen in Figure 4a. Sixty design cycles were needed t prduce the shck free wing shwn in Figure 4b, with an indicated drag reductin f 15 cunts frm.0196 t Figure 5 shws the viscus re-design at a Reynlds number f 12 millin. This was perfrmed n a mesh with cells, with 32 intervals nrmal t the wing cncentrated inside the bundary layer regin. In Figure 5a it can be seen that the Euler design prduces a 20

21 weak shck due t the displacement effects f the bundary layer. Ten design cycles were needed t recver the shck free wing shwn in Figure 5b. It is interesting that the wing sectin mdificatins between the initial wing f Figure 4a and the final wing f Figure 5b are remarkably small. These results were sufficiently prmising that it was decided by Mcnnell uglas t evaluate the methd fr industrial use, and it was used t supprt design studies fr the MXX prject. The results f this experience are discussed in reference 26. It rapidly became apparent that the fuselage effects are t large t be ignred. In viscus design it was als fund that there were discrepancies between the results f the design calculatins, which were initially perfrmed n a relatively carse grid with cells, and the results f subsequent analysis calculatins perfrmed n finer meshes t verify the design. In rder t allw the use f finer meshes with vernight turnarund, the cde was therefre mdified fr parallel cmputatin. Using the parallel implementatin, viscus design calculatins have been perfrmed n meshes with 1.8 millin mesh pints. Starting frm a preliminary inviscid design, 20 design cycles are usually sufficient fr a viscus re-design in inverse mde, with the smthed inviscid results prviding the target pressure. Such a calculatin can be cmpleted in abut hurs using 48 prcessrs f an IM SP2. As an illustratin f the results that culd be btained, Figures 6-10 shw a wing-bdy design with sweep back f abut 38 degrees at the 1/4 chrd. Starting frm the result f an Euler design, the viscus ptimizatin prduced an essentially shck free wing at a cruise design pint f Mach.86, with a lift cefficient f.6 fr the wing bdy cmbinatin at a Reynlds number f 101 millin based n the rt chrd. Figure 6 shws the design pint, while the evlutin f the design is shwn in Figure 7, using Vassberg s COMPPLOT sftware. In this case the pressure cnturs are fr the final design. This wing is quite thick, with a thickness t chrd rati f mre than 14 percent at the rt and 9 percent at the tip. The design ffers excellent perfrmance at the nminal cruise pint. Figures 8 and 9 shw the results f a Mach number sweep t determine the drag rise. It can be seen that a duble shck pattern frms belw the design pint, while there is actually a slight increase in the drag cefficient f abut cunts at Mach.85. Finally, Figure 10 shws a cmparisn f the pressure distributin at the design pint with thse at alternate cruise pints with lwer and higher lift. The tendency t prduce duble shcks belw the design pint is typical f supercritical wings. This wing has a lw drag cefficient, hwever, ver a wide range f cnditins. CONCLUSIONS We have develped a three-dimensinal cntrl thery based design methd fr the Navier Stkes equatins and applied it successfully t the design f wings in transnic flw. The methd represents an extensin f ur previus wrk n design with the ptential flw and Euler equatins. The new methd cmbines the versatility f numerical ptimizatin methds with the efficiency f inverse design. The gemetry is mdified by a grid perturbatin technique which is applicable t arbitrary cnfiguratins. The cmbinatin f cmputatinal efficiency with gemetric flexibility prvides a pwerful tl, with the final gal being t create practical aerdynamic shape design methds fr cmplete aircraft cnfiguratins. Such an accmplishment wuld represent the culminatin f the line f research initiated by Lighthill with his riginal wrk n the inverse prblem 1. ACKNOWLEGMENT This wrk has benefited frm the generus supprt f AFOSR under Grant N. AFOSR , O/URI/ONR/ARPA N J-1796, the NASA-IM erative Research Agreement, EP- SRC and the Rhdes Trust. 1. M.J. Lighthill. A new methd f tw dimensinal aerdynamic design. R & M 1111, Aernautical Research Cuncil, J.L. Lins. Optimal Cntrl f Systems Gverned by Partial ifferential Equatins. Springer- Verlag, New Yrk, Translated by S.K. Mitter. 3. A. Jamesn. Aerdynamic design via cntrl thery. J. Sci. Cmp., 3: ,

22 4. A. Jamesn. Optimum aerdynamic design using CF and cntrl thery. AIAA Paper CP, A. Jamesn. Autmatic design f transnic airfils t reduce the shck induced pressure drag. In Prceedings f the 31st Israel Annual Cnference n Aviatin and Aernautics, Tel Aviv, pages 5 17, February J. Reuther and A. Jamesn. Cntrl based airfil design using the Euler equatins. AIAA paper CP, J. Reuther and A. Jamesn. Aerdynamic shape ptimizatin f wing and wing-bdy cnfiguratins using cntrl thery. AIAA paper , AIAA 33rd Aerspace Sciences Meeting, Ren, Nevada, January J. Reuther, A. Jamesn, J. Farmer, L. Martinelli, and. Saunders. Aerdynamic shape ptimizatin f cmplex aircraft cnfiguratins via an adjint methd. AIAA paper , AIAA 34th Aerspace Sciences Meeting, Ren, Nevada, January O. Pirnneau. Optimal Shape esign fr Elliptic Systems. Springer-Verlag, New Yrk, O. Pirnneau. Optimal shape design fr aerdynamics. In AGAR REPORT 803, S. Ta asan, G. Kuruvila, and M.. Salas. Aerdynamic design and ptimizatin in ne sht. AIAA paper , 30th Aerspace Sciences Meeting and Exibit, Ren, Nevada, January O. aysal and M. E. Eleshaky. Aerdynamic design ptimizatin using sensitivity anaysis and cmputatinal fluid dynamics. AIAA Jurnal, 30(3: , H. Cabuk, C.H. Shung, and V. Mdi. Adjint peratr apprach t shape design fr internal incmpressible flw. In G.S. ulikravich, editr, Prceedings f the 3rd Internatinal Cnference n Inverse esign and Optimizatin in Engineering Sciences, pages , J.C. Huan and V. Mdi. Optimum design fr drag minimizing bdies in incmpressible flw. Inverse Prblems in Engineering, 1:1 25, M. esai and K. It. Optimal cntrls f Navier-Stkes equatins. SIAM J. Cntrl and Optimizatin, 32(5: , N.A. Pierce and M.. Giles. Precnditining cmpressible flw calculatins n stretched meshes. AIAA Paper , 34th Aerspace Sciences Meeting and Exhibit, Ren, NV, N.A. Pierce and M.. Giles. Precnditined multigrid methds fr cmpressible flw calculatins n stretched meshes. Submitted t J. Cmp. Phys., April, A. Jamesn. Optimum aerdynamic design using cntrl thery. Cmputatinal Fluid ynamics Review, pages , L. Martinelli and A. Jamesn. Validatin f a multigrid methd fr the Reynlds averaged equatins. AIAA paper , A. Jamesn, W. Schmidt, and E. Turkel. Numerical slutin f the Euler equatins by finite vlume methds using Runge-Kutta time stepping schemes. AIAA Paper , A. Jamesn. Analysis and design f numerical schemes fr gas dynamics 1, artificial diffusin, upwind biasing, limiters and their effect n multigrid cnvergence. Int. J. f Cmp. Fluid yn., 4: , P.L. Re. Apprximate Riemann slvers, parameter vectrs, and difference schemes. J. Cmp. Phys., 43: , A. Jamesn. Analysis and design f numerical schemes fr gas dynamics 2, artificial diffusin and discrete shck structure. Int. J. f Cmp. Fluid yn., 5:1 38, L. Martinelli. Calculatins f viscus flws with a multigrid methd. Princetn University Thesis, May A. Jamesn. Multigrid algrithms fr cmpressible flw calculatins. In W. Hackbusch and U. Trttenberg, editrs, Lecture Ntes in Mathematics, Vl. 1228, pages Prceedings f the 2nd Eurpean Cnference n Multigrid Methds, Clgne, 1985, Springer-Verlag, A. Jamesn. Re-Engineering the design prcess thrugh camputatin. AIAA paper , January