Aerdynamic Shape Optimizatin Using the Adjint Methd Antny Jamesn epartment f Aernautics & Astrnautics Stanfrd University Stanfrd, Califrnia 94305 USA Lectures at the Vn Karman Institute, russels Febuary 6, 2003 Abstract These Lecture Ntes review the frmulatin and applicatin f ptimizatin techniques based n cntrl thery fr aerdynamic shape design in bth inviscid and viscus cmpressible 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. Representative results are presented fr viscus ptimizatin f transnic wing-bdy cmbinatins. 1 Intrductin: Aerdynamic esign The definitin f the aerdynamic shapes f mdern aircraft relies heavily n cmputatinal simulatin t enable the rapid evaluatin f many alternative designs. Wind tunnel testing is then used t cnfirm the perfrmance f designs that have been identified by simulatin as prmising t meet the perfrmance gals. In the case f wing design and prpulsin system integratin, several cmplete cycles f cmputatinal analysis fllwed by testing f a preferred design may be used in the evlutin f the final cnfiguratin. Wind tunnel testing als plays a crucial rle in the develpment f the detailed lads needed t cmplete the structural design, and in gathering data thrughut the flight envelpe fr the design and verificatin f the stability and cntrl system. The use f cmputatinal simulatin t scan many alternative designs has prved extremely valuable in practice, but it still suffers the limitatin that it des nt guarantee the identificatin f the best pssible design. Generally ne has t accept the best s far by a given cutff date in the prgram schedule. T ensure the realizatin f the true best design, the ultimate gal f cmputatinal simulatin methds shuld nt just be the analysis f prescribed shapes, but the autmatic determinatin f the true ptimum shape fr the intended applicatin. This is the underlying mtivatin fr the cmbinatin f cmputatinal fluid dynamics with numerical ptimizatin methds. Sme f the earliest studies f such an apprach were made by Hicks and Henne [1,2]. The principal bstacle was the large cmputatinal cst f determining the sensitivity f the cst functin t variatins f the design parameters by repeated calculatin f the flw. Anther way t apprach the prblem is t frmulate aerdynamic shape design within the framewrk f the mathematical thery fr the cntrl f systems gverned by partial differential equatins [3]. In this view the wing is regarded as a device t prduce lift by cntrlling the flw, and its design is regarded as a prblem in the ptimal cntrl f the flw equatins by changing the shape f the bundary. If the bundary shape is regarded as arbitrary within sme requirements f smthness, then the full generality f shapes cannt be defined with a finite number f parameters, and ne must use the cncept f the Frechet derivative f the cst with respect t a functin. Clearly such a derivative cannt be determined directly by separate variatin f each design parameter, because there are nw an infinite number f these. Using techniques f cntrl thery, hwever, the gradient can be determined indirectly by slving an adjint equatin which has cefficients determined by the slutin f the flw equatins. This directly crrespnds t the gradient technique fr trajectry ptimizatin pineered by rysn [4]. The cst f slving
the adjint equatin is cmparable t the cst f slving the flw equatins, with the cnsequence that the gradient with respect t an arbitrarily large number f parameters can be calculated with rughly the same cmputatinal cst as tw flw slutins. Once the gradient has been calculated, a descent methd can be used t determine a shape change which will make an imprvement in the design. The gradient can then be recalculated, and the whle prcess can be repeated until the design cnverges t an ptimum slutin, usually within 10-50 cycles. The fast calculatin f the gradients makes ptimizatin cmputatinally feasible even fr designs in three-dimensinal viscus flw. There is a pssibility that the descent methd culd cnverge t a lcal minimum rather than the glbal ptimum slutin. In practice this has nt prved a difficulty, prvided care is taken in the chice f a cst functin which prperly reflects the design requirements. Cnceptually, with this apprach the prblem is viewed as infinitely dimensinal, with the cntrl being the shape f the bunding surface. Eventually the equatins must be discretized fr a numerical implementatin f the methd. Fr this purpse the flw and adjint equatins may either be separately discretized frm their representatins as differential equatins, r, alternatively, the flw equatins may be discretized first, and the discrete adjint equatins then derived directly frm the discrete flw equatins. The effectiveness f ptimizatin as a tl fr aerdynamic design als depends crucially n the prper chice f cst functins and cnstraints. One ppular apprach is t define a target pressure distributin, and then slve the inverse prblem f finding the shape that will prduce that pressure distributin. Since such a shape des nt necessarily exist, direct inverse methds may be ill-psed. The prblem f designing a tw-dimensinal prfile t attain a desired pressure distributin was studied by Lighthill, wh slved it fr the case f incmpressible flw with a cnfrmal mapping f the prfile t a unit circle [5]. The speed ver the prfile is q = 1 h φ, where φ is the ptential which is knwn fr incmpressible flw and 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. A slutin exists fr a given speed q at infinity nly if 1 q dθ = q, 2π and there are additinal cnstraints n q if the prfile is required t be clsed. The difficulty that the target pressure may be unattainable may be circumvented by treating the inverse prblem as a special case f the ptimizatin prblem, with a cst functin which measures the errr in the slutin f the inverse prblem. Fr example, if p d is the desired surface pressure, ne may take the cst functin t be an integral ver the the bdy surface f the square f the pressure errr, I = 1 (p p d 2 d, 2 r pssibly a mre general Sblev nrm f the pressure errr. This has the advantage f cnverting a pssibly ill psed prblem int a well psed ne. It has the disadvantage that it incurs the cmputatinal csts assciated with ptimizatin prcedures. The inverse prblem still leaves the definitin f an apprpriate pressure architecture t the designer. One may prefer t directly imprve suitable perfrmance parameters, fr example, t minimize the drag at a given lift and Mach number. In this case it is imprtant t intrduce apprpriate cnstraints. Fr example, if the span is nt fixed the vrtex drag can be made arbitrarily small by sufficiently increasing the span. In practice, a useful apprach is t fix the planfrm, and ptimize the wing sectins subject t cnstraints n minimum thickness. Studies f the use f cntrl thery fr ptimum shape design f systems gverned by elliptic equatins were initiated by Pirnneau [6]. The cntrl thery apprach t ptimal aerdynamic design was first applied t transnic flw by Jamesn [7 12]. He frmulated the methd fr inviscid cmpressible flws with shck waves gverned by bth the ptential flw and the Euler equatins [8]. Numerical results shwing the methd t be extremely effective fr the design f airfils in transnic ptential flw were presented in [13,14], and fr three-dimensinal wing design using the Euler equatins in [15]. Mre recently the methd has been emplyed fr the shape design f cmplex aircraft cnfiguratins [16,17], using a grid perturbatin apprach t accmmdate the gemetry mdificatins. The methd has been used t supprt the aerdynamic design
studies f several industrial prjects, including the eech Premier and the Mcnnell uglas MXX and lended Wing-dy prjects. The applicatin t the MXX is described in [10]. The experience gained in these industrial applicatins made it clear that the viscus effects cannt be ignred in transnic wing design, and the methd has therefre been extended t treat the Reynlds Averaged Navier-Stkes equatins [12]. Adjint methds have als been the subject f studies by a number f ther authrs, including aysal and Eleshaky [18], Huan and Mdi [19], esai and It [20], Andersn and Venkatakrishnan [21], and Peraire and Ellit [22]. Ta asan, Kuruvila and Salas [23], wh have implemented a ne sht apprach in which the cnstraint represented by the flw equatins is nly required t be satisfied by the final cnverged slutin. In their wrk, cmputatinal csts are als reduced by applying multigrid techniques t the gemetry mdificatins as well as the slutin f the flw and adjint equatins. 2 Frmulatin f the esign Prblem as a Cntrl Prblem 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, fr example, be the drag cefficient r the lift t drag rati, and 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 gradient vectr I α may nw be used t determine a directin f imprvement. The simplest prcedure is t make a step in the negative gradient directin by setting α n1 = α n λδα, s that t first rder I δi = I IT I δα = I λit α α α. Mre sphisticated search prcedures may be used such as quasi-newtn methds, which attempt t estimate the secnd derivative 2 I α iα j f the cst functin frm changes in the gradient I α in successive ptimizatin steps. These methds als generally intrduce line searches t find the minimum in the search directin which is defined at each step. The main disadvantage f this apprach is the need fr a number f flw calculatins prprtinal t the number f design variables t estimate the gradient. The cmputatinal csts can thus becme prhibitive as the number f design variables is increased. Using techniques f cntrl thery, hwever, the gradient can be determined indirectly by slving an adjint equatin which has cefficients defined by the slutin f the flw equatins. The cst f slving the adjint equatin is cmparable t that f slving the flw equatins. Thus the gradient can be determined with rughly the cmputatinal csts f tw flw slutins, independently f the number f design variables, which may be infinite if the bundary is regarded as a free surface. The underlying cncepts are clarified by the fllwing abstract descriptin f the adjint methd. 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 I = I (w, F, and a change in F results in a change [ ] I T δi = w 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 assists in gruping the numerus terms that arise during the derivatin f the full Navier Stkes adjint peratr, utlined later, s that the basic structure f the apprach as it is sketched in the present sectin can easily be recgnized. Suppse that the gverning equatin R which expresses the dependence f w and F within the flwfield dmain can be written as R (w, F = 0. (2 Then δw is determined frm the equatin δr = [ ] R δw w I [ ] R δf = 0. (3 F II Since the variatin δr is zer, it can be multiplied by a Lagrange Multiplier ψ and subtracted frm the variatin δi withut changing the result. Thus equatin (1 can be replaced by Chsing ψ t satisfy the adjint equatin ([ δi = IT IT R δw w F δf ψt w { I T [ ] } { R I T = w ψt δw w I ] δw [ R ] δf F ] } F ψt [ R F II δf. (4 [ ] T R ψ = I w w (5 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. In reference [8] Jamesn derived the adjint equatins fr transnic flws mdeled by bth the ptential flw equatin and the Euler equatins. The thery was develped in terms f partial differential equatins, leading t an adjint partial differential equatin. In rder t btain numerical slutins bth the flw and the adjint equatins must be discretized. Cntrl thery might be applied directly t the discrete flw equatins which result frm the numerical apprximatin f the flw equatins by finite element, finite vlume r finite difference prcedures. This leads directly t a set f discrete adjint equatins with a matrix which is the transpse f the Jacbian matrix f the full set f discrete nnlinear flw equatins. On a three-dimensinal mesh with indices i, j, k the individual adjint equatins may be derived by cllecting tgether all the terms multiplied by the variatin δw i,j,k f the discrete flw variable w i,j,k. The resulting discrete adjint equatins represent a pssible discretizatin f the adjint partial differential equatin. If these equatins are slved exactly they can prvide an exact gradient f the inexact cst functin which results frm the discretizatin f the flw equatins. The discrete adjint equatins derived directly frm the discrete flw equatins becme very cmplicated when the flw equatins are discretized with higher rder upwind biased schemes using flux limiters. On the ther hand any cnsistent discretizatin f the adjint partial differential equatin will yield the exact gradient in the limit as the mesh is refined. The trade-ff between the cmplexity f the adjint discretizatin, the accuracy f the resulting estimate f the gradient, and its impact n the cmputatinal cst t apprach an ptimum slutin is a subject f nging research. The true ptimum shape belngs t an infinitely dimensinal space f design parameters. One mtivatin fr develping the thery fr the partial differential equatins f the flw is t prvide an indicatin in
principle f hw such a slutin culd be apprached if sufficient cmputatinal resurces were available. It displays the character f the adjint equatin as a hyperblic system with waves travelling in the reverse directin t thse f the flw equatins, and the need fr crrect wall and far-field bundary cnditins. It als highlights the pssibility f generating ill psed frmulatins f the prblem. Fr example, if ne attempts t calculate the sensitivity f the pressure at a particular lcatin t changes in the bundary shape, there is the pssibility that a shape mdificatin culd cause a shck wave t pass ver that lcatin. Then the sensitivity culd becme unbunded. The mvement f the shck, hwever, is cntinuus as the shape changes. Therefre a quantity such as the drag cefficient, which is determined by integrating the pressure ver the surface, als depends cntinuusly n the shape. The adjint equatin allws the sensitivity f the drag cefficient t be determined withut the explicit evaluatin f pressure sensitivities which wuld be ill psed. Anther benefit f the cntinuus adjint frmulatin is that it allws grid sensitivity terms t be eliminated frm the gradient, which can finally be expressed purely in terms f the bundary displacement, as will be shwn in Sectin 4. This greatly simplifies the implementatin f the methd fr verset r unstructured grids. The discrete adjint equatins, whether they are derived directly r by discretizatin f the adjint partial differential equatin, are linear. Therefre they culd be slved by direct numerical inversin. In three-dimensinal prblems n a mesh with, say, n intervals in each crdinate directin, the number f unknwns is prprtinal t n 3 and the bandwidth t n 2. The cmplexity f direct inversin is prprtinal t the number f unknwns multiplied by the square f the bandwidth, resulting in a cmplexity prprtinal t n 7. The cst f direct inversin can thus becme prhibitive as the mesh is refined, and it becmes mre efficient t use iterative slutin methds. Mrever, because f the similarity f the adjint equatins t the flw equatins, the same iterative methds which have been prved t be efficient fr the slutin f the flw equatins are efficient fr the slutin f the adjint equatins. 3 esign using the Euler Equatins The applicatin f cntrl thery t aerdynamic design prblems is illustrated in this sectin fr the case f three-dimensinal wing design using the cmpressible Euler equatins as the mathematical mdel. 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 w t f i x i = 0 in, (7 where 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 (8 and δ ij is the Krnecker delta functin. Als, { p = (γ 1 ρ E 1 ( } u 2 2 i, (9 and ρh = ρe p (10 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 [ ] [ ] xi K ij =, J = det (K, Kij 1 ξi =, ξ j x j
and S = JK 1. 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 ɛ ijk we can express the elements f S as Then S ij = 1 2 ɛ jpqɛ irs x p ξ r x q ξ s. (11 where S ij = 1 ( 2 ξ i 2 ɛ x p x q jpqɛ irs x p 2 x q ξ r ξ i ξ s ξ r ξ s ξ i = 0. (12 Als in the subsequent analysis f the effect f a shape variatin it is useful t nte that S 1j = ɛ jpq x p ξ 2 x q ξ 3, S 2j = ɛ jpq x p ξ 3 x q ξ 1, Nw, multiplying equatin(7 by J and applying the chain rule, S 3j = ɛ jpq x p ξ 1 x q ξ 2. (13 J w t R (w = 0 (14 f j R (w = S ij = (S ij f j, (15 ξ i ξ i using (12. We can write the transfrmed fluxes in terms f the scaled cntravariant velcity cmpnents as U i = S ij u j ρu i ρu i u 1 S i1 p F i = S ij 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 ij. 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 (14. 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. (16
This can be split as with δi = δi I δi II, (17 δm = [M w ] I δw δm II, δp = [P w ] I δw δp II, (18 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 (14 specifies the variatin f the state vectr δw by δr = ξ i δf i = 0. (19 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. (20 f i [F iw ] I = S ij 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 ψ T δf i d ξ = 0. (21 ξ i Assuming that ψ is differentiable, the terms with subscript I may be integrated by parts t give n i ψ T ψ T δf ii d ξ δf ii d ξ ψ T δr II d ξ = 0. (22 ξ i 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 (16 t give δi = δi II ψ T [ δr II d ξ δmi n i ψ T ] δf ii dξ ] [δp I ψt δf ii d ξ. (23 ξ i 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 (18 and (20, the variatin δw may be eliminated frm (23 by equating all field terms with subscript I t prduce a differential adjint system gverning ψ ψ T ξ i [F iw ] I [P w ] I = 0 in. (24 Taking the transpse f equatin (24, 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 ξ i = 0 in, (25
where the inviscid Jacbian matrices in the transfrmed space are given by C i = S ij f j w. The crrespnding adjint bundary cnditin is prduced by equating the subscript I bundary terms in equatin (23 t prduce n i ψ T [F iw ] I = [M w ] I n. (26 The remaining terms frm equatin (23 then yield a simplified expressin fr the variatin f the cst functin which defines the gradient δi = δi II ψ T δr II d ξ, (27 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 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 (26. 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 (23. 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 (24 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 (23 and (26 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 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 0 0 S 21 δs 21 δf 2 = δp p. (28 S 22 δs 22 S 23 0 δs 23 0
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 where n j are the cmpnents f the surface nrmal ψ j n j = p p d (29 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. We find finally that δi = W ψ T ξ i δs ij f j d (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 p dξ 1 dξ 3. (30 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 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. 4 The Reduced Gradient Frmulatin Cnsider the case f a mesh variatin with a fixed bundary. Then, but there is a variatin in the transfrmed flux, δi = 0 δf i = C i δw δs ij f j. Here the true slutin is unchanged. Thus, the variatin δw is due t the mesh mvement δx at each mesh pint. Therefre δw = w δx = w x j δx j (= δw
and since it fllws that ξ i δf i = 0, (δs ij f j = (C i δw. (31 ξ i ξ i It is verified belw that this relatin hlds in the general case with bundary mvement. Nw φ T δrd = φ T C i (δw δw d ξ i = φ T C i (δw δw d Here n the wall bundary φ T ξ i C i (δw δw d. (32 C 2 δw = δf 2 δs 2j f j. (33 Thus, by chsing φ t satisfy the adjint equatin (25 and the adjint bundary cnditin (26, we reduce the cst variatin t a bundary integral which depends nly n the surface displacement: δi = ψ T (δs 2j f j C 2 δw dξ 1 dξ 3 W (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 p dξ 1 dξ 3. (34 W Fr cmpleteness the general derivatin f equatin(31 is presented here. Using the frmula(11, and the prperty (12 ξ i (δs ij f j = 1 2 ξ i { ( δxp ɛ jpq ɛ irs ξ r x q x q x p δx q ξ s ξ r ξ s fj = 1 ( 2 ɛ δxp jpqɛ irs x p δx q ξ r ξ s ξ r ξ s = 1 { ( } 2 ɛ x q f j jpqɛ irs δx p ξ r ξ s ξ i 1 { ( } 2 ɛ x p f j jpqɛ irs δx q ξ s ξ r ξ i = ( x q f j δx p ɛ pqj ɛ rsi ξ r ξ s ξ i ξ i f j } Nw express δx p in terms f a shift in the riginal cmputatinal crdinates δx p = x p ξ k δξ k.. (35 Then we btain The term in ξ 1 is ξ i (δs ij f j = ξ r ( x p x q f j ɛ pqj ɛ rsi δξ k. (36 ξ k ξ s ξ i ( x p xq f j ɛ 123 ɛ pqj 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 ɛ jpq x p x q f j. ξ 1 ξ 2 ξ 3 ξ 1 ξ 3 ξ 2 Accrding t the frmulas(13 this may be recgnized as S 2j f 1 ξ 2 S 3j f 1 ξ 3 r, using the quasi-linear frm(15 f the equatin fr steady flw, as S 1j f 1 ξ 1. The terms multiplying δξ 2 and δξ 3 are ( xp x q f j ɛ jpq x p x q f j ξ 2 ξ 2 ξ 3 ξ 2 ξ 3 ξ 2 and Thus the term in ξ 1 is reduced t Finally, with similar reductins f the terms in as was t be prved. = S 1j f 1 ξ 2 ( xp x q f j ɛ jpq x p x q f j f 1 = S 1j. ξ 3 ξ 2 ξ 3 ξ 3 ξ 3 ξ 2 ξ 3 ξ i (δs ij f j = ξ i ( f 1 S 1j δξ k. ξ 1 ξ k ξ 2 and ξ 3, we btain ( S ij f j ξ k δξ k = ξ i (C i δw 5 Optimizatin Prcedure 5.1 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. Suppse that we wish t find the path y(x which minimizes I = b a F (y, y dx with fixed end pints y(a and y(b. Under a variatin δy(x, δi = = b a b a ( F F δy δy dx y y ( F y d F δydx dx y
Thus defining the gradient as g = F y d F dx y and the inner prduct as b (u, v = uvdx a we find that δi = (g, δy. If we nw set δy = λg, λ > 0 we btain a imprvement 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 rachistrne prblem, fr example, which calls fr the determinatin f the path f quickest descent between tw laterally separated pints when a particle falls under gravity, F (y, y 1 y 2 = y and It can be seen that each step g = 1 y 2 2yy 2 (y(1 y 2 3/2 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 [24] u, v = (uv ɛu v dx where ɛ is a parameter that cntrls the weight f the derivatives. We nw define a gradient g such that δi = g, δy Then we have δi = (gδy ɛg δy dx = (g x ɛ g x δydx = (g, δy
where g x ɛ g x = g and g = 0 at the end pints. Thus g can be btained frm g by a smthing equatin. Nw the step gives an imprvement y n1 = y n λ n g n δi = λ n g n, g n but y n1 has the same smthness as y n, resulting in a stable prcess. 5.2 Sblev gradient fr shape ptimizatin 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 ɛ Ḡ = G. (37 ξ 1 ξ 1 In the multi-dimensinal case the smthing is applied in prduct frm. Finally we set with the result that δf = λḡ (38 δi = λ < Ḡ, Ḡ > < 0, unless Ḡ = 0, and crrespndingly G = 0. When secnd-rder central differencing is applied t (37, the equatin at a given nde, i, can be expressed as Ḡ i ɛ ( Ḡ i1 2Ḡi Ḡi 1 = Gi, 1 i n, where G i and Ḡi 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, where A is the n n tri-diagnal matrix such that 1 2ɛ ɛ 0. 0 ɛ.. A 1 = 0...... ɛ. 0 ɛ 1 2ɛ Using the steepest descent methd in each design iteratin, a step, δf, is taken such that δf = λag. (39 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. Our sftware als includes an ptin fr Krylv acceleratin [25]. We have fund this t be particularly useful fr inverse prblems.
5.3 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 esign Cycle until Cnvergence Sblev Gradient Shape & Grid Mdificatin Fig. 1. esign 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 8 have been btained using wellvalidated sftware fr the slutin f the Euler and Navier-Stkes equatins develped ver the curse f many years [26 28]. 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 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.
6 esign using the Navier-Stkes Equatins 6.1 The Navier-Stkes equatins in the cmputatinal dmain The next sectins present the extensin f the adjint methd t the Navier-Stkes equatins. These take the frm w t f i = f vi in, (40 x i x i where the state vectr w, inviscid flux vectr f and viscus flux vectr f v are described respectively by ρ ρu i 0 ρu 1 ρu i u 1 pδ i1 σ ij δ j1 w = ρu 2, f i = ρu i u 2 pδ i2, f vi = σ ij δ j2. (41 ρu 3 ρu i u 3 pδ i3 σ ij δ j3 ρe ρu i H u j σ ij k T x i The viscus stresses may be written as ( ui σ ij = µ u j u k λδ ij, (42 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, (43 where P r is the Prandtl number, c p is the specific heat at cnstant pressure, and R is the gas cnstant. Using a transfrmatin t a fixed cmputatinal dmain as befre, the Navier-Stkes equatins can be written in the transfrmed crdinates as (Jw t (F i F vi ξ i = 0 in, (44 where the viscus terms have the frm F vi = ( Sij f v j. ξ i ξ i 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 (19 t (22, we btain ψ T ( δs 2j f v j S ψ 2jδf v j T ( dξ δsij f v j ξ S ijδf v j dξ, i 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 [14,29]. 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. This simplificatin has been successfully used fr may aerdynamic prblems f interest. Hwever, if the flw variatins culd result in significant changes in the turbulent viscsity, it may be necessary t accunt fr its variatin in the calculatin. 6.2 Transfrmatin t Primitive Variables The derivatin f the viscus adjint terms can be 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 κ ( p. x i ρ where κ = k R = γµ P r(γ 1. The relatinship between the cnservative and primitive variatins is defined by the expressins δw = Mδ w, δ w = M 1 δw which make use f the transfrmatin matrices M = w w transpsed frm fr future cnvenience u 1 u 1 u 2 u iu i 3 2 0 ρ 0 0 ρu 1 M T = 0 0 ρ 0 ρu 2 0 0 0 ρ ρu 3 0 0 0 0 1 γ 1 u2 ρ u3 ρ 1 u1 ρ 1 0 M 1T ρ 0 0 (γ 1u 1 = 1 0 0 ρ 0 (γ 1u 2. 1 0 0 0 ρ (γ 1u 3 0 0 0 0 γ 1 and M 1 = w w. These matrices are prvided in (γ 1u iu i 2 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, with L = M T L, s that after determining the primitive adjint peratr by direct evaluatin f the viscus prtin f (24, 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. 6.3 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 φ k (δs 2j σ kj S 2j δσ kj d ξ (δs ij σ kj S ij δσ kj d ξ. (45 ξ i The velcity derivatives can be expressed as with crrespnding variatins δ u i x j = 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 ξ l δu j λ { [ ( Slj µ δ uk J δ ( S lk ] uj J ξ l λ ξ l ξ l II δ S δ lm jk J [ ( Slj J. ξ l δu m ]}I δ jk δ ( S lm J 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 (45 appear as { ( φ k Slj S ij µ ξ i J δu k S lk ξ l 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 S lm δu j λδ jk ξ l J φ k ξ i φ k ξ i ( S lm S ij λδ jk J φ k ξ i d ξ d ξ d ξ, } δu m d ξ. ξ l 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 S im φ m λδ jk d ξ, ξ l J ξ i J ξ i J ξ i 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. (46 ξ l x j x k x m The bundary cntributins that cntain δu i in equatin (45 may be simplified using the fact that n the bundary s that they becme ( S2j φ k S 2j {µ J ξ l δu i = 0 if l = 1, 3 δu k S 2k ξ 2 J S 2m δu j λδ jk ξ 2 J } δu m d ξ. (47 ξ 2 Tgether, (46 and (47 cmprise the field and bundary cntributins f the mmentum equatins t the viscus adjint peratr in primitive variables. 6.4 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 (43. The cntributin frm the energy equatin can then be written as θ θ (δs 2j Q j S 2j δq j d ξ (δs ij Q j S ij δq j d ξ. (48 ξ i The field cntributins that cntain δu i,δp, and δρ in equatin (48 appear as θ S ij δq j d ξ = ξ i { θ S ij δu k σ kj u k δσ kj κ S lj ξ i J ξ l ( δp ρ p ρ } δρ d ξ. (49 ρ
The term invlving δσ kj may be integrated by parts t prduce { ( } θ θ θ δu k S lj µ u k u j λδ jk u m d ξ, (50 ξ l x j x k x m where the cnditins u i = δu i = 0 are used t eliminate the bundary integral n. Ntice that the ther term in (49 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 (51 ξ i The terms in expressin (49 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 ξ i which may be simplified by transfrming the inner derivative t Cartesian crdinates ( δp ρ p ( δρ S lj κ θ d ξ. (52 ρ ρ ξ l x j The bundary integral becmes ( δp κ ρ p ρ δρ S2j S ij ρ J θ ξ i d ξ. (53 This can be simplified by transfrming the inner derivative t Cartesian crdinates ( δp κ ρ p δρ S2j θ d ξ, (54 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, (55 x j ( δp ρ p ρ δρ, ρ kδt θ n d ξ. (56 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 (48 which has the frm θδ (S 2j Q j d ξ, (57 where since u i = 0. If the wall is adiabatic s that using (55, Q j = k T x j, T n = 0, δ (S 2j Q j = 0,
and bth the δw and δs bundary cntributins vanish. On the ther hand, if T is cnstant T ξ l = 0 fr l = 1, 3, s that Q j = k T ( ( Sl j T S2j T = k = k. x j J ξ l J ξ 2 Thus, the bundary integral (57 becmes { 2 S2j kθ δt δ J ξ 2 ( S2j 2 J T ξ 2 } d ξ. (58 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. Finally the cntributins frm the energy equatin t the viscus adjint peratr are the three field terms (50, (51 and (52, and either f tw bundary cntributins ( 56 r ( 58, depending n whether the wall is adiabatic r has cnstant temperature. 6.5 The Viscus Adjint Field Operatr 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 ξ l S lj κ θ x j ( Lψ i1 = ξ l { ( Lψ 5 { ξ l σ ij S lj θ ( ξ l = 1 ρ ξ l ( S lj [µ φi ( x j S lj [µ S lj κ θ u i θ x j x j. ]} φj φ x i λδ k ij x k ]} θ θ u j x i λδ ij u k x k fr i = 1, 2, 3 The cnservative viscus adjint peratr may nw be btained by the transfrmatin L = M 1T L. 7 Viscus Adjint undary Cnditins It was recgnized in Sectin 3 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. Cnsequently the cntributins f the pressure and viscus stresses need t be merged. As in the derivatin f the field equatins, it prves cnvenient t cnsider the cntributins frm the mmentum equatins and the energy equatin separately. 7.1 undary Cnditins Arising frm the Mmentum Equatins The bundary term that arises frm the mmentum equatins including bth the δw and δs cmpnents (45 takes the frm φ k δ (S 2j (δ kj p σ kj d ξ.
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 ttal 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 (δ kj p σ kj d ξ. τ k = n j (δ kj p 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 δτ k ds, (59 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, (60 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. The frce in a directin with csines q i has the frm C q = q i τ i ds. If we take this as the cst functin (60, this quantity gives N = q i τ i. Cancellatin with the flw variatin terms in equatin (59 therefre mandates the adjint bundary cnditin φ k = q k. Nte that this chice f bundary cnditin als eliminates the first term in equatin (59 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 φ k δτ k = a lk (τ l τ dl δτ k.
This is satisfied by the bundary cnditin φ k = a lk (τ l τ dl. (61 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 τ n = n k n j σ kj p p d, N (τ = 1 2 τ n 2 = 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 (61. efining the viscus nrmal stress as τ vn = n k n j σ kj, the measure can be expanded as 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 vn 2 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, (62 which shuld be cmpared with the single scalar equatin derived fr the inviscid bundary cnditin (29. In the case f an inviscid flw, chsing N (τ = 1 2 (p p d 2 requires φ k n k δp = (p p d n 2 kδp = (p p d δp which is satisfied by equatin (62, but which represents an verspecificatin f the bundary cnditin since nly the single cnditin (29 needs be specified t ensure cancellatin.
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 (56 kδt θ n d ξ, while fr the cnstant temperature case the bundary term is (58. One pssibility is t intrduce a cntributin int the cst functin which depends n 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 (57. 8 Results 8.1 Redesign f the eing 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 eing 747-200 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 aldwin 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 2 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 0.47. 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 3 displays the result f an ptimizatin at Mach 0.90. 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 0.86. The eing 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 4 and 5 verify that the span lading and thickness were nt changed by the redesign, while figures 6 and 7 indicate the required sectin changes at 42 percent and 68 percent span statins.
8.2 Wing 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 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 wing-pyln-nacelle cmbinatins. Sme preliminary results are presented belw. These have been btained with new sftware t implement the adjint methd fr unstructured meshes which is currently under develpment [30]. Figures 8 and 9 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 10, 11, 12, 13, 14, 15, 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. Anther test case fr the inverse design prblem uses the wing frm an airplane (cde named SHARK [31] which has been designed fr the Ren Air Races. The initial and final pressure distributins are shwn the figure 16. As can be seen frm these plts, the initial pressure distributin has a weak shck in the utbard sectins f the wing, while the final pressure distributin is shck-free. The final pressure distributins are cmpared with the target distributins alng three sectins f the wing in figures 17, 18, 19. Again the design prcess captures the target pressure with gd accuracy in abut 50 design cycles. The drag minimizatin prblem has als been studied fr this wing, and the results are shwn in figure 20. As can be seen frm this plt, the final gemetry has a shck-free prfile and the drag cefficient has been slightly reduced. 9 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, 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 F49620-98-1-2002. I have drawn extensively frm the lecture ntes prepared by Luigi Martinelli and myself fr a CIME summer curse in 1999 [32]. I am als indebted t my research assistant Kasidit Leviriyakit fr his assistance in preparing the Latex files fr this text. References 1. R. M. Hicks, E. M. Murman, and G. N. Vanderplaats. An assessment f airfil design by numerical ptimizatin. NASA TM X-3092, Ames Research Center, Mffett Field, Califrnia, July 1974. 2. R. M. Hicks and P. A. Henne. Wing design by numerical ptimizatin. Jurnal f Aircraft, 15:407 412, 1978. 3. J. L. Lins. Optimal Cntrl f Systems Gverned by Partial ifferential Equatins. Springer-Verlag, New Yrk, 1971. Translated by S.K. Mitter. 4. A. E. rysn and Y. C. H. Applied Optimal Cntrl. Hemisphere, Washingtn, C, 1975. 5. M. J. Lighthill. A new methd f tw-dimensinal aerdynamic design. R & M 1111, Aernautical Research Cuncil, 1945. 6. O. Pirnneau. Optimal Shape esign fr Elliptic Systems. Springer-Verlag, New Yrk, 1984.