A COMPARISON OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACH TO AUTOMATIC AERODYNAMIC OPTIMIZATION

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1 AIAA--667 A COMPARISON OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACH TO AUTOMATIC AERODYNAMIC OPTIMIZATION Siva K. Nadarajah and Antny Jamesn Department f Aernautics and Astrnautics Stanfrd University Stanfrd, Califrnia 9435 U.S.A. Abstract This paper cmpares the cntinuus and discrete adjint-based autmatic aerdynamic ptimizatin. The bjective is t study the trade-ff between the cmplexity f the discretizatin f the adjint equatin fr bth the cntinuus and discrete apprach, the accuracy f the resulting estimate f the gradient, and its impact n the cmputatinal cst t apprach an ptimum slutin. First, this paper presents cmplete frmulatins and discretizatin f the Euler equatins, the cntinuus adjint equatin and its cunterpart the discrete adjint equatin. The differences between the cntinuus and discrete bundary cnditins are als explred. Secnd, the results demnstrate tw-dimensinal inverse pressure design and drag minimizatin prblems as well as the accuracy f the sensitivity derivatives btained frm cntinuus and discrete adjint-based equatins cmpared t finite-difference gradients. Intrductin In the 97s several attempts were made t use Cmputatinal Fluid Dynamics (CFD as a design tl. 3 Since then CFD has had a significant impact. Many individuals have refcused their attentin n autmatic aerdynamic ptimizatin, because f accurate numerical schemes and an expnential grwth in cmputatinal speed at affrdable prices. The mathematical thery fr cntrl systems gverned by partial differential equatins has created a framewrk fr the frmulatin f inverse design and general aerdynamic prblems at a reduced cmputatinal cst. 4, 5 Recently, with the help f Graduate Student, Student Member AIAA Thmas V. Jnes Prfessr f Engineering, Stanfrd University, AIAA Fellw Cpyright c 999 by Siva Nadarajah and Antny Jamesn a new generatin f cmputers, autmatic aerdynamic ptimizatin has been revisited. 6 Optimizatin techniques fr the design f aerspace vehicles generally use gradient-based methds in which the vehicle shape is parameterized with a set f design variables. A feasible ptimum shape is nly achievable with an apprpriate cst functin. Typically such cst functins are drag cefficients, lift t drag ratis, target pressure distributins, etc. Sensitivity derivatives f the cst functin with respect t the design variables are calculated by taking small steps in each and every design variable. These sensitivity derivatives are then used t get a directin f imprvement and a step is taken until cnvergence is achieved. Each step requires a cmplete flw slutin, and fr a large number f design variables such methds are cmputatinally cstly. The mathematical thery fr the cntrl systems gverned by partial differential equatins, as develped, fr example, by J.L. Lins, 4 decreases the cst and is mre advantageus than the classical finite-difference methds. In cntrl thery the gradient is calculated indirectly by slving the adjint equatin. The cst f btaining the sensitivity derivatives f the cst functin with respect t each design variable frm the slutin f the adjint equatin is negligible in cmparisn with the cst f the flw calculatin. Cnsequently, the ttal cst t btain these gradients is essentially independent f the number f design variables, amunting t the cst f ne flw slutin and ne adjint slutin, where the adjint equatin is a linear equatin and thus f reduced cmplexity. This methd was first applied t transnic flw by Jamesn. 6 In the last six years autmatic aerdynamic design f cmplete aircraft cnfiguratins has been successful, yielding ptimized slutins f wing 4, 6 and wing-bdy cnfiguratins. The cntinuus adjint apprach thery was develped by cmbining the variatin f the cst func- American Institute f Aernautics and Astrnautics

2 tin and field equatins with respect t the flw-field variables and design variables thrugh the use f Lagrange multipliers als called cstate r adjint variables. Cllecting the terms assciated with the variatin f the flw-field variables prduces the adjint equatin and its bundary cnditin. The terms assciated with the variatin f the design variable prduce the gradient. The field equatins and the adjint equatin with its bundary cnditin must be discretized t btain numerical slutins. As the mesh is refined, the cntinuus adjint yields the exact gradient. The discrete adjint apprach means applying the cntrl thery directly t the set f discrete field equatins. The discrete adjint equatin is derived by cllecting tgether all the terms multiplied by the variatin δw f the discrete flw variable. If the discrete adjint equatin is slved exactly, then the resulting slutin fr the Lagrange multiplier prduces an exact gradient f the inexact cst functin and the derivatives are cnsistent with finite difference gradients independent f the mesh size. A subject f n-ging research is 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. Shubin and Frank presented a cmparisn between the cntinuus and discrete adjint fr quasi-ne-dimensinal flw. A variatin f the discrete field equatins prved t be cmplex fr higher rder schemes. Due t this limitatin f the discrete adjint apprach, early implementatin f the discretizatin f the adjint equatin was nly cnsistent with a first rder accurate flw equatin. Beux and Dervieux 5 used a first rder upwind scheme with Van Leer flux vectr splitting n a tw-dimensinal unstructured grid. Burgreen and Baysal 6 carried a secnd rder implementatin f the discrete adjint n threedimensinal shape ptimizatin f wings fr structured grids. Fr secnd rder accuracy n unstructured grids, Ellit and Peraire 7 slved the Euler equatins by a multistage Runge-Kutta scheme with Re decmpsitin fr the dissipative fluxes n tw and three-dimensinal unstructured grids. They perfrmed ptimizatin n inverse pressure designs f multielement airfils and wing-bdy cnfiguratins in transnic flw. Andersn and Venkatakrishnan 8 cmputed inviscid and viscus ptimizatin n unstructured grids using bth the cntinuus and discrete adjint. Ill, Salas, and Ta saan 9 investigated shape ptimizatin n ne and twdimensinal flws using the cntinuus adjint apprach. Ta saan, Kuruvila, and Salas used a nesht apprach with the cntinuus adjint frmulatins. Kim, Alns, and Jamesn cnducted an extensive gradient accuracy study f the Euler and Navier-Stkes equatins which cncluded that gradients frm the cntinuus adjint methd were in clse agreement with thse cmputed by finite difference methds, and less dependent n the level f cnvergence f the flw slver. Objectives The bjectives f this wrk are:. Review the frmulatin and develpment f the cmpressible adjint equatins fr bth the cntinuus and discrete apprach.. Investigate the differences in the implementatin f bundary cnditins fr each methd. 3. Cmpare the gradients f the tw methds t finite difference gradients fr inverse pressure design and drag minimizatin. 4. Cmpare the cnvergence between the cntinuus and discrete adjint. 5. Study the differences in calculating the exact gradient f the inexact cst functin (discrete adjint r the inexact gradient f the exact cst functin (cntinuus. The Design Prblem as a Cntrl Prblem A simple apprach t ptimizatin is t represent 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 m b m (x. I α m Next, a cst functin I which is a functin f the weight parameters α m is chsen. Such a cst functin can be the difference between the current and target pressure distributin fr inverse design prblems, drag cefficient fr drag minimizatin prblems, r lift t drag rati. The sensitivities may nw be estimated by making a small variatin δα m in each design parameter in turn and recalculating the flw t btain the change in I. Then, using a finite difference frmula, I I(α m δα m I(α m. α m δα m American Institute f Aernautics and Astrnautics

3 The gradient vectr I α may nw be used t determine a directin f imprvement. The simplest prcedure uses the methd f steepest descent and takes a step in the negative gradient directin by setting α n = α n λ I α, s that t first rder I δi = I IT δα = I λ IT α α I α. The main disadvantageus f the finite difference methd are first that N flw calculatins are needed t calculate the sensitivities f N design variables, and secnd that the accuracy is sensitive t the step size δα m. These difficulties are circumvented by the cntrl thery apprach which may be utlined in abstract frm as fllws. Fr flw arund an airfil, the aerdynamic prperties that define the cst functin are functins f the flw-field variables, w, andthephysical 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 δi = IT IT δw δf, ( F in the cst functin. Using cntrl thery the gverning equatins f the flw-field are nw intrduced as a cnstraint in such a way that the final expressin fr the gradient des nt require reevaluatin f the flw-field. In rder t achieve this δw must be eliminated frm (. Suppse that the gverning equatin R which expresses the dependence f w and F within the flw-field dmain D can be written as R (w, F =. ( Then δw is determined frm the equatin [ ] [ ] R R δr = δw δf =. (3 F Next, intrducing a Lagrange Multiplier ψ, wehave ([ ] [ ] I T IT R R δi = δw F δf ψt δw δf F { [ ]} { [ ]} I T R I T R = ψt δw F ψt δf. F Chsing ψ t satisfy the adjint equatin [ ] T R ψ = I (4 the first term is eliminated, and we find that where δi = GδF, (5 G = IT F ψt [ ] R. F Euler Equatins In rder t allw fr gemetric shape changes it is cnvenient t use a bdy fitted crdinate system, s that the cmputatinal dmain is fixed. This requires the frmulatin f the Euler equatins in the transfrmed crdinate system. The Cartesian crdinates and velcity cmpnents are dented by x, x,andu, u. Einstein ntatin simplifies the presentatin f the equatins, where summatin ver k = t is implied by a repeated index k. Then the tw-dimensinal cmpressible Euler equatins may be written as t f k = ind, (6 x k where w = ρ ρu ρu ρe, f k = ρu k ρu k u pδ k ρu k u pδ k ρu k H (7 and δ kl is the Krnecker delta functin. Als, { p =(γ ρ E ( } u k, (8 and ρh = ρe p (9 where γ is the rati f the specific heats. Cnsider a transfrmatin t crdinates ξ, ξ, where [ xk K kl = ξ l and ], J =det(k, K kl = S = JK. [ ξk x l ], The elements f S are the cefficients f K, andin a finite vlume discretizatin they are just the face 3 American Institute f Aernautics and Astrnautics

4 areas f the cmputatinal cells prjected in the x and x directins. Als intrduce scaled cntravariant velcity cmpnents as U k = S kl u l. The Euler equatins can nw be written as where and W t F k ξ k = ind, ( F k = S kl f l = W = Jw, ρu k ρu k u S l p ρu k u S l p ρu k H. ( Assume nw that the new cmputatinal crdinate system cnfrms t the airfil in such a way that the airfil surface B W is represented by ξ =. Then the flw is determined as the steady state slutin f equatin ( subject t the flw tangency cnditin U = nb W. ( At the far field bundary B F, cnditins are specified fr incming waves, as in the tw-dimensinal case, while utging waves are determined by the slutin. When equatin ( is frmulated fr each cmputatinal cell, a system f first-rder rdinary differential equatins is btained. T eliminate dd-even decupling f the slutin and vershts befre and after shck waves, the cnservative flux is added t a diffusin flux. The artificial dissipatin scheme used in this research is a blended first and third rder flux, first intrduced by Jamesn, Schmidt, and Turkel. The artificial dissipatin scheme is defined as, D i,j = ɛ i,j(w i,j w ɛ 4 i,j(w 3w i,j 3w w i,j. (3 The first term in equatin (3 is a first rder scalar diffusin term, where ɛ is scaled by the i,j nrmalized secnd difference f the pressure and serves t damp scillatins arund shck waves. ɛ 4 is the cefficient fr the third derivative f the i,j artificial dissipatin flux. The cefficient is scaled such that it is zer at regins f large gradients, such as shck waves and eliminates dd-even decupling elsewhere. Design using the Euler Equatins This sectin illustrates applicatin f cntrl thery t aerdynamic design prblems fr the case f tw-dimensinal airfil design using the cmpressible Euler equatins as the mathematical mdel. Cntinuus Adjint The weak frm f the Euler equatins fr steady flw is φ T F k dd = n k φ T F k db, (4 D ξ k B where the test vectr φ is an arbitrary differentiable functin and n k is the utward nrmal at the bundary. If a differentiable slutin w is btained t this equatin, then it can be integrated by parts t give φ T F k dd =. ξ k D Since this is true fr any φ the differential frm can be recvered. If the slutin is discntinuus, then (4 may be integrated by parts separately n either side f the discntinuity t recver the shck jump cnditins. Suppse nw that we desire t cntrl the surface pressure by varying the wing shape. Fr this purpse, it is cnvenient t retain a fixed cmputatinal dmain. Variatins in the shape then result in crrespnding variatins in the mapping derivatives defined by K. Intrduce the cst functin I = (p p d ds, B W where p d is the desired pressure. The design prblem is nw treated as a cntrl prblem where the cntrl functin is the wing shape, which is chsen t minimize I subject t the cnstraints defined by the flw equatins (-. A variatin in the shape causes a variatin δp in the pressure and cnsequently a variatin in the cst functin δi = (p p d δp ds (p p d δds. B W B W (5 Since p depends n w thrugh the equatin f state (8 9, the variatin δp is determined frm the variatin δw. Define the Jacbian matrices A k = f k, C k = S kl A l. (6 4 American Institute f Aernautics and Astrnautics

5 The weak frm f the equatin fr δw in the steady state becmes φ T δf k dd = (n k φ T δf k db, ξ k where D B δf k = C k δw δs kl f l, which shuld hld fr any differentiable test functin φ. This equatin may be added t the variatin in the cst functin, which may nw be written as δi = (p p d δp ds (p p d δds B W B W ψ T δf k dd (n k ψ T δf k db (7 ξ k D On the wing surface B W, n =. Thus, it fllws frm equatin ( that S δf = δp δs p. (8 S δp δs p Since the weak equatin fr δw shuld hld fr an arbitrary chice f the test vectr φ, wearefree t chse φ t simplify the resulting expressins. Therefre we set φ = ψ, where the cstate vectr ψ is the slutin f the adjint equatin ψ t CT k B ψ ξ k = ind. (9 At the uter bundary incming characteristics fr ψ crrespnd t utging characteristics fr δw. Cnsequently we can chse bundary cnditins fr ψ such that n k ψ T C k δw =. If the crdinate transfrmatin is such that δs is negligible in the far field, then the nly remaining bundary term is ψ T δf dξ. B W Thus, by letting ψ satisfy the bundary cnditin, ψ j n j = p p d n B W, ( where n j are the cmpnents f the surface nrmal, n j = S j Sj S j we find finally that δi = (p p d δds B W ψ T δs kl f l dd D ξ k (δs ψ δs ψ 3 pdξ. ( B W Numerical Discretizatin The cntinuus adjint equatin is linear and cnsequently it culd be slved by direct numerical inversin. The cst f the assciated matrix inversin can becme prhibitive as the number f mesh cells are increased. Instead, since the equatins are similar t that f the Euler equatins, the same iterative methd is used t slve the cntinuus adjint equatin. In this research, a five stage Runge-Kutta scheme with three evaluatins f the artificial dissipatin scheme is used. We emply the blended first and third rder scalar diffusin scheme used fr the Euler equatins here as well. The fllwing is a secnd rder discretizatin f the cntinuus adjint equatin, ( V ψ [ ] T [ ] T f = y η x η t ( [ ] T [ ] T f y η x η ( [ ] T [ ] T f y ξ x ξ ( [ ] T [ ] T f y ξ x ξ d i,j d i,j d d ( ψi,j ( ψi,j ( ψ ( ψ ( where, V is the cell area and d i,j has the same frm as equatin (3. In the case f the cntinuus adjint bundary cnditin, equatin ( dictates values fr the nrmal adjint velcities. The chice fr ψ, ψ 4,and the tangential adjint velcity are arbitrary, therefre assigning a zer value fr these variables des 5 American Institute f Aernautics and Astrnautics

6 nt vilate equatin (. This results, hwever, in a pr cnvergence fr the adjint equatin since it is an ver-specificatin f the adjint bundary cnditin. A satisfactry bundary cnditin may be frmulated as fllws: ψ i, = ψ i, ( ψ i, = ψ i, n (p pd n ψ i, n ψ 3i, ( ψ 3i, = ψ 3i, n (p pd n ψ i, n ψ 3i, ψ 4i, = ψ 4i, (3 Discrete Adjint The discrete adjint equatin is btained by applying the cntrl thery directly t the set f discrete field equatins. The resulting equatin depends n the type f scheme used t slve the flw equatins. This paper uses, a cell centered multigrid scheme with upwind biased blended first and third rder fluxes as the artificial dissipatin scheme. A full discretizatin f the equatin wuld invlve discretizing every term that is a functin f the state vectr. where, n i = S i Sj S j nx ny ( δi = δi c ψ T δ R (w D (w i= j= (5 The subscript i, andi, in the abve equatins dente cells belw and abve the wall. Here, the first and furth cstate variables belw the wall are set equal t the crrespnding values abve the wall and the tangential adjint velcities abve and belw the wall are equated. Drag Minimizatin If the drag is t be minimized, then the cst functin is the drag cefficient, I = C ( d y = C p c B ( W c ξ dξ B W C p x ξ dξ cs α sin α A variatin in the shape causes a variatin p in the pressure and cnsequently a variatin in the cst functin, δi = c c B W C p B W C p ( y x cs α ξ ( y cs α δ ξ ( δ ξ sin α ( x ξ pdξ sin α dξ (4 As in the inverse design case, the first term is a functin f the state vectr, and therefre is incrprated int the bundary cnditin, where the integrand replaces the pressure difference term in equatin (3. The secnd term is added n t the gradient term. where δi c is the discrete cst functin, R(w isthe field equatin, and D(w is the artificial dissipatin term. Terms multiplied by the variatin δw f the discrete flw variables are cllected and the fllwing is the resulting discrete adjint equatin, V ψ = ( t [ ] T [ ] T f y ηi x ηi,j,j ( [ ] T [ ] T f y ηi x ηi,j,j ( [ ] T [ ] T f x ξ y ξ ( [ ] T [ ] T f x ξ y ξ ( [ ] T [ ] T f y ηi x ηi,j,j ( [ ] T [ ] T f y ηi x ηi,j,j ( [ ] T [ ] T f x ξ y ξ ( [ ] T [ ] T f x ξ y ξ ψ i,j ψ i,j ψ ψ ψ ψ ψ ψ δd i,j δd i,j δd δd. (6 6 American Institute f Aernautics and Astrnautics

7 where, δd i,j = ɛ i,j(ψ i,j ψ ɛ 4 i 3,jψ 3ɛ 4 i,j (ψ i,j ψ ɛ 4 i 3,jψ i,j (7 is the discrete adjint artificial dissipatin term and V is the cell area. The dissipatin cefficients ɛ and ɛ 4 are functins f the flw variables, but t reduce cmplexity they are treated as cnstants. The effect f this partial discretizatin f the artificial dissipatin term is explred in the Results sectin. In the case f an inverse design, δi c is the discrete frm f equatin (5. The δw i, term is added t the crrespnding term frm equatin (6, and the metric variatin term is added t the gradient term. In cntrast t the cntinuus adjint, where the bundary cnditin appears as an update t the cstate variables in the cell belw the wall, the discrete bundary cnditin appears as a surce term in the adjint fluxes. At cell i, the adjint equatin is as fllws, V ψ i, = t [ ] A T i, (ψ i, ψ i, A T i, (ψ i, ψ i, [ ] B T i, (ψ 5 i,3 ψ i, Φ (8 where V is the cell area, Φ is the surce term fr inverse design, Φ= ( y ξ ψ i, x ξ ψ 3i, (p p T s i δpi, and, A T i, = y η i, [ ] T [ ] T f x ηi, i, i, All the terms in equatin (8 except fr the surce term are scaled as the square f x. Therefre, as the mesh width is reduced, the terms within parenthesis in the surce term divided by s i must apprach zer as the slutin reaches a steady state. One then recvers the cntinuus adjint bundary cnditin as stated in equatin (. If a first rder artificial dissipatin equatin is used, then equatin (7 wuld reduce t the term assciated with ɛ. In such a case, the discrete adjint equatins are cmpletely independent f the cstate variables in the cells belw the wall. Hwever, if we use the blended first and third rder equatin, then these values are required. As shwn later, a simple zerth rder extraplatin acrss the wall prduces gd results. Replacing the inverse design bundary cnditin in equatin (8 by the discrete frm f equatin (4 results in a discrete adjint equatin fr drag minimizatin. Optimizatin Prcedure 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 an imprvement can be made with a shape change δf = λg, The gradient G can be replaced by a smthed value G in the descent prcess. This ensures that each new shape in the ptimizatin sequence remains smth and acts as a precnditiner which allws the use f much larger steps. T apply smthing in the ξ directin, the smthed gradient G may be calculated frm a discrete apprximatin t G ɛ G = G ξ ξ where ɛ is the smthing parameter. If the mdificatin is applied n the surface ξ = cnstant, then the first rder change in the cst functin is δi = = λ GδFdξ ( G ɛ ξ ( ( = λ G G ɛ ξ <, G ξ Gdξ dξ assuring an imprvement if λ is sufficiently small and psitive. The smthing leads t a large reductin in the number f design iteratins needed fr cnvergence. An assessment f alternative search methds fr a mdel prblem is given by Jamesn and Vassberg. 3 7 American Institute f Aernautics and Astrnautics

8 Results This sectin presents the results f the inviscid inverse design and drag minimizatin cases. Fr each case, we cmpare the cntinuus and discrete gradients, study the adjint slutins frm each methd, and cmpare the cnvergence f the methds. Grid Size Cnt. Disc. Cnt-Disc 96 x 6 3.6e 3.397e e 4 9 x 3.73e 3.74e 3.3e 4 56 x 64.44e 3.49e e 5 Table : L nrm f the Difference Between Adjint and Finite Difference Gradient Inverse Design The target pressure is first btained using the FLO83flw slver fr the NACA 64A4 airfil at a flight cnditin f M =.74 and a lift cefficient f C l =.63n a 9 x 3 C-grid. At such a cnditin the NACA 64A4 prduces a strng shck n the upper surface f the airfil, thus making it an ideal test case fr the adjint versus finite difference cmparisn. The gradient fr the cntinuus and discrete adjint is btained by perturbing each pint n the airfil. We apply an implicit smthing technique t the gradient, befre it is used t btain a directin f descent fr each pint n the surface f the airfil. Figure ( illustrates an inverse design case f a Krn t NACA 64A4 airfil at fixed lift cefficient. Figure (a shws the slutin fr the Krn airfil at M =.74 and C l =.63. After five design cycles we achieve a general shape f the target airfil as shwn in figure (b. After twenty five design cycles the upper surface shape is btained, and nearly eighty percent f the lwer surface is achieved. Fllwing a few mre iteratins, we btain the desired target pressure except fr a few pints at the trailing edge. Observe the pint-t-pint match-up at the shck. Figures (, (3, and (4 exhibit the values f the gradients btained frm the adjint methds and finite difference fr varius grid sizes. The circles dente values that we btain by using the finite difference methd. The square represents the discrete adjint gradient. The asterisk represents the cntinuus adjint gradient. The gradient is btained with respect t variatins in Hicks-Henne sine bump functins placed alng the upper and lwer surface f the airfil. 3, The figures nly illustrate the values btained frm the upper surface starting frm the leading edge n the left and ending at the trailing edge n the right. In rder t reach an accurate finite difference gradient, we btain gradients fr varius step sizes until the finite difference gradient fr each pint cnverges. The discrete adjint equatin is btained frm the discrete flw equatins but withut taking int accunt the dependence f the dissipa- tin cefficients n the flw variables. Therefre, in rder t eliminate the effect f this n cmparisns with the finite difference gradient we cmpute the flw slutin until attaining a decrease f seven rders f magnitude in the residue. We then freeze the dissipative cefficients and calculate the finite difference value fr each design pint. The figures shw that the nly discrepancies exist in the trailing edge area. Table cntains values f the L nrm f the difference between the adjint and finite difference gradients. The table illustrates three imprtant facts: the difference between the cntinuus adjint and finite difference gradient is slightly greater than that between the discrete adjint and finite difference gradient; the nrm decreases as the mesh size is increased; and the difference between cntinuus and discrete adjint gradients decreases as the mesh size is reduced. The secnd clumn depicts the difference between the cntinuus adjint and finite difference gradient. The third clumn depicts the difference between the discrete adjint and finite difference gradients. The last clumn depicts the difference between the discrete adjint and cntinuus adjint. As the mesh size increases the nrms decrease as expected. Since we derive the discrete adjint by taking a variatin f the discrete flw equatins, we expect it t be cnsistent with the finite difference gradients and thus t be clser than the cntinuus adjint t the finite difference gradient. This is cnfirmed by numerical results, but the difference is very small. As the mesh size increases, the difference between the cntinuus and discrete gradients shuld decrease, and this is reflected in the last clumn f table. Figure (5 presents the effect f the partial discretizatin f the flw slver t btain the discrete adjint equatin. Here we btain the finite difference gradients in the figure withut freezing the dissipative cefficients. A small discrepancy exists in regins clser t the leading edge and arund the shck. Kim, Alns, and Jamesn verified that accurate finite difference gradients require a cnvergence 8 American Institute f Aernautics and Astrnautics

9 f fur t five rders f magnitude in the flw slver. Hwever, bth the cntinuus and discrete adjint gradients nly require a cnvergence f tw rders f magnitude in the flw slver. Figures (6 and (7 illustrate the cntinuus and discrete gradients fr varius flw slver cnvergence. In figure (8 and (9 cntinuus and discrete adjint gradients are pltted fr varius adjint slver cnvergence. The gradients nly require tw rders f magnitude cnvergence in the adjint slver. Figure ( shws a cmparisn f the prfiles f the secnd and third cstate values between the cntinuus and discrete adjint methd in a directin nrmal t the bundary. The slutins agree in the interir pints, differing nly at the cell belw the bundary due t the different treatment f the bundary cnditin. In the cntinuus case the value at cell ne is updated by the bundary cnditin. This is in cntrasts t the discrete case where the bundary cnditin appears as a surce term when the fluxes are accumulated in cell tw and the bundary cnditin des nt depend n the value f the cstate in cell ne. In figure ( bth methds prduce similar cnvergence histries. In figure ( we attempt t design a Krn airfil based n the target pressure f the NACA 64A4 at a Mach number f.78. Bth the initial and target pressures cntain a very strng shck. A cmparisn f the finite difference and adjint gradients reveals an increase in the discrepancy between the tw gradients in the vicinity f the shck. In cntrast t figure (, where the shck lcatin is at mesh pint 75 alng the surface, figure (3 illustrates the discrepancy arund the strnger shck arund mesh pint 8. Drag Minimizatin The cst functin fr drag minimizatin is the pressure drag f the airfil. We perfrm cmputatins n a NACA 64A4 airfil at a flight cnditin f M =.75 and fixed lift cefficient f C l =.63. As befre, the gradients were btained by taking variatins respect t Hicks-Henne sine bump functins placed alng the upper and lwer surface f the airfil. Figure (5a illustrates the initial slutin f the airfil with 3 drag cunts. After tw design cycles, the drag is reduced by a third t 44 drag cunts. The strng shck in the initial slutin is weakened. And after just fur design cycles, this value is further halved. In figure (5d, the final design des nt cntain any shck and the drag cunt is a mere 5. Grid Size Cnt. Disc. Cnt-Disc 96 x 6.9e.75e.9e 9 x 3.49e 7.577e 3 5.7e 3 56 x e e 3.35e 3 Table : L nrm f the Difference Between Adjint and Finite Difference Gradient Figures (6-8 illustrate the values f the gradients btained frm the adjint methds and finite difference fr varius grid sizes. The finite difference gradients are based n the same methd used fr the inverse design case, where the dissipative cefficients are frzen after a cnverged flw slutin is btained t simulate a full discretizatin fr the discrete adjint equatin. We reduce the finite difference step sizes until we gain cnverged values fr each design pint. We plt gradients fr the upper surface frm leading edge t trailing edge. In figure (6 design pints between 5 and 6 are lcated in the vicinity f the leading edge, where the gradient has a psitive slpe. In this regin the discrete adjint gradient agrees better with the finite difference gradient, if cmpared t the cntinuus adjint gradient. The difference reduces as the grid size increases. Apart frm the regin f the leading edge, the adjint and finite difference gradients agree. Table ( cntains values f the L nrm f the difference between the adjint and finite difference gradients. Similar t the inverse design case, the table illustrates three imprtant facts: the discrete adjint gradient is clser than the cntinuus adjint gradient t the finite difference gradient; the nrms decrease as the mesh size increases; and, finally, the difference between the cntinuus and discrete adjint gradient decreases as the mesh size increases. We recalculate the finite difference and adjint gradients in figure (9 fr the medium size mesh f 9 x 3 cells t illustrate the effect f partial discretizatin f the flw slver. The dissipative cefficients are nt frzen during the finite difference calculatins. A very small discrepancy appears in the leading edge and in the shck wave (pints: Figures ( and ( illustrate the cntinuus and discrete gradients fr varius flw slver cnvergence. Only a single rder magnitude drp in the flw slver is required fr the adjint gradients t cnverge. We plt cntinuus and discrete adjint gradients in figure ( and (3 fr varius adjint slver cnvergence. The gradients nly require ne rder f magnitude cnvergence in the adjint slver. 9 American Institute f Aernautics and Astrnautics

10 Figure (4 shws a cmparisn f cnvergence f the bjective functin between the cntinuus and discrete adjint. Bth methds cnverge t the same value fr the bjective functin. Figure (5 presents the secnd and third cstate prfiles nrmal t the bundary fr the cntinuus and discrete adjint slutins. Bth slutins agree in the interir pints but disagree at the cell belw the wall. This is due t the difference between the enfrcement f the bundary cnditin. Figure (6 shws that bth adjint methds prduce the same cnvergence histry. Cnclusin This paper presents a cmplete frmulatin fr the cntinuus and discrete adjint apprach t autmatic aerdynamic design using the Euler equatins. The gradients frm each methd are cmpared t finite difference gradients. We cnclude:. The cntinuus bundary cnditin appears as an update t the cstate values belw the wall fr a cell-centered scheme, and the discrete bundary cnditin appears as a surce term in the cell abve the wall. As the mesh width is reduced, ne recvers the cntinuus adjint bundary cnditin frm the discrete adjint bundary cnditin. (Equatins 3and 8. Discrete adjint gradients have better agreement than cntinuus adjint gradients with finite difference gradients as expected, but the difference is generally small. (Figure 6 3. As the mesh size increases, bth the cntinuus adjint gradient and the discrete adjint gradient apprach the finite difference gradient. (Figures The difference between the cntinuus and discrete gradient reduces as the mesh size increases. (Tables and 5. The cst f deriving the discrete adjint is greater. (Equatin 6 6. With ur search prcedure as utlined, the verall cnvergence f the bjective functin is nt significantly affected when the discrete adjint gradient is used instead f the cntinuus adjint gradient. Cnsequently, we find n particular benefit in using the discrete adjint methd, which requires greater cmputatinal cst. Hwever, we believe it beneficial t use the discrete adjint equatin as a guide fr the discretizatin f the cntinuus adjint equatin. (Figure 4 Acknwledgments This research has benefited greatly frm the generus supprt f the AFOSR under grant number AF F The first authr wuld like t thank Juan Alns and James Reuther, fr many useful discussins. References [] F. Bauer, P. Garabedian, D. Krn, and A. Jamesn Supercritical Wing Sectins II. Springer-Verlag, New Yrk, 975 [] P. Garabedian, and D. Krn Numerical Design f Transnic Airfils Prceedings f SYNSPADE 97. pp 53-7, Academic Press, New Yrk, 97. [3] R. M. Hicks and P. A. Henne. Wing Design by Numerical Optimizatin. Jurnal f Aircraft. 5:47-4, 978. [4] J.L. Lins. Optimal Cntrl f Systems Gverned by Partial Differential Equatins. Springer-Verlag, New Yrk, 97. Translated by S.K. Mitter. [5] O. Pirnneau. Optimal Shape Design fr Elliptic Systems. Springer-Verlag, New Yrk, 984. [6] A. Jamesn. Aerdynamic design via cntrl thery. In Jurnal f Scientific Cmputing, 3:33-6,988. [7] A. Jamesn. Autmatic design f transnic airfils t reduce the shck induced pressure drag. In Prceedings f the 3st Israel Annual Cnference n Aviatin and Aernautics, Tel Aviv, pages 5 7, February 99. [8] A. Jamesn. Optimum aerdynamic design using CFD and cntrl thery. AIAA paper 95-79, AIAA th Cmputatinal Fluid Dynamics Cnference, San Dieg, CA, June 995. [9] A. Jamesn., N. Pierce, and L. Martinelli. Optimum aerdynamic design using the Navier- Stkes equatins. In AIAA 97-, 3 5th. Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January 997. American Institute f Aernautics and Astrnautics

11 [] A. Jamesn., L. Martinelli, and N. Pierce Optimum aerdynamic design using the Navier- Stkes equatins. In Theretical Cmputatinal Fluid Dynamics, :3-37, 998. [] G. R. Shubin and P. D. Frank A Cmparisn f the Implicit Gradient Apprach and the Variatinal Apprach t Aerdynamic Design Optimizatin. Being Cmputer Services Reprt AMS-TR-63, April 99. [] J. Reuther and A. Jamesn. Aerdynamic shape ptimizatin f wing and wing-bdy cnfiguratins using cntrl thery. AIAA 95-3, 33rd Aerspace Sciences Meeting and Exibit, Ren, Nevada, January 995. [3] J. Reuther, A. Jamesn, J. Farmer, L. Martinelli, and D. Saunders. Aerdynamic shape ptimizatin f cmplex aircraft cnfiguratins via an adjint frmulatin. AIAA 96-94, AIAA 34th Aerspace Sciences Meeting and Exhibit, Ren, NV, January 996. [] S. Ta asan, G. Kuruvila, and M. D. Salas. Aerdynamic design and ptimizatin in ne sht. AIAA 9-5, 3th Aerspace Sciences Meeting and Exibit, Ren, Nevada, January 99. [] S. Kim,J. J. Alns, and A. Jamesn A Gradient Accuracy Study fr the Adjint-Based Navier-Stkes Design Methd. AIAA 99-99, AIAA 37th. Aerspace Sciences Meeting and Exhibit, Ren, NV, January 999. [] A. Jamesn Slutin f the Euler Equatins fr Tw Dimensinal Transnic Flw By a Multigrid Methd. Applied Mathematics and Cmputatin, 3:37-355, 983. [3] A. Jamesn Studies f Alternative Numerical Optimizatin Methds Applied t the Brachistchrne Prblem. [4] J. Reuther, A. Jamesn, J. J. Alns, M. J Rimlinger, and D. Saunders. Cnstrained multipint aerdynamic shape ptimizatin using an adjint frmulatin and parallel cmputers. AIAA 97-3, AIAA 35th Aerspace Sciences Meeting and Exhibit, Ren, NV, January 997. [5] F. Beux and A. Dervieux. Exact-Gradient Shape Optimizatin f a -D Euler Flw. Finite Elements in Analysis and Design, Vl., 99, 8-3. [6] G. W. Burgreen and O. Baysal. Three- Dimensinal Aerdynamic Shape Optimizatin f Wings Using Discrete Sensitivity Analysis. AIAA Jurnal, Vl. 34, N.9, September 996, pp [7] J. Ellit and J. Peraire. Aerdynamic Design Using Unstructured Meshes. AIAA 96-94, 996. [8] W. K. Andersn and V. Venkatakrishnan Aerdynamic Design Optimizatin n Unstructured Grids with a Cntinuus Adjint Frmulatin. AIAA 96-94, 996. [9] A. Ill, M. Salas, and S. Ta asan. Shape Optimizatin Grvened by the Euler Equatins Using and Adjint Methd. ICASE reprt 93-78, Nvember 993. American Institute f Aernautics and Astrnautics

12 Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E a: Initial Slutin f Krn Airfil b: After 5 Design Iteratins Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E c: After 5 Design Iteratins d: Final Design Figure : Inverse Design f Krn t NACA 64A4 at Fixed C l Grid - 9 x 3, M =.74, C l =.63, α = degrees American Institute f Aernautics and Astrnautics

13 Magnitude f Gradient.. Magnitude f Gradient Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient.6 Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient.8 cnt fdg = 3.6e 3 disc fdg =.397e Design Pint.8 cnt fdg =.73e 3 disc fdg =.74e Design Pint Figure : Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Carse Grid - 96 x 6, M =.74, C l =.63 Figure 3: Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Medium Grid - 9 x 3, M =.74, C l = Magnitude f Gradient Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.44e 3 disc fdg =.49e Design Pint Magnitude f Gradient Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg = 3.88e 3 disc fdg = 3.3e Design Pint Figure 4: Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Fine Grid - 56 x 64, M =.74, C l =.63 Figure 5: Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Dissipative Cefficients Nt Frzen Medium Grid - 9 x 3, M =.74, C l =.63 3 American Institute f Aernautics and Astrnautics

14 .5.5.e Order.e Order.e 3 Order.e 4 Order.e Order.e Order.e 3 Order.e 4 Order.5.5 Gradient Gradient Design Pints Design Pints Figure 6: Cntinuus Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inverse Design Case Figure 7: Discrete Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inverse Design Case.8.e Order.e Order.e 3 Order.e 4 Order.8.e Order.e Order.e 3 Order.e 4 Order Gradient Gradient Design Pints Design Pints Figure 8: Cntinuus Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inverse Design Case Figure 9: Discrete Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inverse Design Case 4 American Institute f Aernautics and Astrnautics

15 Secnd Cstate Variable Third Cstate Variable Cnt Adjint Disc Adjint 4 4 Cntinuus Discrete 8 6 Cntinuus Discrete 8 6 Lg(Errr Value f Cstate Value f Cstate Figure : Cmparisn f Cstate Values Between the Cntinuus and Discrete Adjint Methd fr the Inverse Design f Krn t NACA 64A4 at Fixed C l. Medium Grid - 9 x 3, M =.74, C l = Number f Iteratins Figure : Cnvergence Histry fr the Cntinuus and Discrete Adjint fr the Inverse Design f Krn t NACA 64A4 at Fixed C l. M =.74, C l =.63 5 American Institute f Aernautics and Astrnautics

16 Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E a: Initial Slutin f Krn Airfil b: Final Design Figure : Inverse Design f Krn t NACA 64A4 at Fixed C l Grid - 9 x 3, M =.78, C l =.63, α = degrees Magnitude f Gradient.4.. Magnitude f Gradient.. Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.369e 3 disc fdg =.8e 3.4 Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.3e 3 disc fdg =.895e Design Pint Design Pint Figure 3: Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Carse Grid - 96 x 6, M =.78, C l =.63 Figure 4: Adjint Versus Finite Difference Gradients fr Inverse Design f Krn t NACA 64A4 at Fixed C l. Medium Grid - 9 x 3, M =.78, C l =.63 6 American Institute f Aernautics and Astrnautics

17 Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E 5a: Initial Slutin f NACA 64A4. M =.75, C l =.63, C d =.3, α =.34 degrees 5b: After Design Iteratins. M =.75, C l =.66, C d =.44, α =.8 degrees Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E Cp.E.8E.4E -.E-5 -.4E -.8E -.E -.E 5c: After 4 Design Iteratins M =.75, C l =.66, C d =., α =.8 degrees 5d: After Design Iteratins M =.75, C l =.69, C d =.5, α =.79 degrees Figure 5: Drag Minimizatin f NACA 64A4 at Fixed C l Grid - 9 x 3, M =.75, C l =.63 7 American Institute f Aernautics and Astrnautics

18 Magnitude f Gradient Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.9e disc fdg =.75e Magnitude f Gradient 4 6 Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.49e disc fdg = 7.577e Design Pint Design Pint Figure 6: Adjint Versus Finite Difference Gradients fr Drag Minimizatin f NACA 64A4 at Fixed C l. Carse Grid - 96 x 6, M =.75, C l =.63 Figure 7: Adjint Versus Finite Difference Gradients fr Drag Minimizatin f NACA 64A4 at Fixed C l. Medium Grid - 9 x 3, M =.75, C l =.63 Magnitude f Gradient 4 6 Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg = 6.4e 3 disc fdg = 5.54e 3 Magnitude f Gradient Cnt Adjint Gradient Disc Adjint Gradient Finite Difference Gradient cnt fdg =.366e disc fdg =.93e Design Pint Figure 8: Adjint Versus Finite Difference Gradients fr Drag Minimizatin f NACA 64A4 at Fixed C l. Fine Grid - 56 x 64, M =.75, C l = Design Pint Figure 9: Adjint Versus Finite Difference Gradients fr Drag Minimizatin f NACA 64A4 at Fixed C l. Dissipative Cefficients Nt Frzen Medium Grid - 9 x 3, M =.75, C l =.63 8 American Institute f Aernautics and Astrnautics

19 .e Order.e Order.e 4 Order.e 6 Order.e Order.e Order.e 4 Order.e 6 Order Gradient 3 4 Gradient Design Pints Design Pints Figure : Cntinuus Adjint Gradients fr Varying Flw Slver Cnvergence fr the Drag Minimizatin Case Figure : Discrete Adjint Gradients fr Varying Flw Slver Cnvergence fr the Drag Minimizatin Case.e Order.e Order.e 3 Order.e 4 Order.e Order.e Order.e 3 Order.e 4 Order Gradient 3 4 Gradient Design Pints Design Pints Figure : Cntinuus Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Drag Minimizatin Case Figure 3: Discrete Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Drag Minimizatin Case 9 American Institute f Aernautics and Astrnautics

20 Cntinuus Adjint Discrete Adjint Objective Functin Design Cycles Figure 4: Cmparisn f Cnvergence f the Objective Functin Between the Cntinuus and Discrete Adjint Methd fr Drag Minimizatin Secnd Cstate Variable Third Cstate Variable 8 6 Cntinuus Discrete 8 6 Cntinuus Discrete Cnt Adjint Disc Adjint Lg(Errr Value f Cstate Value f Cstate Figure 5: Cmparisn f Cstate Values Between the Cntinuus and Discrete Adjint Methd fr Drag Minimizatin f NACA 64A4 at Fixed C l. Medium Grid - 9 x 3, M =.75, C l = Number f Iteratins Figure 6: Cnvergence Histry fr the Cntinuus and Discrete Adjint fr Drag Minimizatin f NACA 64A4 at Fixed C l. M =.75, C l =.63 American Institute f Aernautics and Astrnautics