Analysis of the Hessian for Aero elastic Optimization Ey al Arian ICASE Mail Stop C NASA Langley Researc h Cen ter Hampton V A Abstract The sym b ol of the Hessian for an aero elastic optimization mo del problem is ana lyzed The o w is mo deled b y the small disturbance full p oten tial equation and the structure is mo deled b y an isotropic v on K arm an plate equation The cost function consists of b oth aero dynamic and structural terms In the new analysis the sym b ol of the cost function Hessian near the minim um is computed The result indicates that under some conditions whic h are lik ely ful lled in most applications the system is decoupled for the non smo oth comp onen ts The result also sho ws that the structure part in the Hessian is w ell conditioned while the aero dynamic part is ill conditioned Applications of the result to optimization strategies are discussed This researc h w as supp orted b y the National Aeronautics and Space Administration under NASA Con tract No NAS while the author w as in residence at the Institute for Computer Applications in Science and Engineering ICASE Mail Stop C NASA Langley Researc h Cen ter Hampton V A i In tro duction Lately there is a gro wing in terest in Multidisciplinary Design and Optimization MDO An imp ortan t problem in that eld is the aero elastic optimal design problem for example In this problem there are t w o coupled disciplines aero dynamics and structural analysis The problem is to compute the aero dynamic shap e and structural rigidit y suc h that some giv en cost function is minim ize d The purp ose of this pap er is to demonstrate new analysis of Hessians for MDO problems on the ab o v e aero elastic optimization problem and to dra w some practical conclusions The approac h is to consider a simple mo del problem and compute the sym b ol of the Hessian near the minim um for the non smo oth frequencies The Hessian con tains curv ature information whic h is essen tial for the solution of ill conditioned optimization problems Hessian sym b ols w ere previously computed for smo othing predictions in the dev elopmen t of m ultigrid one shot metho ds and lately for the analysis of in viscid aero dynamic optimization problems The analysis in this pap er indicates that for the non smo oth comp onen ts the system is decoupled under certain conditions whic h are lik ely ful lled in most applications The analysis also sho ws that the structures part in the Hessian is w ell conditioned while the aero dynamics part is ill conditioned One consequence of this result is that if the decoupling condition holds the solution of suc h problems can b e ac hiev ed in t w o stages In the rst stage the MDO approac h should b e tak en on a coarse mo del that is the o w and the structure equations are considered sim ultaneously during the minim izati on whic h is a more complex problem than optimizing the decoupled individual disciplines problems In the second stage a re ned CFD co de for the o w and a detailed nite elemen t co de for structure should b e used in a serial algorithm in whic h the shap e is optimized relativ e to aero dynamic considerations follo w ed b y structural optimization limite d to a giv en shap e that under o w conditions it will t wist and b end to the aero dynamic optimal shap e This approac h should result in a go o d appro ximation of the m ultidisci pli nary optimal solution A second consequence is that if the decoupling condition do es not hold and th us the t w o disciplines are coupled a m ultidiscipl inary algorithm is necessary and a preconditioner for the coupled system is necessary to obtain e ectiv e con v ergence The pap er outline is as follo ws In Section the optimization problem is form ulated In Section the necessary conditions for a minim um are deriv ed with the adjoin t metho d and their relation with the Hessian is discussed In Section the sym b ol of the Hessian for the non smo oth frequencies is deriv ed b y using lo cal mo de analysis In Section decoupling conditions and m ultidisci plinary preconditioners are deriv ed In Section applications of the result are nally discussed Problem F orm ulation In this section the aero elastic analysis problem and the optimal design problem are presen ted The aero elastic analysis problem couples the full p oten tial o w equation with the isotropic v on K arm an plate equation to giv e the pressure distribution o v er the plate p and the plate deformation W for a giv en plate shap e and rigidit y distribution D The design problem is to compute the b est shap e and structural rigidit y so that a giv en cost function is minimi ze d The cost function is comp osed of aero dynamic and structure parts The aero dynamic cost function estimates p erformance b y measuring the di erence in L norm of the pressure distribution from a desired one The structure cost function giv es a measure of the structural w eigh t and p enalizes structural deformation Since our in terest is in a lo cal mo de analysis of the Hessian near the minim um w e consider the small disturbance equations of o w o v er a at plate The Flo w Mo del W e c ho ose the full p oten tial equation as a mo del for the o w It appro ximates in viscid o w c haracteristics and is used in applications for aero dynamic optimal design for example F or the analysis of the cost function s Hessian in the vicinit y of the minim um it is enough to consider small p erturbations of the shap e from the optimal solution The resulting c hanges in the p oten tial satisfy the steady state small disturbance full p oten tial equation The geometry is tak en to b e half space x y z where the x axis is the stream wise co ordinate y is the co ordinate p erp endicular to the stream and parallel to the plate span wise direction and z is in the normal direction to the plate The Aero dynamic State Equation L z B C W z with the follo wing de nitions of the in terior op erator L and b oundary op erators B and C L M xx y y z z B z C U x where U is the free stream v elo cit y M is the Mac h n um b er with the follo wing far eld b oundary conditions In o w Boundary Condition Subsonic U x Sup ersonic U and w e assume that the normal free stream v elo cit y V is x zero Out o w Boundary Condition Subsonic U x Sup ersonic No Boundary Condition The missing lo w order terms in the b oundary condition of v anish if the analysis is p erformed around a at shap e The Structural Mo del The structural mo del consists of the isotropic v on K arm an plate equation for the displace men t W G D W C z with the follo wing de nition of the the op erator G h i G D W D W D W D W D W D W xx xx y y y y y y xx xx y y xy xy where D is the plate rigidit y distribution is the o w densit y and is the P oisson ratio In t w o space dimensions Eq reduces to the b eam equation D W U z xx xx x The b oundary conditions for the plate are that it is clamp ed at y or at x for a b eam and free at the other b oundaries W x W x y Ho w ev er Eq is elliptic so the e ect of a high frequency error comp onen t in the de ec tion W is lo cal and therefore the plate b oundary conditions do not pla y a role in the lo cal mo de analysis The Cost F unction Mo del The de nition of the cost function is not unique and dep ends on the sp eci c application under consideration In general the requiremen t of the aero elastic optimal design is that it ha v e maxim um aero dynamic p erformance and minim um structural w eigh t and deformation Some of the desired features of the nal design are in man y cases mo deled b y a set of inequalit y constrain ts as is the case for the minim um deformation requiremen t Ho w ev er for the purp ose of this pap er w e will a v oid inequalit y constrain ts b y adding a term to the cost function whic h p enalizes deformation In the follo wing the di eren t terms comp osing the cost function are discussed The Aero dynamic P erformance T erm A common aero dynamic cost function is drag or drag o v er lift Ho w ev er in in viscid aero dynamic optimization mo dels a commonly used cost function is pressure matc hing for example In p oten tial mo dels the pressure p is related to the p oten tial b y the Bernoulli relation p U x This results in the follo wing cost function term Z aer o F f d x where d is an in tegration elemen t on the shap e The target distribution f x y L is related to the desired pressure distribution p x y b y the relation p x y f x y U The Structural W eigh t T erm Another imp ortan t factor in aero elastic design is the resulting w eigh t of the structure In practice the w eigh t is measured b y the sum of the w eigh ts of all the comp onen ts comp osing the structure In plate mo dels the w eigh t is related with the plate rigidit y D and is giv en in the follo wing table where E is the Y oung mo dulus of elasticit y b and h are the cross section D w eigh t R E bh b eam bhdx p R E t plate tdxdy p comp onen ts of the b eam is the structural densit y and t is the plate s thic kness p d In b oth cases the w eigh t of the structure is prop ortional to D where d is the space dimension Z w eig ht d F D d The Structural Deformation T erm As a result of the pressure p exerted on the plate b y the o w the structure will deform its shap e b y W b end and t wist In practice the structure is designed so that the amoun t of deformation will b e constrained not to exceed some giv en limits In this mo del w e accoun t for this requiremen t b y p enalizing the deformation whic h is measured b y the w ork of the aero dynamic pressure on the plate pW This will add to the cost function the follo wing term using the Bernoulli relation Z def or m F U W d x The Optimizati on Problem W e de ne the cost function F F W D to b e Z Z Z d F W D f d D d W d x x where are parameters The cost function is a map from a function space to I R The minimi zation problem is to nd a shap e function and rigidit y distribution D suc h that the cost function is minim iz ed sub ject to Eqs and W e assume the existence of a solution for b oth the state equations and for the optimization problem a rigorous treatmen t of existence and uniqueness of solutions is b ey ond the scop e of this pap er Adjoin t F orm ulation and the Hessian In this section the necessary conditions for a minim um are deriv ed with the adjoin t metho d The necessary conditions are giv en as a set of state equations the analysis prob lem costate equations the adjoin t problem and design equations optimalit y conditions Then the relation b et w een the design equation residuals and the Hessian of the cost function is discussed This relation will b e used in the next section to deriv e the Hessian s sym b ol The Necessary Conditions for a Minim um The Lagrangian is a functional de ned b y R L W D F W D B C W d R R L d G D W C d where x y x y and x y z are Lagrange m ultiplie rs The rst order necessary conditions for a minim um are deriv ed b y the requiremen t that the rst order v ariation of the Lagrangian v anish this is kno wn as the adjoin t metho d and the resulting conditions are kno wn as the Kuhn T uc k er conditions When considering the v ariation of the structure state equation a linearization is p er formed G D D W W G D W G W D G D W h o t D W where D and W are small p erturbations of the displacemen t and rigidit y from the optimal solution W and D resp ectiv ely and where the linearized op erators G and G are de ned D W as follo ws h i G W D D W D W D W D W D W xx y y xx y y xy D xx y y y y xx xy G D W G D W W F ormally W and D serv e as non constan t co e cien ts in the linearized structure op erator In the follo wing the costate and design equations are giv en Costate Equations L z B C F z G D C F z W W In o w Boundary Condition Subsonic x Sup ersonic No Boundary Condition Out o w Boundary Condition Subsonic x Sup ersonic and x Design Equations C z G W F z D D where F f W x x x F W x d d F D D d and where the op erators in the adjoin t and design equations satisfy L L C C G D G D W W G W G W D D The adjoin t b oundary op erator B corresp onds to the normal deriv ativ e applied to a z solution of the in terior costate PDE when using the adjoin t far eld b oundary conditions W e assume the existence of a solution to the costate equations The Relation of the Hessian with the Necessary Conditions If the state and costate equations are satis ed in the strong sense then the v ariation of the Lagrangian is equal to the v ariation in the cost function and is giv en b y Z Z L C d D G W F d D D where and D are v ariations in the design v ariables Therefore the quan tities m ultiplying and D in are the sensitivit y gradien ts of the cost function with resp ect to the design v ariables when computed on the constrain t manifold g C g G W F D D The state and costate equations and giv e an implici t relation b et w een the costate v ariables and the design v ariables D D Using equations and w e can write the follo wing relation whic h holds near the minim um and D denote the optimal v alue of the design v ariables and D resp ectiv ely g D D H H D h o t g D D D D H H D h o t where at the minim um g D g D D W e conclude that on the constrain t manifold near the minim um the Hessian of the cost function relates the errors in the design v ariables with the residuals of the design equations sensitivit y gradien ts In the next section w e will use this fact to calculate the sym b ol of the Hessian Deriv ation of the Hessian s Sym b ol In the follo wing section w e compute the sym b ol of the Hessian with lo cal mo de analysis Hessian sym b ols w ere previously computed for smo othing prediction in the dev elopmen t of m ultigrid one shot metho d and lately for the analysis of in viscid aero dynamic optimization problems In the follo wing the lo cal mo de analysis is outlined The analysis is p erformed in the vicinit y of the minim um where the design v ariables are assumed to ha v e an error and D W e assume that the state and costate equations are satis ed and consider the errors in the state and costate v ariables W with resp ect to their v alue at the optimal solution These errors are assumed to satisfy homogeneous equations similar to Eqs and a linearization of Eq W e then consider the high frequency errors in the design v ariables and compute an explicit solution of the problem in terms of exp onen tial functions in a half space Then with a standard pro cedure the problem in a half space is reduced to the b oundary On the b oundary w e study the mapping from the transformed design v ariables errors to the residuals of the design equations g g The sym b ol of this mapping giv es the eigen v alues of the Hessian F ourier Represen tation W e start with the F ourier represen tation of the solution in a half space and p erform lo cal mo de analysis Consider errors in the solution of the form i x y x y e i x y D x y D e As a result the errors in the state and costate v ariables are assumed to ha v e the follo wing form i x y z x y z e i x y z x y z e i x y W x y W e i x y x y e Before computing the relation b et w een the state and costate error sym b ols W and the design error sym b ols D w e reduce the problem to the b oundary b y eliminating from the sym b ols and Reduction to the Boundary The reduction to the b oundary is done b y eliminating from the sym b ol expressions using the in terior equations The follo wing discussion regarding the c hoice of w as done in and is rep eated here for completeness The term satis es the in terior equation for i x y z L e Assuming a non trivial solution Eq results in an algebraic relation b et w een and M The c hoice of should b e done suc h that it will result in a ph ysical solution W e di eren tiate b et w een subsonic and sup ersonic o ws Subsonic Flo w In the subsonic regime M the ph ysical solution is giv en b y q i M whic h corresp onds to deca ying solutions p i x y M z x y z e e p i x y M z x y z e e In that case the sym b ols of the b oundary op erators B and B are giv en b y recall that B and B are the normal deriv ativ es applied to a solution of the in terior state and costate PDE resp ectiv ely q B B M Sup ersonic Flo w W e di eren tiate b et w een t w o sup ersonic cases whic h are determined b y a o w sp eed denoted M and giv en b y c s M c The case M M results in the same sym b ols for B and B as for the subsonic o w case c Eq In the case M M b oth signs of in corresp ond to ph ysical solutions The c p ositiv e ro ot corresp ond to the c haracteristic whic h propagates in to the shap e and the negativ e ro ot corresp ond to the c haracteristic whic h propagates out of the shap e and a similar expression for p p i x y j M jz i x y j M jz x y z e e Since the in o w information do es not c hange as a result of a shap e p erturbation the follo wing holds x y z In the same manner the out o w c haracteristic of the adjoin t is not c hanging as a result of a shap e p erturbation x y z Therefore p i x y j M jz x y z e p i x y j M jz x y z e W e conclude that for o w sp eeds M M the b oundary op erator B is an tisymme tric c with resp ect to the adjoin t op eration and the sym b ols B and B are giv en b y q B i j M j q B i j M j In all o w conditions the m ultipli cation B B results in the same expression B B M By eliminating from the transformed equations the state and costate o w equations can b e written on the surface whic h corresp onds to the b oundary x y T reatmen t of the Structure Equations In this subsection w e giv e a short note concerning the transformation of the structure state and costate equations The structure state and costate equations con tain non constan t co e cien ts whic h should b e frozen prior to the lo cal mo de analysis The structure state and costate error equations are giv en b y see Eqs G W D G D W C z D W G D C F z W W where D W and denote the error v ariables F denotes the error in F and the W W op erators G and G are de ned in Since Eqs ha v e v ariable co e cien ts D D W and W it is necessary to freeze them around a p oin t on the b oundary This pro cedure is justi ed as long as the errors in the design v ariables are highly oscillatory compared to W and D W e denote the v alues of W x y and D x y at a p oin t x y on the b oundary b y W and D resp ectiv ely The Coupled State and Costate Equations in F ourier Space In terms of their F ourier represen tation on the b oundary the state and costate error equations are giv en b y the follo wing matrix form B C C B C B C B C B C B C B C C G D W W G W D B C D B C B C B C B C B C B C F F B C A A W A F C G D W W The v arious sym b ols are giv en explicitly b y F F i W F i W G D D l o t W G W W W W W W D xx y y y y xx xy C iU Note that the terms originating in the cost function serv e as a coupling symmetri c blo c k b et w een the state and costate systems The Sym b ol of the Hessian The design equations residuals in the transformed space are giv en b y g C g G W F D D D D D where d d d F D D D d and the sym b ols g and g are the sym b ols of the sensitivit y gradien ts g and g resp ectiv ely see In v erting the system and substituting and in the sym b ol of the design residuals results in a relation b et w een the residuals of the design equations and the errors in the design v ariables In F ourier space g H H g D H H where the matrix H is the sym b ol of the Hessian as discussed in Sec H is the sym b ol ij of the aero dynamic optimization Hessian H of the structural optimization Hessian and H H are the coupling terms In the follo wing the terms H are giv en explicitly ij C G F G C F W W W H B G C B G C W W C G C F G B F G C F D W W W W H B G C B G C W W C G C F G B F G C F D W W W W H B G C B G C W W C G F C B B F W D H F D D B G C B G C W W Since G is a fourth order p olynomial in G F B and B are of second order F W D W and C are rst order and F is of zero order the principal parts of the ab o v e expressions D the asymptotic limits of high frequencies are giv en b y C F H U j M j B B W W W W W C G F U xx y y y y xx xy D H H D j M j B B G W d d d H F D D D d Note that for simplicit y w e assumed a complex represen tation of the errors and obtained a complex Hessian sym b ol If considering a real represen tation i e i x y conj i x y x y e e conj where is the complex conjugate of and a similar expression for D then the resulting Hessian sym b ol is real and symmetri c Discretiz ati on and the Condition Num b er In practice the problem is solv ed n umericall y and th us discretization is in tro duced Therefore the analysis should b e p erformed in the discrete space and the Hessian will dep end on the sp eci c discretization F or the ideal discretization the sym b ol of the Hessian is equal to the di eren tial one with the substitution h h where h h are the mesh sizes in the x y directions resp ectiv ely and where and v ary in the domain Note that high frequencies are those whic h ob ey c for some constan t c whic h is i determined b y the di eren t parameters in the problem In the discrete space this corresp onds to ch Since the constan t c is indep enden t of the mesh size h as the grid is re ned the i i p ortion of high frequencies in the sp ectrum increases and therefore the appro ximation tak en b y the lo cal mo de analysis ab o v e is more accurate for a larger part of the sp ectrum This is not surprising since as the grid is re ned its resolution increases while the resolution of the smo oth comp onen ts remains unc hanged The maxim um eigen v alue of eac h of the disciplinary Hessians is estimated b y H max ii h Unfortunately the lo w est eigen v alue cannot b e estimated b y the pro cedure ab o v e since this is precisely the sp ectrum range in whic h the appro ximation tak en b y the lo cal mo de analysis do es not hold Ho w ev er the lo w est eigen v alue is a xed n um b er indep enden t of the mesh size and therefore the condition n um b er of the Hessian is prop ortional to F or a max t w o dimensional o w o v er a b eam w e get for the aero dynamic part of the Hessian see Eq U max j M j h W e conclude that the aero dynamic part of the Hessian is ill conditioned and its condition n um b er is increasing as the grid is re ned see for further discussion The structure s sym b ol approac hes a constan t for the high frequencies indep enden t of the mesh size W e therefore conclude that the structural optimization problem is w ell conditioned On the Coupling Bet w een Aero dynamic and Struc tural Design In the previous section w e computed explicitly the Hessian s sym b ol The coupling b et w een the t w o disciplines during optimization is determined b oth b y the o diagonal terms in the sym b ol of the Hessian and b y the smo othness of the design v ariables W e sa y that the optimization problem can b e decoupled if see Eq jH j jH D j or equiv alen tly if jH g j jH g j If condition holds then the aero dynamic opti mization problem can b e solv ed at the rst stage indep enden t of the structural optimization problem setting the structural deformation to zero W Then at the second stage the resulting shap e and p oten tial are giv en as inputs to the structural optimization problem see more details in the discussion The solution of this t w o stage approac h will b e a go o d appro ximation of the m ultidisci plinary solution in the high frequencies W e emphasize that if condition do es not hold then this pro cedure will result in a p o or appro ximation of the m ultidiscipl inary optimal solution In that case a m ultidisciplinary preconditioner should b e constructed to giv e a corrected disciplinary searc h directions This is done b y transforming Eq bac k to the PDE lev el Note that if jH j jH D j holds and condition do es not hold the pro cess of rst computing an optimal structure and then computing an optimal shap e is not w ell de ned In the follo wing w e examine condition explicitly for t w o and three dimensional o ws and illustrate the deriv ation of a precondi tioner for a simple case Tw o Space Dimensions In a t w o dimensional o w o v er a b eam the principal part of the Hessian is giv en b y see Eqs W U U xx j M j j M j D B C H A W U xx D j M j D In practice the curv ature of the de ection in the stream wise direction W is negligible xx with resp ect to the rigidit y D as is the case for airfoils Setting W in results in xx a diagonal matrix whic h implies a complete decoupling b et w een aero dynamic and structure design In case W is not negligible let us assume the optimal solution exists and b elongs to xx the follo wing spaces s k C and D C where s and k are in tegers The matrix implies s H k H D and therefore the decoupling condition is satis ed if k s s i e as long as the rigidit y D is not m uc h more rough than the shap e D C or rougher In case condition is not satis ed a preconditioner should b e constructed The disciplinary searc h directions and D are solutions of a PDE whic h is obtained b y the in v erse transform of Eq ac b cg bg xx acD b D bg a g xx xx with U a j M j U W xx b j M j D c D Using these directions as the searc h directions in a m ultidisci plinary optimization algorithm should result in a m uc h more e ectiv e con v ergence prop erties Three Space Dimensions In a three dimensional con guration w e di eren tiate b et w een the stream wise and span wise directions Let us assume that the curv ature of the de ection in the stream wise direction is negligible i e set W As a result the coupling term H has the follo wing form xx W W W U y y y y xy H W xx D j M j F or the design of the structure in the span wise direction only i e freezing the stream wise design as done in practice for aircraft wing design the o diagonal terms in the Hessian v anish and therefore the problem is decoupled F or errors in the stream wise direction only i e the o diagonal terms in the Hessian reduce to U W y y H D j M j s s By a similar argumen t to the t w o dimensional case if C and D C or smo other then the problem can b e decoupled If the structure design v ariables b elong to a s m uc h rougher space than the shap e v ariables D C or rougher then a preconditioner should b e applied similar to Eq with U a j M j W U y y b j M j D c D s s The case C and D C needs a more careful examination In this sp ecial case the decoupling condition implies W y y jH j jH j D Let us assume a wing lik e geometry where a plate of length L is clamp ed at y and free at the other b oundaries Assume that at the optimal solution the plate is b en t to w ards the tip y L as a quadratic in y y tip W x y W L tip where W is the maxim um de ection at the tip of the plate In that case the decoupling condition b ecomes tip W E t L where E is the Y oung mo dulus of elasticit y is the P oisson ratio and t is the plate thic k ness In case condition is not satis ed then the problem can not b e decoupled and a preconditioner is required Discussion and Concluding Remarks The sym b ol of the Hessian for aero elastic optimization mo del problem w as computed Eqs The result indicates that for the non smo oth comp onen ts the system is decoupled under certain conditions In t w o dimensions it is enough to assume that the curv ature of the de ection is negligible as is the case for airfoils to obtain decoupling see Sec for further discussion of the decoupling condition In three dimensions this requiremen t is not enough and unless the rigidit y is not m uc h rougher than the shap e the system is coupled see Sec The result also sho ws that the aero dynamic optimization problem is ill conditioned and therefore second order information is essen tial for e cien tly solving this part of the problem while the structural optimization problem is w ell conditioned Th us it is an ticipated that the n um b er of optimization iterations required to solv e the m ultidiscipl inary problem is determined b y the aero dynamic optimization part of the problem W e no w discuss the application of this result to optimization strategies for the solution of the problem W e di eren tiate b et w een t w o basic approac hes the disciplinary and the m ultidiscipli nary In the disciplinary approac h the solution of the problem is divided so that one discipline optimization problem is solv ed at eac h stage decoupled from the other discipline In the m ultidiscipl inary approac h b oth the analysis and optimization solutions are p erformed in a tigh tly coupled manner These t w o approac hes are no w presen ted in more detail The Disciplinary Approac h W eak Coupling A common practical strategy used to solv e large aero elastic shap e optimization prob lems is the disciplinary approac h i e design the aero dynamic optimal shap e to giv e the b est p erformance and then design a minim um w eigh t structure restricted to the aero dynamic shap e that under o w will t wist and b end to the aero dynamic optimal shap e The Disciplinary Algorithm The aerodynamic shape optimizat ion problem is solved setting W R min f d x subject to L z B C z The structure minimum weight problem is solved given a fixed shape and pressure potential distribut ion i e R R d min D d W d D x subject to G D W C z where and are parameter s The final shape is computed such that under cruise conditions given pressure distribut ion p the shape will deform into the aerodynami c optimal shape The Multidisciplinary Approac h Tigh t Coupling Lately there has b een an e ort to dev elop new optimization strategies whic h couple the t w o disciplines tigh tly during the analysis and optimization computation This is kno wn as the MDO approac h According to this approac h after eac h call to the optimizer the analysis and adjoin t equations are relaxed or solv ed exactly dep ending on the feasibilit y c hoice Multi Disciplinary F easibilit y MDF Individual Discipline F easibilit y IDF or All at Once AA O The MDO Algorithm The coupled aerodynam ic shape and structure minimum weight optimizatio n problems are solved simultane ous ly In order to achieve efficient conve rgence both disciplin es gradients play a role in each disciplina ry optimizer see Eq R R R d min f dx D dx W dx D x x subject to L z B C W z G D W C z The aim of the MDO approac h is to couple a re ned CFD co de with a detailed nite elemen t structural analysis co de to compute the aero elastic states prior to eac h optimization iteration The computational complexit y of the MDO algorithm is m uc h greater than that of the disciplinary algorithm since at eac h m ultidiscipli nary iteration b oth the aero dynamic and structural optimization problems ha v e to b e solv ed the MDO approac h also in tro duces a tec hnical di cult y of joining together t w o large co des The result of this pap er sho ws that under certain conditions giv en in Sec whic h are most lik ely met for aircraft wing design the system is decoupled for the non smo oth frequencies Therefore under these conditions the MDO approac h applied on a ne scale mo del migh t not b e necessary to obtain a go o d appro ximation of the optimal solution The e ect of the smo oth comp onen ts can b e captured b y a coarse mo del con taining a relativ ely small n um b er of design v ariables and th us can b e solv ed b y the MDO approac h with a relativ ely lo w computational cost This will require simple mo dels for the o w panel metho d or small disturbance full p oten tial on a coarse grid coupled with a plate mo del or coarse nite elemen t mo del for the structure If indeed the decoupling condition holds as discussed w e prop ose that the problem b e solv ed in t w o stages as illustrated in Fig In the rst stage the MDO approac h will b e applied on a coarse mo del The second stage starts with the solution of the MDO algorithm and the re ned problem is solv ed with the disciplinary algorithm th us a v oiding the enormous complexit y of the MDO algorithm when applied on the ne scale mo del W e claim that the resulting design will b e a go o d appro ximation of the optimal solution W e emphasize that this is p ossible due to the lo ose coupling b et w een the t w o disciplines otherwise the prop osed approac h will result in a p o or appro ximation of the optimal design since the result will retain high residuals of the m ultidiscipl inary optimalit y conditions In that case the MDO approac h should b e applied also on ne scales and a preconditioner should b e used as illustrated in Eq to ac hiev e e ectiv e con v ergence Finally ho w far can w e extrap olate the conclusions from this mo del problem to a more realistic mo del As for the aero dynamic mo del it is sho wn in that an iden tical sym b ol for the aero dy namic part of the Hessian is obtained when using Euler equations instead of the full p oten tial The analysis for the Na vier Stok es equations has not b een completed y et Sho c ks w ere also neglected in the aero dynamic mo del but w e p ostulate that they are not going to c hange the main conclusion since sho c ks ha v e a global e ect and are not lik ely to a ect the conditioning of the Hessian As for the sp eci c mo deling whic h w e ha v e c hosen to analyze since there are man y di eren t mo dels for the cost function and di eren t constrain ts dep ending on the application it is impractical to analyze them all In the mo del discussed in this pap er w e used a p enalt y term in the cost function to accoun t for constrain ts on the structure deformation Ho w ev er if using inequalit y constrain ts instead of p enalt y terms it is not clear ho w the coupling of the t w o disciplines will b e a ected In practice most of the constrain ts are not binding at the solution and therefore are e ectiv ely of small n um b er When in tro duced in small n um b ers w e an ticipate that they are not going to c hange the basic structure of the Hessian near the minim um Ac kno wledgmen t The author w ould lik e to thank R T Haftk a J W Hou J A Burns and R M Lewis for fruitful discussions during di eren t stages of this w ork References J S Sobieszczanski Sobieski Sensitivit y of Complex In ternally Coupled 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Conference Sep A Iollo and M Salas Con tribution to the Optimal Shap e Design of Tw o Dimensional In ternal Flo ws with Em b edded Sho c ks ICASE Rep ort No P E Gill W Murra y and M H W righ t Practical Optimization Academic Press M D Gunzburger and L S Hou Finite Dimensional Appro ximation of a Class of Constrained Nonlinear Optimal Con trol Problems ICASE Rep ort No - COARSE MODEL H FINE MODEL - h MULTIDISCIPLINARY ANALYSIS , h h h h AERODYNAMIC , STRUCTURAL , FINAL h h OPTIMIZATION D,W OPTIMIZATION D,W DESIGN D,W MULTIDISCIPLINARY OPTIMIZATION Figure An optimization strategy to solv e aero elastic optimization problems in case of decoupling as de ned in Sec Apply the MDO approac h on a coarse mo del follo w ed b y a disciplinary serial approac h on ne scales The result should b e a go o d appro ximation of the m ultidiscipl inary optimal solution