Jameson-Aerodynamic Shape Optimization Using the Adjoint Method

8/9/2019 Jameson-Aerodynamic Shape Optimization Using the Adjoint Method

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Aerodynamic Shape OptimizationUsing the Adjoint Method

Antony Jameson

Department of Aeronautics & AstronauticsStanford UniversityStanford, California 94305 USA

Lectures at the Von Karman Institute, BrusselsFebuary 6, 2003

Abstract

These Lecture Notes review the formulation and application of optimization techniques based on controltheory for aerodynamic shape design in both inviscid and viscous compressible ow. The theory is appliedto a system dened by the partial differential equations of the ow, with the boundary shape acting asthe control. The Frechet derivative of the cost function is determined via the solution of an adjoint partial

differential equation, and the boundary shape is then modied in a direction of descent. This process isrepeated until an optimum solution is approached. Each design cycle requires the numerical solution of boththe ow and the adjoint equations, leading to a computational cost roughly equal to the cost of two owsolutions. Representative results are presented for viscous optimization of transonic wing-body combinations.

1 Introduction: Aerodynamic Design

The denition of the aerodynamic shapes of modern aircraft relies heavily on computational simulationto enable the rapid evaluation of many alternative designs. Wind tunnel testing is then used to conrmthe performance of designs that have been identied by simulation as promising to meet the performancegoals. In the case of wing design and propulsion system integration, several complete cycles of computationalanalysis followed by testing of a preferred design may be used in the evolution of the nal conguration.Wind tunnel testing also plays a crucial role in the development of the detailed loads needed to complete

the structural design, and in gathering data throughout the ight envelope for the design and vericationof the stability and control system. The use of computational simulation to scan many alternative designshas proved extremely valuable in practice, but it still suffers the limitation that it does not guarantee theidentication of the best possible design. Generally one has to accept the best so far by a given cutoff date inthe program schedule. To ensure the realization of the true best design, the ultimate goal of computationalsimulation methods should not just be the analysis of prescribed shapes, but the automatic determinationof the true optimum shape for the intended application.

This is the underlying motivation for the combination of computational uid dynamics with numericaloptimization methods. Some of the earliest studies of such an approach were made by Hicks and Henne [1,2].The principal obstacle was the large computational cost of determining the sensitivity of the cost functionto variations of the design parameters by repeated calculation of the ow. Another way to approach theproblem is to formulate aerodynamic shape design within the framework of the mathematical theory for thecontrol of systems governed by partial differential equations [3]. In this view the wing is regarded as a deviceto produce lift by controlling the ow, and its design is regarded as a problem in the optimal control of theow equations by changing the shape of the boundary. If the boundary shape is regarded as arbitrary withinsome requirements of smoothness, then the full generality of shapes cannot be dened with a nite numberof parameters, and one must use the concept of the Frechet derivative of the cost with respect to a function.Clearly such a derivative cannot be determined directly by separate variation of each design parameter,because there are now an innite number of these.

Using techniques of control theory, however, the gradient can be determined indirectly by solving anadjoint equation which has coefficients determined by the solution of the ow equations. This directly cor-responds to the gradient technique for trajectory optimization pioneered by Bryson [4]. The cost of solving


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the adjoint equation is comparable to the cost of solving the ow equations, with the consequence that thegradient with respect to an arbitrarily large number of parameters can be calculated with roughly the samecomputational cost as two ow solutions. Once the gradient has been calculated, a descent method can beused to determine a shape change which will make an improvement in the design. The gradient can thenbe recalculated, and the whole process can be repeated until the design converges to an optimum solution,usually within 10 - 50 cycles. The fast calculation of the gradients makes optimization computationally fea-sible even for designs in three-dimensional viscous ow. There is a possibility that the descent method couldconverge to a local minimum rather than the global optimum solution. In practice this has not proved a diffi-culty, provided care is taken in the choice of a cost function which properly reects the design requirements.Conceptually, with this approach the problem is viewed as innitely dimensional, with the control being theshape of the bounding surface. Eventually the equations must be discretized for a numerical implementationof the method. For this purpose the ow and adjoint equations may either be separately discretized fromtheir representations as differential equations, or, alternatively, the ow equations may be discretized rst,and the discrete adjoint equations then derived directly from the discrete ow equations.

The effectiveness of optimization as a tool for aerodynamic design also depends crucially on the properchoice of cost functions and constraints. One popular approach is to dene a target pressure distribution,and then solve the inverse problem of nding the shape that will produce that pressure distribution. Sincesuch a shape does not necessarily exist, direct inverse methods may be ill-posed. The problem of designing atwo-dimensional prole to attain a desired pressure distribution was studied by Lighthill, who solved it forthe case of incompressible ow with a conformal mapping of the prole to a unit circle [5]. The speed overthe prole is

q =1h

| | ,

where is the potential which is known for incompressible ow and h is the modulus of the mapping function.The surface value of h can be obtained by setting q = qd , where qd is the desired speed, and since the mappingfunction is analytic, it is uniquely determined by the value of h on the boundary. A solution exists for agiven speed q at innity only if

12 q d = q ,

and there are additional constraints on q if the prole is required to be closed.The difficulty that the target pressure may be unattainable may be circumvented by treating the inverse

problem as a special case of the optimization problem, with a cost function which measures the error in the

solution of the inverse problem. For example, if pd is the desired surface pressure, one may take the costfunction to be an integral over the the body surface of the square of the pressure error,

I =12 B ( p pd )2dB,

or possibly a more general Sobolev norm of the pressure error. This has the advantage of converting apossibly ill posed problem into a well posed one. It has the disadvantage that it incurs the computationalcosts associated with optimization procedures.

The inverse problem still leaves the denition of an appropriate pressure architecture to the designer.One may prefer to directly improve suitable performance parameters, for example, to minimize the drag at agiven lift and Mach number. In this case it is important to introduce appropriate constraints. For example,if the span is not xed the vortex drag can be made arbitrarily small by sufficiently increasing the span. Inpractice, a useful approach is to x the planform, and optimize the wing sections subject to constraints onminimum thickness.

Studies of the use of control theory for optimum shape design of systems governed by elliptic equationswere initiated by Pironneau [6]. The control theory approach to optimal aerodynamic design was rst appliedto transonic ow by Jameson [712]. He formulated the method for inviscid compressible ows with shockwaves governed by both the potential ow and the Euler equations [8]. Numerical results showing the methodto be extremely effective for the design of airfoils in transonic potential ow were presented in [13,14], andfor three-dimensional wing design using the Euler equations in [15]. More recently the method has beenemployed for the shape design of complex aircraft congurations [16,17], using a grid perturbation approachto accommodate the geometry modications. The method has been used to support the aerodynamic design


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studies of several industrial projects, including the Beech Premier and the McDonnell Douglas MDXX andBlended Wing-Body projects. The application to the MDXX is described in [10]. The experience gained inthese industrial applications made it clear that the viscous effects cannot be ignored in transonic wing design,and the method has therefore been extended to treat the Reynolds Averaged Navier-Stokes equations [12].Adjoint methods have also been the subject of studies by a number of other authors, including Baysal andEleshaky [18], Huan and Modi [19], Desai and Ito [20], Anderson and Venkatakrishnan [21], and Peraireand Elliot [22]. Taasan, Kuruvila and Salas [23], who have implemented a one shot approach in whichthe constraint represented by the ow equations is only required to be satised by the nal convergedsolution. In their work, computational costs are also reduced by applying multigrid techniques to the geometrymodications as well as the solution of the ow and adjoint equations.

2 Formulation of the Design Problem as a Control Problem

The simplest approach to optimization is to dene the geometry through a set of design parameters, whichmay, for example, be the weights i applied to a set of shape functions bi (x ) so that the shape is representedas

f (x) = i bi (x ).

Then a cost function I is selected which might, for example, be the drag coefficient or the lift to drag ratio,and I is regarded as a function of the parameters i . The sensitivities I i may now be estimated by makinga small variation i in each design parameter in turn and recalculating the ow to obtain the change in I .Then

I i

I ( i + i ) I ( i )

i.

The gradient vector I may now be used to determine a direction of improvement. The simplest procedureis to make a step in the negative gradient direction by setting

n +1 = n ,

so that to rst orderI + I = I

I T

= I

I T

I

.

More sophisticated search procedures may be used such as quasi-Newton methods, which attempt to estimate

the second derivative 2

I i j of the cost function from changes in the gradient I in successive optimizationsteps. These methods also generally introduce line searches to nd the minimum in the search direction whichis dened at each step. The main disadvantage of this approach is the need for a number of ow calculationsproportional to the number of design variables to estimate the gradient. The computational costs can thusbecome prohibitive as the number of design variables is increased.

Using techniques of control theory, however, the gradient can be determined indirectly by solving anadjoint equation which has coefficients dened by the solution of the ow equations. The cost of solving theadjoint equation is comparable to that of solving the ow equations. Thus the gradient can be determinedwith roughly the computational costs of two ow solutions, independently of the number of design variables,which may be innite if the boundary is regarded as a free surface. The underlying concepts are claried bythe following abstract description of the adjoint method.

For ow about an airfoil or wing, the aerodynamic properties which dene the cost function are functionsof the ow-eld variables ( w) and the physical location of the boundary, which may be represented by thefunction F , say. Then

I = I (w, F ) ,

and a change in F results in a change

I =I T

w I w +

I T

F II F (1)

in the cost function. Here, the subscripts I and II are used to distinguish the contributions due to thevariation w in the ow solution from the change associated directly with the modication F in the shape.


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This notation assists in grouping the numerous terms that arise during the derivation of the full NavierStokes adjoint operator, outlined later, so that the basic structure of the approach as it is sketched in thepresent section can easily be recognized.

Suppose that the governing equation R which expresses the dependence of w and F within the owelddomain D can be written as

R (w, F ) = 0 . (2)

Then w is determined from the equation

R =Rw I

w +R F II

F = 0 . (3)

Since the variation R is zero, it can be multiplied by a Lagrange Multiplier and subtracted from thevariation I without changing the result. Thus equation (1) can be replaced by

I =I T

ww +

I T

F F

T Rw

w +R F

F

=I T

w

T Rw

I

w +I T

F

T R F

II

F . (4)

Choosing to satisfy the adjoint equation

Rw

T

=I w

(5)

the rst term is eliminated, and we nd that

I = GF , (6)

where

G =I T

F T

R F

.

The advantage is that (6) is independent of w , with the result that the gradient of I with respect to an

arbitrary number of design variables can be determined without the need for additional ow-eld evaluations.In the case that (2) is a partial differential equation, the adjoint equation (5) is also a partial differentialequation and determination of the appropriate boundary conditions requires careful mathematical treatment.

In reference [8] Jameson derived the adjoint equations for transonic ows modeled by both the potentialow equation and the Euler equations. The theory was developed in terms of partial differential equations,leading to an adjoint partial differential equation. In order to obtain numerical solutions both the ow and theadjoint equations must be discretized. Control theory might be applied directly to the discrete ow equationswhich result from the numerical approximation of the ow equations by nite element, nite volume or nitedifference procedures. This leads directly to a set of discrete adjoint equations with a matrix which is thetranspose of the Jacobian matrix of the full set of discrete nonlinear ow equations. On a three-dimensionalmesh with indices i , j ,k the individual adjoint equations may be derived by collecting together all the termsmultiplied by the variation w i,j,k of the discrete ow variable wi,j,k . The resulting discrete adjoint equationsrepresent a possible discretization of the adjoint partial differential equation. If these equations are solvedexactly they can provide an exact gradient of the inexact cost function which results from the discretizationof the ow equations. The discrete adjoint equations derived directly from the discrete ow equations becomevery complicated when the ow equations are discretized with higher order upwind biased schemes using uxlimiters. On the other hand any consistent discretization of the adjoint partial differential equation will yieldthe exact gradient in the limit as the mesh is rened. The trade-off between the complexity of the adjointdiscretization, the accuracy of the resulting estimate of the gradient, and its impact on the computationalcost to approach an optimum solution is a subject of ongoing research.

The true optimum shape belongs to an innitely dimensional space of design parameters. One motivationfor developing the theory for the partial differential equations of the ow is to provide an indication in


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principle of how such a solution could be approached if sufficient computational resources were available. Itdisplays the character of the adjoint equation as a hyperbolic system with waves travelling in the reversedirection to those of the ow equations, and the need for correct wall and far-eld boundary conditions. It alsohighlights the possibility of generating ill posed formulations of the problem. For example, if one attempts tocalculate the sensitivity of the pressure at a particular location to changes in the boundary shape, there is thepossibility that a shape modication could cause a shock wave to pass over that location. Then the sensitivitycould become unbounded. The movement of the shock, however, is continuous as the shape changes. Thereforea quantity such as the drag coefficient, which is determined by integrating the pressure over the surface, alsodepends continuously on the shape. The adjoint equation allows the sensitivity of the drag coefficient tobe determined without the explicit evaluation of pressure sensitivities which would be ill posed. Anotherbenet of the continuous adjoint formulation is that it allows grid sensitivity terms to be eliminated fromthe gradient, which can nally be expressed purely in terms of the boundary displacement, as will be shownin Section 4. This greatly simplies the implementation of the method for overset or unstructured grids.

The discrete adjoint equations, whether they are derived directly or by discretization of the adjointpartial differential equation, are linear. Therefore they could be solved by direct numerical inversion. Inthree-dimensional problems on a mesh with, say, n intervals in each coordinate direction, the number of unknowns is proportional to n 3 and the bandwidth to n 2 . The complexity of direct inversion is proportionalto the number of unknowns multiplied by the square of the bandwidth, resulting in a complexity proportionalto n 7 . The cost of direct inversion can thus become prohibitive as the mesh is rened, and it becomes moreefficient to use iterative solution methods. Moreover, because of the similarity of the adjoint equations tothe ow equations, the same iterative methods which have been proved to be efficient for the solution of theow equations are efficient for the solution of the adjoint equations.

3 Design using the Euler Equations

The application of control theory to aerodynamic design problems is illustrated in this section for the caseof three-dimensional wing design using the compressible Euler equations as the mathematical model. Itproves convenient to denote the Cartesian coordinates and velocity components by x 1 , x2 , x3 and u1 , u2 ,u 3 , and to use the convention that summation over i = 1 to 3 is implied by a repeated index i . Then, thethree-dimensional Euler equations may be written as

w

t+

f i

x i= 0 in D, (7)

where

w =

u 1u 2u 3E

, f i =

u iu i u 1 + pi1u i u 2 + pi2u i u 3 + pi3

u i H

(8)

and ij is the Kronecker delta function. Also,

p = ( 1) E 12

u 2i , (9)

andH = E + p (10)

where is the ratio of the specic heats.In order to simplify the derivation of the adjoint equations, we map the solution to a xed computational

domain with coordinates 1 , 2 , 3 where

K ij =x i j

, J = det( K ) , K 1ij = ix j

,


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andS = JK 1 .

The elements of S are the cofactors of K , and in a nite volume discretization they are just the face areasof the computational cells projected in the x 1 , x2 , and x3 directions. Using the permutation tensor ijk wecan express the elements of S as

S ij =12 jpq irs

x p r

x q s . (11)

Then

i

S ij =12 jpq irs

2x p r i

x q s

+x p r

2xq s i

= 0 . (12)

Also in the subsequent analysis of the effect of a shape variation it is useful to note that

S 1j = jpqx p 2

x q 3

,

S 2j = jpqx p

3

x q

1,

S 3j = jpqx p 1

x q 2

. (13)

Now, multiplying equation(7) by J and applying the chain rule,

J wt

+ R (w) = 0 (14)

whereR (w) = S ij

f j i

=

i(S ij f j ) , (15)

using (12). We can write the transformed uxes in terms of the scaled contravariant velocity components

U i = S ij u jas

F i = S ij f j =

U iU i u 1 + S i1 pU i u 2 + S i2 pU i u 3 + S i3 p

U i H

.

For convenience, the coordinates i describing the xed computational domain are chosen so that eachboundary conforms to a constant value of one of these coordinates. Variations in the shape then resultin corresponding variations in the mapping derivatives dened by K ij . Suppose that the performance ismeasured by a cost function

I =

B

M (w, S ) dB +

D

P (w, S ) dD ,

containing both boundary and eld contributions where dB and dD are the surface and volume elementsin the computational domain. In general, M and P will depend on both the ow variables w and the metricsS dening the computational space. The design problem is now treated as a control problem where theboundary shape represents the control function, which is chosen to minimize I subject to the constraintsdened by the ow equations (14). A shape change produces a variation in the ow solution w and themetrics S which in turn produce a variation in the cost function

I = B M (w, S ) dB + D P (w, S ) dD . (16)


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This can be split asI = I I + I II , (17)

with

M = [Mw ]I w + M II ,P = [P w ]I w + P II , (18)

where we continue to use the subscripts I and II to distinguish between the contributions associated withthe variation of the ow solution w and those associated with the metric variations S . Thus [M w ]I and[P w ]I represent

Mw and

P w with the metrics xed, while M II and P II represent the contribution of the

metric variations S to M and P .In the steady state, the constraint equation (14) species the variation of the state vector w by

R =

iF i = 0 . (19)

Here also, R and F i can be split into contributions associated with w and S using the notation

R = R I + R II F i = [ F i w ]I w + F i II . (20)

where[F i w ]I = S ij

f iw

.

Multiplying by a co-state vector , which will play an analogous role to the Lagrange multiplier introducedin equation (4), and integrating over the domain produces

D T i F i dD = 0 . (21)Assuming that is differentiable, the terms with subscript I may be integrated by parts to give

B n i T F i I dB D T

iF i I dD + D T R II dD = 0 . (22)

This equation results directly from taking the variation of the weak form of the ow equations, where istaken to be an arbitrary differentiable test function. Since the left hand expression equals zero, it may besubtracted from the variation in the cost function (16) to give

I = I II D T R II dD B M I n i T F i I dB+ D P I +

T

iF i I dD . (23)

Now, since is an arbitrary differentiable function, it may be chosen in such a way that I no longerdepends explicitly on the variation of the state vector w . The gradient of the cost function can then beevaluated directly from the metric variations without having to recompute the variation w resulting fromthe perturbation of each design variable.

Comparing equations (18) and (20), the variation w may be eliminated from (23) by equating all eldterms with subscript I to produce a differential adjoint system governing

T

i[F i w ]I + [P w ]I = 0 in D. (24)

Taking the transpose of equation (24), in the case that there is no eld integral in the cost function, theinviscid adjoint equation may be written as

C T i i

= 0 in D, (25)


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where the inviscid Jacobian matrices in the transformed space are given by

C i = S ijf jw

.

The corresponding adjoint boundary condition is produced by equating the subscript I boundary termsin equation (23) to produce

n i T

[F i w ]I = [ Mw ]I on B. (26)The remaining terms from equation (23) then yield a simplied expression for the variation of the costfunction which denes the gradient

I = I II + D T R II dD , (27)which consists purely of the terms containing variations in the metrics, with the ow solution xed. Hencean explicit formula for the gradient can be derived once the relationship between mesh perturbations andshape variations is dened.

The details of the formula for the gradient depend on the way in which the boundary shape is parame-terized as a function of the design variables, and the way in which the mesh is deformed as the boundaryis modied. Using the relationship between the mesh deformation and the surface modication, the eldintegral is reduced to a surface integral by integrating along the coordinate lines emanating from the surface.

Thus the expression for I is nally reduced to the form of equation (6)

I = B GF dBwhere F represents the design variables, and G is the gradient, which is a function dened over the boundarysurface.

The boundary conditions satised by the ow equations restrict the form of the left hand side of theadjoint boundary condition (26). Consequently, the boundary contribution to the cost function M cannotbe specied arbitrarily. Instead, it must be chosen from the class of functions which allow cancellation of all terms containing w in the boundary integral of equation (23). On the other hand, there is no suchrestriction on the specication of the eld contribution to the cost function P , since these terms may alwaysbe absorbed into the adjoint eld equation (24) as source terms.

For simplicity, it will be assumed that the portion of the boundary that undergoes shape modications isrestricted to the coordinate surface 2 = 0. Then equations (23) and (26) may be simplied by incorporatingthe conditions

n 1 = n 3 = 0 , n 2 = 1 , dB = d1 d3 ,

so that only the variation F 2 needs to be considered at the wall boundary. The condition that there is noow through the wall boundary at 2 = 0 is equivalent to

U 2 = 0 ,

so thatU 2 = 0

when the boundary shape is modied. Consequently the variation of the inviscid ux at the boundary reduces

to

F 2 = p

0

S 21

S 22

S 23

0

+ p

0

S 21

S 22

S 23

0

. (28)


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Since F 2 depends only on the pressure, it is now clear that the performance measure on the boundaryM (w, S ) may only be a function of the pressure and metric terms. Otherwise, complete cancellation of theterms containing w in the boundary integral would be impossible. One may, for example, include arbitrarymeasures of the forces and moments in the cost function, since these are functions of the surface pressure.

In order to design a shape which will lead to a desired pressure distribution, a natural choice is to set

I =1

2 B

( p pd )2 dS

where pd is the desired surface pressure, and the integral is evaluated over the actual surface area. In thecomputational domain this is transformed to

I =12 B w ( p pd )2 |S 2 | d1 d3 ,

where the quantity|S 2 | = S 2j S 2jdenotes the face area corresponding to a unit element of face area in the computational domain. Now, to

cancel the dependence of the boundary integral on p, the adjoint boundary condition reduces to

j n j = p pd (29)where n j are the components of the surface normal

n j =S 2j|S 2 |

.

This amounts to a transpiration boundary condition on the co-state variables corresponding to the momen-tum components. Note that it imposes no restriction on the tangential component of at the boundary.

We nd nally that

I = D T

iS ij f j dD

B W (S 21 2 + S 22 3 + S 23 4) pd1d3 . (30)

Here the expression for the cost variation depends on the mesh variations throughout the domain whichappear in the eld integral. However, the true gradient for a shape variation should not depend on the wayin which the mesh is deformed, but only on the true ow solution. In the next section we show how theeld integral can be eliminated to produce a reduced gradient formula which depends only on the boundarymovement.

4 The Reduced Gradient Formulation

Consider the case of a mesh variation with a xed boundary. Then,

I = 0

but there is a variation in the transformed ux,

F i = C i w + S ij f j .

Here the true solution is unchanged. Thus, the variation w is due to the mesh movement x at each meshpoint. Therefore

w = w x =wx j

x j (= w )


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and since

iF i = 0 ,

it follows that

i(S ij f j ) =

i

(C i w ) . (31)

It is veried below that this relation holds in the general case with boundary movement. Now

D T Rd D = D T i C i (w w ) dD= B T C i (w w ) dB D

T

iC i (w w ) dD. (32)

Here on the wall boundaryC 2w = F 2 S 2j f j . (33)

Thus, by choosing to satisfy the adjoint equation (25) and the adjoint boundary condition (26), we reduce

the cost variation to a boundary integral which depends only on the surface displacement:

I = B W T (S 2j f j + C 2w ) d1d3 B W (S 21 2 + S 22 3 + S 23 4) pd1d3 . (34)

For completeness the general derivation of equation(31) is presented here. Using the formula(11), and theproperty (12)

i (S ij f j )

=12

i

jpq irsx p r

x q s

+x p r

x q s

f j

=12 jpq irs

x p r

x q s

+x p r

x q s

f j i

=12 jpq irs

r

x px q s

f j i

+12 jpq irs

s

x qx p r

f j i

=

rx p pqj rsi

x q s

f j i

. (35)

Now express x p in terms of a shift in the original computational coordinates

x p = x p k k .

Then we obtain

i(S ij f j ) =

r

pqj rsix p k

x q s

f j i

k . (36)

The term in 1 is

123 pqjx p k

x q 2

f j 3

x q 3

f j 2

k .


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Here the term multiplying 1 is

jpqx p 1

x q 2

f j 3

x p 1

x q 3

f j 2

.

According to the formulas(13) this may be recognized as

S 2jf 1 2 + S 3j

f 1 3

or, using the quasi-linear form(15) of the equation for steady ow, as

S 1jf 1 1

.

The terms multiplying 2 and 3 are

jpqx p 2

x q 2

f j 3

x p 2

x q 3

f j 2

= S 1jf 1 2

and

jpqx p 3

x q 2

f j 3

x p 3

x q 3

f j 2 = S 1j

f 1 3 .

Thus the term in 1 is reduced to

1S 1j

f 1 k

k .

Finally, with similar reductions of the terms in 2 and

3 , we obtain

i

(S ij f j ) =

iS ij

f j k

k =

i(C i w )

as was to be proved.

5 Optimization Procedure

5.1 The need for a Sobolev inner product in the denition of the gradient

Another key issue for successful implementation of the continuous adjoint method is the choice of an appro-priate inner product for the denition of the gradient. It turns out that there is an enormous benet fromthe use of a modied Sobolev gradient, which enables the generation of a sequence of smooth shapes. Thiscan be illustrated by considering the simplest case of a problem in the calculus of variations.

Suppose that we wish to nd the path y(x) which minimizes

I =b

a F (y, y )dxwith xed end points y(a ) and y(b). Under a variation y (x),

I =b

aF y

y +F y

y dx

=b

aF y

d

dxF y

ydx


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Thus dening the gradient as

g =F y

d

dxF y

and the inner product as

(u, v ) =b

a uvdxwe nd that

I = ( g,y ).

If we now set

y = g, > 0

we obtain a improvement

I = (g, g) 0unless g = 0, the necessary condition for a minimum.

Note that g is a function of y, y , y ,g = g(y, y , y )

In the well known case of the Brachistrone problem, for example, which calls for the determination of thepath of quickest descent between two laterally separated points when a particle falls under gravity,

F (y, y ) = 1 + y 2yand

g = 1 + y 2 + 2 yy

2 (y(1 + y 2)) 3/ 2

It can be seen that each stepyn +1 = yn n gn

reduces the smoothness of y by two classes. Thus the computed trajectory becomes less and less smooth,leading to instability.

In order to prevent this we can introduce a weighted Sobolev inner product [24]

u, v = (uv + u v )dxwhere is a parameter that controls the weight of the derivatives. We now dene a gradient g such that

I = g,y

Then we have

I = (gy + g y )dx= (g x gx )ydx= ( g,y )


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whereg

x

gx

= g

and g = 0 at the end points. Thus g can be obtained from g by a smoothing equation. Now the step

yn +1 = yn n gn

gives an improvement I = n gn , gn

but yn +1 has the same smoothness as yn , resulting in a stable process.

5.2 Sobolev gradient for shape optimization

In applying control theory to aerodynamic shape optimization, the use of a Sobolev gradient is equallyimportant for the preservation of the smoothness class of the redesigned surface. Accordingly, using theweighted Sobolev inner product dened above, we dene a modied gradient G such that

I = < G, F > .

In the one dimensional case G is obtained by solving the smoothing equation

G

1

1

G = G. (37)

In the multi-dimensional case the smoothing is applied in product form. Finally we set

F = G (38)

with the result thatI = < G, G > < 0,

unless G = 0, and correspondingly G = 0.When second-order central differencing is applied to (37), the equation at a given node, i , can be expressed

asGi Gi+1 2Gi + Gi 1 = Gi , 1 i n,

where Gi and Gi are the point gradients at node i before and after the smoothing respectively, and n is thenumber of design variables equal to the number of mesh points in this case. Then,

G = AG,

where A is the n n tri-diagonal matrix such that

A 1 =

1 + 2 0 . 0 . .

0 . . .. . .

0 1 + 2

.

Using the steepest descent method in each design iteration, a step, F , is taken such that

F = A G. (39)

As can be seen from the form of this expression, implicit smoothing may be regarded as a preconditionerwhich allows the use of much larger steps for the search procedure and leads to a large reduction in the numberof design iterations needed for convergence. Our software also includes an option for Krylov acceleration [25].We have found this to be particularly useful for inverse problems.


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5.3 Outline of the design procedure

The design procedure can nally be summarized as follows:

1. Solve the ow equations for , u 1 , u 2 , u3 , p.

2. Solve the adjoint equations for subject to appropriate boundary conditions.3. Evaluate G and calculate the corresponding Sobolev gradient G.4. Project G into an allowable subspace that satises any geometric constraints.5. Update the shape based on the direction of steepest descent.6. Return to 1 until convergence is reached.

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification

Fig. 1. Design cycle

Practical implementation of the design method relies heavily upon fast and accurate solvers for boththe state ( w) and co-state ( ) systems. The result obtained in Section 8 have been obtained using well-validated software for the solution of the Euler and Navier-Stokes equations developed over the course of many years [2628]. For inverse design the lift is xed by the target pressure. In drag minimization it isalso appropriate to x the lift coefficient, because the induced drag is a major fraction of the total drag,and this could be reduced simply by reducing the lift. Therefore the angle of attack is adjusted during eachow solution to force a specied lift coefficient to be attained, and the inuence of variations of the angleof attack is included in the calculation of the gradient. The vortex drag also depends on the span loading,which may be constrained by other considerations such as structural loading or buffet onset. Consequently,the option is provided to force the span loading by adjusting the twist distribution as well as the angle of attack during the ow solution.


15/30

6 Design using the Navier-Stokes Equations

6.1 The Navier-Stokes equations in the computational domain

The next sections present the extension of the adjoint method to the Navier-Stokes equations. These takethe form

w

t

+f i

x i=

f vi

x iin D, (40)

where the state vector w, inviscid ux vector f and viscous ux vector f v are described respectively by

w =

u 1u 2u 3E

, f i =

u iu i u 1 + pi 1u i u 2 + pi 2u i u 3 + pi 3

u i H

, f v i =

0 ij j 1 ij j 2 ij j 3

u j ij + k T x i

. (41)

The viscous stresses may be written as

ij = u ix j

+u jx i

+ iju kx k

, (42)

where and are the rst and second coefficients of viscosity. The coefficient of thermal conductivity andthe temperature are computed as

k =c pP r

, T =p

R, (43)

where P r is the Prandtl number, c p is the specic heat at constant pressure, and R is the gas constant.Using a transformation to a xed computational domain as before, the Navier-Stokes equations can be

written in the transformed coordinates as

(Jw )t

+ (F i F vi )

i= 0 in D, (44)

where the viscous terms have the form

F v i i =

i S

ij f v j .

Computing the variation w resulting from a shape modication of the boundary, introducing a co-statevector and integrating by parts, following the steps outlined by equations (19) to (22), we obtain

B T S 2j f v j + S 2j f v j dB D T

iS ij f v j + S ij f v j dD ,

where the shape modication is restricted to the coordinate surface 2 = 0 so that n 1 = n 3 = 0, and n 2 = 1.Furthermore, it is assumed that the boundary contributions at the far eld may either be neglected or elseeliminated by a proper choice of boundary conditions as previously shown for the inviscid case [14,29].

The viscous terms will be derived under the assumption that the viscosity and heat conduction coefficients and k are essentially independent of the ow, and that their variations may be neglected. This simplicationhas been successfully used for may aerodynamic problems of interest. However, if the ow variations couldresult in signicant changes in the turbulent viscosity, it may be necessary to account for its variation in thecalculation.

6.2 Transformation to Primitive Variables

The derivation of the viscous adjoint terms can be simplied by transforming to the primitive variables

wT = ( , u 1 , u 2 , u 3 , p)T ,


16/30

because the viscous stresses depend on the velocity derivatives u ix j , while the heat ux can be expressed as

x i

p

.

where = kR =

P r ( 1) . The relationship between the conservative and primitive variations is dened bythe expressions

w = M w, w = M 1

wwhich make use of the transformation matrices M = w w and M

1 = ww . These matrices are provided intransposed form for future convenience

M T =

1 u 1 u2 u 3 u i u i20 0 0 u 10 0 0 u 20 0 0 u 30 0 0 0 1 1

M 1 T =

1 u 1 u 2

u 3

( 1) u i u i2

0 1 0 0 ( 1)u 10 0 1 0 ( 1)u 20 0 0 1 ( 1)u 30 0 0 0 1

.

The conservative and primitive adjoint operators L and L corresponding to the variations w and w arethen related by

D wT L d D = D wT L d D ,with

L = M T L,

so that after determining the primitive adjoint operator by direct evaluation of the viscous portion of (24),the conservative operator may be obtained by the transformation L = M 1 T L . Since the continuity equationcontains no viscous terms, it makes no contribution to the viscous adjoint system. Therefore, the derivation

proceeds by rst examining the adjoint operators arising from the momentum equations.

6.3 Contributions from the Momentum Equations

In order to make use of the summation convention, it is convenient to set j +1 = j for j = 1 , 2, 3. Thenthe contribution from the momentum equations is

B k (S 2j kj + S 2j kj ) dB D k i (S ij kj + S ij kj ) dD . (45)The velocity derivatives can be expressed as

u ix j

=u i l

lx j

=S ljJ

u i l

with corresponding variations

u ix j

=S ljJ I

l

u i +u i l II

S ljJ

.

The variations in the stresses are then

kj = S ljJ

l u k +S lkJ

l u j + jk

S lmJ

l u m I

+ S ljJ u k l +

S lkJ

u j l + jk

S lmJ

u m l II

.


17/30

As before, only those terms with subscript I , which contain variations of the ow variables, need be consideredfurther in deriving the adjoint operator. The eld contributions that contain u i in equation (45) appear as

D k i S ij S ljJ l u k + S lkJ l u j + jk S lmJ l u m dD .This may be integrated by parts to yield

D u k l S lj S ij J k i dD+ D u j l S lk S ij J k i dD+ D u m l S lm S ij jkJ k i dD ,

where the boundary integral has been eliminated by noting that u i = 0 on the solid boundary. By exchangingindices, the eld integrals may be combined to produce

D

u k

lS lj

S ijJ

k i

+S ikJ

j i

+ jkS im

J m i

dD ,

which is further simplied by transforming the inner derivatives back to Cartesian coordinates

D u k l S lj kx j + jx k + jk mx m dD . (46)The boundary contributions that contain u i in equation (45) may be simplied using the fact that

l

u i = 0 if l = 1 , 3

on the boundary B so that they become

B k S 2j

S 2jJ

2 u

k +S 2kJ

2 u

j + jkS 2m

J

2 um dB . (47)

Together, (46) and (47) comprise the eld and boundary contributions of the momentum equations to theviscous adjoint operator in primitive variables.

6.4 Contributions from the Energy Equation

In order to derive the contribution of the energy equation to the viscous adjoint terms it is convenient to set

5 = , Q j = u i ij +

x j

p

,

where the temperature has been written in terms of pressure and density using (43). The contribution fromthe energy equation can then be written as

B (S 2j Q j + S 2j Q j ) dB D i (S ij Q j + S ij Q j ) dD . (48)The eld contributions that contain u i ,p, and in equation (48) appear as

D i S ij Q j dD = D i S ij u k kj + uk kj + S ljJ l p p dD . (49)


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The term involving kj may be integrated by parts to produce

D u k l S lj uk x j + u j x k + jk u m x m dD , (50)where the conditions u i = u i = 0 are used to eliminate the boundary integral on B. Notice that the otherterm in (49) that involves u k need not be integrated by parts and is merely carried on as

D u k kj S ij i dD (51)The terms in expression (49) that involve p and may also be integrated by parts to produce both a

eld and a boundary integral. The eld integral becomes

D p p l S lj S ij J i dDwhich may be simplied by transforming the inner derivative to Cartesian coordinates

D

p

p

l

S lj

x jdD . (52)

The boundary integral becomes

B p p S 2j S ijJ i dB . (53)This can be simplied by transforming the inner derivative to Cartesian coordinates

B p p S 2jJ x j dB , (54)and identifying the normal derivative at the wall

n= S 2j

x j, (55)

and the variation in temperature

T =1R

p

p

,

to produce the boundary contribution

B kT n dB . (56)This term vanishes if T is constant on the wall but persists if the wall is adiabatic.

There is also a boundary contribution left over from the rst integration by parts (48) which has theform

B

(S 2j Q j ) dB , (57)

whereQ j = k

T x j

,

since u i = 0. If the wall is adiabaticT n

= 0 ,

so that using (55), (S 2j Q j ) = 0 ,


19/30

and both the w and S boundary contributions vanish.On the other hand, if T is constant T l = 0 for l = 1 , 3, so that

Q j = kT x j

= kS l jJ

T l

= kS 2jJ

T 2

.

Thus, the boundary integral (57) becomes

B k S 2j2

J

2T +

S 2j 2

J T 2

dB . (58)

Therefore, for constant T , the rst term corresponding to variations in the ow eld contributes to theadjoint boundary operator, and the second set of terms corresponding to metric variations contribute to thecost function gradient.

Finally the contributions from the energy equation to the viscous adjoint operator are the three eldterms (50), (51) and (52), and either of two boundary contributions ( 56) or ( 58), depending on whetherthe wall is adiabatic or has constant temperature.

6.5 The Viscous Adjoint Field Operator

Collecting together the contributions from the momentum and energy equations, the viscous adjoint operatorin primitive variables can be expressed as

(L )1 = p 2

l S lj

x j

(L ) i+1 = l S lj ix j +

jx i + ij

kx k

+ l S lj u i

x j + u j

x i + ij u k

x k for i = 1 , 2, 3 ij S lj l

(L )5 = 1

l S lj

x j .

The conservative viscous adjoint operator may now be obtained by the transformation

L = M 1T L.

7 Viscous Adjoint Boundary Conditions

It was recognized in Section 3 that the boundary conditions satised by the ow equations restrict the formof the performance measure that may be chosen for the cost function. There must be a direct correspondencebetween the ow variables for which variations appear in the variation of the cost function, and those variablesfor which variations appear in the boundary terms arising during the derivation of the adjoint eld equations.Otherwise it would be impossible to eliminate the dependence of I on w through proper specication of the adjoint boundary condition. Consequently the contributions of the pressure and viscous stresses needto be merged. As in the derivation of the eld equations, it proves convenient to consider the contributionsfrom the momentum equations and the energy equation separately.

7.1 Boundary Conditions Arising from the Momentum Equations

The boundary term that arises from the momentum equations including both the w and S components(45) takes the form

B k (S 2j (kj p + kj )) dB .


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Replacing the metric term with the corresponding local face area S 2 and unit normal n j dened by

|S 2 | = S 2j S 2j , n j =S 2j|S 2 |

then leads to

B

k (|S 2 | n j (kj p + kj )) dB .

Dening the components of the total surface stress as

k = n j (kj p + kj )

and the physical surface elementdS = |S 2 | dB ,

the integral may then be split into two components

B k k |S 2 | dB + B k k dS, (59)where only the second term contains variations in the ow variables and must consequently cancel the wterms arising in the cost function. The rst term will appear in the expression for the gradient.

A general expression for the cost function that allows cancellation with terms containing k has the formI = B N ( )dS, (60)

corresponding to a variation

I = B N k k dS,for which cancellation is achieved by the adjoint boundary condition

k = N k

.

Natural choices for N arise from force optimization and as measures of the deviation of the surface stressesfrom desired target values.

The force in a direction with cosines qi has the form

C q = B qi i dS.If we take this as the cost function (60), this quantity gives

N = qi i .

Cancellation with the ow variation terms in equation (59) therefore mandates the adjoint boundary condi-tion

k = qk .

Note that this choice of boundary condition also eliminates the rst term in equation (59) so that it need

not be included in the gradient calculation.In the inverse design case, where the cost function is intended to measure the deviation of the surfacestresses from some desired target values, a suitable denition is

N ( ) =12

a lk ( l dl ) ( k dk ) ,

where d is the desired surface stress, including the contribution of the pressure, and the coefficients a lkdene a weighting matrix. For cancellation

k k = a lk ( l dl ) k .


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This is satised by the boundary condition

k = a lk ( l dl ) . (61)

Assuming arbitrary variations in k , this condition is also necessary.In order to control the surface pressure and normal stress one can measure the difference

n j {kj + kj ( p pd)} ,

where pd is the desired pressure. The normal component is then

n = n k n j kj + p pd ,

so that the measure becomes

N ( ) =12

2n

=12

n l n m n k n j { lm + lm ( p pd )} {kj + kj ( p pd )} .

This corresponds to settinga lk = n l n k

in equation (61). Dening the viscous normal stress as

vn = n k n j kj ,

the measure can be expanded as

N ( ) = 12

n l n m n k n j lm kj +12

(n k n j kj + n ln m lm ) ( p pd ) +12

( p pd)2

=12

2vn + vn ( p pd) +12

( p pd)2 .

For cancellation of the boundary terms

k (n j kj + n k p) = n ln m lm + n 2l ( p pd) n k (n j kj + n k p)

leading to the boundary conditionk = n k ( vn + p pd ) .

In the case of high Reynolds number, this is well approximated by the equations

k = n k ( p pd) , (62)

which should be compared with the single scalar equation derived for the inviscid boundary condition (29).

In the case of an inviscid ow, choosing

N ( ) =12

( p pd )2

requiresk n k p = ( p pd ) n 2k p = ( p pd ) p

which is satised by equation (62), but which represents an overspecication of the boundary condition sinceonly the single condition (29) needs be specied to ensure cancellation.


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Boundary Conditions Arising from the Energy Equation

The form of the boundary terms arising from the energy equation depends on the choice of temperatureboundary condition at the wall. For the adiabatic case, the boundary contribution is (56)

B

kT n

dB ,

while for the constant temperature case the boundary term is (58). One possibility is to introduce a contri-bution into the cost function which depends on T or T n so that the appropriate cancellation would occur.Since there is little physical intuition to guide the choice of such a cost function for aerodynamic design, amore natural solution is to set

= 0

in the constant temperature case orn

= 0

in the adiabatic case. Note that in the constant temperature case, this choice of on the boundary wouldalso eliminate the boundary metric variation terms in (57).

8 Results

8.1 Redesign of the Boeing 747 wing

Over the last decade the adjoint method has been successfully used to rene a variety of designs for ight atboth transonic and supersonic cruising speeds. In the case of transonic ight, it is often possible to producea shock free ow which eliminates the shock drag by making very small changes, typically no larger thanthe boundary layer displacement thickness. Consequently viscous effects need to be considered in order torealize the full benets of the optimization.

Here the optimization of the wing of the Boeing 747-200 is presented to illustrate the kind of benetsthat can be obtained. In these calculations the ow was modeled by the Reynolds Averaged Navier-Stokes

equations. A Baldwin Lomax turbulence model was considered sufficient, since the optimization is for thecruise condition with attached ow. The calculation were performed to minimize the drag coefficient at axed lift coefficient, subject to the additional constraints that the span loading should not be altered andthe thickness should not be reduced. It might be possible to reduce the induced drag by modifying the spanloading to an elliptic distribution, but this would increase the root bending moment, and consequently requirean increase in the skin thickness and structure weight. A reduction in wing thickness would not only reducethe fuel volume, but it would also require an increase in skin thickness to support the bending moment. Thusthese constraints assure that there will be no penalty in either structure weight or fuel volume.

Figure 2 displays the result of an optimization at a Mach number of 0.86, which is roughly the maximumcruising Mach number attainable by the existing design before the onset of signicant drag rise. The liftcoefficient of 0.42 is the contribution of the exposed wing. Allowing for the fuselage to total lift coefficientis about 0.47. It can be seen that the redesigned wing is essentially shock free, and the drag coefficient isreduced from 0.01269 (127 counts) to 0.01136 (114 counts). The total drag coefficient of the aircraft at thislift coefficient is around 270 counts, so this would represent a drag reduction of the order of 5 percent.

Figure 3 displays the result of an optimization at Mach 0.90. In this case the shock waves are noteliminated, but their strength is signicantly weakened, while the drag coefficient is reduced from 0.01819(182 counts) to 0.01293 (129 counts). Thus the redesigned wing has essentially the same drag at Mach 0.9as the original wing at Mach 0.86. The Boeing 747 wing could apparently be modied to allow such anincrease in the cruising Mach number because it has a higher sweep-back than later designs, and a ratherthin wing section with a thickness to chord ratio of 8 percent. Figures 4 and 5 verify that the span loadingand thickness were not changed by the redesign, while gures 6 and 7 indicate the required section changesat 42 percent and 68 percent span stations.


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8.2 Wing design using an unstructured mesh

A major obstacle to the treatment of arbitrarily complex congurations is the difficulty and cost of meshgeneration. This can be mitigated by the use of unstructured meshes. Thus it appears that the extension of the adjoint method to unstructured meshes may provide the most promising route to the optimum shapedesign of key elements of complex congurations, such as wing-pylon-nacelle combinations. Some preliminaryresults are presented below. These have been obtained with new software to implement the adjoint method for

unstructured meshes which is currently under development [30]. Figures 8 and 9 shows the result of an inversedesign calculation, where the initial geometry was a wing made up of NACA 0012 sections and the targetpressure distribution was the pressure distribution over the Onera M6 wing. Figures 10, 11, 12, 13, 14, 15,show the target and computed pressure distribution at six span-wise sections. It can be seen from theseplots the target pressure distribution is well recovered in 50 design cycles, verifying that the design processis capable of recovering pressure distributions that are signicantly different from the initial distribution.This is a particularly challenging test, because it calls for the recovery of a smooth symmetric prole froman asymmetric pressure distribution containing a triangular pattern of shock waves.

Another test case for the inverse design problem uses the wing from an airplane (code named SHARK) [31]which has been designed for the Reno Air Races. The initial and nal pressure distributions are shown thegure 16. As can be seen from these plots, the initial pressure distribution has a weak shock in the outboardsections of the wing, while the nal pressure distribution is shock-free. The nal pressure distributions arecompared with the target distributions along three sections of the wing in gures 17, 18, 19. Again the designprocess captures the target pressure with good accuracy in about 50 design cycles. The drag minimizationproblem has also been studied for this wing, and the results are shown in gure 20. As can be seen from thisplot, the nal geometry has a shock-free prole and the drag coefficient has been slightly reduced.

9 Conclusion

The accumulated experience of the last decade suggests that most existing aircraft which cruise at transonicspeeds are amenable to a drag reduction of the order of 3 to 5 percent, or an increase in the drag rise Machnumber of at least .02. These improvements can be achieved by very small shape modications, which aretoo subtle to allow their determination by trial and error methods. The potential economic benets aresubstantial, considering the fuel costs of the entire airline eet. Moreover, if one were to take full advantageof the increase in the lift to drag ratio during the design process, a smaller aircraft could be designed toperform the same task, with consequent further cost reductions. It seems inevitable that some method of this type will provide a basis for aerodynamic designs of the future.

Acknowledgment

This work has beneted greatly from the support of the Air Force Office of Science Research under grantNo. AF F49620-98-1-2002. I have drawn extensively from the lecture notes prepared by Luigi Martinelliand myself for a CIME summer course in 1999 [32]. I am also indebted to my research assistant KasiditLeoviriyakit for his assistance in preparing the Latex les for this text.

References

1. R. M. Hicks, E. M. Murman, and G. N. Vanderplaats. An assessment of airfoil design by numerical optimization.NASA TM X-3092 , Ames Research Center, Moffett Field, California, July 1974.

2. R. M. Hicks and P. A. Henne. Wing design by numerical optimization. Journal of Aircraft , 15:407412, 1978.3. J. L. Lions. Optimal Control of Systems Governed by Partial Differential Equations . Springer-Verlag, New York,

1971. Translated by S.K. Mitter.4. A. E. Bryson and Y. C. Ho. Applied Optimal Control . Hemisphere, Washington, DC, 1975.5. M. J. Lighthill. A new method of two-dimensional aerodynamic design. R & M 1111 , Aeronautical Research

Council, 1945.6. O. Pironneau. Optimal Shape Design for Elliptic Systems . Springer-Verlag, New York, 1984.


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7. A. Jameson. Optimum aerodynamic design using CFD and control theory. AIAA paper 95-1729 , AIAA 12thComputational Fluid Dynamics Conference, San Diego, CA, June 1995.

8. A. Jameson. Aerodynamic design via control theory. Journal of Scientic Computing , 3:233260, 1988.9. A. Jameson and J.J. Alonso. Automatic aerodynamic optimization on distributed memory architectures. AIAA

paper 96-0409 , 34th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1996.10. A. Jameson. Re-engineering the design process through computation. AIAA paper 97-0641 , 35th Aerospace

Sciences Meeting and Exhibit, Reno, Nevada, January 1997.11. A. Jameson, N. Pierce, and L. Martinelli. Optimum aerodynamic design using the Navier-Stokes equations. AIAA

paper 97-0101 , 35th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1997.12. A. Jameson, L. Martinelli, and N. A. Pierce. Optimum aerodynamic design using the Navier-Stokes equations.

Theoret. Comput. Fluid Dynamics , 10:213237, 1998.13. A. Jameson. Computational aerodynamics for aircraft design. Science , 245:361371, July 1989.14. A. Jameson. Automatic design of transonic airfoils to reduce the shock induced pressure drag. In Proceedings of

the 31st Israel Annual Conference on Aviation and Aeronautics, Tel Aviv , pages 517, February 1990.15. A. Jameson. Optimum aerodynamic design via boundary control. In AGARD-VKI Lecture Series, Optimum

Design Methods in Aerodynamics . von Karman Institute for Fluid Dynamics, 1994.16. J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, and D. Saunders. Constrained multipoint aerodynamic

shape optimization using an adjoint formulation and parallel computers. AIAA paper 97-0103 , 35th AerospaceSciences Meeting and Exhibit, Reno, Nevada, January 1997.

17. J. Reuther, J. J. Alonso, J. C. Vassberg, A. Jameson, and L. Martinelli. An efficient multiblock method foraerodynamic analysis and design on distributed memory systems. AIAA paper 97-1893 , June 1997.

18. O. Baysal and M. E. Eleshaky. Aerodynamic design optimization using sensitivity analysis and computationaluid dynamics. AIAA Journal , 30(3):718725, 1992.19. J.C. Huan and V. Modi. Optimum design for drag minimizing bodies in incompressible ow. Inverse Problems

in Engineering , 1:125, 1994.20. M. Desai and K. Ito. Optimal controls of Navier-Stokes equations. SIAM J. Control and Optimization , 32(5):1428

1446, 1994.21. W. K. Anderson and V. Venkatakrishnan. Aerodynamic design optimization on unstructured grids with a con-

tinuous adjoint formulation. AIAA paper 97-0643 , 35th Aerospace Sciences Meeting and Exhibit, Reno, Nevada,January 1997.

22. J. Elliott and J. Peraire. 3-D aerodynamic optimization on unstructured meshes with viscous effects. AIAApaper 97-1849 , June 1997.

23. S. Taasan, G. Kuruvila, and M. D. Salas. Aerodynamic design and optimization in one shot. AIAA paper 92-0025 ,30th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 1992.

24. A. Jameson, L. Martinelli, and J. Vassberg. Reduction of the adjoint gradient formula in the continuous limit.

AIAA paper , 41st

AIAA Aerospace Sciences Meeting, Reno, NV, January 2003.25. A. Jameson and J.C. Vassberg. Studies of alternative numerical optimization methods applied to the brachis-tochrone problem. Computational Fluid Dynamics , 9:281296, 2000.

26. A. Jameson, W. Schmidt, and E. Turkel. Numerical solutions of the Euler equations by nite volume methodswith Runge-Kutta time stepping schemes. AIAA paper 81-1259 , January 1981.

27. L. Martinelli and A. Jameson. Validation of a multigrid method for the Reynolds averaged equations. AIAApaper 88-0414 , 1988.

28. S. Tatsumi, L. Martinelli, and A. Jameson. A new high resolution scheme for compressible viscous ows withshocks. AIAA paper To Appear, AIAA 33nd Aerospace Sciences Meeting, Reno, Nevada, January 1995.

29. A. Jameson. Optimum aerodynamic design using control theory. Computational Fluid Dynamics Review , pages495528, 1995.

30. A. Jameson, Sriram, and L. Martinelli. An unstructured adjoint method for transonic ows. AIAA paper , 16th

AIAA CFD Conference, Orlando, FL, June 2003.31. E. Ahlstorm, R. Gregg, J. Vassberg, and A. Jameson. The design of an unlimited class reno racer. AIAA

paper-4341 , 18th

AIAA Applied Aerodynamics Conference, Denver, CO, August 2000.32. A. Jameson and L. Martinelli. Aerodynamic shape optimization techniques based on control theory. Lecturenotes in mathematics #1739, proceeding of computational mathematics driven by industrial problems, CIME(International Mathematical Summer (Center), Martina Franca, Italy, June 21-27 1999.


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SYMBOL SOURCESYN107 DESIGN 50SYN107 DESIGN 0

ALPHA2.2582.059

CD0.011360.01269

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.860 , CL = 0.419

Solution 1Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C

p

X / C41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C

p

X / C89.3% Span

Fig. 2. Redesigned Boeing 747 wing at Mach 0.86, Cp distributions


ALPHA1.7661.536

CD0.012930.01819

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Solution 1Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C89.3% Span

Fig. 3. Redesigned Boeing 747 wing at Mach 0.90, Cp distributions


26/30

t 1 :

1.11.11.11.11.11.11.11.11.11.1

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

S P A N L O A D &

S E C T C L

PERCENT SEMISPAN

C*CL/CREF

CL


ALPHA1.7661.536

CD0.012930.01819

COMPARISON OF SPANLOAD DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Fig. 4. Span loading, Redesigned Boeing 747 wing atMach 0.90

FOILDAGNSpanwise Thickness Distributions

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

Percent Semi-Span

H a l

f - T h i c k n e s s

SYMBOL AIRFOILWing 1.030Wing 2.030

Maximum Thickness

Average Thickness

Fig. 5. Spanwise thickness distribution, Redesigned Boe-ing 747 wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions

Antony Jameson19:02 Sun

2 Jun 02

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0-10.0

0.0

10.0

Percent Chord

A i r f o i l

-1.0

0.0

1.0

2.0

C a m

b e r

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

H a l



ETA42.042.0

R-LE0.3820.376

Tavg2.782.78

Tmax4.094.08

@ X44.5044.50

Fig. 6. Section geometry at = 0.42, redesigned Boe-ing 747 wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions

Antony Jameson19:02 Sun

2 Jun 02

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0-10.0

0.0

10.0

Percent Chord

A i r f o i l

-1.0

0.0

1.0

2.0

C a m

b e r

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

H a l



ETA69.169.1

R-LE0.3790.390

Tavg2.792.78

Tmax4.064.06

@ X43.0241.55

Fig. 7. Section geometry at = 0.68, redesigned Boe-ing 747 wing at Mach 0.90


27/30

Mach: 0.840 Alpha: 3.060CL: 0.325 CD: 0.02319 CM: 0.0000Design: 0 Residual: 0.2763E-02Grid: 193X 33X 33

: : :-Root Section: 9.8% Semi-Span

Cp = -2.0

: : :-Mid Section: 48.8% Semi-Span

Cp = -2.0

Cl: 0.262 Cd:-0.00437 Cm:-0.0473Tip Section: 87.8% Semi-Span

-

Fig. 8. Initial pressure distribution over a NACA 0012wing

Mach: 0.840 Alpha: 3.060CL: 0.314 CD: 0.01592 CM: 0.0000Design: 50 Residual: 0.1738E+00Grid: 193X 33X 33


Cp = -2.0


Cp = -2.0

Cl: 0.291 Cd:- 0.00239 Cm: -0.0489Tip Section: 87.8% Semi-Span

-

Fig. 9. Final pressure distribution and modied sectiongeometries along the wing span

NACA 0012 WING TO ONERA M6 TARGETMACH 0.840 ALPHA 3.060 Z 0.00

CL 0.2814 CD 0. 0482 CM -0.1113

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+

+

+

+ +

+ +

+ + +

+ + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ +

+ +

+

+

+

+

+

+

+

+++

+

+

+

+

+

+

+

++

+

+

++++

+++++

+++++++++

++++++++

++++++

++++ ++ +

+ ++

+

++ + +

++

++

+oo

oo

ooooo

oooooo

oooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

oooooooo

oooooooo

ooooo o

o o o oo o o

oo o

o

o

oo o o o

oo

oo

Fig. 10. Final computed and target pressure distribu-tions at 0 % of the wing span


CL 0.2814 CD 0. 0482 CM -0.1113

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+

+

+

+ +

+ +

+ + +

+ + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ +

+ +

+

+

+

+

+

+

+

+++

+

+

+

+

+

+

+

++

+

+

++++

+++++

+++++++++

++++++++

++++++

+++ + ++ +

+ ++

+

++ + +

++

++

+oo

oo

ooooo

oooooo


oooo

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

oooooooo

oooooooo

ooooo o

o o o oo o o

oo o

o

o

oo o o o

oo

oo



28/30


CL 0.3269 CD 0. 0145 CM -0.0865

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+

+

+

+ +

+ +

+ +

+ +

+ +

+ +

+ + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + +

+

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

+++++++++

+

++++++++++++++

++++++++

+++++

+++

+

+

+ ++ + + + + + + + +

++

+oo

oo

oooo

oooooo


oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooo

ooooooooooooooo

oooooooo

oo ooo o

oo

o

o

o

o o o o o o o o o oo

oo

o



CL 0.3356 CD 0. 0081 CM -0.0735

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+

+

+

+ +

+ +

+ +

+ +

+ +

+ +

+ + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + +

+

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

++++++++++++++++

+++++++++++

++++++

++

++

+

+

+ ++ + + + + + + + + + + +

++

+oo

oo

oooo

oooooo

ooooooo

oooooooooooooooooooooooooooooooooooo

o

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooooooooooo

ooooooooooo

oooooo

oo

o o

o

o

o o oo o o o o o o o o o o

oo

o



CL 0.3176 CD 0. 0011 CM -0.0547

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+

+

+

+ +

+ +

+ + +

+ + +

+ +

+ +

+ + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + +

+

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

++++++++++++++++++++++++

+++++

++

+

+

+

+++

+++ + + + + + + + + + + + + +

++

+oo

ooo

ooooo

ooooo

ooooooo

ooooooooooooooooooooooooooooooooooo

o

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooooooooooooooooooo

ooooo

oo

o

o

o

oo o

o oo o o o o o o o o o o o o o

oo

o



CL 0.4846 CD 0.0178 CM -0.1518

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E +

C p

+ +

+ +

+ + + +

+ + + + +

+ + + +

+ + + +

+ + + + +

+ + + +

+ + + + + + + + + + + + + + + + + + + + + + + + +

+ +

+

+

+

+

+ +

++

++

+

+

+

+

+

+

+

+++++++++++++++++++++++

+

+

+

+

+++

+++++++++ + + + + + + + + + + + + + +

+ +

+oooooo

ooooooooo

ooooooooo

ooooooooooooooooooooooooooooooooooo

o

o

o

oooo

o

o

o

o

o

o

o

o

oooooooooooooooooooooo

o

o

o

o

ooo

oooo o o o o o o o o o o o o o o o o o o o

oo

o



29/30

:Mach: 0.780 Alpha: 1.400CL: 0.280 CD: 0.00624 CM: 0.0000Design: 60 Residua l: 0.1528E+00Grid: 193X 33X 49


Cp = -2.0


Cp = -2.0


-

Fig. 16. Initial and nal pressure and section geometries

SHARKX6 (JCV: 1 6 DEC 99)MACH 0.780 ALPHA 1.400 Z 16.548

CL 0.2787 CD 0. 0120 CM -0.1352

-

0 . 1

E + 0 1

0 . 8 E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + +

+ +

+ +

+ +

+

+ +

+ +

+ +

+ +

+ +

+ + + + + + + + + + + + + +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+

+

+

+

+

+ ++

+

+

+

+

+++++++

++++++

++++

+++++

+++++++++++++++ + + + + + + + +

++

++

++

++

++

+ooooo

oo

oo

oo

oo

ooo

oooo

oooooooooooooooooooooooooooooooooooooo

o

oooo

o

o

o

oooooooooo

oooo

ooooo

oooooooo

ooooo oo o o o o o o o o o oo

oo

oo

oo

oo

oo

Fig. 17. Initial and nal pressure distributions at 5 % of the wing span

SHARKX6 (JCV: 16 DEC 99)MACH 0.780 ALPHA 1.400 Z 66.191

CL 0.4341 CD 0. 0018 CM -0.2010

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ + +

+ + + + + + + + + + + + + + + + + + +

+ +

+ +

+ +

+ +

+ +

+ +

+

+

+

+

+

+

+++

+

+

+

+

+++++++++++++

+++++++

++++++++++++++++++ + + + + +

++

++

++

++

++

++

+oooooo

oo

oo

oo

oooo

oooo


ooooo

o

o

o

o

o

oooooooooooo

ooooooo

oooooooooo oo o o o o o o o o o o o

oo

oo

oo

oo

oo

oo

o

Fig. 18. Initial and nal pressure distributions at 50 %of the wing span

SHARKX6 (JCV: 16 DEC 99)MACH 0.780 ALPHA 1.400 Z 115.834

CL 0.3122 CD -0.0139 CM -0.1244

-

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 3

E - 0

7

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + + +

+ +

+ +

+ +

+ +

+ +

+ +

+ + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

+ +

+ +

+ +

+ +

+

+

+

+

+

+++

+

+

+

+

+

+++++++++++

+++++++++++++++++++++++++ + + + + + + + + + + + ++

+ + + ++

+oooooo

oo

ooo

oooo

ooooo


oooooo

o

o

o

o

o

oooooooo

ooooooooooooooooooooooo o o o o o o o o o o o o o o o oo

oo o

oo

o

Fig. 19. Initial and nal pressure distributions at 95 %of the wing span


30/30

Mach: 0.780 Alpha: 1.421CL: 0.279 CD: 0.00613 CM:-1.2454Design: 10 Residual: 0.1887E-01Grid: 193X 33X 49

Root Section: 6.6% Semi-Span

Cp = -2.0

Mid Section: 49.2% Semi-Span

Cp = -2.0


Fig. 20. Drag minimization for the SHARK wing

Documents

Jameson-Aerodynamic Shape Optimization Using the Adjoint Method