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Copyright ©1997, American Institute of Aeronautics and
Astronautics, Inc.
AIAA Meeting Papers on Disc, January 1997A9715189,
AF-AFOSR-91-0391, AIAA Paper 97-0101
Optimum aerodynamic design using the Navier-Stokes equations
A. JamesonPrinceton Univ., NJ
N. A. PierceOxford Univ., United Kingdom
L. MartinelliPrinceton Univ., NJ
AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV,
Jan. 6-9, 1997
This paper describes the formulation of optimization techniques
based on control theory for aerodynamic shape design inviscous
compressible flow, modelled by the Navier-Stokes equations. It
extends previous work on optimization for inviscidflow. The theory
is applied to a system defined by the partial differential
equations of the flow, with the boundary shapeacting as the
control. The Frechet derivative of the cost function is determined
via the solution of an adjoint partialdifferential equation, and
the boundary shape is then modified in a direction of descent. This
process is repeated until anoptimum solution is approached. Each
design cycle requires the numerical solution of both the flow and
the adjointequations, leading to a computational cost roughly equal
to the cost of two flow solutions. The cost is kept low by
usingmultigrid techniques, in conjunction with preconditioning to
accelerate the convergence of the solutions. The power of themethod
is illustrated by designs of wings and wingbody combinations for
long range tranport aircraft. Satisfactory designsare usually
obtained with 20-40 design cycles. (Author)
Page 1
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Optimum Aerodynamic Designusing the Navier-Stokes Equations
A.JAMESON*, N.A. PIERCE t AND L. MARTINELLI §f' 3 Department of
Mechanical and Aerospace Engineering
Princeton UniversityPrinceton, New Jersey 08544 USA
and
T Oxford University Computing LaboratoryNumerical Analysis
Group
Oxford OX1 3QD UK
ABSTRACT
This paper describes the formulation of optimiza-tion techniques
based on control theory for aerody-namic shape design in viscous
compressible flow,modelled by the Navier-Stokes equations. It
ex-tends previous work on optimization for inviscidflow. The theory
is applied to a system defined bythe partial differential equations
of the flow, with theboundary shape acting as the control. The
Frechetderivative of the cost function is determined via
thesolution of an adjoint partial differential equation,and the
boundary shape is then modified in a di-rection of descent. This
process is repeated until anoptimum solution is approached. Each
design cyclerequires the numerical solution of both the flow andthe
adjoint equations, leading to a computationalcost roughly equal to
the cost of two flow solutions.The cost is kept low by using
multigrid techniques,in conjunction with preconditioning to
acceleratethe convergence of the solutions. The power of themethod
is illustrated by designs of wings and wing-body combinations for
long range tranport aircraft.Satisfactory designs are usually
obtained with 20-40design cycles.
Copyright ©1997 by the Authors.Published by the AIAA Inc. with
permission
* James S. McDonnell Distinguished University Professor
ofAerospace Engineering, AIAA Fellow
t Doctoral Candidate, Student Member AIAA§ Assistant Professor,
Member AIAA
1 INTRODUCTION
The ultimate success of an aircraft design dependson the
resolution of complex multi-disciplinarytrade-offs between factors
such as aerodynamic effi-ciency, structural weight, stability and
control, andthe volume required to contain fuel and payload.
Adesign is finalized only after numerous iterations,cycling between
the disciplines. The developmentof accurate and efficient methods
for aerodynamicshape optimization represents a worthwhile
inter-mediate step towards the eventual goal of full
multi-disciplinary optimal design.
Early investigations into aerodynamic optimizationrelied on
direct evaluation of the influence of eachdesign variable. This
dependence was estimatedby separately varying each design parameter
andrecalculating the flow. The computational cost ofthis method is
proportional to the number of designvariables and consequently
becomes prohibitive asthe number of design parameters is
increased.
An alternative approach to design relies on the factthat
experienced designers generally have an intu-itive feel for the
type of pressure distribution thatwill provide the desired
aerodynamic performance.The resulting inverse problem amounts to
deter-mination of the shape corresponding to a specifiedpressure
distribution. This approach has the advan-tage that only one flow
solution is required to obtainthe desired design. However, the
problem must beformulated carefully to ensure that the target
pres-
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sure distribution corresponds to a physically realiz-able
shape.
The problems of optimal and inverse design canboth be
systematically treated within the mathemat-ical theory for the
control of systems governed bypartial differential equations [I] by
regarding thedesign problem as a control problem in which
thecontrol is the shape of the boundary. The inverseproblem then
becomes a special case of the opti-mal design problem in which the
shape changes aredriven by the discrepancy between the current
andtarget pressure distributions.
The control theory approach to optimal aerody-namic design, in
which shape changes are basedon gradient information obtained by
solution of anadjoint problem, was first applied to transonic
flowbyjameson [2,3]. He formulated the method for in-viscid
compressible flows with shocks governed byboth the potential
equation and the Euler equations[2, 4, 5]. With this approach, the
cost of a designcycle is independent of the number of design
vari-ables and the method has been employed for wingdesign in the
context of complex aircraft configura-tions [6, 7], using a grid
perturbation technique toaccommodate the geometry
modifications.
Pironneau had earlier studied the use of control the-ory for
optimum shape design of systems governedby elliptic equations [8].
Ta'asan, Kuruvila and Salashave proposed a one shot approach in
which theconstraint represented by the flow equations needonly be
satisfied by the final converged design so-lution [9]. Adjoint
methods have also been used byBaysal and Eleshaky [10], and by
Cabuk and Modi[11,12].
The objective of the present work is the extensionof adjoint
methods for optimal aerodynamic de-sign to flows governed by the
compressible Navier-Stokes equations. While inviscid formulations
haveproven useful for the design of transonic wings atcruise
conditions, the inclusion of boundary layerdisplacement effects
with viscous design providesincreased realism and alleviates shocks
that wouldotherwise form in the viscous solution over the
finalinviscid design. Accurate resolution of viscous ef-fects such
as separation and shock/boundary layerinteraction is also essential
for optimal design en-compassing off-design conditions and
high-lift con-figurations.
The computational costs of viscous design are atleast an order
of magnitude greater than for designusing the Euler equations
because a) the number of
mesh points must be increased by a factor of two ormore to
resolve the boundary layer, b) there is theadditional cost of
computing the viscous terms anda turbulence model, and c)
Navier-Stokes calcula-tions generally converge much more slowly
thanEuler solutions due to discrete stiffness and direc-tional
decoupling arising from the highly stretchedboundary layer cells.
The computational feasibil-ity of viscous design therefore hinges
on the de-velopment of a rapidly convergent Navier-Stokesflow
solver. Pierce and Giles have developed apreconditioned multigrid
method that dramaticallyimproves convergence of viscous
calculations byensuring that all error modes inside the
stretchedboundary layer cells are either damped or expelled[13,14].
The same acceleration techniques are appli-cable to the adjoint
calculation, so that a substantialreduction in the cost of each
design cycle is achiev-able.
2 GENERAL FORMULATION OF THE AD-JOINT APPROACH TO OPTIMAL
DESIGN
Before embarking on a detailed derivation of theadjoint
formulation for optimal design using theNavier-Stokes equations, it
is helpful to summa-rize the general abstract description of the
adjointapproach which has been thoroughly documentedin references
[2, 3].
The progress of the design procedure is measuredin terms of a
cost function /, which could be, forexample the drag coefficient or
the lift to drag ratio.For flow about an airfoil or wing, the
aerodynamicproperties which define the cost function are func-tions
of the flow-field variables (w) and the physicallocation of the
boundary, which may be representedby the function T', say. Then
and a change in T results in a change
61 = dw 6w + —— ' (1)
in the cost function. Here, the subscripts I and // areused to
distinguish the contributions due to the vari-ation 5w in the flow
solution from the change associ-ated directly with the modification
67 in the shape.This notation is introduced to assist in grouping
thenumerous terms that arise during the derivation ofthe full
Navier-Stokes adjoint operator, so that it re-mains feasible to
recognize the basic structure of theapproach as it is sketched in
the present section.
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Using control theory, the governing equations of theflow field
are introduced as a constraint in such away that the final
expression for the gradient doesnot require multiple flow
solutions. This corre-sponds to eliminating 6w from (1).
Suppose that the governing equation R which ex-presses the
dependence of w and T within the flow-field domain D can be written
as
(2)
Then 6w is determined from the equation
..57 = -^— 6waw
T \9R]\- ^ \^-\([aw] J /
(3)
Next, introducing a Lagrange Multiplier if>, we have
(4)
(5)
(6)
(7)
(8)
(9)
Choosing i[> to satisfy the adjoint equation
dRdw
dl
the first term is eliminated, and we find that
SI =
where
aiT T dR-The advantage is that (9) is independent of 5w, withthe
result that the gradient of / with respect to anarbitrary number of
design variables can be deter-mined without the need for additional
flow-fieldevaluations. In the case that (2) is a partial
differ-ential equation, the adjoint equation (8) is also apartial
differential equation and determination ofthe appropriate boundary
conditions requires care-ful mathematical treatment.
The computational cost of a single design cycle isroughly
equivalent to the cost of two flow solutionssince the the adjoint
problem has similar complex-ity. When the number of design
variables becomeslarge, the computational efficiency of the
control
theory approach over traditional approach, whichrequires direct
evaluation of the gradients by indi-vidually varying each design
variable and recom-puting the flow field, becomes compelling.
Once equation (3) is established, an improvementcan be made with
a shape change
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In these definitions, p is the density, 111,112,113 arethe
Cartesian velocity components, E is the totalenergy and
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SFvl = (20)
The inviscid contributions arc easily evaluated as
[Fiw]r -dfi.
*dw'&Fvin =
The details of the viscous contributions are compli-cated by the
additional level of derivatives in thestress and heat flux terms
and will be derived inSection 6. Multiplying by a co-state vector
?/>, whichwill play an analogous role to the Lagrange
mul-tiplier introduced in equation (7), and integratingover the
domain produces
(21)
If tj) is differentiable this may be integrated by partsto
give
(22)
(23)
Since the left hand expression equals zero, it maybe subtracted
from the variation in the cost function(17) to give
61 =
+
f [6MJB
S (Ft - Fvi)}
. (24)
Now, since t/J is an arbitrary differentiable function,it may be
chosen in such a way that 51 no longer de-pends explicitly on the
variation of the state vectorSw. The gradient of the cost function
can then beevaluated directly from the metric variations with-out
having to recompute the variation 6w resultingfrom the perturbation
of each design variable.
Comparing equations (18) and (20), the variation 6wmay be
eliminated from (24) by equating all fieldterms with subscript "/"
to produce a differentialadjoint system governing ifj
- Fvjw] r + Pw = 0 in V. (25)
The corresponding adjoint boundary condition isproduced by
equating the subscript "/" boundaryterms in equation (24) to
produce
(26)
The remaining terms from equation (24) then yielda simplified
expression for the variation of the costfunction which defines the
gradient
SI = I {6Mn - n.^r (F, - Fvi}}
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ness. Taking the transpose of equation (25), theinviscid adjoint
equation may be written as
C'f— = 0 in£>, (28)' d&
where the inviscid Jacobian matrices in the trans-formed space
are given by
The transformed velocity components have theform
where the quantity
denotes the face area corresponding to a unit el-ement of face
area in the computational domain.Now, to cancel the dependence of
the boundary in-tegral on Sp, the adjoint boundary condition
reducesto
1/j.jrij =p-pd (30)where rij are the components of the surface
normal
and the condition that there is no flow through thewall boundary
at £2 = 0 is equivalent to
so that
when the boundary shape is modified. Conse-quently the variation
of the inviscid flux at theboundary reduces to
5F2 = Sp <
0
S2j5225230
55225523
0
(29)
Since 6F2 depends only on the pressure, it is nowclear that the
performance measure on the bound-ary M (w, S) may only be a
function of the pressureand metric terms. Otherwise, complete
cancellationof the terms containing 5w in the boundary inte-gral
would be impossible. One may, for example,include arbitrary
measures of the forces amd mo-ments in the cost function, since
these are functionsof the surface pressure.
In order to design a shape which will lead to a de-sired
pressure distribution, a natural choice is to set
= 5/0-z JB PdfdS
where p,i is the desired surface pressure, and theintegral is
evaluated over the actual surface area. Inthe computational domain
this is transformed to
I=\JI (P-P
This amounts to a transpiration boundary condi-tion on the
co-state variables corresponding to themomentum components. Note
that it imposes norestriction on the tangential component of if} at
theboundary.
In the presence of shock waves, neither p nor pj. arenecessarily
continuous at the surface. The bound-ary condition is then in
conflict with the assump-tion that i/i is differentiable. This
difficulty can becircumvented by the use of a smoothed
boundarycondition [15].
6 DERIVATION OF THE VISCOUS ADJOINTTERMS
In computational coordinates, the viscous terms inthe
Navier-Stokes equations have the form
Computing the variation 5w resulting from a shapemodification of
the boundary, introducing a co-statevector t/} and integrating by
parts following the stepsoutlined by equations (19) to (23)
produces
- I ̂JT> "w
where the shape modification is restricted to thecoordinate
surface £2 — 0 so that n\ — ris = 0,and ri2 = 1. Furthermore, it is
assumed that theboundary contributions at the far field may
eitherbe neglected or else eliminated by a proper choiceof boundary
conditions as previously shown for theinviscid case [4, 15].
The viscous terms will be derived under the as-sumption that the
viscosity and heat conduction
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coefficients \i and k are essentially independent ofthe flow,
and that their variations may be neglected.In the case of turbulent
flow, if the flow variationsare found to result in significant
changes in the tur-bulent viscosity, it may eventually be necessary
toinclude its variation in the calculations.
Transformation to Primitive Variables
The derivation of the viscous adjoint terms is sim-plified by
transforming to the primitive variables
wr = (p,u,v,w,p)T,
because the viscous stresses depend on the velocityderivatives ̂
, while the heat fluxes can be ex-pressed as
"''n —OXi /9,
The relationship between the conservative andprimitive
variations are defined by the expressions
Sw = M5w, 6w = M~l6w
which make use of the transformation matricesM ~ if and M~l =
if- Th686 matrices are pro-vided in transposed form for future
convenience
MT =
" 10000
' 1 iap0 1p0 00 00 0
Mlp000
Hip
01p00
'20p00
U300p0
U3p
001p0
2 'pUl
pU2
PU31-y-1 -
(l-l)uiUi -\2
— (7 — !)MI\ / I— ("y — 1)«2
-(7 - 1)U3
7-1 .The conservative and primitive adjoint operators Land L
corresponding to the variations 6w and 6ware then related by
T> T>with
L = MTL,so that after determining the primitive adjoint
op-erator by direct evaluation of the viscous portion of(25), the
conservative operator may be obtained bythe transformation L — M~l
L. There is no contri-bution from the continuity equation so the
deriva-tion proceeds by first examining the adjoint opera-tors
arising from the momentum equations.
Contributions from the Momentum Equations
In order to make use of the summation convention,it is
convenient to set i/Jj+i — j f°r J = 1)2,3. Thenthe contribution
from the momentum equations is
/ k ('Jo
_[dfrJv dti
+ (31)
The velocity derivatives in the viscous stresses canbe expressed
as
Jwith corresponding variations
The variation in the stresses are then
As before, only those terms with subscript 7, whichcontain
variations of the flow variables, need be con-sidered further in
deriving the adjoint operator. Thefield contributions that contain
6m in equation (31)appear as
, d „ Sik d „
S'"> d x i ,m-^-dum 1-d^.
This may be integrated by parts to yield
d
f a d/
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which is further simplified by transforming the in-ner
derivatives back to Cartesian coordinates
af/Jv T —dxm
(32)
The boundary contributions that contain 6ui inequation (31)
maybe simplified using the fact that
5 « i = 0 if 1 = 1,3oti
on the boundary B so that they become
f2j < /IB
(33)
Together, (32) and (33) comprise the field andboundary
contributions of the momentum equa-tions to the viscous adjoint
operator in primitivevariables.
Contributions from the Energy Equation
In order to derive the contribution of the energyequation to the
viscous adjoint terms it is convenientto set
A; "» "-**_'t JJ dxj
and identifying the normal derivative at the wall
d ^s odn 3dxj'
and the variation in temperature
6T= k (— -5^"\7-1 V P P P / '
(40)
(41)
to produce the boundary contribution
L ~dn (42)This term vanishes if T is constant on the wall
butpersists if the wall is adiabatic.
There is also a boundary contribution left over fromthe first
integration by parts (34) which has the form
/ 95 (SyQj) dBs, (43)JB
where
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since w; = 0. Notice that for future convenience indiscussing
the adjoint boundary conditions result-ing from the energy
equation, both the 5w and SSterms corresponding to subscript
classes / and // areconsidered simultaneously. If the wall is
adiabatic
so that using (41),
= 0
and both the Sw and SS boundary contributionsvanish.
On the other hand, if T is constant then it is moreconvenient to
expand (43) into
(SS2jQj + Sy
where, since 1̂ = 0 for / - 1,3,
j dbj v JThus, the boundary integral (43) becomes
(44)
Therefore, for constant T, the first term correspond-ing to
variations in the flow field contributes to theadjoint boundary
operator and the second set ofterms corresponding to metric
variations contributeto the cost function gradient.
All together, the contributions from the energyequation to the
viscous adjoint operator are the threefield terms (36), (37) and
(38), and either of twoboundary contributions ( 42) or (44),
depending onwhether the wall is adiabatic or has constant
tem-perature.
7 THE VISCOUS ADJOINT FIELD OPERATOR
Collecting together the contributions from the mo-mentum and
energy equations, the viscous adjointoperator in primitive
variables can be expressed as
'
r (oe 09 \ ., de ] \•j k br~ + TT- + xs» ir~ f[ \dXj dxij
dxk\\
(LB) =(7 - 1)
The conservative viscous adjoint operator may nowbe obtained by
the transformation
L = M~lTL.
8 VISCOUS ADJOINT BOUNDARY CONDI-TIONS
It was recognized in Section 4 that the boundaryconditions
satisfied by the flow equations restrictthe form of the performance
measure that may bechosen for the cost function. There must be a
di-rect correspondence between the flow variables forwhich
variations appear in the variation of the costfunction, and those
variables for which variationsappear in the boundary terms arising
during thederivation of the adjoint field equations. Otherwiseit
would be impossible to eliminate the dependenceof 51 on Sw through
proper specification of the ad-joint boundary condition. As in the
derivation of thefield equations, it proves convenient to consider
thecontributions from the momentum equations andthe energy equation
separately.
Boundary Conditions Arising from the Momen-tum Equations
The boundary term that arises from the momentumequations
including both the Sw and 55 components(31) takes the form
/ faSJB
dBe.
Replacing the metric term with the correspondinglocal face area
£2 and unit normal rij defined by
then leads to
S (\S2\njakj)
Defining the components of the surface stress as
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and the physical surface element
dS= \S2\dBf,
the integral may then be split into two components
/ 04 T* \SS2\ dB,. + I k \S2 5rkdS, (45)JB JBwhere only the
second term contains variations inthe flow variables and must
consequently cancel the6w terms arising in the cost function. The
first termwill appear in the expression for the gradient.
A general expression for the cost function that al-lows
cancellation with terms containing STk has theform
/= I M(T}dS, (46)JB
corresponding to a variation
f 9AT/ "5—QTkdo,JB ork
for which cancellation is achieved by the adjointboundary
condition
dtf*rk = ——— •
Natural choices for A/" arise from force optimiza-tion and as
measures of the deviation of the surfacestresses from desired
target values.
For viscous force optimization, the cost functionshould measure
friction drag. The friction force inthe Xi direction is
CDfi = I ffijdSj = I S2jffij•IB JBBso that the force in a
direction with cosines n-L hasthe form
= IJB
Expressed in terms of the surface stress T;, this cor-responds
to
C,nf ~ I niTidS,JBso that basing the cost function (46) on this
quantitygives
-A/" = HiTi.
Cancellation with the flow variation terms in equa-tion (45)
therefore mandates the adjoint boundarycondition
4>k = nk.
Note that this choice of boundary condition alsoeliminates the
first term in equation (45) so that itneed not be included in the
gradient calculation.
In the inverse design case, where the cost functionis intended
to measure the deviation of the surfacestresses from some desired
target values, a suitabledefinition is
- rdk) ,
where Td is the desired surface stress, including
thecontribution of the pressure, and the coefficients a//tdefine a
weighting matrix. For cancellation
t
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leading to the boundary condition
k = nk (rvn +p~ p,i).
In the case of high Reynolds number, this is wellapproximated by
the equations
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Suppose that a locally minimum value of the costfunction/* -
I(F*) is attained when T = f . Thenthe gradient Q* = G(F*) must be
zero, while theHessian matrix A* — A(F*) must be positive
defi-nite. Since Q* is zero, the cost function can be ex-panded as
a Taylor series in the neighborhood of T*with the form
= r + (.F - F*) A (F - F*)
Correspondingly,
As T approaches T" ', the leading terms becomedominant. Then,
setting T = (F — F*), the searchprocess approximates
dt -Also, since A" is positive definite it can be expandedas
A* = RMRT,where M is a diagonal matrix containing the
eigen-values of A", and
RRT = RTR = /.
Settingv = RTF,
the search process can be represented as
— = —Mv.dtThe stability region for the simple forward
Eulerstepping scheme is a unit circle centered at -1 onthe negative
real axis. Thus for stability we mustchoose
while the asymptotic decay rate, given by the small-est
eigenvalue, is proportional to
In order to improve the rate of convergence, one canset
6F = -XPG,where P is a preconditioner for the search. An
idealchoice is P = A"~l, so that the corresponding timedependent
process reduces to
for which all the eigenvalues are equal to unity, andT is
reduced to zero in one time step by the choiceAZ = 1. Quasi-Newton
methods estimate ^4* fromthe change in the gradient during the
search pro-cess. This requires accurate estimates of the gradi-ent
at each time step. In order to obtain these, boththe flow solution
and the adjoint equation must befully converged. Most quasi-Newton
methods alsorequire a line search in each search direction,
forwhich the flow equations and cost function mustbe accurately
evaluated several times. They haveproven quite robust for
aerodynamic optimization[6].An alternative approach which has also
proved suc-cessful in our previous work [15], is to smooth
thegradient and to replace Q by its smoothed value Qin the descent
process. This both acts as a precondi-tioner, and ensures that each
new shape in the opti-mization sequence remains smooth. It turns
out thatthis approach is tolerant to the use of approximatevalues
of the gradient, so that neither the flow solu-tion nor the adjoint
solution need be fully convergedbefore making a shape change. This
results in verylarge savings in the computational cost. For
inviscidoptimization it is necessary to use only 15 multigridcycles
for the flow solution and the adjoint solutionin each design
iteration. For viscous optimization,about 100 multigrid cycles are
needed. This is partlybecause convergence of the lift coefficient
is muchslower, so about 20 iterations must be made beforeeach
adjustment of the angle of attack to force thetarget lift
coefficient. The new preconditioner forthe flow and adjoint
calculations allows the numberof iterations to be substantially
reduced in both theflow and the adjoint simulation.
The numerical tests so far have focused on theviscous design of
wings for optimum cruise, forwhich the flow remains attatched, and
the main vis-cous effect is due to the displacement thickness ofthe
boundary layer. While some tests have beenmade with the viscous
adjoint terms included, ithas been found that the optimization
process con-verges when the viscous terms are omitted from
theadjoint system. This may reflect the tolerance of thesearch
process to inexact gradients.
10 RESULTS
Preconditioned Inverse Design
The first demonstration is an application of the
pre-conditioning technique for inverse design with theEuler
equations. The ONERA M6 (Figure Ib) wing
12
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is recovered for a lifting case starting from a wingwith a
NACA0012 section (Figure la) and usingthe ONERA M6 pressure
distributions computed ata = 3.0 and M = 0.84 as the target (Fig.
2). Thus, asymmetric wing section is to be recovered from
anasymmetric pressure distribution. The calculationswere performed
on a 192x32x48 C-H mesh with294,912 cells. Each design cycle
required 3 multi-grid cycles for the flow solver using
characteristic-based matrix dissipation with a matrix
precondi-tioner and 12 multigrid cycles for the adjoint solverusing
scalar dissipation and a variable local timestep (scalar
preconditioner). Compared to a test inwhich the 3 multigrid cycles
using the matrix pre-conditioner were replaced by 15 multigrid
cyclesusing a standard scalar preconditioner, and 15 cy-cles were
used in adjoint solver, each design cyclerequired about 3/8 as much
computer time, whilethe number of design cycles required to reach
thesame level of error also fell from 100 to about 50.Use of the
matrix preconditioner therefore reducedthe total CPU time on an IBM
590 workstation from97,683 sec (-27 hours) to 18,222 sec (-5 hours)
forroughly equivalent accuracy.
Viscous Design
Due to the high computational cost of viscous de-sign, a
two-stage design strategy is adopted. In thefirst stage, a design
calculation is performed withthe Euler equations to minimize the
drag at a givenlift coefficient by modifying the wing sections
witha fixed planform. In the second stage, the pres-sure
distribution of the Euler solution is used as thetarget pressure
distribution for inverse design withthe Navier-Stokes equations.
Comparatively smallmodifications are required in the second stage,
sothat it can be accomplished with a small number ofdesign
cycles.
In order to test this strategy it was used for the re-design of
a wing representative of wide body trans-port aircraft. The results
are shown in Figures 3and 4. The design point was taken as a lift
coeffi-cient of .55 at a Mach number of .83. Figure 3 illus-trates
the Euler redesign, which was performed ona mesh with 192x32x48
cells, displaying both thegeometry and the upper surface pressure
distribu-tion, with negative Cp upwards. The initial wingshows a
moderately strong shock wave across mostof the top surface, as can
be seen in Figure 3a. Sixtydesign cycles were needed to produce the
shock freewing shown in Figure 3b, with an indicated dragreduction
of 15 counts from .0196 to .0181. Figure
4 shows the viscous redesign at a Reynolds num-ber of 12
million. This was performed on a meshwith 192x64x48 cells, with 32
intervals normal tothe wing concentrated inside the boundary layer
re-gion. In Figure 4a it can be seen that the Euler designproduces
a weak shock due to the displacement ef-fects of the boundary
layer. Ten design cycles wereneeded to recover the shock free wing
shown in Fig-ure 4b. It is interesting that the wing section
modi-fications between the initial wing of Figure 3a andthe final
wing of Figure 4b are remarkably small.
These results were sufficiently promising that itwas decided by
McDonnell Douglas to evaluate themethod for industrial use, and it
was used to sup-port design studies for the MDXX project. The
re-sults of this experience are discussed in an accompa-nying paper
[21]. It rapidly became apparent thatthe fuselage effects are too
large to be ignored. Inviscous design it was also found that there
were dis-crepancies between the results of the design
calcula-tions, which were initially performed on a relativelycoarse
grid with 192x64x48 cells, and the results ofsubsequent analysis
calculations performed on finermeshes to verify the design.
In order to allow the use of finer meshes withovernight
turnaround, the code was therefore mod-ified for parallel
computation. Using the parallelimplementation, viscous design
calculations havebeen performed on meshes with 1.8 million
meshpoints. Starting from a preliminary inviscid design,20 design
cycles are usually sufficient for a viscousre-design in inverse
mode, with the smoothed in-viscid results providing the target
pressure. Sucha calculation can be completed in about 7| hoursusing
48 processors of an IBM SP2.
As an illustration of the results that could be ob-tained,
Figures 5 - 9 show a wing-body designwith sweep back of about 38
degrees at the 1/4chord. Starting from the result of an Euler
design,the viscous optimization produced an essentiallyshock free
wing at a cruise design point of Mach.86, with a lift coefficient
of .6 for the wing bodycombination at a Reynolds number of 101
millionbased on the root chord. Figure 5 shows the designpoint,
while the evolution of the design is shown inFigure 6, using
Vassberg's COMPPLOT software.In this case the pressure contours are
for the finaldesign. This wing is quite thick, with a thicknessto
chord ratio of more than 14 percent at the rootand 9 percent at the
tip. The design offers excellentperformance at the nominal cruise
point. Figures 7and 8 show the results of a Mach number sweep
13
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to determine the drag rise. It can be seen that adouble shock
pattern forms below the design point,while there is actually a
slight increase in the dragcoefficient of about 1 \ counts at Mach
.85. Finally,Figure 9 shows a comparison of the pressure
dis-tribution at the design point with those at alternatecruise
points with lower and higher lift. The ten-dency to produce double
shocks below the designpoint is typical of supercritical wings.
This wing hasa low drag coefficient, however, over a wide rangeof
conditions.
CONCLUSIONS
We have developed a three-dimensional control the-ory based
design method for the Navier Stokesequations and applied it
successfully to the designof wings in transonic flow. The method
representsan extension of our previous work on design withthe
potential flow and Euler equations. The newmethod combines the
versatility of numerical op-timization methods with the efficiency
of inversedesign. The geometry is modified by a grid per-turbation
technique which is applicable to arbitraryconfigurations. The
combination of computationalefficiency with geometric flexibility
provide a pow-erful tool, with the final goal being to create
practicalaerodynamic shape design methods for completeaircraft
configurations.
ACKNOWLEDGMENT
This work has benefited from the generous supportof AFOSR under
Grant No. AFOSR-91-0391, theNASA-IBM Cooperative Research
Agreement, andalso the Rhodes Trust.
REFERENCES
[1] J.L. Lions. Optimal Control of Systems Gov-erned by Partial
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[2] A. Jameson. Aerodynamic design via controltheory. /. Sci.
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[3] A. Jameson. Optimum aerodynamic design us-ing CFD and
control theory. AIAA Paper 95-1729-CP, 1995.
[4] A. Jameson. Automatic design of transonicairfoils to reduce
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Annual
Conference on Aviation and Aeronautics, Tel Aviv,pages 5-17,
February 1990.
[5] J. Reuther and A. Jameson. Control based air-foil design
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[6] J. Reuther and A. Jameson. Aerodynamic shapeoptimization of
wing and wing-body configu-rations using control theory. AIAA paper
95-0123, AIAA 33rd Aerospace Sciences Meeting,Reno, Nevada, January
1995.
[7] J. Reuther, A. Jameson, J. Farmer, L. Martinelli,and D.
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configurationsvia an adjoint method. AIAA paper 96-0094,AIAA 34th
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[8] O. Pironneau. Optimal Shape Design for EllipticSystems.
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[9] S. Ta'asan, G. Kuruvila, and M. D. Salas. Aero-dynamic
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optimization using sensitivity anaysisand computational fluid
dynamics. AIAA jour-nal, 30(3):718-725,1992.
[11] H. Cabuk, C.H. Shung, and V. Modi. Adjointoperator approach
to shape design for internalincompressible flow. In G.S.
Dulikravich, edi-tor, Proceedings of the 3rd International
Conferenceon Inverse Design and Optimization in Engineer-ing
Sciences, pages 391-404,1991.
[12] J.C. Huan and V. Modi. Optimum designfor drag minimizing
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for compressible flow cal-culations on stretched meshes. Submitted
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14
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[16] L. Martinelli and A. Jameson. Validation of amultigrid
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15
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Initial Wing. Cp on Upper Surface.
Figure la: M = .84, C, = .3000, Cd = .0205, Q = 2.935°.
Redisigned wing. Cp on Upper Surface.
Figure Ib: M = .84, C, = .2967, Cd = .0141, a = 2.935°
Figure 1: Redesign of the Onera M6 Wing. 1000 design cycles in
inverse mode.
16
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.•\
6 H
•'•-...
2a: span station z = 0.297
'
c2b: span station z = 0.484
* J
2c: span station 2 = 0.672
S J
2d: span station z = 0.859
Figure 2: Target and Computed Pressure Distributions of
Redesigned Onera M6 Wing.M = 0.84, CL = 0.2967, CD = 0.0141, Q =
2.935°.
17
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Initial Wing. Cp on Upper Surface.
Figure 3a: M = .83, Ct = .5498, Cd = .0196, a = 2.410°.
Redisigned wing. Cp on Upper Surface.
Figure 3b: M = .83, Cf = .5500, Cd = .0181, a = 1.959°.
Figure 3: Redesign of the wing of a wide transport aircraft.
Stage 1 Inviscid design : 60 design cycles indrag reduction mode
with forced lift.
18
-
Initial Wing. Cp on Upper Surface.
Figure 4a: M = 0.83, Ci = .5506, Cd = .0199, a = 2.317°
Redisigned wing. Cp on Upper Surface.iFigure 4b: M = 0.83, Ci =
.5508, Cd = .0194, a = 2.355°
Figure 4: Redesign of the wing of a wide transport aircraft.
Stage 2: Viscous re-design. 10 design cycles ininverse mode.
19
-
-1.0-0.8
-0.5
-O.J
U ().()0.2
0.5
0.8
1.0
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X
WING-BODY
REN = 101.00 , MACH = 0.860
SYMBOL SOURCE ALPHA CL CD——————— SYN»)7PI)hSIGN4« 2.0V4 (Ktlll O
O l l ' f i
0.8 •
0.5 •
-0.2 -
Figure 5: Pressure distribution of the MPX5X at its design
point.
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X
WING-BODY
REN = 101.00 , MACH = 0.860 , CL = 0.610 ,
SYMBOL SOURCE
SYNIDTPIJKSIGNO
ALPHA
2.251
CD
0.01 131
-0.8 •-0.5 •-0.2 '
J 0.00.2-0.5 '0.8
1.0
Figure 6: Optimization Sequence in the design of the MPX5X.
20
-
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X
WING-BODY
REN= lOl.(X) , CL =0.610
MHX5X [IKSIGN'10MI'XSX DKSIGN
40MPX5XUHSIGN4I)MPX5XDKSICN4I)MHX5XDKSIGN4I)
Figure 7: Off design performance of the MPX5X below the design
point.
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X
WING-BODY
REN = 101.00 . CL = 0.610
Figure 8: Off design performance of the MPX5X above the design
point.
21
-
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X
WING-BODYREN= 101.00 , MACH=0.«60
SYMBOL
—— •- —
SOURCE
MPX5X DHS1GN 40
ALPHA
2.3BO
CL
O.fifil
CD
0.01314 /
Figure 9: Comparison of the MPX5X at its design point and at
lower and higher lift.
22