Two-Dimensional Aerodynamic Optimization Using

the Discrete Adjoint Method with or without

Parameterization

M. Bompard, J. Peter∗

ONERA, 92322 Chatillon Cedex, France

G. Carrier †

ONERA, 92190 Meudon, France

J.-A. Desideri ‡

INRIA, Sophia-Antipolis Mediterranee, 06905 Sophia-Antipolis, France

An optimization method based on the use of the derivatives of functional outputs withrespect to (w.r.t.) solid body mesh nodes is presented. These derivatives are obtained bya discrete adjoint method that first computes the derivatives of functional outputs w.r.t.all volume mesh nodes. They are smoothed before being used in a numerical optimizationalgorithm. The procedure is demonstrated for a 2D flow governed by the compressibleReynolds-Averaged Navier-Stokes equations (RANS) completed by the Spalart-Allmarasturbulence model. Discrete derivatives are computed with or without making the frozeneddy-viscosity assumption. The design algorithm is compared with a more classical oneusing design variables related to B-splines on the four test cases introduced by Kim et al.1

Nomenclature

AoA Angle of AttackCD Drag coefficientCL Lift coefficientJ (Gm) Aerodynamic objective (constraint) function as function of design parametersJ (Gm) Aerodynamic objective (constraint) function as function of flow field and volume mesh

J (Gm) Aerodynamic objective (constraint) function as function of volume meshnS Vector of wall normalsS Wall meshW Flow fieldX Volume meshα Vector of design parametersΛJ (Λm)adjoint vector of J (Gm)for scheme R

I. Introduction

Numerical optimization for airplane design was introduced almost as soon as mature codes appeared.The aerodynamic optimizations carried out by R. Hicks and G.N. Vanderplaats at NASA in the mid 70’s2, 3

illustrate this early interest in optimization. At that time 2D and simple 3D configurations were considered,simplex and descent methods were used and the gradients required by descent methods were estimatedby finite differences. Since then, the framework of aerodynamic optimization has known at least threedrastic extensions: (1) several global optimization methods have been considered and intensively used;4–7 (2)surrogate functions have been used for a part of the evaluations needed by global optimization methods;8–11

∗CFD and aeroacoustic department, 29 av. de la division Leclerc†Applied aerodynamics department, 8 rue des Vertugadins‡Head Opale Project-Team, 2004 route des Lucioles

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(3) adjoint-vectors and direct-differentiation methods have been introduced, studied and more and moreoften used to compute the gradients necessary for descent algorithms.12

The cost of an adjoint sensitivity analysis (proportional to the number of functions to be derived, almostindependent of the number of design parameters) has been its greatest asset and the main reason of itssuccess. Besides, since the landmark article of Jameson,12 the adjoint mode of CFD codes became notonly popular and commonly-used for shape optimization, but also in other related subjects.16 In particular,it appeared that the adjoint vector could be useful to error estimation and mesh refinement. Pierce andGiles,17, 18 Venditti and Darmofal19–21 and then Dwight22, 23 made important contributions to this topic.Adjoint-based local shape optimization has become very popular since the early 90’s. Nevertheless, someissues still need to be addressed. The first one is the accuracy of the sensitivities: The full linearization of the(RANS) and turbulence model equations results in a Jacobian matrix to be inverted with exceptionally poorconditioning. Hence “frozen eddy viscosity” linearization is today more or less the norm. Unfortunately thecorresponding sensitivities suffer from a 20/30% bias for many classical aerospace design problems. A secondlimitation is due to shape parametrization which may limit dramatically the space of reachable shapes. Forthese reasons the four local optimization test cases proposed by Kim et al1 are here run with exact andapproximate discrete adjoints and with and without shape parameterization for comparison.The article is organized as follows. In section two, the equations of adjoint sensitivity analysis and theformulas of total derivative of aerodynamic functions w.r.t. volume mesh nodes and wall mesh nodes arerecalled. In section three, the modules of the local optimization chain are described. In section four, thecomputational chain is assessed in the four cases considered.

II. Discrete adjoint gradient optimization

II.A. Foreword and notations

Let us define the main notations of the article: W is the flow field, X the volume mesh, and S (part of X)the mesh of the solid body (wing, aircraft...). In many cases, CFD computations are run in order to computeaerodynamic functions like drag, lift, rolling or pitching moment. These functions depend explicitly on theflow field, W , and the mesh, X . When carrying out a shape optimization, an objective function – denotedJ = J(W, X) – is distinguished from constraint functions – denoted Gm = Gm(W, X) k ∈ [1, nc]. Drag andtotal pressure losses are classical examples of objectives to be minimized whereas lift and pitching momententer classical constraint functions.The state variables W (size nW ) are linked to volume mesh by the discrete equations for fluid dynamics.Actually only the case of a finite-difference/finite-volume scheme will be considered here. Hence the schemeequations read

R(W, X) = 0, (1)

which represents a set of nW nonlinear equations. By application of the implicit function theorem, equation(1) defines the flow field W as a function of the volume mesh X in the vicinity of (Wi, Xi), satisfying steadystate equations R(Wi, Xi) = 0 under condition

det(∂R

∂W)(Xi, Wi) 6= 0 (2)

This property states that a single steady state flow field corresponds to a single geometry. It will be supposedto be true for all considered meshes ; this allows us to define J(X) = J(W (X), X), Gm(X) = J(W (X), X).

II.B. Sensitivity analysis using discrete direct and adjoint method

When carrying out a shape optimization, a vector of design parameters denoted α (size nd) defines therelevant changes of the solid shape. Under assumption (2), not only the volume mesh X , but also the flowfield W can be seen as functions of the design parameters. Moreover, if X(α) and R(W, X) are C1 regularfunctions of their arguments, then W (α) is also a C1 function.From the designer point of view, the aerodynamic outputs are considered as functions of the design variables.The notations for the objective, as a function of α will be J (α), the corresponding notation for the constraintsGm(α). Nevertheless, the evaluation of the functions of interest at a given point of the design space is basedon a steady state computation and the link between J and Gm and the functions J and Gm, introduced

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before, is:J (α) = J(W (α), X(α)) Gm(α) = Gm(W (α), X(α))

At each step of a local shape optimization based on a descent method, the derivatives of the objective andthe active constraints w.r.t. the components of α are required. As the flow field W and X are linked bythe set of nonlinear equations (1), the differentiation of J and Gm is not straightforward. In the frameworkof the discrete approach, the derivatives of interest can be computed by the direct differentiation method(DD)24, 25 or the discrete adjoint (AV ) method:26

(DD) ∀l ∈ [1, nd]∂R

∂W

dW

dαl

= −∂R

∂X

dX

dαl

(3)

dJ

dαl

=∂J

∂X

dX

dαl

+∂J

∂W

dW

dαl

dGm

dαl

=∂Gm

∂X

dX

dαl

+∂Gm

∂W

dW

dαl

(4)

(AV ) (∂R

∂W)T ΛJ = −(

∂J

∂W)T ∀m ∈ [1, nc] (

∂R

∂W)T Λm = −(

∂Gm

∂W)T (5)

dJ

dαl

=∂J

∂X

dX

dαl

+ ΛTJ (

∂R

∂X

dX

dαl

)dGm

dαl

=∂Gm

∂X

dX

dαl

+ ΛTm(

∂R

∂X

dX

dαl

) (6)

The four derivatives given in equations (4)(6) are sums of two terms. The first is a geometrical sensitivity(change in the function of interest due to the change of shape steered by the design parameter αl). Thesecond term is the aerodynamic sensitivity (change in the function of interest due to the change in theflow-field caused by the change of shape).The CPU costly operation of (DD) and (AV ) is the solution of the large linear systems (equations (3),(5), size nW ). The number of linear systems to be solved is equal to the number of design parameters(nd) for (DD) as it is equal to the number of functions to be differentiated (1+nc) for (AV ). For almost allindustrial shape optimizations, the number of design parameters is by far larger than the number of functionsof interest. This is why the adjoint vector method raises much more interest than the direct differentiationmethod.Concerning the memory requirement, both methods need the storage of the mesh sensitivities (dX/dαl i ∈[1, nd]) which becomes the main memory requirement for very large number of design parameters. Thesemesh sensitivities are needed at least for the computation of the geometrical part of the function sensitivities((∂J/∂X)(dX/dαl)). Concerning the geometrical sensitivity of the explicit residual (∂R/∂X)(dX/dαl), it ismost often estimated by second order finite differences using one of the two following formulas:

∂R

∂X

dX

dαl

≃R(W (α), X(α + δαl)) − R(W (α), X(α − δαl))

2δαl

∂R

∂X

dX

dαl

≃R(W (α), X(α) + δαl

dXdαl

) − R(W (α), X(α) − δαldXdαl

)

2δαl

This avoids the tedious differentiation of the numerical scheme w.r.t. metric terms. Conversely, this requireseither the storage of shifted meshes X(α + δαl) or, once again, the storage of mesh sensitivities (dX/dαl).Our purpose is now to discuss ways of computing the function sensitivities that do not suffer from thesedemanding memory requirements

II.C. Total derivative of aerodynamic functions w.r.t. volume mesh nodes or solid body meshnodes

Let us now suppose that the numerical scheme R has been differentiated (by hand, using automatic differen-tiation...) w.r.t. metric terms. The geometrical sensitivity of the scheme can then be expressed as a productof two Jacobian matrices and the sensitivity computed in adjoint mode (equation (6)) can be rewritten as:

dJ

dαl

=( ∂J

∂X+ ΛT

J

∂R

∂X

)dX

dαl

dGm

dαl

=(∂Gm

∂X+ ΛT

m

∂R

∂X

)dX

dαl

(7)

From this, the total derivative of the functions of interest w.r.t. the mesh coordinates are clearly identified.

dJ

dX=

∂J

∂X+ ΛT

J

∂R

∂X

dGm

dX=

∂J

∂X+ ΛT

m

∂R

∂X(8)

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The meaning of the two terms of the function sensitivity w.r.t. a design parameter has been discussed before.A similar analysis can be done for equation (8): the first term, (∂J/∂Xl) (resp.(∂Gm/∂Xl) ), correspondsto the direct dependency of the function J (resp. Gm) on the location of node l, whereas the second term –ΛT

J (∂R/∂X) (resp. ΛTm(∂R/∂X)) – corresponds to changes of the flow field on the support of the function

J (resp.Gm), due to the change of node l location.The first advantage of this choice is that it makes possible to carry out the sensitivity computation usingtwo successive computers (a) a fast, low memory computer to perform the computation of ΛJ and dJ/dXonly, which does not require the storage of the mesh sensitivities dX/dαl; (b) a possibly slower computerwith higher memory resources to carry out the product (dJ/dX)(dX/dαl).The second advantage of this method is also related to shape optimization. Let the surface mesh be denotedS(α). If the volume mesh X is assumed to be an explicit function of S (via an algebraic mesh deformationmethod for example), another expression of the functions of interest can be defined: J(S) = J(X(S)) (resp.Gm(S) = Gm(X(S)) ) The derivative of J and Gm w.r.t. the nodes of S can be computed:

dJ

dS=

dJ

dX

dX

dS

dGm

dS=

dGm

dX

dX

dS

This gives an insight into the changes in shape that are the most effective in shape optimization independentlyof any set of the design parameters (and, moreover, allows an even cheaper computation of the sensitivitiesmultiplying last equation by (dS/dαl)).A third and related advantage is that several types of geometrical parameterization of X or S can veryquickly be tested without re-running the large super-computer that solves the adjoint equations.

II.D. Fully linearized or frozen turbulence modeling

Ten years ago, frozen eddy-viscosity in RANS sensitivity computations seemed to be the norm. Recentlyseveral teams have linearized turbulence models by hand or using AD. To our knowledge two classes oftransport equations models have been linearized in the literature:– The one-equation model of Spalart-Allmaras62 was linearized by under others Giles et al.,30 and theteam of Anderson, Nielsen and Bonhaus at NASA Langley27, 29, 33 who compared resulting gradients withfrozen eddy-viscosity gradients.28 The complete linearized code was applied to the 3d optimization of anisolated wing. Further examples include Nemec and Zingg31 and Brezillon et al.35 who both presented airfoiloptimizations.– The two-equation transport models k− ǫ, k−ω SST and Wilcox k−ω have been differentiated and appliedto sub- and transonic airfoil design, as well as high-lift profile and setting optimization by Kim et al.32, 34

The former model has also been adjointed in the context of turbomachinery by Renac et al.37

There is however a notable lack of linearized transport equation turbulence models applied to configurationssignificantly more complex than isolated wings with fully attached flow28 or 2d high-lift profiles.34 Wesuspect this is a consequence, not of the difficulty of performing the linearization itself or accounting for thecoupling with the mean-flow, but of the problems associated with the solution of the resulting linear system,which may be exceptionally poorly conditioned.36

Both frozen turbulence and fully linearized Spalart-Allmaras model are presently considered.

III. Modules of the parameter free optimization chain

In this section, the optimization chain of a parameterization-free design procedure is described. Thenumerical optimization method is iterative and its steps can be briefly described as follows:

1. Solve the (RANS) equations around the current shape.

2. Extract the relevant aerodynamic coefficient and evaluate the objective function.

3. Compute the total derivative of the objective function (w.r.t. mesh coordinates) by the discrete adjointmethod.

4. Compute the derivatives of the objective function w.r.t. surface nodes.

5. Update the shape using a descent algorithm.

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III.A. Flow analysis, adjoint computation and aerodynamic function evaluation

The two-dimensional Reynolds averaged Navier-Stokes (RANS) equations are considered. The turbulent vis-cosity is defined by the Spalart-Allmaras model.62 The six-equation non-linear system is solved numericallyby the ONERA finite-volume cell-centered code for structured mesh, elsA.44 Second order Roe’s flux39 (usingthe MUSCL approach40 with the Van Albada limiting function41) is used for the mean flow convective terms.Centered fluxes with interface-centered evaluation of gradients are used for both diffusive terms. Flow andadjoint equations are solved by an implicit backward-Euler method. Two approaches for the computationof the gradient are compared in this study (a) discrete adjoint method with frozen turbulence model; (b)discrete adjoint method with linearization of the Spalart-Allmaras turbulence model with respect to the statevariables. The ONERA far-field drag extraction code FFD7259, 60 is used for the evaluation of aerodynamicfunctions.

III.B. Volume mesh deformation

The shape is not parameterized and its deformation is directly steered by the normal displacement of wallmesh nodes. Within the optimization process, the volume mesh is updated by a distance-based algebraicmethod, developed by Meaux et al.53 Let δS (resp. δX) denote the displacement of the wall mesh (resp.volume mesh) nodes. The displacement of a point pi of the volume mesh is computed as:

δX(pi) = ν(pi)

∑

pj∈S δS(pj)((−−→pj pi,−→nj) + 1.)

||pj pi||daj

∑

pj∈S

((−−→pj pi,−→nj) + 1.)||pj pi||

daj

, (9)

where daj and nj denote the area and outward normal vector associated with the node pj on the wall. Theν function is defined so that the deformation is smoothly reduced as the distance to the body increases. Thevalue of ν is 1 for near field points defined as those nodes where distance to the body is less than 4 chords;ν = 0 for far field points (distance to the airfoil greater than 8 chords). Assuming that the perturbations ofthe mesh are small during the optimization process, all distances can be computed on the initial mesh. Thedeformation function is then linear and its Jacobian can be computed easily:

d δX(pi)

d δS(pl)= ν(pi)

(−−→plpi,−→nl) + 1.)||pl pi||

dal

∑

pj∈S

(−−→pjpi,−→nj) + 1.)||pj pi||

daj

(10)

As the current volume mesh and wall mesh are updated by

X = X0 + δX S = S0 + δS

(X0, S0 being the initial volume mesh and wall mesh), equation (10) also defines the Jacobian dX/dS. Thederivative of the objective function J w.r.t. the wall nodes can then be computed by:

dJ

dS=

dJ

dX

dX

dS(11)

III.C. Numerical optimization method

The optimization method is the CONMIN iterative method developed by Vanderplaats.57 This program,largely used for shape optimization, can solve multi-variable, linear or nonlinear, unconstrained and con-strained problems.

If some constraints are active, the descent direction is determined by Zoutendijk’s method of FeasibleDirections.58 Otherwise, the conjugate direction method of Fletcher and Reeves49 can be used. In ourapplications, a steepest descent algorithm is used for the unconstrained cases. After the descent direction iscomputed, a one-dimensional minimization is carried out by polynomial interpolation.

The horizontal and vertical coordinates of each node of S are used as design parameters. The evaluationof the objective function for a given set of design variables is managed by a python script, that successivelycalculates the corresponding mesh and executes the elsA flow solver and the FFD72 postprocessing code.Another script calculates the gradient of the function with respect to wall mesh nodes by solving adjointequation and using 11.

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III.D. Smoothing of the descent direction

Considering each solid surface node as a design parameter results in a very large space of reachable shapes.Therefore, it increases the ability to improve the initial shape. Li and Padula52 have given an example ofhow the output of an optimization can be affected by an insufficient number of shape parameters. Inversely,it is known that a high resolution design space tends to generate high frequency noise during the airfoil shapemodification, ultimately leading to unsmooth geometry profiles.55 This effect may reduce the aerodynamicperformance of the airfoil15 because the regularity of the pressure distribution is directly related to the airfoilcurvature.46 Consequently, a common practice in airfoil optimization is to smooth the shapes variationsgenerated by the optimization process. The CDISC56 inverse design process uses a polynomial interpolationof the airfoil to get rid of undesirable curvature features. Unfortunately, this approach can also diminish thebenefit of the optimization. A second approach is to smooth the descent direction before each deformation.Jameson proposed and demonstrated an implicit residual smoothing method (IRS) in previous works.14

Mohammadi introduces a similar smoothing operator in its CAD-free framework.54

In this paper, a new smoothing process is introduced. The smoothing is performed by spline interpolationof the normal component of the gradient. The interpolation algorithm, introduced by Dierckx,48 minimizesthe second derivatives variation. It has been used successfully by Li and Krist51 for the smoothing of theairfoil curvature. In this study, it is used to reduce the jumps in curvature distribution of the airfoils duringthe optimization process.

III.D.1. Dierckx’s spline interpolation

The method of Dierckx is based on the search of a balance between fitting the data and minimizing asmoothness measure. It ensures a data fitting error lower than a given parameter ǫS and seeks the smoothestcurve subject to this constraint. In our method, the smoothness measure is chosen as the sum of the squaresof the third derivative jumps at the knots.

III.D.2. Iterative smoothing process

In preparation of the smoothing process, the descent direction is set to the opposite of the normal componentof the objective function derivative w.r.t. each node of the current airfoil S:

d1 = −

⟨

dJ

dS, nS

⟩

. (12)

The ǫS value, which defines the acceptable error in the spline interpolation, is initially set to 0.The smoothing procedure is iterative : iteration l is made of the following steps:

1. The field dl is interpolated by the Dierckx’s spline interpolation with the parameter value ǫS . Theresulting field is set in sl.

2. An estimated step corresponding to sl is chosen as the step that would reduce the objective functionof a given proportion βOBJ if its variation was linear:

tl = βOBJ

J

〈sl, sl〉. (13)

3. A target airfoil is computed using the estimated step tl and the current descent direction sl:

Sl = S0 + tlslnS . (14)

4. The curvature of the target airfoil kSl is computed:

• The smoothing process stops if the curvature total variation is bounded by a positive constant γ:∫

x∈Sl

|kSl(x)| ≤ γ. (15)

• Else, the interpolation parameter ǫS is augmented and the process restarts at step 1.

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III.D.3. Curvature computation

The curvature is computed by spline interpolation of the discrete coordinates of the airfoil, by an interpolationprocedure, based on Akima’s splines45 which interpolated field with reduced variation. This method allowsto compute the curvature in area of the mesh where the discretization nodes tend to cluster.

III.E. Geometric constraints

The application of our algorithm has led in some cases to unacceptable shapes. For example, the maxi-mization of the lift coefficient tends to reduce the thickness profile at the trailing edge. If the shape is notcontrolled, the lower side grid line may overlap the upper side.

To avoid this behavior, geometric constraints have been added to better control nodes near the trailingedge. At each point, the thickness of the airfoil is constrained to remain larger than a given fraction of theinitial thickness.

Geometric constraints are taken into account by the CONMIN method. Their gradients are computedanalytically and smoothed by the process presented above.

IV. Parameter-based optimization chain

IV.A. Parameterization

The parameterization consists of a B-spline based perturbation of the lower and upper sides of the airfoil. Two20-control-point, second-order, B-splines are used to parameterize independently the shape modificationsapplied to the lower and upper sides. The first and last control points correspond to the leading and trailingedge (x/c = 0 and x/c = 1, respectively).

The remaining 18 control points are spread between the leading and trailing edges according to a hy-perbolic tangent distribution, as illustrated in figure 1. Note that the control points at the leading andtrailing edges for the upper and lower sides are identical, resulting in a total number of degree of freedom of2 ∗ 18 + 2 = 38 (see figure 1).

One design variable (shape parameter) is used to control the vertical displacement of each control pointindependently, with the exception of the two control points placed at x/c = 0 (in addition to the controlpoint linked to the leading edge), on the upper and lower sides respectively, which move simultaneously inopposite directions and are linked by a unique shape parameter. These two control points and the associateddesign parameters give a direct control on the leading edge radius of curvature.

Parametrization: 38 control points (SeAnDef)

Figure 1. Control points used for parameterization of the RAE2822 airfoil geometry

IV.B. Surface and volume mesh deformation

The ONERA shape parameterization and deformation tool SeAnDef(Sequential Analytical Deformation)is used to deform the airfoil geometry as well as the corresponding body-fitted CFD mesh. This softwareproceeds through a composition of 3D analytical deformation fields associated to a set of elementary possibledeformation mode. Typically, the possible elementary deformation modes cover deformations of a wingplanform (sweep, span, chord variations), wing dihedral and twist variations, airfoil thickness and cambermodification. Each defomation mode is specified through paramaters piloting the amplitude of deformationat different spanwise control sections. Different interpolation of the deformations fields are possible betweenthe spanwise control sections. In addition to the simple camber/thickness deformation mode, wing airfoilscan be modified with more flexibilty by adding independently to the upper and lower airfoil part a chord-normal displacement defined with a B-spline. Finally, thanks to the approach based on composition of

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deformation modes and to the simplicity of each deformation modes, SeAnDef also calculates the analyticalmesh sensitivities w.r.t. each deformation parameter.

IV.C. Numerical optimization method

The optimization were conducted with the CONMIN optimizer.

V. Results

V.A. Validation experiment of the RAE2822 airfoil using the RANS equations and theSpalart-Allmaras turbulence model

The basic test case corresponds to Cook’s et al47 ninth experiment related to the RAE2822 airfoil. The flowcharacteristics are: M∞ = 0.730, AoA = 3.19, Re = 6.5 106 (this Reynolds number is based on the density,velocity, eddy viscosity at infinity and the chord of the 2D profile). The project EUROVAL50 suggestedthat the influence of the wind tunnel walls could be corrected in the calculations by correcting the angle ofattack (AoA = 2.79 instead of AoA = 3.19). This correction, that has been widely accepted by the CFDcommunity, is retained for the computations.

A two-domain mesh, including 32832 points, is used (see figure 2).The pressure distribution on the airfoil is given and compared to experimental data of Cook et al. on

figure 3. The agreement between experimental data and numerical results is consistent with that found inthe literature.61

x

z

-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

Figure 2. Two-domain mesh of RAE2822

x

-cp

0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

Figure 3. Validation of flow solver on RAE 2822 withflight conditions of EUROVAL

V.B. Design test cases

In every test, the computation is initiated with the RAE2822 airfoil geometry and carried out at the flightconditions of the ninth experiment of Cook et al (M∞ = 0.730 and Re = 6.5 106). Four design cases,similar to those introduced by Kim et al,1 have been considered :

1. drag minimization at fixed angle of attack AoA = 2.79;

2. lift maximization at fixed angle of attack AoA = 2.79;

3. drag minimization at fixed lift CL0 = 0.8;

4. lift maximization at fixed drag CD0 = 0.0110.

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In the study of Kim et al, the surface of the airfoil was parameterized using 50 Hicks-Henne bumpfunctions spread evenly on the upper and lower sides. In this paper, the same computations are made bythe two design procedures presented above.

V.C. Airfoil design based on dJdS

(without parameterization)

In this section, the design procedure without parameterization, described in the third section, is applied tothese test cases.

For the unconstrained cases, the optimization process is run twice:

• initially, the descent direction is computed from a gradient obtained by the adjoint method with makingthe frozen turbulence assumption;

• in a second step, the sensitivities are obtained by the adjoint method with linearization of the Spalart-Allmaras turbulence model w.r.t. the state variables.

The optimization procedure is interrupted when the objective function no longer evolves. The geometricconstraints are set to allow reduction of the thickness of the airfoil until 80 percent of the thickness on theinitial airfoil.

V.C.1. Drag minimization at fixed angle of attack

The sum of the wave (CDw) and viscous pressure drag (CDvp) coefficients as calculated by the ONERAfar-field drag post processor FFD72 is minimized. The design calculation is carried out without changingthe angle of attack of the airfoil (2.79 deg), which does not guarantee that lift coefficient is maintained.The far-field drag corresponding to the sum of the wave, viscous pressure drag and friction drag (CDf)coefficients, denoted by CDff, is also computed.

The normal component of the gradient of the drag on each node of the airfoil surface is plotted in figures4 and 5 for the two cases. The field is very disturbed. Therefore, the smoothing procedure is necessary sincea smooth geometry airfoil is desirable. A second difficulty is that the normal component of the gradient ishigh at the edges while the position of leading and trailing edge should be preserved during the optimizationprocess. This justifies that the gradient is set to zero at the edges.

Figure 4. Normal component of the gradient of the dragon the lower side of the initial shape where making thefrozen turbulence assumption (red) and with linearizationof the Spalart-Allmaras turbulence model w.r.t. the statevariables (green)

Figure 5. Normal component of the gradient of the dragon the upper side of the initial shape where making thefrozen turbulence assumption (red) and with linearizationof the Spalart-Allmaras turbulence model w.r.t. the statevariables (green)

The smoothed results are given in figures 6 and 7. Low frequency variations of the field is retained whilethe high frequency components were eliminated by the smoothing procedure. The field is gradually reducedto zero at the edges. The descent direction thus obtained can be used to update the current shape.

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Figure 6. Smoothed descent direction for minimizationof the drag on the lower side of the initial shape wheremaking the frozen turbulence assumption (red) and withlinearization of the Spalart-Allmaras turbulence modelw.r.t. the state variables (green)

Figure 7. Smoothed descent direction for minimizationof the drag on the upper side of the initial shape wheremaking the frozen turbulence assumption (red) and withlinearization of the Spalart-Allmaras turbulence modelw.r.t. the state variables (green)

The optimum designs and the pressure distributions on the surface of the airfoil are given in figures 8and 9.

Figure 8. Pressure distribution on the optimum (solid)and initial (pointed) design of drag minimization withfixed angle of attack AoA = 2.79 and making the frozenturbulence assumption

Figure 9. Pressure distribution on the optimum (solid)and initial (pointed) design of drag minimization withfixed angle of attack AoA = 2.79 and linearization of theSpalart-Allmaras turbulence model w.r.t. the state vari-ables

As a result of the optimization, the far-field drag has been reduced from 0.0143 to 0.0098 when the adjointgradient is computed with making the frozen turbulence assumption (respectively 0.0102 when the Spalart-Allmaras turbulence model is linearized w.r.t. the state variables). At the same time, the lift diminishedfrom 0.757 to 0.225 (respectively 0.258). The reduction of the drag is essentially due to the elimination of thewave drag which drops from 0.0027 to 0.000002 (respectively 0.000000). Note that the friction drag, whichis not considered in the optimization but is included in CDff increases of 5 points for the two optimizations.

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V.C.2. Lift maximization at fixed angle of attack

In this test case, we attempt to maximize the lift coefficient (CL) of the RAE2822 airfoil by altering theshape without changing the angle of attack of the configuration. This coefficient is calculated by the postprocessor FFD72 as the sum of the pressure component (CLp) and the friction component (CLf). On thisconfiguration, the pressure component is predominant (CLp=0.75705391 and CLf=0.00000385 on the initialairfoil).

Figures 10 and 11 show the results of the design calculations where the objective function is chosen tobe the opposite of the lift coefficient.

Figure 10. Pressure distribution on the optimum (solid)and initial (pointed) design of lift maximization with fixedangle of attack AoA = 2.79 with making the frozen turbu-lence assumption

Figure 11. Pressure distribution on the optimum (solid)and initial (pointed) design of lift maximization withfixed angle of attack AoA = 2.79 with linearization of theSpalart-Allmaras turbulence model w.r.t. the state vari-ables

The lift coefficient increases from 0.757 to 0.901 (respectively 0.916), while the drag has dropped from0.0143 to 0.0146 (respectively has increased to 0.0163). The increase of CL is mainly due to the pressurecomponent.

V.C.3. Drag minimization at fixed lift

The minimization of the sum of the wave (CDw) and viscous pressure drag (CDvp) coefficients is performedwith a minimum bound on the lift coefficient CL0 = 0.8. The angle of attack is added to the designparameters and its initial value is AoA = 3.09. Figure 12 shows the results of the design calculations.

During the optimization process, the drag coefficient is reduced from 0.0169 to 0.0120. In the same time,the lift coefficient is keeped to the initial value and the angle of attack is modified from 3.09 to 3.068.

V.C.4. Lift maximization at fixed drag

In this test case, we attempt to maximize the lift coefficient (CL) and constraint the drag coefficient to belower than the initial value CD0 = 0.0110. The angle of attack is added to the design parameters and itsinitial value is AoA = 3.09. The result of the design calculations is given in figure 13.

The lift coefficient increases from 0.800 to 0.998 and the final angle of attack is AoA = 3.088.

V.D. Airfoil design with B-spline parameterization

The same design computations using B-spline parameterization are presented in this part.

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Figure 12. Pressure distribution on the optimum (solid)and initial (pointed) design of drag minimization s.t. liftconstraint CL ≥= 0.8 with making the frozen turbulenceassumption

Figure 13. Pressure distribution on the optimum (solid)and initial (pointed) design of lift maximization s.t. dragconstraint CD ≤= 0.0110 with making the frozen turbu-lence assumption

V.D.1. Drag minimization at fixed angle of attack

The objective function is again the sum of the wave and viscous pressure drag coefficients. The angle ofincidence is fixed during the optimization and the lift coefficient is not constrained.

As a result of the optimization, the coefficient of drag of the airfoil has been diminished from 0.0143 to0.0104. The initial and optimum designs are given in figure 14. Similarly to the case without parameter-ization, the initial strong shock wave has disappeared and the wave drag is eliminated (less than 0.0001).

CL

CM

CD

5 100.7

0.75

0.8

0.85

0.9 -0.12

-0.1

-0.08

-0.06 0

0.005

0.01

0.015

0.02

CD

ff

CD

p

CD

w

CD

f

CD

vp

CD

s

CD

nf

design variable number

cont

rolp

oint

∆Z

Ang

leof

atta

ck(d

eg)

5 10 15 20 25 30 35

-0.4

-0.2

0

0.2

0.4

1

1.5

2

2.5

3

X/C

CP

0 0.2 0.4 0.6 0.8 1

-1

0

1InitialCurrent

Figure 14. Pressure distribution on the optimum designfrom CDff minimization with fixed angle of attack AoA =

2.79 (compared to initial RAE2822 results)

CL

CM

CD

5 100.7

0.75

0.8

0.85

0.9 -0.12

-0.1

-0.08

-0.06 0

0.005

0.01

0.015

0.02

CD

ff

CD

p

CD

w

CD

f

CD

vp

CD

s

CD

nf

design variable number

cont

rolp

oint

∆Z

Ang

leof

atta

ck(d

eg)

5 10 15 20 25 30 35

-0.4

-0.2

0

0.2

0.4

1

1.5

2

2.5

3

X/C

CP

0 0.2 0.4 0.6 0.8 1

-1

0

1InitialCurrent

Figure 15. Pressure distribution on the optimum designfrom CDff minimization under lift constraint CL > 0.8(compared to initial RAE2822 results)

V.D.2. Drag minimization at fixed lift

This test case is similar to the previous one, except for the fact that the optimization procedure takes intoaccount a constraint on the lift. This constraint is achieved by using the angle of attack as an additionaldesign parameter. The minimum value of the lift is fixed to 0.8 and the incidence of the initial configuration

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is modified to satisfy this constraint. The initial incidence is chosen to be 3.1 degrees, which corresponds toa lift coefficient of 0.804.

The coefficient of the far-field drag is reduced from 0.0172 to 0.0115. The optimum design is very farfrom the constraint (CL=0.798). The initial and optimum design are given in figure 15.

V.E. Comparison of numerical results

The results of application of the two present design procedures to the four design test cases are given intable 1. They are compared with the results given by Kim et al. (50 Hicks-Henne functions) on similar testcases.

min CD max CL min CD max CL

s.t. CL ≥ CL0 s.t. CD ≤ CD0

Parameter-free

frozen µt 0.0143 ⇒ 0.0098 0.756 ⇒ 0.901 0.0169 ⇒ 0.0120 0.8 ⇒ 0.998

CD0 = 0.0110

CL0 = 0.8

Parameter-free

lin. of S.A. 0.0143 ⇒ 0.0102 0.756 ⇒ 0.916 0.0169 ⇒ 0.0118 0.8 ⇒ 0.818

CD0 = 0.0110

CL0 = 0.8

38 B-Spline

frozen µt 0.0143 ⇒ 0.0104 0.0172 ⇒ 0.0115

CL0 = 0.8

50 Hicks-Henne

lin. of S.A. 0.0152 ⇒ 0.0100 0.799 ⇒ 0.857 0.0167 ⇒ 0.0109 0.799 ⇒ 0.976

CD0 = 0.0143

CL0 = 0.83

Table 1. Results of parameter-free and parameter-based optimization chain on the four test cases

In all the cases, the parameter-free method has produced results comparable to parameter-based ap-proach. In particular, the lift maximisation designs seem to benefit of the deformation flexibility of theparameter-free procedure. Conversely, the constrained cases are difficult for this procedure, because thesmoothing approximation is applied twice (on the objective and constraint function).

VI. Conclusion

In this paper, a parameterization-free design procedure has been presented. A curvature variation-basedcriterion has been introduced to define the geometric quality of an airfoil. This criterion is used to terminatethe process during the smoothing of the sensitivities computed by total adjoint method. The smoothingalgorithm is a spline interpolation method which minimizes the second derivative variation.

This procedure, coupled with geometric constraints, leads to optimized airfoil which are acceptable in ageometric sense. It has been applied to four design test cases using the classical RAE2822 airfoil as initialgeometry. The results have been compared with more classical parameter-based approach, using B-Splinesor Hicks-Henne functions. The parameter-free optimization chain gives equivalent results, depending of thetest cases.

References

1S. K. Kim, J. J. Alonso, A. Jameson, Two-Dimensional High-Lift Aerodynamic Optimization Using the ContinuousAdjoint Method, AIAA Paper 2000-4741, 2000.

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American Institute of Aeronautics and Astronautics

2G.N Vanderplaats, R. Hicks. Numerical airfoil optimization using a reduced number of design coordinates. TechnicalReport TMX 73151, NASA. 1976.

3R. Hicks, E. Murman, G.N Vanderplaats. An assessment of airfoil design by numerical optimization. Technical ReportTMX 3092, NASA. 1974.

4J.H. Holland. Adaptation in natural and artificial systmes. Ann-Arbor, MI: University of Michigam Press, 1975.5D.E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989.6R. Storn, K. Price. Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces.

Journal of Global Optimization Vol.11. pp. 341-349. 1997.7R. Storn. On the usage of differential evolution for function optimization. In Proceedings of the biennial conference of

the north american fuzzy information processing society (NAFIPS), pp. 519-523. 1996.8J. Sacks, W.J. Welch, T.J. Mitchell, H.P. Wynn. Design and analysis of computer experiments. Journal of statistical

science Vol.11. pp. 409-423. 1989.9D.R. Jones. A Taxonomy of Global Optimization Methods Based on Response Surfaces. Journal of global optimization

Vol.21. pp. 345-383. 2001.10Giannakoglou K. Acceleration of genetic algorithms using artificial neural network. Theoretical background. Von-Karman

Institute Lecture Series 2000-07, May 2000.11Giannakoglou K, Giotis A. Acceleration of genetic algorithms using artificial neural networks. Application of the method.

Von-Karman Institute Lecture Series 2000-07, May 2000.12A. Jameson. Aerodynamic design via control theory. Journal of Scientific Computing Vol.3. pp. 233-260. 1988.13A. 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, pp. 5-17. 1990.14A. Jameson. Optimum aerodynamic design via boundary control. In AGARD-VKI Lecture Series, Optimum Design

Methods in Aerodynamics. von Karman Institute for Fluid Dynamics. 1994.15A. Jameson and J. Vassberg. Computational Fluid Dynamics for Aerodynamic Design: It’s Current and Future Impact.

AIAA-2001-0538. 2001.16J. Peter, R.P. Dwight. Numerical sensitivity analysis: a survey of approaches. Computers and Fluids Vol.39 (3). pp.

373-391. 201017N.A. Pierce, M.B. Giles. Adjoint recovery of superconvergent functionals for PDE approximations. SIAM ReviewVol. 42

pp.247–264. 2000.18N.A. Pierce, M.B. Giles. Adjoint and defect error bounding and correction for functional estimates. AIAA Paper 03-3846.

2003.19D.A. Venditti, D.L. Darmofal. Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-

one-dimensional flow. Journal of Computational Physics Vol. 164 pp. 40–69. 2000.20D.A. Venditti, D.L. Darmofal. Grid adaptation for functional outputs: Application to two-dimensional inviscid flows.

Journal of Computational Physics Vol. 176 pp. 40–69. 2002.21D.A. Venditti, D.L. Darmofal. Anisotropic grid adaptation for functional outputs: Application to two-dimensional viscous

flows. Journal of Computational Physics Vol. 187 pp. 22–46. 2003.22R.P. Dwight. Goal-oriented mesh adaptation using a dissipation based error indicator Internation Journal for Numerical

Methods in Fluids Vol.56(8) pp. 1193-2000. 2007.23R.P. Dwight. Heuristic a posteriori estimation of error due to dissipation in finite volumes schemes and application to

mesh adaptation. Journal of Computational Physics Vol. 227 pp. 2845–2863. 2008.24A. Taylor III, V. Korivi, G. Hou. Sensitivity analysis applied to the Euler equations: a feasibility study withemphasis on

variation of geometric shape. AIAA Paper 91-0173. 1991.25O. Baysal, M. Eleshaky. Aerodynamic design sensitivity analysis methods for the compressible Euler equations. Journal

of Fluids Engineering Vol. 113(4) pp. 681–688. 1991.26P. Frank, G. Shubin. A comparison of optimisation based approaches for a model computational aerodynamics design

problem. Journal of Computational Physics Vol.98 pp74–89. 1992.27W.K. Anderson, D.L. Bonhaus. Aerodynamic design on unstructured grids for turbulent flows NASA TM 112867 1997.28E.J. Nielsen, W.K. Anderson. Aerodynamic design optimization on unstructured meshes using the Navier-Stokes equa-

tions. AIAA Journal Vol.37(11) pp 1411–1419. 1999.29W.K. Anderson, D.L. Bonhaus. Airfoil design optimization on unstructured grids for turbulent flows AIAA Journal

Vol.37(2) pp 185–191. 1999.30M.B. Giles, M.C. Duta, J.D. Muller. Adjoint code developments using the exact discrete approach. AIAA Paper 2001-

2596, 2001.31N. Nemec, D.W. Zingg. Towards Efficient Aerodynamic Shape Optimization Based on the Navier-Stokes Equations. AIAA

Paper 2001-2532. 2001.32C.S. Kim, C. Kim, O.H. Rho. Sensitivity analysis for the Navier-Stokes equations with two equations turbulence models

AIAA Journal Vol.39(5) pp 838–845. 2001.33Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes. AIAA Journal Vol.40(6) pp 1155–

1163. 2002.34C.S. Kim, C. Kim, O.H. Rho Effects of Constant Eddy Viscosity Assumption on Gradient-Based Design Optimization.

AIAA Paper 2002-0262. 2002.35J. Brezillon, R.P. Dwight Discrete Adjoint of the Navier-Stokes Equations for Aerodynamic Shape Optimization in

Proceedings of EUROGEN 2005, Evolutionary and Deterministic Methods for Design, Munich. 2005.36R.P. Dwight, J. Brezillon. Adjoint Algorithms for the Optimization of 3d Turbulent Configurations. in Proceedings of

the 15th STAB Symposium. 2006.

14 of 15

American Institute of Aeronautics and Astronautics

37F. Renac, C.-T. Pham, J. Peter. Sensitivity Analysis for the RANS Equations Coupled with a Linearized TurbulenceModel. AIAA Paper 2007-3948. 2007.

38A. Jameson, W. Schmidt, E. turkel. Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes. AIAA Paper 1981-1259. 1981.

39PL. Roe. Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys., 1981, 43, 357-37240B. van Leer. Towards the ultimate conservative difference scheme. V - A second order sequel to Godunov’s method. J.

Comput. Phys., 1979, 32, 101-13641G.D. van Albada, B. van Leer, W.W. Roberts. A comparative study of computational methods in cosmic gas dynamics.

Astronomy and Astrophysics, 1982, 108, 76-8442I.H. Abbott, A.E. von Doenhoff. Theory of wing sections. Dover, New York, 1959. ISBN 0-486-60586-8.43J.C. Vassberg, A. Jameson. In pursuit of grid convergence, part I: two-dimensional Euler solution. AIAA paper 2009-4114,

San Antonio, USA, 2009.44L. Cambier, J.-P. Veuillot. Status of the elsA CFD software for flow analysis and multidisciplinary applications. AIAA

Paper 08-664. 2008.45H. Akima. A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures, JACM, Vol. 17, No.

4, pp. 589–602. 1970.46R. L. Barger. Streamline Curvature Design Procedure for Subsonic and Transonic Ducts, NASA TN D-7368. December

1973.47P. H. Cook, M. A. McDonald and M. C. P. Firmin, Aerofoil RAE 2822 - Pressure Distributions, and Boundary Layer

and Wake Measurements, Experimental Data Base for Computer Program Assessment, AGARD Report AR 138. 1979.48P. Dierckx. An Algorithm for Surface-Fitting with Spline Functions, Journal of Numerical Analysis, Vol. 1, pp. 267–283.

1980.49R. Fletcher and C. M. Reeves. Function Minimization by Conjugate Gradients, British Computer Journal, Vol. 7, No. 2,

1964.50W. Haase, F. Bradsma, E. Elsholz, M. Leschziner and D. Schwamborn, EUROVAL - an european initiative on validation

of CFD codes, Notes on numerical fluid mechanics, Vol. 42. 1993.51W. Li and S. Krist. Spline-Based Airfoil Curvature Smoothing and Its Applications, Journal of Aircraft, Vol. 42, No. 4.

2005.52W. Li and S. Padula. Using High Resolution Design Spaces for Aerodynamic Shape Optimization under Uncertainty,

NASA TP-2004-213004. 2004.53M. Meaux, M. Cormery, G. Voizard. Viscous aerodynamic shape optimization based on the discrete adjoint state for 3D

industrial configurations. in Proceedings of ECCOMAS 2004.54B. Mohammadi and O. Pironneau. Applied Shape Design for Fluids. Oxford Univ. Press, Oxford. 2001.55J. Reuther, A. Jameson, J. Alonso, M. Rimlinger and D. Saunders, Constrained Multipoint Aerodynamic Shape Opti-

mization Using an Adjoint Frormulation and Parallel Computers. Journal of Aircraft, Vol. 36, pp. 51–74. 1999.56L. A. Smith and R. L. Campbell. A Method for the Design of Transonic Flexible Wings, NASA TP 3045. December 1990.57G. N. Vanderplaats, CONMIN - A Fortran Program for Constrained Function Minimization - User’s Manual. NASA TM

X-62,282. August 1973.58G. Zoutendijk. Methods of Feasible Directions, Elsevier. 1960.59D. Destarac, Far-Field / Near-Field Drag Balance and Applications of Drag Extraction in CFD, VKI Lecture Series 2003,

CFD-based Aircraft Drag Prediction and Reduction, Rhode Saint Genese (Belgium), February 3-7, 2003, National Institute ofAerospace, Hampton (VA), November 3-7, 2003.

60S. Esquieu, M. Meheut, H.F. Von Geyr and O. Brodersen, Far-field drag extraction from numerical solutions and wakesurveys. 7th ONERA-DLR Aerospace Symposium, Toulouse, France, 2006.

61J.C. Jouhaud, P. Sagaut and B. Labeyrie A Kriging Approach for CFD/Wind-Tunnel Data Comparison, Journal ofFluids Engineering, Vol. 128, pp. 847–855, 2006,

62P. R. Spalart and S. R. Allmaras, A one-equation turbulence model for aerodynamic flows, AIAA Paper 1992-0439, 1992.

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American Institute of Aeronautics and Astronautics