An Efficient Method for Aerodynamic Shape Optimization

Subhendu Bikash Hazra

Department of Mathematics, University of Trier, D-54286 Trier, Germany

Abstract. We present simultaneous pseudo-timestepping as an efficient method for aerodynamic shapeoptimization. In this method, instead of solving the necessary optimality conditions by iterative tech-niques, pseudo-time embedded nonstationary system is integrated in time until a steady state is reached.The main advantages of this method are that it requires no additional globalization techniques and that apreconditioner can be used for convergence acceleration which stems from the reduced SQP method. Theimportant issue of this method is the trade-off between the accuracy of the forward and adjoint solverand its impact on the computational cost to approach an optimum solution is addressed. The methodis applied to a test case of drag reduction for an RAE2822 airfoil, keeping it’s thickness constant. Theoptimum overall cost of computation that is achieved in this method is less than 4 times that of theforward simulation run.

Nomenclature

(x, y) ∈ R2 :cartesian coordinates H :total enthalpy

(ξ, η) ∈ [0, 1]2 :generalized coordinates M :Mach number

Ω :flow field domain )∞ :values at free stream

∂Ω :flow field boundary γ :ratio of specific heats

~n :=(nx

ny

):unit outward normal Cref :chord length

α :angle of attack CD :drag coefficient

ρ :density I :cost unction

u :x-component of velocity w :vector of state variables

v :y-component of velocity q :vector of design variables

p :pressure λ :vector of adjoint variables

E :total energy J :Jacobian

Cp :pressure coefficient B :reduced Hessian

I. Introduction

Computational fluid dynamics (CFD) and numerical optimization techniques are being used widelyin the field of aerodynamic shape optimization. This growing interest is due to the fact that CFD ismaking fast progress using advanced computer technology and available efficient numerical algorithms.This helps faster computation of the flow field which is necessary for the optimization. Also, efficientnumerical algorithms for optimization are available so that the combined effort makes it possible to

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compute optimum solution in considerable time. This helps saving the cost occurred in experimentalmethods.

Among various methods used in this field, gradient methods are used mostly in practical applica-tions. This method requires the gradient of the objective function with respect to the shape parameters.These gradients can be computed using direct methods or adjoint methods. Adjoint methods receivedconsiderable attention since its derivation by A. Jameson in,12 because this is independent of numberof design variables, unlike the direct methods which depend on number of design variables. In adjointmethods, additional system of PDEs are to be solved together with the flow equations. In each designupdate of this method, these equations are to be solved accurately to get the accurate gradient infor-mation which is used to find the direction of the optimum. This leads to high cost of this method.Fast numerical techniques for solving the flow and adjoint equations, e.g., multigrid or preconditionedGMRES iterative techniques for solving linear equation, helps saving the cost upto certain extent, butstill the overall cost is quite high. Among many others, computational results based on these methodsare presented in6, 8, 13, 14, 23, 24 on structured grid. An application on unstructured grid has been presentedin.1 This approach with a less accurate state and costate solution has been performed in Iollo et. al. 22

In,9 we proposed a new method for solving problems in this class, and applied to an academictest problem (boundary control problem in elliptic equations). It is based on simultaneous pseudo-timestepping. Instead of using an iterative technique to solve the necessary optimality conditions consistingof state, costate and design equations, pseudo-time embedded nonstationary system is considered. Thisformulation is advantageous since the steady-state flow is obtained by integrating the pseudo-unsteadyEuler (or Navier-Stokes) equations in this problem class. Therefore, one can use the simultaneous time-stepping for the whole set of equations. The pseudo-time embedded nonstationary system of state, costateand design equations is usually stiff system of ODEs and explicit time stepping schemes may convergevery slowly or may even diverge. Preconditioning is necessary to avoid this problem. Preconditionersused in9 stems from SQP methods, whose mathematical background is well studied. In,10 we appliedthis method to an aerodynamic shape optimization problem of drag reduction with constant thicknessfor RAE2822 airfoil using Euler equations. Accurate Hessian approximation and its impact on theoptimization convergence is discussed. In the present paper, we discuss the accuracy issues of the stateand costate solutions in each optimization iteration and its impact on overall cost of computation. Thenumber of iterations required for the full optimization problem is less than 4 times that of the analysisproblem.

The paper is organized as follows. In the next section we discuss briefly about the optimizationstrategy. In section 3, we present the state, costate and design equations. Numerical results are presentedin section 4. We draw our conclusions in section 5.

II. Pseudo-timestepping for optimization problems

The optimization problem that we are dealing within this study can be written in abstract form as

min I(w, q)

s. t. c(w, q) = 0,(1)

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where (w, q) ∈ X × P (X, P are appropriate Hilbert spaces), I : X × P → R and c : X × P → Y are

twice Frechet-differentiable (with Y an appropriate Banach space). The Jacobian, J =∂c

∂w, is assumed

to be invertible. Theoretical works on the methodology for solving such problems are presented in.18–21

Here, the equations c(w, q) = 0 represent the steady-state flow equations (in our case Euler equations)together with the boundary conditions, w is the vector of dependent variables and q is the vector ofdesign variables. The objective I(w, q) is the drag of an airfoil for the purposes of this paper.

The necessary optimality conditions are

c(w, q) = 0, (State equation) (2)

∇wL(w, q, λ) = 0, (Costate equation) (2a)

∇qL(w, q, λ) = 0, (Design equation) (2b)

where

L(w, q, λ) = I(w, q) − λ∗c(w, q), (3)

is the Lagrangian functional and λ is the Lagrange multiplier or the adjoint variable from the dual Hilbertspace. Usually, system of equations (2) is solved using iterative methods for solution of the optimizationproblem.

Instead, we use simultaneous pseudo-time stepping in the proposed new method for solving theabove system (2). It is well known that there is a strong correlation between iterative methods andpseudo-time stepping which has been exploited for the construction of a time-stepping method in thespirit of reduced SQP-methods. That is, to determine the solution of (2), we look for the steady statesof the following pseudo-time embedded evolution equations

dw

dt+ c(w, q) = 0,

dλ

dt+ ∇wL(w, q, λ) = 0, (4)

dq

dt+ ∇qL(w, q, λ) = 0.

The pseudo-time embedded system (4) is usually a stiff system of ODEs (after semi-discretization).Therefore explicit time-stepping schemes may converge very slowly or may even diverge. In order toaccelerate convergence, this system needs some preconditioning. The preconditioner that we use stemsfrom reduced SQP method (see for example25, 26 ). A step of this method can also be interpreted as anapproximate Newton step for the necessary conditions of finding the extremum of problem (1), since theupdates of the variables are computed according to the linear system

0 0 A∗

0 B

(∂c

∂q

)∗

A∂c

∂q0

∆w

∆q

∆λ

=

−∇wL

−∇qL

−c

, (5)

where A is some approximation of the Jacobian J and B is the reduced Hessian.

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We use the inverse of the matrix in equation (5) as a preconditioner for the time-stepping process.The pseudo-time embedded system that we consider is

w

q

λ

=

0 0 A∗

0 B

(∂c

∂q

)∗

A∂c

∂q0

−1

−∇wL

−∇qL

−c

. (6)

This seems natural since equation (5) can be considered as an explicit Euler discretization for the cor-responding time-stepping that we envision. Also, due to its block structure, it is computationally inex-pensive. The preconditioner employed is similar to the preconditioners for KKT-systems discussed in2, 3

in the context of Krylov subspace methods and in4 in the context of Lagrange-Newton-Krylov-Schurmethods.

Within the inexact reduced SQP-preconditioner, one has to look for an appropriate approximationof the reduced Hessian, B. In particular, when dealing with partial differential equations constitutingthe state equations, the reduced Hessian can often be expressed as a pseudo-differential operator, thesymbol of which can be computed and exploited for preconditioning purposes as in.9 In10 we used twodifferent approximations of B and shown that a better approximation leads to faster convergence of theoptimization algorithm. In this study we address the accuracy issues of the state and costate solutions.

III. Detailed equations of the aerodynamic shape optimization problem

In this section we explain briefly the state, costate, and design equations represented in equations(2) for the aerodynamic shape optimization problem.State equations: Since we are interested in steady flow, a proper approach for numerical modeling isto integrate the unsteady Euler equations in time until a steady state is reached. These equations incartesian coordinates (x, y) for two-dimensional flow can be written in integral form for the region Ω withboundaries ∂Ω as

∂

∂t

∫

Ω

w dΩ +∫

∂Ω

F · ~nds = 0, (7)

where ~n denotes the unit outward normal to ∂Ω and

w :=

ρ

ρu

ρv

ρE

, F := [f, g] , f :=

ρu

ρu2 + p

ρuv

ρuH

and g :=

ρv

ρuv

ρv2 + p

ρvH

.

For a perfect gas the pressure and total enthalpy is given by

p = (γ − 1)ρ

E − 12(u2 + v2)

, H = E +

p

ρ,

respectively. The boundary conditions used to solve these equations are the zero normal velocity on thesolid wall, and the farfield boundary is treated by considering the incoming and outgoing characteristicsbased on the one dimensional Riemann invariants.

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The cost function that we choose in the present optimization problem is drag reduction (with thegeometric constraint of constant thickness of the airfoil). Hence, the cost function reads as

I(w, q) := CD =1

Cref

∫

C

Cp (nx cosα + ny sin α) ds, (8)

where the surface pressure coefficient is defined by

Cp :=2(p − p∞)γM2

∞p∞. (9)

The other constraint of constant thickness is maintained as we replace the airfoil by its camberlinerepresentation.Costate equations: The costate or adjoint Euler equations are given by (see, for example,7)

∂

∂t

∫

Ω

λ dΩ +∫

∂Ω

F · ~nds = 0, (10)

where the vector λ contains the components of the adjoint variable and F is the matrix of adjoint fluxdensity, defined as

λ :=

λ1

λ2

λ3

λ4

, F :=

[(∂f

∂w

)T

λ ,

(∂g

∂w

)T

λ

].

The boundary conditions for the adjoint Euler equations on the solid body are of Neumann-type and forthe above mentioned cost function they are given by

nxλ2 + nyλ3 = − 2γM2

∞p∞Cref(nx cosα + ny sin α), on the solid body. (11)

The farfield boundary conditions are based upon incoming and outgoing characteristics and free-streamconditions apply there as well. It is important to note that the adjoint Euler equations are linear in λ

and the wall boundary conditions depend on the cost function.

Design equation: For the design equation (2b), we need an expression for the derivative of the La-grangian with respect to the geometry of the airfoil. All the computations are carried out in a generalizedcoordinate system. Therefore, a transformation is used to transform the physical (x, y)-domain to thecomputational (ξ, η)-domain. In the computational domain, the components of the gradient ∂L

∂q can bedetermined by integrating the adjoint solutions multiplied by the metric sensitivities as follows

(∂L(q + εq

∂q

)∣∣∣∣ε=0

= −∫

C

p(−λ2qyξ + λ3q

xξ )ds −

∫

Ω

(λT

ξ

(qyηf − qx

ηg)

+ λTη

(−qy

ξ f + qxξ g

))dΩ

+1

Cref

∫

C

Cp

((q⊥)x cosα + (q⊥)y sin α

)ds, (12)

where q is the variation in the geometry of the airfoil and qx, qy are its x- and y-components,((q⊥)x, (q⊥)y

)

are the components of the unit normal to q.

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Discretization: The state and costate equations are discretized using a cell centered finite volumescheme. The semidiscrete equations are augmented with 1st and 3rd order artificial dissipations and thensolved using a 5-stage Runge-Kutta type time stepping scheme.

Surface Parameterization: The airfoil is represented by its camberline so that the constant thickness ismaintained during the optimization (otherwise the drag reduction problem will result in a flat geometry).In the current study, the geometry is modeled by Hicks-Henne functions.11 In this representation they-coordinates of the surface are written in parametric form. These parameters are the design variablesof the optimization problem.

Gradient Computation: As an efficient method of calculating the gradient (δI)m=1,...,n := ∇qI , weuse, as in,7 the so called ’grid moving technique’ based on J. Reuther’s approach (s.15) for evaluating theintegrals in (12), which are dependent on the adjoint field vector λ and the metric sensitivities generatedby the perturbation of the geometry (by the design variables).

Grid-Perturbation Strategy: As the shape of the airfoil changes during the optimization process, thelocation of the grid nodes has to be adjusted. This can be done by generating a new grid after eachdesign iteration or by using a grid-perturbation strategy after each design iteration. Here we followed agrid-perturbation strategy.

’n’ RK steps of Flow solver

’n’ RK steps of Adjoint solver

Gradient Computation

1 Euler step of Design Eqn

Approximation of B

Update Airfoil Geometry

Modify Grid

ConvergenceRepeated UntilDesign Cycle

Figure 1. Optimization Cycle

Details of discretization, surface parameterization, gradient computation and grid-perturbationstrategies can be found in.10 The algorithmic overview is presented in Figure 1. The block matrices A

and A∗ corresponding to the state and costate equations in the preconditioner are just identity matricesin the current implementation.

IV. Numerical results and discussion

The optimization method is applied to a test case of the RAE 2822 airfoil. The physical do-main is discretized using an algebraically generated (193 × 33) C-grid. On this grid, the preconditionedpseudo-stationary equations are solved. Camberline representation of the airfoil is parameterized by 21Hicks-Henne parameters. Complete optimization cycle is performed under the optimization platform

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Test Case RK-steps/opt. Iter. Opt. Iter. total RK-steps

T1 n = 1 1225 2450

T2 n = 2 1000 4000

T3 n = 3 925 5550

T4 n = 5 880 8800

T5 n = 10 860 17200

T6 n = 20 860 34400

Table 1. Number of optimization iterations and total number of Runge-Kutta steps

SynapsPointerPro.5 The optimization iteration is started with initial state and costate values (i.e., w0

and λ0) as those obtained after 500 time steps of the state and costate equations. We use the FLOWercode16, 17 of the German Aerospace Center (DLR) for solving the forward and adjoint equations.

The design equation is integrated in time using an explicit Euler scheme. Therefore, the time stepsused for the three sets of equations are not the same. In the current implementation of FLOWer, thetime steps are not same even in each discretization cell as they are determined independently accordingto the local stability criterion. However, this has no effect on the steady state solution.

One of the main issues of using this kind of preconditioned pseudo-timestepping is the approxima-tion of the reduced Hessian. As it is shown in,10 a better approximation will lead to faster convergenceof the optimization algorithm. In the current study, the reduced Hessian approximation is based on thecurvature information, as in the case of BFGS optimization methods. We define sk := (qk+1 − qk) andzk := (∇Ik+1 −∇Ik), where k represents the iteration number. Then, the curvature in the direction sk isobtained from the product (zT

k sk). If the curvature is positive, the reduced Hessian is approximated by

Bk = βzT

k sk

zTk zk

δij ,

where β is a constant. Otherwise, it is approximated by βδij , where β is a different constant. Additionally,we impose upper and lower limits on the factor so that

βmin < βzT

k sk

zTk zk

< βmax.

This prevents the optimizer from taking too small or too large steps. The constants βmin and βmax canbe chosen, e.g., depending on the accuracy achieved in one time step by the forward and adjoint solver.This gives the flexibility of using different codes (e.g. a multigrid forward and adjoint solver).

As convergence criterion of the optimization iteration, we use the discrete 2-norm of the incrementsof the profile parameters (||qk−qk−1||2) to be less than 0.002. Six case studies are made in regard to stateand costate solution accuracies and its effect on the optimization convergence (see Table 1). The residual(in logarithmic scale) of the state equations lies between 10−3 − 10−5 and the residual (in logarithmicscale) of costate equations lies between 10−1 − 10−3 depending on different Runge-Kutta(RK) steps usedper optimization cycle (Figure 2). The optimization algorithm requires between 1225− 860 iterations toconverge (Table 1). The convergence history of drag for these cases are presented in Figure 3.

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cycle

Lo

g(re

sidu

al)

500 1000 1500 200010­6

10­5

10­4

10­3

10­2

10­1

100

Main(T1)Adjoint(T1)

Opt

.beg

in

Opt

.en

d

cycle

Log

(res

idu

al)

0 1000 2000 300010­6

10­5

10­4

10­3

10­2

10­1

100

Main(T2)Adjoint(T2)

Op

t.b

egin

Opt

.end

cycle

Log

(res

idu

al)

0 1000 2000 300010­6

10­5

10­4

10­3

10­2

10­1

100

Main(T3)Adjoint(T3)

Op

t.b

egin

Op

t.en

d

cycle

Log

(res

idu

al)

0 1500 3000 450010­6

10­5

10­4

10­3

10­2

10­1

100

Main(T4)Adjoint(T4)

Op

t.en

d

Op

t.b

egin

cycle

Log

(res

idu

al)

0 2500 5000 7500

10­6

10­5

10­4

10­3

10­2

10­1

100

Main(T5)Adjoint(T5)

Op

t.b

egin

Op

t.en

d

cycle

Log

(res

idu

al)

1 5001 10001 15001

10­7

10­6

10­5

10­4

10­3

10­2

10­1

100

Main(T6)Adjoint(T6)

Opt

.beg

in

Op

t.en

d

Figure 2. Convergence history of the optimization iterations

cycle

Cd

500 1000 1500 20000

0.003

0.006

0.009

0.012Drag(T1)

Op

t.b

egin

Op

t.en

d

cycle

Cd

0 1000 2000 30000

0.003

0.006

0.009

0.012Drag(T2)

Opt

.beg

in

Op

t.en

d

cycle

Cd

0 1000 2000 30000

0.003

0.006

0.009

0.012Drag(T3)

Opt

.beg

in

Op

t.en

d

cycle

Cd

0 1500 3000 45000

0.003

0.006

0.009

0.012Drag(T4)

Opt

.beg

in

Op

t.en

d

cycle

Cd

0 2500 5000 75000

0.003

0.006

0.009

0.012Drag(T5)

Opt

.beg

in

Op

t.en

d

cycle

Cd

1 5001 10001 150010

0.003

0.006

0.009

0.012Drag(T6)

Op

t.b

egin Opt

.en

d

Figure 3. Convergence history of the optimization iterations

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The prominent oscillations in residual, for case T6, is due to the fact that we start the new iteration(on the new grid) with the solution of last iteration (on the old grid). Therefore, interpolation is used toget the initial solution on the new grid from the old solution on old grid. Due to this interpolation error,residual is deterioreted at the beginning of the optimization iteration. Since, we use 20 Runge-Kuttasteps, the residual comes down considerably below the other cases and we see prominent oscillations inthis case. In T5, the oscillations are smaller, and in T1-T4, they are almost not visible.

X/C

Y

0 0.25 0.5 0.75 1

­0.08

­0.04

0

0.04

0.08

Initial airfoilOptimized (T1)Optimized(T2)Optimized(T3)Optimized(T4)Optimized (T5)Optimized (T6)

X/C

Cam

ber

0 0.25 0.5 0.75 1

­0.006

0

0.006

0.012

InitialOpt.(T1)Opt.(T2)Opt.(T3)Opt.(T4)Opt.(T5)Opt.(T6)

Figure 4. Comparison of initial and final airfoils(left) and camberlines(right)

RK­steps/Opt. Cycle

Opt

.ite

ratio

n(x

100)

Tot

alR

K­s

tep

s(x

1000

)

5 10 15 20

3

6

9

12

20

40

60

80Opt. CyclesTotal RK­steps

Figure 5. Number of optimization iterations and toal number of RK-steps

Figure 4 presents the comparison of initial and final airfoils (left) and camber lines (right) computedin all six cases. We see there is almost no difference in the optimized airfoils and camberlines obtained bysix different cases. However, the total number of Runge-Kutta steps required are not same in all the cases.Figure 5 presents the plot of number of RK-steps/optimization cycle versus total number of optimizationiterations (left vertical axis) and total number of RK-steps (right vertical axis). We see that, with thesame Hessian approximation, increasing the state and costate solution accuracy decreases the numberof optimization iterations, but the total effort remains quite high. Therefore, in this one-shot approachone has to make the important decesions of wheather to decrease the number of optimization iterations

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(which involve other processes like obtaining new grid, gradient computation, read/write solutions etc.)or to decrease the total number of RK-steps (which might involve more computational effort) in thecomplete solution process.

Design Variable

­gra

d

5 10 15 20

­0.25

­0.125

0

Initial sensitivitiesFinal sensitivities

X/C

cp

0 0.25 0.5 0.75 1

­1.5

­1

­0.5

0

0.5

1

1.5

InitialOptimized (T1)Opt. (Steepest descent)

Figure 6. Comparison of initial and final gradients(left) and surface pressure distributions(right)

In Figure 6 (left) the initial and final gradient is presented for case T1. The optimized surfacepressure distribution obtained using the current method (case T1) and that obtained with steepestdescent method are also compared in the same Figure (right). Both optimized surface pressures almostcoincide. However, the steepest descent method required 23 forward runs and 6 adjoint runs togetherwith a line search. Each forward and adjoint run requires approximately 1500 iterations in time. Thatmeans that the steepest descent method needs an effort of about 29 forward runs, whereas the presentmethod needs an effort of little more than 3 forward runs. Additionally, the simultaneous pseudo-timemethod needs time integration of the design equation using explicit Euler time step, approximation ofthe Hessian, a new grid, read/write solutions after each optimization iteration. However, the total timerequired for this overhead is negligible compared to one complete forward run. If we add all these effortstogether, the time taken is still less than 4 forward simulation runs. In terms of CPU time, the completeoptimization cycle needs about 40 minutes on an Intel(R) Xeon(TM) CPU 1700MHz machine in thiscase.

In the pseudo-time optimization iteration, the initial drag of 0.0081012 is reduced to 0.0025996in the optimization process which is a reduction of about 68%. Since there is no constraint on lift andpitching moment coefficient, they are also reduced by about 10% and 20% respectively.

V. Conclusions

Simultaneous pseudo-timestepping is used to solve aerodynamic shape optimization problem forcompressible inviscid flow. The preconditioned pseudo-stationary state, costate and design equations areintegrated simultaneously in time until a steady state is reached. Accuracy issues of state and costate so-lutions per optimization iteration are discussed. Improving these accuracies alone, in the present context,does not reduce the overall cost of computation. The best overall cost of computation is approximately

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15% of that of a straight forward application of the steepest descent method. Generalization of theproposed strategy to problems with state constraints (e.g., drag reduction with constant lift and pitchingmoment) and to applications in 3D is our future goal.

VI. Acknowledgments

The author wishes to thank V. Schulz for all his discussions and cooperations to carry out thiswork as a member in his group. Thanks are due to J. Brezillon and N. Gauger for their discussions onFLOWer code.

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