aerodynamic design of classical and innovative...

ERCOFTAC 2004 ATHENS

Aerodynamic Design of Classical and Innovative Configurations

Using Advanced Multiobjective Algorithms

Domenico Quagliarella, Pierluigi Iannelli, Pier Luigi Vitagliano, Giorgio ChinniciCentro Italiano Ricerche Aerospaziali, Italy

[email protected], [email protected], [email protected], [email protected]

ERCOFTAC DESIGN OPTIMIZATION:

METHODS & APPLICATIONS

International Conference

&

Advanced Course Program

Athens, GREECE March 31–April 2, 2004

March 31, April 2, 2004 1


Introductory notes

This talk is focused on the application of evolutionary optimizers to aerodynamicdesign problems that require the use of high fidelity CFD solvers, and hence a hugeamount of computational resources.

The challenge related to these problems is the development of techniques for im-proving the efficiency of evolutionary optimizers.

Various techniques will be discussed along with some optimization and design exam-ples.



Outline of the presentation

• Genetic algorithms and efficiency:

• Fitness function approximation used with an asymmetric MOGA

• Hybrid GA using adjoint techniques

• Hybrid GA for constrained optimization

• Application examples (AEROSHAPE project)

• A design challenge:

• NLF wing for high speeds (Piaggio, VITAS Project)

• Final remarks.



Part IUse of approximate fitness evaluators

• Optimization techniques based on evolutionary computing are very attractive interms of robustness and global extremum location capabilities, but often lackefficiency.

• This is a problem in many engineering applications, where fitness evaluationoften requires a substantial amount of computational resources.

• This makes attractive the use of approximate fitness evaluators, with lowercomputational requirements, whenever it is possible.

• In aerodynamic optimization design various approaches have been experimented,such as the use of Neural Network-based interpolators to reduce the number oftrue flow field evaluations.

• Another option is the mixed use of solvers with different levels of approximationin a hierarchical organization.



Use of approximate fitness evaluators: advantages and limits

• Neural Networks are able to improve the quality of their approximation when thenumber of exact evaluations available increases.

• Their performance may improve in real time during the optimization process.

• Their more serious drawback is the fast decrease in approximation fidelity whenthe parameter space dimension increases.

• Low fidelity solvers (e.g. Euler+boundary layer versus Navier-Stokes), do nothave this problem, but it is not always easy to understand the limits of theapproximate model.

• How high-fidelity and low-fidelity fitness approximations can be mixed?



An example helps to understand the basic aspects of the problem

Let us have:

f(x) = [x2 − 10cos(2πx) + 10] x ∈ [−1,3]

and let g(x) a polynomial of degree 10 that minimizes∑

xi∈X0

[f(xi) − g(xi)]2 to best fit

f(x) for a given training set X0.

-5

0

5

10

15

20

25

30

-1 -0.5 0 0.5 1 1.5 2 2.5 3

f, g

x

f(x)g(x)



What this example points out

This example show us two important aspects that have to be considered when usingan approximated model for fitness evaluation:

1. Shift of global extremum points between the two models;

because this is what determines the global optimum location capability of a searchprocedure that uses fitness function approximation.

2. Value shift between exact and approximate functions;

because this shift has to be very little to allow a free mix of exact and approximatedvalues.

Summing up, the lesser are these two kinds of shifts, the better and more reliablethe optimization results obtained using the approximate model will be.



These two kinds of shifts are exactly what the Pareto front (f, g) put inevidence very clearly.

For optimization purposes, the the relationship between f and its approximatingfunction g described by the Pareto front is all that we need.

-3

-2

-1

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5

g(x)

f(x)

The smaller is the front, the better is the accuracy of g in approximating f . If thefront is a single point, then the global extremum of f and g is located at the samevalue of x∗. Furthermore, if f(x∗) = g(x∗), the values of f and g can be freely mixed.



Analysis of an example CFD problem

To better fix the concepts illustrated, let us now consider an aerodynamic optimiza-tion problem:

The RAE 2822 airfoil is assigned as starting configuration, and the optimizationgoal is the reduction of the drag coefficient cd at given Mach (M = 0.78) and liftcoefficient cl = 0.60. The maximum airfoil thickness has to remain unchanged.

A two-objective optimization run is carried out using two different flow solvers:

1. Euler + Boundary layer, interactive (high fidelity solver, objective f).

2. Full potential, non-conservative formulation (low fidelity solver, objective g).



Euler + Boundary Layer - Full Potential comparison

-1

-0.5

0

0.5

1

1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c p

x/c

Euler + BLFull potential

-1

-0.5

0

0.5

1

1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c p

x/c

Euler + BLFull potential

cd x 102 (Euler + BL)

c dwx

102

(Ful

lPot

entia

l)

0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94

0

0.005

0.01

0.015

0.02

full potential: cdw=0.000281

Euler + BL: cdw=0.000561full potential: cdw=0.000

Euler + BL: cdw=0.000006

The figure shows how these two solvers lead to different solution for the same designproblem, and that a free mix of both kinds of results can hamper the efficiency ofthe optimization process.

March 31, April 2, 2004 10


Summing up

• What really matters when trying to use g instead of f are the shifts in theextremum point.

• In general, we are quite far from the ideal situation of a Pareto front reduced toa point.

• The multi-objective optimization framework can be used to analyze the impli-cations of mixing fitness evaluators with different levels of precision because therelations between f and its approximated model g are well described by thistwo-objective problem:

min f(x), g(x)x ∈ X

where f(x), g(x) ∈ < are defined in X ⊆ <n

March 31, April 2, 2004 11


The analysis of the Pareto front trade-off between exact and approximatedfunctions gives an immediate indication about the usability of the

approximation during each step of the optimization

IV

g

f

I

g global minimum

glob

al m

inim

umf

III

II

March 31, April 2, 2004 12


Pareto front trade-off analysis

Given a generic point P (f, g), four different zones can be identified in the objectivefunction plane, that are characterized by different behavior of g and f in movingtowards their global optimum points, namely F and G, when a search algorithm withbias towards G is used.

• In zone (I), a solution P ′ that has a better g is very likely to dominate the initialone (P ) and hence it is very likely to be closer to both F and G global minimumpoints.

• In zone (II), a solution that improves g has less chance to be also closer to F .

• Conversely, zone (IV) has an opposite behavior, as a solution point closer to Ghas a good chance of being also closer to F , similarly to zone (I).

• In zone (III), finally, even if it is still possible to find a P ′ that dominates P , analgorithm biased towards G has a very low probability of producing candidatesolutions with better values of f .

March 31, April 2, 2004 13


Pareto front trade-off analysis

• Given this framework, we want figure out an interleaving strategy that minimizesthe number of exact function evaluations.

• The limit of this approach is that in a real optimization run the Pareto front isunknown and only a rough estimate of the collocation of a particular solution inone of the four zones is possible.

• To overcome this limit an Asymmetric multi-objective optimization algo-rithm “AMOGAe” is introduced.

• This algorithm is characterized by an asymmetry in the evaluation of the objec-tives, so that the cheap approximation is used much more times than the “highfidelity” evaluation.

March 31, April 2, 2004 14


AMOGAe — Simple asymmetric MOGA with elitism

1. A small initial population is generated, and evaluated with respect to both ob-jectives.

2. Classical crossover and bit mutation operators are applied

3. Each population element xi0 is used as starting point of a single-objective evo-

lutionary procedure with g as fitness (internal level evolutionary procedure). Anassigned number of evolution steps is performed in order to have a given ratiobetween exact n(f) and approximate m(g) solutions.

4. The best element of the single-objective run (xif) is selected and inserted in the

external level MOGA.

5. The population of the external MOGA is evaluated, the non-dominated solutionsare saved and, eventually, re-introduced in the next generations if an elitistapproach is used.

March 31, April 2, 2004 15


Scheme illustrating the two levels of the AMOGAe

g

evolution step

promoted element

sub−pop element

global pop. element

glob

al m

inim

umf

f

g global minimum

March 31, April 2, 2004 16


Relationship between external and internal level evolutionary procedures

• The internal level is a very simple (1+1)-ES.

• The external MOGAe is based on Local Pareto Selection enhanced with elitismand with secondary ranking criterion to choose between non dominated individ-uals.

March 31, April 2, 2004 17


Multi-point RAE2822 optimization example

• The minimization of viscous drag is required in three design points:

α1 = 2.8, M1 = 0.734, Re1 = 6.5 × 106

α2 = 2.8, M2 = 0.754, Re2 = 6.2 × 106

α3 = 1.8, M3 = 0.680, Re3 = 5.7 × 106

• This means: minOBJ =3∑

i=1

wicd(αi, Mi) with w1 = 2, w2 = 1, w3 = 1.

• Transition is at x/c = 0.03 in the first two points and at x/c = 0.11 in the thirdone.

• The starting airfoil is the classical RAE2822.

• Lift is constrained as follows: cl1 ≥ 0.8, cl2 ≥ 0.74, cl3 ≥ 0.56.

• Airfoil thickness and leading edge radius cannot decrease.

March 31, April 2, 2004 18


Flow solvers

• Two flow solvers with different level of accuracy have been adopted for theobjective function computation:

1. The in-home developed ZEN Navier-Stokes solver with k − ω turbulencemodel.

2. Drela’s MSES code, based on a finite-volume discretization of the Eulerequations on a streamlined grid. The viscous region is computed using anintegral boundary layer based on a multi-layer velocity profile representation.The inviscid and viscous regions are coupled using displacement thickness.

March 31, April 2, 2004 19


Airfoil shape

• Defined as a linear combination of an initial geometry y0(x) and some modifi-cation functions, fi(x), i = 1, . . . , n:

y(x) = k

(y0(x) +

n∑

i=1

wifi(x)

)

where wi are the design variables and k is a scaling factor used to explicitlysatisfy the constraint on thickness.

• 20 design variables are used to modify the airfoil shape (10 for the upper surfaceand 10 for the lower one).

March 31, April 2, 2004 20


Asymmetric GA setup and optimization run

• External level MOGA:

– 100% crossover probability.

– 3% bit-mutation probability.

– 8 population elements.

– 5 generations.

• Asymmetric internal level (1 + 1)−ES evolutionary step:

– 4% bit-mutation rate.

– 15 iterations for each call.

March 31, April 2, 2004 21


Asymmetric GA setup and optimization run

• Each objective function evaluation required 3 Navier-Stokes computations or 3Euler+BL computations.

• The constraint on lift coefficient is taken into account by introducing a penaltyin the objective function.

• A single Navier-Stokes evaluation requires, for each design point, approximately650 seconds on a vector computer NEC-SX6 for 2000 multigrid iterations on agrid of 14336 (256x56) cells.

• Approximately 20 seconds per design point are required by Euler+BL solver.

• The AMOGAe with these settings obtained the reported results after 48 Navier-Stokes and 600 Euler+BL objective function calculations.

March 31, April 2, 2004 22


+

OBJ Euler+BL

OB

JN

avie

r-S

toke

s

0.060 0.065 0.070 0.075

0.068

0.070

0.072

0.074

0.076

0.078

0.080

0.082

0.084

population elements

Pareto front

starting airfoil

line of exact approximation+

PF and population elements evaluated using both NS and Euler+BL

March 31, April 2, 2004 23


OBJNS convergence history

evaluations

OB

J NS

0 10 20 30 40 50

0.068

0.072

0.076

0.08

0.084

March 31, April 2, 2004 24


OBJEuler+BL convergence history

evaluations

OB

J EU

+B

L

100 200 300 400 5000.058

0.062

0.066

0.07

0.074

0.078

0.082

0.086

March 31, April 2, 2004 25


Starting and optimized airfoil comparison

Cl1 Cd1 Cm1 Cl2 Cd2 Cm2 Cl3 Cd3 Cm3

RAE2822 0.8122 0.020 -0.0941 0.7875 0.0295 -0.1076 0.5621 0.0096 -0.0829OPT. 0.8126 0.016 -0.0969 0.8107 0.0261 -0.1147 0.5601 0.0099 -0.0913

March 31, April 2, 2004 26


x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

x/cc f

0 0.25 0.5 0.75 1

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

OPTIMIZED cfRAE 2822 cf

cp comparison in the first design point cf comparison in the first design point

March 31, April 2, 2004 27


x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

x/cc f

0 0.25 0.5 0.75 1

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008


cp comparison in the second designpoint

cf comparison in the second designpoint

March 31, April 2, 2004 28


x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

x/cc f

0 0.25 0.5 0.75 1

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008


cp comparison in the third design point cf comparison in the third design point

March 31, April 2, 2004 29


Part IIGAs and adjoint methods

Joint work with A. Iollo, D. D’Ambrosio, Politecnico di Torino

• Some remarks on adjoint problems.

• The use of computer algebra techniques for adjoint equation derivation.

• An example with a simple 2D problem.

• The adjoint and the gradient for 3D Navier-Stokes.

• Numerical solution of the adjoint equations.

• Features of Hybrid Genetic Algorithms.

• An application example.

March 31, April 2, 2004 30


Optimization of a functional

Let

J : f ∈ U → c ∈ <

where U is an assigned function space; a particular f is sought so that:

minf∈U

J(f)

possibly subject to constraints.

If one uses a gradient based method,the first variation of J has to be computed!

The adjoint method is used to simplify calculations!

March 31, April 2, 2004 31


Adjoint method

The adjoint method uses Lagrange multipliers to find constrained minima of J.

The main mathematical tool needed to build the adjoint equations is the

Lagrange Identity

(Av, w) = (v, A∗w)

v ∈ D(A), w ∈ D(A∗)

with:

A: linear or non-linear operator

A∗: adjoint operator

( · , · ): scalar product defined in an appropriate Hilbert space

March 31, April 2, 2004 32


Let

F = F (b,u) objective function

R(b,u) = 0 state equations

with b = b(x), u = u(b,x) and R a generic differential operator.Introducing the Lagrange multipliers λ(x), one obtains:

L(b,u, λ) = F (b,u) + (R(b,u), λ) (Lagrangian equation)

where, now, u = u(x).

If bv = b + εb, uv = u + εu, λv = λ + ελ

δL = L(b + εb,u + εu,λ + ελ) − L(b,u, λ) =

F (b + εb,u + εu) − F (b,u) +(R(b + εb,u + εu), λ + ελ

)− (R(b,u), λ)

March 31, April 2, 2004 33


Neglecting O(ε2) terms, we can write:

δL =(Lλ, ελ

)+(Lb, εb

)+ (Lu, εu)

where Lλ, Lb and Lu are the first variations of L with respect to λ, b and u.

The three addenda of the equation above must be equal to zero separately, that is:(Lλ, ελ

)= R(b,u) = 0

(Lb, εb

)=(Fb, εb

)+[(

Rb, εb)

, λ]= 0

(Lu, εu) = (Fu, εu) + [(Ru, εu) , λ] = 0

Note that the first equation is the state equation.The last two equations will be treated using the Lagrange identity.

March 31, April 2, 2004 34


Thus, let’s apply the Lagrange identity:(Lb, εb

)=(Fb, εb

)+[(

Rb, εb)

, λ]=(Fb, εb

)+[(R∗

b, λ) , εb]= 0

(Lu, εu) = (Fu, εu) + [(Ru, εu) , λ] = (Fu, εu) + [(R∗u, λ) , εu] = 0

Rearranging, we finally obtain:([Fb + (R∗

b, λ)] , εb)

= 0

([Fu + (R∗u, λ)] , εu) = 0

and since εb and εu are arbitrary small perturbations, it must be

Fb + (R∗b, λ) = 0 gradient

Fu + (R∗u, λ) = 0 adjoint equation

March 31, April 2, 2004 35


Using the present approach, one can avoid to compute εu. In fact, applying theTaylor expansion:

εu = u(b + εb) − u(b) =du

dbεb + O(ε2)

Problem:

du

dbis computationally expensive!

Solution:

Choose λ so that

Fu + R∗u(λ) = 0

ADJOINT EQUATION SET

March 31, April 2, 2004 36


A symbolic computation algorithm was developed to automatically compute theadjoint equation set and related gradient in continuous form using MAPLE ComputerAlgebra software package, and the following criteria were used in the developmentphase:

• A compromise was sought between output conciseness and ease of implemen-tation: the equations have been specified term by term, but the volume andsurface integrals of the Lagrangian have been left unexpanded.

• Classical manipulations for adjoint derivation were obtained using classical Greenformulas, while the variation with respect to geometry were obtained throughHadamard formulas.

• This approach required the capability of discriminating between the various pos-sible structures of a given Lagrangian, in order to apply the proper transforma-tions.

• This was obtained applying recursively the pattern matching functions of theCA system.

March 31, April 2, 2004 37


Symbolic computation algorithm

variation function

Input

1. functional L

2. function to be variated u

3. geometric deformation vector h(if needed)

Output

1. variation of the functional L

∂L

∂uδu +

∂L

∂uxδux +

∂L

∂uyδuy +

∂L

∂uzδuz +

∂L

∂uxxδuxx +

∂L

∂uxyδuxy + ...

March 31, April 2, 2004 38


Variation with respect to geometry

Formulas adopted for the treatment of surface and volume integral variation:

Surface integrals

δF =

∫∫

S

∇f ·∂h

∂gδg dσ +

∫∫

S

fH ·∂h

∂gδg dσ +

∫∫

S

∂f

∂gδg dσ (1)

Volume integrals

δF =

∫∫∫

T

∂f

∂gδg dV +

∫∫

FT

f∂h

∂g· nδg dσ (2)

This formulas are exactly equivalent to the Hadamard formulas. Pattern matchingwas used to individuate and select surface and volume integrals.

March 31, April 2, 2004 39


intparts3D function

Input

1. variated functional δL

Output

1. variated functional δL were the terms containing derivatives of the variationare moved to surface integrals.

Comment: this function works applying recursively integration by parts and Green’stheorem.

March 31, April 2, 2004 40


extract adj eq function

Input1. manipulated variation of the functional (δL)Output1. adjoint equations

extract adj bceq function

Input1. manipulated variation of the functional (δL)Output1. adjoint equations boundary conditions

Comment: pattern recognition is used

March 31, April 2, 2004 41


An application example

2D symmetric potential flow with blowing

W = ∇φ · n

V∞

A symmetric airfoil is immersed in a potential flow at zero angle of attack and asymmetric transpiration velocity W is assigned in the direction normal to the airfoilsurface.

March 31, April 2, 2004 42


> alias_dependencies([y,W,dW],[s1]):

> varlist := [x1,x2,x3,s1,s2]:

> alias_dependencies([phi,dphi,n1,n2,n3,h1,h2,h3],varlist):

> alias_dependencies([Phi],[s1,s2]):

> Obj:=Int(Int(1/2*(phi-Phi)^2,s1),s2);

Obj :=

∫∫1

2(φ − Φ)2 ds1 ds2

> Potential_eq:=diff(phi,‘$‘(x1,2))+diff(phi,‘$‘(x2,2)):autoAlias(%);

φx1 , x1 + φx2 , x2

> n:=[n1,n2];

n := [n1 , n2 ]

> Neumann_bc:=dotprod(grad(phi,[x1,x2]),n,’orthogonal’)-W: autoAlias(%);

φx1 n1 + φx2 n2 − W

> alias_dependencies([eta,beta],varlist):

March 31, April 2, 2004 43


> Lagrangian:=Obj+Int(Int(Int(eta*Potential_eq,x1),x2),x3)+

> Int(Int(beta*Neumann_bc,s1),s2);

Lagrangian :=

∫∫1

2(φ − Φ)2 ds1 ds2 +

∫∫∫η (φx1 , x1 + φx2 , x2) dx1 dx2 dx3

+

∫∫β (φx1 n1 + φx2 n2 − W) ds1 ds2

> adj_dphi:=compute_adj_eq(Lagrangian, phi, dphi, 2, [h1, h2, h3], 0,true);

adj dphi := ηx1 , x1 + ηx2 , x2

> adj_dphi_bc:=compute_adj_bc(Lagrangian, phi, dphi, 2, [n1, n2, n3],

> [h1, h2, h3], 0, true);

adj dphi bc :=

φ − Φ +η dphi x1 n1

dphi− ηx1 n1 +

η dphi x2 n2

dphi− ηx2 n2 +

β n1 dphi x1

dphi+

β n2 dphi x2

dphi

> c6:=variation(Lagrangian,[W=W+dW],[s1,y,s2],[s1,s2],[x1,x2,x3]);

−

∫∫β dW ds1 ds2

March 31, April 2, 2004 44


Results are now reported related to Navier-Stokes equations

State equations

Conservation of mass

∂ρvi

∂xi= 0 (3)

Conservation of momentum

∂

∂xj

ρvivj + pδi j − µ

[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]= 0 (4)

Conservation of energy

∂

∂xi

ρviH − k

∂T

∂xi− µ

[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]vj

= 0 (5)

March 31, April 2, 2004 45


Lagrangian

L = I +

∫

Ω

η∂ρvi

∂xidΩ

+

∫

Ω

λi∂

∂xj


[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]dΩ

+

∫

Ω

ζ∂

∂xi

ρviH − k

∂T

∂xi− µ

[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]vj

dΩ

+

∫

Σ

ξividΣ

+

∫

Σ

τΘ(T )dΣ (6)

I is the performance functional, where the flow variables are implicitly dependent onthe boundary; L is the Lagrangian, it depends explicitly on the flow variables, theLagrange multipliers and the geometry.

March 31, April 2, 2004 46


Adjoint equations(with respect to conservative variables variation)

−γ − 1

2vjvj

∂λi

∂xi+ vivj

∂λi

∂xj−

(γ − 1

2V 2 − H

)vi

∂ζ

∂xi+

vj(hj + gj)

ρ−

(γ − 1

2ρRV 2 −

T

ρ

)∂

∂xi

(k

∂ζ

∂xi

)= 0 (7)

[(γ − 1) vivj − Hδi j]∂ζ

∂xj−

∂η

∂xi− vj

∂λi

∂xj− vj

∂λj

∂xi+ (γ − 1)vi

∂λj

∂xj−

hi + gi

ρ+

γ − 1

ρRvi

∂

∂xi

(k

∂ζ

∂xi

)= 0 (8)

−(γ − 1)∂λi

∂xi− γvi

∂ζ

∂xi−

γ − 1

ρR

∂

∂xi

(k

∂ζ

∂xi

)= 0 (9)

with

hi =∂

∂xj

µ

[∂λi

∂xj+

∂λj

∂xi−

2

3

∂λk

∂xk

δi j

]

gj =

∂

∂xi

[µ

(vj

∂ζ

∂xi+ vi

∂ζ

∂xj−

2

3vk

∂ζ

∂xk

δi j

)]− µ

∂ζ

∂xi

[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]

Note the self-adjointness of the linear part.

March 31, April 2, 2004 47


Boundary conditions

ζ = 0 Prescribed wall temperature OR∂ζ

∂xini = 0 Adiabatic wall (10)

λi = −Ji (11)

where

I =

∫

Σ

Jifi dΣ with fi =


[∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

]nj

Terms needed for gradient evaluation

τ = −k∂ζ

∂xini Prescribed wall temperature OR τ = kζ Adiabatic wall (12)

ξi = −

[ρHni − µ

(∂vi

∂xj+

∂vj

∂xi−

2

3

∂vk

∂xk

δi j

)nj

]ζ − µ

(∂λi

∂xj+

∂λj

∂xi−

2

3

∂λk

∂xk

δi j

)nj − ρniη

(13)

March 31, April 2, 2004 48


Gradient

The gradient is the variation of L with respect to the geometry. If I is a field integral,and if ε ω is variation in the direction of the normal to the boundary, we have

δI =

∫

Ω

∂F

∂ωω dΩ +

∫

Σ

Fω dΣ (14)

where F is the integrand of I and ε is small. When I is defined on the boundary wehave

δI =

∫

Σ

∂F

∂ωω dΣ +

∫

Σ

∂F

∂xiniω dΣ +

∫

Σ

HF ω dΣ (15)

and H is the local surface curvature.

For the other terms of L the derivation rules are the same, but some simplificationsoccur since the governing and adjoint equations with the respective boundary con-ditions are satisfied on Σ. The only terms left are those relative to the boundaryterms and the gradient is

δL = δI +

∫

Σ

∂ξivi

∂xjnjω dΣ +

∫

Σ

∂τΘ

∂xiniω dΣ (16)

March 31, April 2, 2004 49


In the particular case under examination the functional to be minimized is

L = ω1D + ω2(L − L∗)2

2

where D is the drag (to be minimized), L is the lift, L∗ is the desired lift and the ωi

are weights. Consequently, the gradient expression is:

δL = ω1

∫

σ

−

(−

∂p ni

∂yδy + ∂τij nj

∂yδy)

t∞i dS+

ω2 (L − L∗)

∫

σ

−

(−

∂p ni

∂yδy + ∂τij nj

∂yδy)

n∞i dS+

∫

σ

ξi∂vi

∂yδydS +

∫

σ

τ ∂Θ∂y

δydS

March 31, April 2, 2004 50


Adjoint equation numerical solution

• The numerical solution of the adjoint equations is obtained by using a first-ordertime-dependent technique based on a finite volume discretization.

• The solver computes the fluxes at cell interfaces using a flux-vector splittingtechnique.

• In a similar way, the boundary conditions are imposed on the numerical fluxesat the computational field edges.

March 31, April 2, 2004 51


Adjoint optimizer scheme

March 31, April 2, 2004 52


Adjoint optimizer features

• The gradient obtained with the adjoint equations is used by a conjugate gradientoptimizer.

• The functional to be minimized is: L = ω1D + ω2(L−L∗)2

2with D= drag L= lift,

L∗= desired lift and ωi weights.

• Multi-point optimization cannot be directly faced by this adjoint solver.

• Each grid point on the airfoil boundary is a design variable.

• The point-wise gradient is smoothed using a Fourier filter. After such filteringthe conjugate gradient method is able to nicely decrease the functional L.

March 31, April 2, 2004 53


Multi-point hybrid optimizer

• An evolutionary computing algorithm is used for multi point design of transonicairfoil.

• The optimizer is coupled to a turbulent Navier-Stokes flow field solver.

• The peculiarity of the approach is in the computation of the modification func-tions.

• They are computed using the adjoint flow solver and are strictly dependent onthe initial airfoil shape and on the design point considered.

March 31, April 2, 2004 54


Multi-point hybrid optimizer

• Each design exercise is characterized by a different and customized set of mod-ification functions.

• The first phase of the design exercise is the computation of the modificationfunctions using the developed adjoint solver and gradient evaluator.

• The second phase is the run of the evolutionary optimization algorithm. Herefurther geometric and aerodynamic constraints can be handled.

• Here the optimization algorithm is a very simple (1+1)-ES evolutionary strategywith elitism.

March 31, April 2, 2004 55


Multi-point hybrid optimizer scheme

DP 1: Cl, Mach, Reynolds, ...

DP N: Cl, Mach, Reynolds, ...

OPTIMIZED AIRFOIL

DP 1: Cl, Mach, Reynolds, ...

......

DP N: Cl, Mach, Reynolds, ...

MULTI−POINT DESIGN PROBLEM

......

modification

OPTIMIZERMULTI−POINT

(1+1)−ES

OPTIMIZERADJOINT

function set N

ADJOINTfunction set 1

modification

OPTIMIZER

March 31, April 2, 2004 56


Multi-point airfoil design: modification functions

• Six modification functions are used, three for upper surface and three for lower.

• The table shows the results of the adjoint run for each design point with cl

unconstrained:

Cl1 Cd1 Cl2 Cd2 Cl3 Cd3

RAE2822 0.8122 0.02 0.7875 0.0295 0.5621 0.0096DCl% DCd%

OPT DP1 0.8063 0.0187 0.726422 6.5OPT DP2 0.7663 0.0276 2.692063 6.440678OPT DP3 0.5643 0.0095 -0.39139 1.041667

March 31, April 2, 2004 57


Multi-point airfoil design: modification functions

x/c

f mod

0 0.25 0.5 0.75 1

0.0000

0.0004

0.0008

0.0012

DP1 - upper surfaceDP1 - lower surface

x/c

f mod

0 0.25 0.5 0.75 1

-0.0012

-0.0008

-0.0004

0.0000

0.0004

0.0008


x/c

f mod

0 0.25 0.5 0.75 1

0.0000

0.0004

0.0008

0.0012


The obtained modification functions in the three different design points arereported.

March 31, April 2, 2004 58


iterations

Res

idua

l

0 500000 1E+0610-5

10-3

10-1

101

103

105

RMSMAX

Average and maximum residuals for the 10 Navier-Stokes iterations needed forgenerating the modification functions related to the first design point.

March 31, April 2, 2004 59


Evolutionary strategy setup and optimization run

• The mutation probability of the (1+1)-ES was set to 6%.

• Two different runs were performed, and each run required 23 objective functionevaluations.

• Each objective function evaluation required 3 Navier-Stokes computations.

• A single Navier-Stokes evaluation requires, for each design point, approximately650 seconds on a vector computer NEC-SX6 for 2000 multigrid iterations on agrid of 14336 (256x56) cells.

March 31, April 2, 2004 60


evaluations

OB

J NS

0 5 10 15 20

0.076

0.078

0.080

0.082

0.084ES RUN 1ES RUN 2

Convergence history for the two ES runs performed

March 31, April 2, 2004 61


Airfoil obtained in the best run

Cl1 Cd1 Cm1 Cl2 Cd2 Cm2 Cl3 Cd3 Cm3

RAE2822 0.8122 0.0200 -0.0941 0.7875 0.0295 -0.1076 0.5621 0.0096 -0.0829OPTIMIZED 0.8019 0.0185 -0.0926 0.7758 0.0280 -0.1050 0.5590 0.0096 -0.0840

March 31, April 2, 2004 62


cp comparison in the first design point

x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

March 31, April 2, 2004 63


cp comparison in the second design point

x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

March 31, April 2, 2004 64


cp comparison in the third design point

x/c

c p

0 0.25 0.5 0.75 1

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPTIMIZEDRAE 2822

March 31, April 2, 2004 65


Part IIIHybrid GAs for constrained optimization

• Hybrid optimization techniques.

• A gradient-based constrained optimization procedure, based on the feasible direc-tion (FD) search technique used as hybrid operator

• The hybrid GA coupled with a 3D aerodynamic flow solver: configuration gener-ation, parameterization and handling problems.

• Multi-point optimization of a wing-body configuration in transonic cruise condi-tions.

March 31, April 2, 2004 66


Hybrid optimization techniques

• Hybridization consists in the combination of the evolutionary procedure with anoptimization technique of different nature;

• Different hybrid strategies can be developed; the success of a strategy is the ex-ploitation of the most favorable characteristics of the methods which are coupled,and the overcoming of the relative drawbacks;

• A proper hybridization strategy may lead to better solutions than those obtainableusing the two methods individually;

March 31, April 2, 2004 67


Hybridization between a genetic algorithm and a hill climber

current

final populationinitial population

mutation hill climber

crossover

selection

generation

selectionintermediate

March 31, April 2, 2004 68


Hybrid techniques

• Variable encoding requires special care when using hybrid techniques.• A specialized optimization technique may relay on a particular encoding

that has a key role in the efficiency of the method: when possible, it isworthwhile to extend to the genetic algorithm the encoding of thespecialized technique.

• Binary encoding used by many genetic algorithms may cause problemsto the specialized algorithms if the conversion of data is not carefullyhandled.If a binary GA does not use enough bits for encoding, the improvementobtained using a gradient based hill climber may be lost whenre-encoding the variables in binary form.

March 31, April 2, 2004 69


Hybrid GA implementation

• GA is hybridized with two different gradient based operators:

– BFGS (Broyden, Fletcher, Goldfarb, Shanno) for unconstrained problems.

– Feasible Directions for constrained ones.

• Constrained problems are treated in a special way:

– At GA level constraints are taken into account as penalties.

– At gradient operator level the objective function does not include penalties,and active inequality constraints and their gradients are explicitly considered.

March 31, April 2, 2004 70


Feasible directions search

constrained optimum point

feasible region

region feasible and usable

usable region

∇f(x)

f(x) = constant

∇g(x)

∇g(x)

∇f(x)

g(x) = 0

March 31, April 2, 2004 71


Hybridization between a binary coded GA and a FD algorithm

March 31, April 2, 2004 72


Configuration definition and parameterization

• The optimizer can define and automatically modify a configuration like a wing-body or an isolated wing, by setting only a few geometrical parameters.

• The computational grid is also automatically generated.

• A configuration is defined by a reference wing and an optional reference body.

• A wing is defined by a plan-form and one or more wing sections.

• Airfoils are delivered as a sequence of points, and they can be modified by addingshape functions (B-Spline) to the reference geometry.

March 31, April 2, 2004 73


AERODYNAMIC ANALYSIS

• The design optimization procedure uses the ZEN system of codes for the aero-dynamic analysis.

• The sequence of operations performed during the evaluation phase is:

– Geometry generation

– Domain modeling and geometry handling

– Grid generation

– Flow solution

– Post-processing

– Objective and constraint functions evaluation

March 31, April 2, 2004 74


WING-BODY OPTIMIZATION EXAMPLE

• Two point optimization of a wing-body configuration of a transonic commercialaircraft.

• Drag minimization is required at two different flow conditions:

OBJ = 10 (2Cd1 + Cd2)

• Design conditions defined in terms of lift coefficient and Mach number: Cl1 =0.50, M1 = 0.85; Cl2 = 0.46, M2 = 0.88.

• The maximum thickness/chord is constrained to be at least that of the baselinewing

• The absolute value of the pitch moment coefficient Cm should not be greaterthan that of the baseline configuration.

March 31, April 2, 2004 75



• The flow is modeled by Euler equations.

• Hybrid GA settings:

– Crossover activation probability is 100%.

– Bit-mutation has a 2% probability

– Maximum number of FD operator iterations is 4, while its activation proba-bility is 4%.

March 31, April 2, 2004 76



• A population of 16 element evolved for 16 generations

• The FD operator was triggered 10 times (leading to 673 fitness evaluations).

• The total CPU time was approximately 52 hours on a NEC SX-6 computer.Each design point, indeed, required 137 seconds for the evaluation with a com-putational GRID of 33152 cells (47914 points).

• 17 design variables were used, namely the angle of attack at each design point,5 twist angles and 10 b-spline coefficients for the crank airfoil section, linearlyblended from root to tip.

March 31, April 2, 2004 77


Objective function convergence history

Best solution: OBJ = 0.558195, initial value: OBJ = 0.563150, total number of evaluations: 673

March 31, April 2, 2004 78


Global aerodynamic coefficient comparison.

α1 Cl1 Cd1 Cm1

BASE 1.0500 0.50052 0.018302 -0.15867OPTIM. 1.0728 0.49918 0.018170 -0.15686

α2 Cl2 Cd2 Cm2

BASE 0.3300 0.46000 0.019711 -0.18809OPTIM. 0.3496 0.45932 0.019439 -0.18633

March 31, April 2, 2004 79


BaselineOptimized

η

twis

tang

le

0 0.2 0.4 0.6 0.8 1

BaselineOptimized

Basis and optimized x, z coordinatesat crank section normalized by the lo-cal chord.

Basis and optimized twist distributionalong the wing.

March 31, April 2, 2004 80


cp: -0.96 -0.83 -0.67 -0.57 -0.35 -0.07 0.04

BASELINE configuration - DP 1

cp: -0.96 -0.83 -0.67 -0.57 -0.35 -0.07 0.04

OPTIMIZED configuration - DP 1

Baseline and optimized cp distribution in the first design point.

March 31, April 2, 2004 81


cp: -0.96 -0.83 -0.67 -0.57 -0.35 -0.07 0.04

BASELINE configuration - DP 2

cp: -0.96 -0.83 -0.67 -0.57 -0.35 -0.07 0.04

OPTIMIZED configuration - DP 2

Baseline and optimized cp distribution in the second design point.

March 31, April 2, 2004 82


x/c

c p

0 0.25 0.5 0.75 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Baseline DP 1Optimized DP 1

η = 0.45

x/cc p

0 0.25 0.5 0.75 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


η = 0.45

cp comparison in the first design pointat η = 0.45.

cp comparison in the second designpoint at η = 0.45.

March 31, April 2, 2004 83


η

c lse

ct.

0 0.25 0.5 0.75 10.3

0.35

0.4

0.45

0.5

0.55

0.6


ηc l

sect

.0 0.25 0.5 0.75 1

0.3

0.35

0.4

0.45

0.5

0.55

0.6


Section lift coefficient comparison in the first and second design point.

March 31, April 2, 2004 84


Design and Optimization of a Transonic NLF WingUsing a Multiobjective Genetic Algorithm

Joint work with U. Cella, Piaggio Aero Industries, within the VITAS project

• The competition in the modern market of the civil aviation calls for the designof high performance products able to fly in a range close to M = 0.8.

• In order to reach such high capabilities, it is fundamental to focus on bothfriction and pressure (wave) drag reduction. As regards the first contribute,one of the most challenging solution is to obtain a natural laminar flow over thewing, chord-wise extended as much as possible, in transonic flight conditions. Atthe same time a substantial reduction of the wave drag (due to high free-streamMach number) is desired.

• The optimization procedure of a transonic laminar (tapered and swept) wing,based on a genetic algorithm is illustrated in the following slides: to evaluatethe laminar flow extension over the wing, a 3D boundary layer and stability codewas used, coupled with a 3D Euler flow solver.

March 31, April 2, 2004 85


Design problem setup

• A two-objectives approach was used, in order to separate two different physicalproblems: laminar flow extension and pressure drag.

• The first objective function is related to the pressure drag of the wing and iscomputed using an Euler flow solver.

• The second objective measures the goodness of the laminar flow extension overthe wing, and the transition line is computed using a database stability codecoupled with a 3D boundary layer.

• two design points are considered, corresponding to two different possible cruiseconditions

Id Mach CL Rey area [m2] FL [103ft]1 0.75 0.50 8317617 17.5 452 0.78 0.46 8650165 17.5 45

March 31, April 2, 2004 86


Constraint and shape handling

The geometric constraints of the optimization procedure are:

• Sweep angle, span and taper are constant

• Maximum thickness is constant at each section

Five stream-wise sections are chosen along the span to control the wing geometryand for each section the following design variables are introduced:

• Twist angle.

• Section shape: 10 control points needed to define the airfoili using a B-splinecurve.

March 31, April 2, 2004 87


Genetic algorithm setup

A standard binary-coded, symmetric MOGA was used with the following parameters:

• Population size: 40

• Number of generations: 150

• Crossover probability: 100%

• Mutation probability: 2.5%

Initial solution

The base wing has a quite regular pressure distribution; the transition line location isat about 70%, but it has s very high pressure drag. Strong improvements are neededbecause of the poor CD behavior.

M CL CD

0.75 0.5000 0.0152470.78 0.4600 0.019319

March 31, April 2, 2004 88


Final Pareto front

obj 1: (CD1+CD2)

obj 2:

(tra

nsiti

on)

0.027 0.028 0.029 0.03 0.031

4

8

12

best compromise

March 31, April 2, 2004 89


Best compromise solution:Transition and laminar separation at the two design points

-5

-4

-3

-2

-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10

Design point 001 --- WING 018 --- o1 = 0.028350, o2 = 5.131435, cd1 = 0.013279

transition upperseparation upper

transition lowerseparation lower

-5

-4

-3

-2

-1

0

1

-10 -8 -6 -4 -2 0 2 4 6 8 10

Design point 002 --- WING 018 --- o1 = 0.028350, o2 = 5.131435, cd2 = 0.015071

transition upperseparation upper

transition lowerseparation lower

March 31, April 2, 2004 90


Best compromise solution:Mach distribution at the two design points

X

YZ

Mach_number: 0.08 0.18 0.28 0.38 0.48 0.58 0.68 0.78 0.88 0.98 1.08 1.18

lower surface upper surface

X

YZ

Mach_number: 0.09 0.19 0.29 0.39 0.49 0.59 0.69 0.79 0.89 0.99 1.09 1.19 1.29 1.39

lower surface upper surface

March 31, April 2, 2004 91


Best compromise solution:cp distribution at the two design points

PRESSURE_COEFFICIENT: -1.06 -0.46 0.14 0.74 PRESSURE_COEFFICIENT: -1.32 -0.72 -0.12 0.48

March 31, April 2, 2004 92


Final remarks

• This talk dealt with some advanced evolutionary computing techniques aimed atfacing aerodynamic design problems requiring a huge amount of computationalresources.

• The techiques illustrated range from the use of approximation to hybrid algorithmsbased on gradient based techniques.

• The adjoint based gradient computation technique was also introduced in thehybrid optimization loop.

• All these techniques were illustrated with application examples that required asubstantial amount of computational resources.

• Finally, a problem of current industrial interest, namely the design of a transoniclaminar wing, was introduced and discussed. Some first optimization results wereobtained using a standard multiobjective GA coupled to an Euler solver and to adatabase + boundary layer method for transition estimation. The next progressstep in solving this problem will require the use of higher fidelity solver and hence,the use of the advanced evolutionary techniques here described.

March 31, April 2, 2004 93

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