AERO-STRUCTURAL WING DESIGN OPTIMIZATION USING ...aero-comlab.stanford.edu/jmartins/doc/ceas2001-slides.pdfHigh-Fidelity Wing Design Optimization--- Cp* ---Optimized Aeroelastic Euler

AERO-STRUCTURAL WING DESIGN OPTIMIZATIONUSING HIGH-FIDELITY SENSITIVITY ANALYSIS

Joaquim R. R. A. MartinsJuan J. AlonsoJames Reuther

Department of Aeronautics and AstronauticsStanford University

CEAS Conference on Multidisciplinary Aircraft Design and Optimization, Koln, Germany, June 2001 1

Outline

• Introduction

– High-fidelity wing design optimization– The need for aero-structural sensitivities– Sensitivity analysis methods

• Theory

– Adjoint sensitivity equations– Lagged aero-structural adjoint equations– Complex-step derivative approximation

• Results

– Aero-structural analysis and sensitivities– Wing optimization

• Conclusions


High-Fidelity Wing Design Optimization

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 -0.133 1.45E+00 -0.37054 -0.00714 180.000 0.40967 -0.00324 -0.19520 0.311810.820 -0.052 1.45E+00 -0.37057 -0.01020 180.000 0.35576 -0.00297 -0.15234 0.27077

• Start from a conceptual design with abaseline geometry.

• Used for preliminary design of the 3Dshape of the wing.

• For transonic configurations shocks arepresent and must be analyzed using CFD.

• High-fidelity analysis needs high-fidelityparameterization, e.g. to smooth shocks.

• Gradient-based optimization is the mostefficient.


Aero-Structural Wing Design Optimization

• Aerodynamics traditionally has used ashape corresponding to the flying shapeof the wing, assuming that shape can bereproduced.

• Wing shape depends on aerodynamicsolution, so need to couple aerodynamicand structural analyses to obtain the truesolution, specially for unusual designs.

• Want to optimize the structure as well,since there is a trade-off betweenaerodynamic performance and structuralweight:

Range ∝ L

Dln

(Wi

Wf

)


The Need for Aero-Structural Sensitivities

OptimizationStructural

OptimizationAerodynamic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Spanwise coordinate, y/b

Lift Aerodynamic optimum

(elliptical distribution)

Aero−structural optimum (fixed weight)

Student Version of MATLAB

Aerodynamic Analysis

Optimizer

Structural Analysis

• Sequential optimization does not lead tothe true optimum.

• Aero-structural optimization requirescoupled sensitivities.

• Add structural element sizes to the designvariables.

• Including structures in the high-fidelitywing optimization will allow largerchanges in the design.


Methods for Sensitivity Analysis

• Finite-Difference: very popular; easy, but lacks robustness andaccuracy; run solver n times.

f ′(x) ≈ f(x + h)− f(x)h

+O(h)

• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver n times.

f ′(x) ≈ Im [f(x + ih)]h

+O(h2)

• Algorithmic/Automatic/Computational Differentiation: accurate;ease of implementation and cost varies.

• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of n.


Objective Function and Governing Equations

Want to minimize scalar objective function,

I = I(x, y),

which depends on:

• x: vector of design variables, e.g. structural plate thickness.

• y: state vector, e.g. structural displacements.

Physical system is modeled by a set of governing equations:

R (x, y (x)) = 0,

where:

• Same number of state and governing equations, nR

• nx design variables.


Variational Equations

R =0

x

Iy

Total variation of the objective function:

δI =∂I

∂xδx +

∂I

∂yδy.

Variation of the governing equations,

δR =∂R

∂xδx +

∂R

∂yδy = 0.


Adjoint Sensitivity Equations

Since the variation of the governing equations must be zero, we can add itto the total variation of the objective,

δI =∂I

∂xδx +

∂I

∂yδy + ψT

(∂R

∂xδx +

∂R

∂yδy

),

where ψ is a vector of arbitrary components known as adjoint variables.

Re-arrange terms,

δI =(

∂I

∂x+ ψT ∂R

∂x

)δx +

(∂I

∂y+ ψT ∂R

∂y

)δy.

If term in blue were zero, term in red would represent the total variation ofthe objective with respect to the design variables.


Adjoint Sensitivity Equations

Since the adjoint variables are arbitrary, we can find a set such that,

[∂R

∂y

]T

︸︷︷︸(nR×nR)

ψ︸︷︷︸(nR×1)

= −∂I

∂y

T

︸︷︷︸(nR×1)

Adjoint valid for all design variables.

Now the total sensitivity of the objective is:

dI

dx︸︷︷︸(1×nR)

=∂I

∂x︸︷︷︸(1×nx)

+ ψT︸︷︷︸

(1×nR)

∂R

∂x︸︷︷︸(nR×nx)

The partial derivatives are inexpensive, since they don’t require the solutionof the governing equations.


Aero-Structural Adjoint Equations

R =0

x

C

wA R =0

uS

D

Two coupled disciplines: Aerodynamics (A) and Structures (S).

R =[

RA

RS

], y =

[wu

]and ψ =

[ψA

ψS

]

Flow variables, w, five for each grid point.

Structural displacements, u, three for each structural node.


Aero-Structural Adjoint Equations

∂RA

∂w

∂RA

∂u

∂RS

∂w

∂RS

∂u

T

[ψA

ψS

]=

∂CD

∂w

∂CD

∂u

• ∂RA/∂w: a change in one of the flow variables affects only the residualsof its cell and the neighboring ones.

• ∂RA/∂u: wing deflections cause the mesh to warp, affecting theresiduals.

• ∂RS/∂w: since RS = Ku− f , this is equal to −∂f/∂w.

• ∂RS/∂u: equal to the stiffness matrix, K.

• ∂CD/∂w: can be obtained from the integration of pressures thatcomputes the total CD.

• ∂CD/∂u: wing displacement changes the surface boundary over whichdrag is integrated.


Lagged Aero-Structural Adjoint Equations

Since the factorization of the complete residual sensitivity matrix isimpractical, decouple the system and lag the adjoint variables,

[∂RA

∂w

]T

ψA =∂CD

∂w︸︷︷︸Aerodynamic adjoint equations

−[∂RS

∂w

]T

ψS

[∂RS

∂u

]T

ψS =∂CD

∂u︸︷︷︸Structural adjoint equations

−[∂RA

∂u

]T

ψA.

Lagged adjoint equations are the single discipline ones with an added forcingterm that takes the coupling into account.

System is solved iteratively, much like the aero-structural analysis.


Total Sensitivity

The aero-structural sensitivities of the drag coefficient with respect to wingshape perturbations are,

dCD

dx=

∂CD

∂x− ψT

A

∂RA

∂x− ψT

S

∂RS

∂x.

• ∂CD/∂x: a change in the geometry will change the boundary over whichthe pressures are integrated.

• ∂RA/∂x: the shape perturbations affect the grid, which in turn changesthe residuals.

• ∂RS/∂x: shape perturbations only affect the stiffness matrix, so this is∂K/∂x · u.


Complex-Step Derivative Approximation

Complex variable defined as z = x + iy, where x, y are real, i =√−1.

Complex function defined as f(z) = u(z) + iv(z), assumed to be analytic.

Can also be derived from a Taylor series expansion about x with a complexstep ih:

f(x + ih) = f(x) + ihf ′(x)− h2f′′(x)2!

− ih3f′′′(x)3!

+ . . .

⇒ f ′(x) =Im [f(x + ih)]

h+ h2f

′′′(x)3!

+ . . .

⇒ f ′(x) ≈ Im [f(x + ih)]h

No subtraction! Second order approximation.


Simple Numerical Example

��

�� e

"!$#&%('*),+.-0/213)$%465$798;:$7=<(>@?BADCEC=F$7GF�H,IJFKBL$MONQP=R$SUT@VBWDXEX=L$P=L$M,YJL

Estimate derivative atx = 1.5 of the function,

f(x) =ex

√sin3x + cos3x

Relative error defined as:

ε =

∣∣∣f ′ − f ′ref

∣∣∣∣∣∣f ′ref

∣∣∣


3D Aero-Structural Design Optimization Framework

• Aerodynamics: SYN107-MB, aparallel, multiblock Navier-Stokesflow solver.

• Structures: detailed finite elementmodel with plates and trusses.

• Coupling: high-fidelity, consistentand conservative.

• Geometry: centralized databasefor exchanges (jig shape, pressuredistributions, displacements.)

• Coupled-adjoint sensitivity analysis


Aero-Structural Analysis Results

SYMBOL

SOURCE AERO-STRUCTURAL

RIGID

ALPHA 0.526

-0.039

CD 0.01186

0.01119

W25 WING ANALYSYSMACH = 0.820 , CL = 0.352

COMPPLOTJCV 1.13

�COMPPLOT

JCV 1.13�

COMPPLOTJCV 1.13

�COMPPLOT

JCV 1.13�

COMPPLOTJCV 1.13

�COMPPLOT

JCV 1.13�

COMPPLOTJCV 1.13

�COMPPLOT

JCV 1.13�

COMPPLOTJCV 1.13

�COMPPLOT

JCV 1.13�

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 0.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 14.9% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 30.9% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

p

X / C 46.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 62.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.5% Span


Structural Deflections

260 280 300 320 340 360 0

50

100

150

200

60

80

z (in)

x (in)

y (in

)

AeroelasticRigid


345 350 355 360 365

65

70

75

80

85

x (in)

z (in

)

AeroelasticRigid



Sensitivity Results: Spanwise Bumps

1 2 3 4 5 6 7 8 9

0

0.02

0.04

0.06

0.08

0.1Coupled adjoint Coupled complex step Coupled finite difference Aerodynamic adjoint Aerodynamic complex step Aerodynamic finite difference

10 11 12 13 14 15 16 17 18−0.01

−0.005

0

0.005

0.01

d C

D /

d x j

Shape variable, xj

Upper Surface

Lower Surface



Sensitivity Results: Chordwise Bumps

1 2 3 4 5 6 7 8 9−0.05

0

0.05

0.1

0.15 Coupled adjoint Coupled complex step Coupled finite difference Aerodynamic adjoint Aerodynamic complex step Aerodynamic finite difference

10 11 12 13 14 15 16 17 18−0.02

0

0.02

0.04

d C

D /

d x j

Shape variable, xj

Lower Surface

Upper Surface



Computational Cost Comparison

Aero-structural Solution 1.0Finite difference 14.2Complex step 34.4Coupled adjoint 7.5

Aerodynamic Solution 0.8Finite difference 13.3Complex step 32.1Adjoint 3.3

• Aero-structural solution takes 25% longer.

• Finite-difference and complex step dependent on the number of designvariables.

• Complex-step calculation is about 2.5 times slower, but no step sizeguessing.

• Adjoint method is much more efficient. This would be even more obviousfor more design variables.


Aerodynamic Optimization Results

SYMBOL

SOURCE OPTIMIZED

BASELINE

ALPHA 1.837

-0.039

CD 0.00814

0.01119

AERODYNAMIC W25 WING OPTIMIZATIONMACH = 0.820 , CL = 0.352



0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 0.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 14.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 30.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

p

X / C 46.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 62.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.7% Span


Aero-Structural Optimization Results

SYMBOL

SOURCE OPTIMIZED

BASELINE

ALPHA 2.398

0.526

CD 0.00810

0.01186

AERO-STRUCTURAL W25 WING OPTIMIZATION MACH = 0.820 , CL = 0.352

COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�COMPPLOTJCV 1.13

�

MCDONNELL DOUGLAS�



0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 0.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 14.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 30.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

p

X / C 46.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 62.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.6% Span


Conclusions

• New method for high-fidelity aero-structural sensitivity analysis.

• Lagged-coupled adjoint was used to compute sensitivities of the dragcoefficient with respect to wing shape perturbations.

• The sensitivities given by the aero-structural adjoint were accurate whencompared to the reference values.

• The coupled-adjoint method was shown to be very efficient, makingoptimization with respect to hundreds design variables viable.


Future Work

• Add structural sizes to the set of design variables.

• Compute sensitivities of the structural stresses to add stress constraints.

• Optimized initial cruise weight for fixed range to get correct aero-structural trade-off.

• Implement multiple load cases.

• Use this design framework for unusual configurations such as a supersonicbusiness jet.


Documents

AERO-STRUCTURAL WING DESIGN OPTIMIZATION USING ...aero-comlab.stanford.edu/jmartins/doc/ceas2001-slides.pdfHigh-Fidelity Wing Design Optimization--- Cp* ---Optimized Aeroelastic Euler