HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF …aero-comlab.stanford.edu/jmartins/doc/denver2002-slides.pdf · HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF A SUPERSONIC

HIGH-FIDELITY AERO-STRUCTURAL DESIGNOPTIMIZATION OF A SUPERSONIC BUSINESS JET

Joaquim R. R. A. MartinsJuan J. Alonso

James J. Reuther

Department of Aeronautics and AstronauticsStanford University

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 1

Outline

• Introduction

– High-fidelity aircraft design optimization– The need for aero-structural sensitivities– Sensitivity analysis methods– Optimization problem statement

• Theory

– Adjoint sensitivity equations– Lagged aero-structural adjoint equations

• Results

– Aero-structural sensitivity validation– Optimization results

• Conclusions and Future Work


High-Fidelity Aircraft Design Optimization

• Start from a baseline geometry providedby a conceptual design tool.

• Required for transonic configurationswhere shocks are present.

• Necessary for supersonic, complexgeometry design.

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

• Gradient-based optimization is the mostefficient and requires accurate sensitivityinformation.


Aero-Structural Aircraft 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 thesolution, 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 (maximum range)

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.


Supersonic Business Jet Optimization Problem

Natural laminar flowsupersonic business jet

Mach = 1.5, Range = 5,300nm1 count of drag = 310 lbs of weight

minimize I = αCD + βW

subject to CL = CLT

KS ≥ 0

xS ≥ xSmin

Lump stress constraints

gi = 1− σiV M

σyield≥ 0,

using the Kreisselmeier-Steinhauserfunction

KS (gi(x)) = −1ρ

ln

[∑

i

e−ρgi(x)

].


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)

• (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

I

wA R =0

uS

Two coupled disciplines: Aerodynamics (RA) and Structures (RS).

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

]= −

∂I

∂w

∂I

∂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.

• ∂I/∂w: for CD, obtained from the integration of pressures; for KS, itszero.

• ∂I/∂u: for CD wing displacement changes the surface boundary overwhich drag is integrated; for KS, related to σ = Su.


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 = − ∂I

∂w︸︷︷︸Aerodynamic adjoint equations

−[∂RS

∂w

]T

ψ̃S

[∂RS

∂u

]T

ψS = −∂I

∂u︸︷︷︸Structural adjoint equations

−[∂RA

∂u

]T

ψ̃A.

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,

dI

dx=

∂I

∂x+ ψT

A

∂RA

∂x+ ψT

S

∂RS

∂x.

• ∂I/∂x: CD changes when the boundary over which the pressures areintegrated is perturbed; stresses change when nodes are moved.

• ∂RA/∂x: the shape perturbations affect the grid, which in turn changesthe residuals; structural variables have no effect on this.

• ∂RS/∂x: shape perturbations affect the structural equations, so this is∂K/∂x · u− ∂f/∂x.


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 sensitivityanalysis: aerodynamic andstructural design variables.


Sensitivity of CD wrt Shape

1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0

2

4

6

8

10x 10

−3

Shape variable, xj

d C

D /

d x A

Coupled adjoint Complex step Avg. rel. error = 3.5%



Sensitivity of CD wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08d

CD /

d x S

Structural variable, xj




Sensitivity of KS wrt Shape

1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Shape variable, xj

d C

D /

d x A




Sensitivity of KS wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20−10

0

10

20

30

40

50

60

70

80

d C

D /

d x S

Structural variable, xj




Rigid Wing Aerodynamic Optimization Results

SYMBOL

SOURCE Rigid Baseline

Rigid, constrained, aero only design

ALPHA 3.440

4.113

CD 0.00743

0.00700

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AERODYNAMIC DESIGN

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25C

p

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span


Aeroelastic Wing Aerodynamic Optimization Results

SYMBOL

SOURCE Baseline Aero-elastic

Aero-elastic, Constrained, aero only design

ALPHA 3.330

4.095

CD 0.00741

0.00701

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AEROELASTIC DESIGN - OML ONLY

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS



0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25C

p

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span


Simplified Optimization Problem

minimize I = αCD + βW + γ max(0,−KS)2

subject to CL = CLT

xS ≥ xSmin

α = 104, β = 3.226× 10−3, γ = 103


Aero-Structural Optimization Convergence History

2 4 6 8 10 12

80

100

120

140

160

180

200

220

0 2 4 6 8 10 12

70

70.5

71

71.5

72

72.5

73

73.5

74

74.5

75

2 4 6 8 10 123000

3200

3400

3600

3800

4000

4200

4400

4600

Objective function

Weight (lbs)

Drag counts

Iteration number


Aero-Structural Optimization Results

SYMBOL

SOURCE Aero-elastic Baseline

Aero-elastic, Aero-structural design

ALPHA 3.330

3.689

CD 0.00741

0.00685

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF FULL AERO-STRUCTURAL DESIGN

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS



0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25C

p

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span


Aero-Structural Optimization Results


Conclusions

• Presented a framework for high-fidelity aero-structural design.

• The sensitivities given by the aero-structural adjoint were extremelyaccurate when compared to the reference results.

• The coupled-adjoint method was shown to be extremely efficient, makingviable design optimization with respect to hundreds of design variables.

• Demonstrated the usefulness of the coupled-adjoint by optimizing asupersonic business jet configuration with aerodynamic and structuralvariables.


Future Work

• Implement multiple load cases, each one with a KS adjoint.

• Use the flow solver in Navier-Stokes mode with a full-configuration mesh.

• Use a better structural model and a more established structural solver.

• Explore the design space for the supersonic business jet configurationand provide useful results to industry.


Documents

HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF …aero-comlab.stanford.edu/jmartins/doc/denver2002-slides.pdf · HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF A SUPERSONIC