complete configuration aero-structural optimization...

COMPLETE CONFIGURATION AERO-STRUCTURALOPTIMIZATION USING A COUPLED SENSITIVITY

ANALYSIS METHOD

Joaquim R. R. A. MartinsJuan J. Alonso

James J. Reuther

Department of Aeronautics and AstronauticsStanford University

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 1

Outline

• Introduction

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

• Theory

– Adjoint sensitivity equations– Lagged aero-structural adjoint equations

• Results

– Optimization problem statement– Aero-structural sensitivity validation– Optimization results

• Conclusions


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.

• Large number of design variables andcomplex models inccur a large cost.


Aero-Structural Aircraft Design Optimization

• Aerodynamics and structures are coredisciplines in aircraft design and are verytightly coupled.

• For traditional designs, aerodynamicistsknow the spanload distributions that leadto the true optimum from experience andaccumulated data. What about unusualdesigns?

• Want to simultaneously optimize theaerodynamic shape and structure, sincethere is a trade-off between aerodynamicperformance and structural weight, e.g.,

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.


Methods for Sensitivity Analysis

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

df

dxn≈ f(xn + h)− f(x)

h+O(h)

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

df

dxn≈ Im [f(xn + ih)]

h+O(h2)

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


Objective Function and Governing Equations

Want to minimize scalar objective function,

I = I(xn, yi),

which depends on:

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

• yi: state vector, e.g. flow variables.

Physical system is modeled by a set of governing equations:

Rk (xn, yi (xn)) = 0,

where:

• Same number of state and governing equations, i, k = 1, . . . , NR

• Nx design variables.


Sensitivity Equations

� �

� � ��

Total sensitivity of the objective function:

dI

dxn=

∂I

∂xn+

∂I

∂yi

dyi

dxn.

Total sensitivity of the governing equations:

dRk

dxn=

∂Rk

∂xn+

∂Rk

∂yi

dyi

dxn= 0.


Solving the Sensitivity Equations

Solve the total sensitivity of the governing equations

∂Rk

∂yi

dyi

dxn= −∂Rk

∂xn.

Substitute this result into the total sensitivity equation

dI

dxn=

∂I

∂xn− ∂I

∂yi

− dyi/ dxn︷︸︸︷[∂Rk

∂yi

]−1∂Rk

∂xn,

︸︷︷︸−Ψk

where Ψk is the adjoint vector.


Adjoint Sensitivity Equations

Solve the adjoint equations

∂Rk

∂yiΨk = − ∂I

∂yi.

Adjoint vector is valid for all design variables.

Now the total sensitivity of the the function of interest I is:

dI

dxn=

∂I

∂xn+ Ψk

∂Rk

∂xn

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


Aero-Structural Adjoint Equations

��

� � � � ��

�

Two coupled disciplines: Aerodynamics (Ak) and Structures (Sl).

Rk′ =[ Ak

Sl

], yi′ =

[wi

uj

], Ψk′ =

[ψk

φl

].

Flow variables, wi, five for each grid point.

Structural displacements, uj, three for each structural node.


Aero-Structural Adjoint Equations

∂Ak∂wi

∂Ak∂uj

∂Sl∂wi

∂Sl∂uj

T [ψk

φl

]= −

∂I∂wi∂I∂uj

.

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

• ∂Ak/∂uj: wing deflections cause the mesh to warp, affecting theresiduals.

• ∂Sl/∂wi: since Sl = Kljuj − fl, this is equal to −∂fl/∂wi.

• ∂Sl/∂uj: equal to the stiffness matrix, Klj.

• ∂I/∂wi: for CD, obtained from the integration of pressures; for stresses,its zero.

• ∂I/∂uj: for CD, wing displacement changes the surface boundary overwhich drag is integrated; for stresses, related to σm = Smjuj.


Lagged Aero-Structural Adjoint Equations

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

∂Ak

∂wiψk = − ∂I

∂wi︸︷︷︸Aerodynamic adjoint

−∂Sl

∂wiφ̃l,

∂Sl

∂ujφl = − ∂I

∂uj︸︷︷︸Structural adjoint

−∂Ak

∂ujψ̃k,

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

dxn=

∂I

∂xn+ ψk

∂Ak

∂xn+ φl

∂Sl

∂xn.

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

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

• Sl/∂xn: shape perturbations affect the structural equations, so this termis equal to ∂Klj/∂xnuj − ∂fl/∂xn.


3D Aero-Structural Design Optimization Framework

• Aerodynamics: FLO107-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 10Design variable, n

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

dCD /

dxn

Complex step, fixed displacements

Coupled adjoint, fixed displacements

Complex step

Coupled adjoint


Sensitivity of CD wrt Structural Thickness


-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

dCD /

dxn

Complex step

Coupled adjoint


Structural Stress Constraint Lumping

To perform structural optimization, we need the sensitivities of all thestresses in the finite-element model with respect to many design variables.

There is no method to calculate this matrix of sensitivities efficiently.

Therefore, lump stress constraints

gm = 1− σm

σyield≥ 0,

using the Kreisselmeier–Steinhauser function

KS (gm) = −1ρ

ln

(∑m

e−ρgm

),

where ρ controls how close the function is to the minimum of the stressconstraints.


Sensitivity of KS wrt Shape


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dKS

/ dx

n

Complex, fixed loads

Coupled adjoint, fixed loads

Complex step

Coupled adjoint


Sensitivity of KS wrt Structural Thickness


-10

0

10

20

30

40

50

60

70

80

dKS

/ dx

n

Complex, fixed loads

Coupled adjoint, fixed loads

Complex step

Coupled adjoint


Computational Cost vs. Number of Variables

0 400 800 1200 1600 2000Number of design variables (Nx)

0

400

800

1200

Nor

mal

ized

tim

e

Complex step

2.1

+ 0.

92 ·

Nx

Finite difference

1.0 + 0.38 · N x

Coupled adjoint

3.4 + 0.01 · Nx


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

where CD is that of the cruisecondition.

Subject to:

KS(σm) ≥ 0where KS is taken from a maneuvercondition.

With respect to: external shapeand internal structural sizes.


Baseline Design


Aero-Structural Optimization Convergence History

0 10 20 30 40 50Major iteration number

3000

4000

5000

6000

7000

8000

9000W

eigh

t (lb

s)

Weight

60

65

70

75

80

Dra

g (c

ount

s)

Drag

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

KS

KS


Aero-Structural Optimization Results


Comparison with Sequential Optimization

CD

(counts)σmaxσyield

Weight(lbs)

Objective

All-at-once approachBaseline 73.95 0.87 9, 285Optimized 69.22 0.98 5, 546 87.12Sequential approachAerodynamic optimization

Baseline 74.04Optimized 69.92

Structural optimizationBaseline 0.89 9, 285Optimized 0.98 6, 567

Aero-structural analysis 69.95 0.99 6, 567 91.14


Conclusions

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

• The computation of sensitivities using the aero-structural adjoint isextremely accurate and efficient.

• Demonstrated the usefulness of the coupled adjoint by optimizing asupersonic business jet configuration with external shape and internalstructural variables.


complete configuration aero-structural optimization...

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