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Gradient-based MultifidelityOptimisation for Aircraft Design using Bayesian Model Calibration

2nd Aircraft Structural Design ConferenceRoyal Aeronautical Society

October 28, 2010

Andrew March, Karen Willcox, & Qiqi Wang

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Multifidelity Surrogates• Definition: High-Fidelity

– The best model of reality that is available and affordable, the analysis that is used to validate the design.

• Definition: Low(er)-Fidelity– A method with unknown accuracy that estimates metrics of interest

but requires lesser resources than the high-fidelity analysis.

Coarsened Mesh

Hierarchical Models

Reduced Physics

x

f(x)

x1

f(x1)

x1x2

f(x)

Regression ModelReduced Order Model

Approximation Models

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Main Messages• Bayesian model calibration enables

synthesizing multifidelity sources of information to speed optimising designs with costly analyses.– Enables multiple low-fidelity models– Can benefit from gradient information when available– Can prove convergence to a high-fidelity optimum

• Adjoint solutions are common in engineering software and provide an inexpensive way of computing gradients.

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Adjoint-based Gradient

• Optimisation problem:

• Implicit function theorem and adjoint:

• Everything expensive is in the objective function.

0uxrux

x=

ℑℜ∈

),(..),(min

high

high

tsn

))(,(min xuxx highf

nℜ∈ xr

xx ∂∂

−∂

∂ℑ= highThighhigh

ddf

ψ

Computational Model

xx

ddfhigh )(

)(xhighf

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Trust-Region DemonstrationDemonstration

• Approximate high-fidelity function:

• Optimise the surrogate model:

• Evaluate Performance:

• Update trust region)()(

)()(

kkkkk

kkhighkhighk mm

ffsxx

sxx+−+−

=ρ

kk

kkk

ts

mn

k

∆≤

+

∞

ℜ∈

s

sxs

..

)(min

)()( xx highk fm ≈

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Bayesian Model Calibration• Define a surrogate model of the

high-fidelity function:

• Cokriging error model, ek(x):– Interpolates fhigh(x)-flow(x) and

∇fhigh(x)-∇flow(x) exactly at all calibration points

• Trust-region algorithms are globally convergent to a stationary point of fhigh(x) if:(i) fhigh(xk)=mk(xk)(ii) ∇fhigh(xk)=∇mk(xk)(iii) ||∇2mk(x)||2≤κbhm– Cokriging models can satisfy

these requirements by construction.

)()()()( xxxx highklowk fefm ≈+≡

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Combining Multiple Lower-Fidelities

• Calibrate all lower-fidelity models to the high-fidelity function using radial basis function error model

• Use a maximum likelihood estimator to predict the high-fidelity function value (Essentially a Kalman Filter)

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Structural Design Problem• Minimize deflection of a 2-

Dimensional hook

• Linear Finite-Elements

• 26 Design variables– 25 shape parameters– Thickness

• 32 Constraints– 31 Geometric constraints– Maximum weight constraint

0)(0)(

)(..)(min

31

max

26

≤≤−

=

=ℑℜ∈

xx

fuxux

x

gww

KtsC

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Multifidelity Structural Models

• Low-fidelity model is coarsely discretized– High-fidelity model: 1770 dof– Low-fidelity model: 276 dof

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Adjoint-Based Derivative• Problem:

• Adjoint:

• Derivative:

– Satisfies constraint

• Derivative calculation:– Two forward solves or

one K-1 calculation

• Finite differences– Requires n+1 forward

solves

• Adjoint-based gradient nearly independent of the number of design variables.

0)(0)(

)(..)(min

max

≤≤−

=

=ℑℜ∈

xx

fuxux

x

gww

KtsC

n

TT CK )()( xx =ψ

uxx

xx

x⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂=

)()( KCd

df Thigh ψ

fux =)(K

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Adjoint Gradient Verification

• Compare the adjoint-based derivative:

• With finite difference derivative:

xxxx

x δδ )()( highhighhigh ff

ddf −+

≈

1.54%

0.00574

10-5

6.57%1.73%1.51%Relative Error

0.02450.006450.00562Max Error

10-810-710-6Finite-difference step, δx

uxx

xx

x⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂=

)()( KCd

df Thigh ψ

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Optimisation Results

• 31.6% Error between high- and low-fidelity deflection estimate

10.212.7171Standard deviation27.5 (-88%)40 (-83%)232 (-)Average high-fidelity function calls

Calibration ApproachFirst-Order TRSQP

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Total Effort

• Nastran solution effort– Conjugate gradient, incomplete Choleksy factorization

preconditioner– Solution: O(dof2)

• Multifidelity effort:– 30 high-fidelity evaluations– 3,200 low-fidelity evaluations, ~80 high-fidelity

evaluations– Total effort: ~110 High-fidelity evaluations

• SQP effort: 232– Multifidelity savings ~50%

(ignoring surrogate model construction)

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Supersonic Airfoil Design

• Minimize drag of a supersonic airfoil– 11 Design variables

• Angle of attack• 5 upper-surface spline points• 5 lower-surface spline points

– M∞=1.5– Constraints

• t/cmin≥0• t/cmax≥5%

%5)(/0)(/

0),(..),(min

11

≤−≤−

=

=ℑℜ∈

xx

uxrux

x

ctct

tsDrag

Euler SolutionPanel Method

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Adjoint Gradient Computation

• Using a mesh generator with a known derivative a gradient can be computed with:

– 1 flow solve– 1 adjoint solve– 1 flow iteration/design variable

(1/500 flow solve)– Total: 2+n/500 solutions

• Finite difference gradient requires n+1 flow solutions

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂ℑ

+∂∂ℑ

=x

rxr

xxx ddM

MddM

Mddf Thigh ψ

Gradient of objective

function with respect to design

variables

Change in objective due

to x at constant pressure

Change in objective

function due to mesh change

Adjointsolution to the Euler equations

Change in residual due to mesh change

Change in residual due to x

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Adjoint Gradient Verification

• Simple diamond airfoil with 3 parameters.

• Compare with finite difference:

0.53%0.53%0.56%0.65%1.6%Relative Error

0.02070.02050.02180.02520.0617Max Error

10-810-710-610-510-4Finite-difference step, δx

;⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂ℑ

+∂∂ℑ

=x

rxr

xxx ddM

MddM

Mddf Thigh ψ

xxxx

x δδ )()( highhighhigh ff

ddf −+

≈

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Optimisation Results

• Multifidelity Results:

• Single-fidelity Results:

4.192.8614.0Standard deviation14.7 (-82%)12.9 (-84%)81.5 (-)Average high-fidelity function calls

Calibration ApproachFirst-Order TRSQPflow(x)=panel method

21.861.214.0Standard deviation64.2 (-21%)91.1 (+12%)81.5 (-)Average high-fidelity function calls

Calibration ApproachFirst-Order TRSQPflow(x)=0

Momentum AdjointPressure

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Conclusions

• Shown that adjoint methods can provide inexpensive gradient estimates for aerodynamic and structural design problems.

• Presented a gradient-based multifidelityoptimisation method using Bayesian model calibration.– Proven convergence– Shown comparable or better performance than other

multifidelity methods– Recommended the method for problems with few

design variables and “poor” low-fidelity models

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Acknowledgements

• The authors gratefully acknowledge support from NASA Langley Research Center contract NNL07AA33C technical monitor Natalia Alexandrov.

• A National Science Foundation graduate research fellowship.

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Questions?