AEROSPACE COMPUTATIONAL DESIGN LABORATORY 1 Gradient-based Multifidelity Optimisation for Aircraft Design using Bayesian Model Calibration 2 nd Aircraft Structural Design Conference Royal Aeronautical Society October 28, 2010 Andrew March, Karen Willcox, & Qiqi Wang
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AEROSPACE COMPUTATIONAL DESIGN LABORATORY11
Gradient-based MultifidelityOptimisation for Aircraft Design using Bayesian Model Calibration
2nd Aircraft Structural Design ConferenceRoyal Aeronautical Society
– 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-
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.