Next Generation of Algorithms for Aerodynamic Design ...arrow.utias.utoronto.ca/ncss13/NCSA Presentations/Nadarajah... · Cross Validation (CV) More accurate than MSE Computationally

Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms

Error-Norm Oriented Adaptation for High-Order MethodsConclusion

Next Generation of Algorithms for Aerodynamic DesignOptimization: Current Status and Future Challengers

Siva Nadarajah

Department of Mechanical EngineeringMcGill University

Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers



Table of contents

1. Current Status of Aerodynamic Design OptimizationAerodynamic Design Optimization using Drag DecompositionAerodynamic Shape Optimization Through Drag DecompositionSensitivity-Based Sequential Sampling for Surrogate ModelsDesign for Predominantly Laminar Flow WingsConstrained Optimization of Multistage TurbomachineryDesign Optimization to Alleviate Gust Response and Flutter

2. Next Generation of Algorithms

3. Error-Norm Oriented Adaptation for High-Order MethodsCurrent Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil

4. Conclusion




Aerodynamic Shape Optimization Through Drag DecompositionSensitivity-Based Sequential Sampling for Surrogate ModelsDesign for Predominantly Laminar Flow WingsConstrained Optimization of Multistage TurbomachineryDesign Optimization to Alleviate Gust Response and Flutter

Adjoint-Based Aerodynamic Shape Optimization Using the DragDecomposition Method (François Bisson)

Key Features

- Phenological breakdown of drag

- Reduce mesh dependancy

- Understand the sensitivity of the design variable for eachdrag component for the full flight envelope

- Potential application to non-planar wings design

DPW-W1 at M = 0.76 and CL = 0.500 for fine grid (inviscid)

Mid-Field Drag Decomposition

Simplified control volume for the mid-field method

(Courtesy [Kusunose et al.,2002])

D =1

γM2∞

∫∫SWake

(∆s

R

)ρu · n dS

+ρ∞

2

∫∫SWake

(ψζ)dS +O(∆2)

• 1st term related to entropy generation processes (shockwaves and viscous/artificial dissipations)

• 2nd term is the Maskell’s induced dragζ - x-vorticity & ψ - stream function (∇2ψ = −ζ)





Induced Drag Minimization - DPW-W1 Wing (François Bisson)

X

CP

0.4 0.45 0.5 0.55

-0.5

0

0.5

Initial DesignFinal Design

94.0% Semi-Span

X

CP

0.15 0.2 0.25 0.3 0.35 0.4 0.45

-1

-0.5

0

0.5

34.3% Semi-Span

X

CP

0.25 0.3 0.35 0.4 0.45 0.5

-1

-0.5

0

0.5

63.6% Semi-Span

CP

0.60.50.40.30.20.10

-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9-1-1.1

Initial Design Final Design

Pressure distribution for DPW-W1 wing induced drag minimization

Fully subsonic (M = 0.60)Lift Constrained (CL = 0.40)Local surface parametrization

→ Camber optimized to approach ellipticalloading

→ 3.43% Induced Drag Reduction

2(z/b)C

l/CL

0 0.2 0.4 0.6 0.8 1

0.4

0.6

0.8

1

1.2

EllipticalInitial DesignFinal Design

DPW-W1 Span Loading





Sensitivity-Based Sequential Sampling for Surrogate Models (ArthurPaul-Dubois-Taine)

Motivation

- Aircraft design involves a large number of parameters.

- High fidelity CFD results remain expensive at a conceptual designstage.

- Surrogate models→ use existing flow solutions to approximatecontinuous response surface

0.70.75

0.8

−5

0

50

0.05

0.1

0.15

Mach Number (M)

Drag Response Surface for DLR−F6

Angle of Attack (α)

Dra

g C

oeffi

cien

t (C

D)

Question: what criteria do we use to decide on newsnapshot locations?

Existing error criteria:

• Mean Square Error (MSE) estimate built inKriging

• Cross Validation (CV)• More accurate than MSE• Computationally expensive• ⇒ Objective: find low cost alternative

to CV

New approach: Sensivity analysis (S)• Uses the mathematical form of Kriging model• Incorporates gradient information in the

sampling process





Sensitivity-Based Sequential Sampling for Surrogate Models (ArthurPaul-Dubois-Taine)Pressure distribution for DPW-W1 at various camber, thickness, and α

X

Y

Z

2.3

2.2

2.1

21.9

1.8

1.7

1.61.5

1.4

1.3

1.2

1.11

0.9

0.8

0.7

0.60.5

0.4

0.3

0.2

3D NACA 0048 Wing, α=0 Deg -- Euler Solution (256x64x64) Pressure Distribution

X

Y

Z

1.25

1.2

1.15

1.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55


X

Y

Z1.25

1.2

1.151.1

1.05

1

0.95

0.9

0.85

0.8

0.75

0.70.65

0.6

0.55

0.5

0.45

0.4


→ Three parameters: Camber, Thickness ratioand Angle of Attack α

→ M∞ = .81, Camber Location = 40%.

Camber [in %]

Ang

le o

f Atta

ck (

α)

Drag Coefficient [Drag Counts] on 3D Wing: RS at t/c=11%

0 2 4 6 80

1

2

3

4

500

1000

1500

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

30

35Drag on 3D Wing: Cokriging RS Convergence vs. CPU Time

CPU Time [s]

L 1 N

orm

of t

he E

rror

[in

Dra

g C

ount

s]

MSE Sequential SamplingSensitivity Sequential SamplingCV Sequential Sampling


Current Status of Aerodynamic Design OptimizationError-Norm Oriented Adaptation for High-Order Methods

Conclusion


Aerodynamic Optimization of Natural Laminar Flow (NLF) Airfoils

NLF(1)-0416: Minimizing total drag at constant lift, Ma=0.1, Re =2.0 million

X

Y

0.26 0.28 0.3 0.32 0.34 0.36 0.38

0.102

0.104

0.106

0.108

0.11

0.1120-0.0001-0.0002-0.0003-0.0004-0.0005-0.0006-0.0007-0.0008-0.0009-0.001

X

Y

0.35 0.352 0.354 0.356 0.358 0.36 0.362 0.364

0.1035

0.104

0.1045

Adjoint of Intermittency Factor

X/C

Y/C

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

0.3

0.4 NLF airfoilOptimized airfoil_ClT=1.5Optimized airfoil_ClT=1.8

Shape modifications

X/C

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1.5

2

2.5

NLF airfoilOptimized airfoil_ClT=1.5Optimized airfoil_ClT=1.8

Pressure distribution

• Improvements to γ−Reθ model:• Khayatzadeh, P and Nadarajah, S, ”Laminar-Turbulent Flow Simulation for Wind

Turbine Profiles Using the γ−Reθ Transition Model”, Wind Energy (2013), (In-press)

• Developed Adjoint counterpart for γ−Reθ model.• Khayatzadeh, P and Nadarajah, S, ”Aerodynamic Shape Optimization of Natural

Laminar Flow (NLF) Airfoils”, 50th AIAA Aerospace Sciences Meeting (2011)

Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current




Constrained Aero Optimization for Multistage Turbomachinery(Benjamin Walther)

2.5-stage transonic compressor. Total pressure ratio: π = 3.0

Mrel, baseline design First adjoint variable ψ1− contours

Walther, B and Nadarajah, S, Constrained Adjoint-Based Aerodynamic Shape Optimization of aTransonic Compressor Stage, ASME Journal of Turbomachinery April, 2013.

Walther, B and Nadarajah, S, Constrained Adjoint-Based Aerodynamic Shape Optimization in aMultistage Turbomachinery Environment, AIAA ASM 2012 January, 2012.

Contributions

1. Effect of constraint violation on the performance of the compressor stage.

2. High-lift airfoil design → reduced number of blades/stages3. Unsteady multistage optimization → rotor-stator interactions using Non-Linear Frequency

Domain schemes.





Design Optimization to Alleviate Gust Response and Flutter(Ali Mosahebi)

Fast Implicit Adaptive Time Spectral Schemes[1

∆t∗+∂f

∂x

](xn+1,s+1,g+1 − xn+1,s+1,g) =

−{xn+1,s+1,g − xn

∆t∗+ f

n+1,s+∂f

∂X· (Xn+1,s+1,g −Xn+1,s)

}

Mode distribtuion pattern

-1 0 1 2 3 4 5 6 7 8-2

-1

0

1

2

8

7

6

5

4

3

2

1

Kachra, F and Nadarajah, S, Aeroelastic Solutions Using the Non-Linear Frequency DomainMethod, AIAA Journal, 46(9), September 2008. Mosahebi, A and Nadarajah, S, ”An Adaptive

NonLinear Frequency Domain Method for Viscous Flows”, Computers and Fluids, Elsevier (InPress)


airfoil3.aviMedia File (video/avi)



Next-Generation of Algorithms for Aerodynamic Design Optimization

Future

• Potential increase in computing power. Greater parallelism.• High-order methods.• Novel approaches to explore the design space. (Multimodality)• Evaluate sensitivity of design variables on aerodynamic performance for off-design

certification cases and include it within the design loop.

• Higher geometric detail during the optimization.• Quantify uncertainty.• Evaluating the Hessian.




Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil

Current Status of Goal-Oriented Mesh Adaptation (J-S. Cagnone)

Adaptation based on Lift Adaptation based on Drag

Current State-of-the-Art

• Wide variety of phenomenon / scale lengths• Reliable CFD requires a-priori knowledge of the flow• Meshing best practices may be insufficient in some cases...






• Adjoint provides sensitivity of scalar quantity of interest• Formally J (u) =

∫Γ p(u)ds

• In practice, interested in Cl, Cd or Cm

• Can be used to guide mesh refinement• Theory well understood

• FEM: Becker & Rannacher (1996,1998)• FV: Venditti & Darmofal (2002,2003)• DG: Houston & Hartmann (2001,2002)

Fidkowski & Darmofal (2007)

Some limitations

• What if there is no obvious output?• What if interested in actual 3D flow field?• What if interested in a pressure distribution?







• Current approaches ask, ”What is the error in the integrated functional?”

• Perhaps we should ask, ”What is the integral of the error on the surface orvolume?”







• Current approaches ask, ”What is the error in the integrated functional?”• Perhaps we should ask, ”What is the integral of the error on the surface orvolume?”






Adaptation based on Lift (left) and Drag (right)

Our Contribution

• Developed a new p-adaptive differential-form of the DG Scheme (Based on HTHuynh and ZJ Wang’s CPR formulation).

1. Cagnone, JS and Nadarajah, S, A Stable Interface Element Scheme for the p-AdaptiveLifting Collocation Penalty Formulation, Journal of Computational Physics, 231(4),February, 2012.

2. Cagnone, JS, Vermiere, B, and Nadarajah, S, A p-adaptive LCP formulation for the

compressible Navier-Stokes equations, Journal of Computational Physics (2012), doi:

http://dx.doi.org/10.1016/ j.jcp.2012.08.053.

• A new error-norm oriented adjoint-based mesh adaptation.1. Cagnone, JS and Nadarajah, S, An error-norm oriented adaptation procedure for the

discontinuous Galerkin method





Error-norm adjoint problem (J-S. Cagnone)

Primal flow problem∇ · F(u) = r(u) = 0 in Ω

Cost-function

J (u) = 12

∫Ω

(u− ũ)2 dΩ

Incorporate PDE constraint into Lagrangian

L(u, ψ) = 12

∫Ω

(u− ũ)2 dΩ +∫

Ω

ψT r(u) dΩ

Enforce stationary w.r.t. δu

δL =∫

Ω

(u− ũ) δu dΩ +∫

Ω

ψT r′δu dΩ = 0






Replace r(u) = ∇ · F(u)

δL =∫

Ω

(u− ũ) δu dΩ−∫

Ω

∇ψT · F ′δu dΩ +∫∂Ω

ψTn · F ′δu ds = 0

Achieved by choosing ψ s.t.

{(F ′)T · ∇ψ = u− ũ&in Ω(F ′ · n)Tψ = 0&on ∂Ω

Summary

• Adjoint field equation• Adjoint boundary condition• Connection with L2 error-norm ũ← uh






Taylor expansion of r(u)

r(ũ) ≈��r(u)− r′(u− ũ) ≈ −r′δu.

Thus we find∫Ω

ψT r(ũ) dΩ ≈ −∫

Ω

ψT r′[u]δu dΩ (from Taylor exp.)

=

∫Ω

(u− ũ)δu dΩ (from stationarity)

=

∫Ω

(u− ũ)2 dΩ

= ‖u− ũ‖2Ω

Summary

• Inner product is an error-norm estimate• Adjoint variables quantify sensitivity to residual perturbations• Element-wise indicator ηk ≡

∫ΩkψTk r(uh,k) dΩ





Full algorithm (J-S. Cagnone)

1 Solve flow

rh(uh) = 0

2 Solve linear error eqns

r′(u− uh) = −r(uh)

3 Solve adjoint

{(F ′)T · ∇ψ = u− uh in Ω(F ′ · n)Tψ = 0 on ∂Ω

4 Evaluate error indicator ηk + Adapt mesh

In practice, r(u) is approximated by a p-refined discretization





Steady 1D linear advection (J-S. Cagnone)

∂u

∂x= f(x) x ∈ [−1; 1]

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

u

Exact solution

Numerical solution

DG-P4 solution

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

|Num

erical solu

tion e

rror|

Error profile

Summary

• Error magnitude ! = Accurate indicatorSiva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers




Adjoint results (J-S. Cagnone)

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

ψ

Adjoint solution

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

Err

or in

dica

tor

Refinement indicator

Summary

• Adjoint identifies prime contribution to ‖u− uh‖Ω• Adequate refinement indicator


M∞ = 2 Prandtl-Meyer expansion fan, β = 15o

(a) Mach number

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

−0.2

0

0.2

0.4

0.6

0.8

1

Pressure

Y

Exact solutionNumerical solution

(b) Outflow profile

Compare goal/norm-oriented adjoints

• J1 =∫

Γ+p(u)ds

• J2 =∫

Γ+(u− uh)

2ds

Adaptively refined meshes

X

Y

-0.5 0 0.5 1

-0.2

0

0.2

0.4

0.6

0.8

1

(c) Pressure integral adjoint

XY

-0.5 0 0.5 1

-0.2

0

0.2

0.4

0.6

0.8

1

(d) Outflow error-norm adjoint

First adjoint component

(e) Pressure integral adjoint (f) Outflow error-norm adjoint

103

104

105

10−9

10−8

10−7

10−6

10−5

10−4

10−3

DOF

Inte

grat

ed p

ress

ure

erro

r

Uniform refinementIntegrated pressure adjointSurface norm adjoint

(g) Pressure integral error

103

104

105

10−3

10−2

10−1

DOF

Sur

face

err

or−

norm

Uniform refinementIntegrated pressure adjointSurface norm adjoint

(h) Solution error

Conclusion

• Each adjoint is optimal for its respective cost function

• Error-norm adjoint is useful to minimize actual solution error

M∞ = 1.5 Supersonic diamond airfoil

X

Y

-2 0 2 4 6 8 10

-2

0

2

4

6

8

Compare goal/norm-oriented adjoints

• J1 =∫

Sp(u)ds

• J2 =∫

S(u− uh)

2dΩ

(c) Pressure integral adjoint (d) Surface norm adjoint




104

105

10−2

10−1

L2−

err

or

norm

DOF

Surface norm error

Summary

• Each adjoint is optimal for its respective cost function• Norm-oriented predicts more accurate pressure signature




Contributions

• Novel norm-oriented adjoint method• Verification on supersonic flow problems

Conclusion

• Adjoint enables identification of error sources• Correctly accounts for physics & error transport• Useful of accurate signal capture

Future work

• Algorithmic cost reductions• hp-Adaptation• Viscous problems




Acknowledgements

• Graduate Students:• Benjamin Walther (PhD)• Brian Vermeire (PhD)• Peyman Khayatzadeh (PhD)• Jean-Sebastien Cagnone (PhD)• Mostafa Najafiyazdi (PhD)

• François Bisson (Masters)• Jeremy Schembri (Masters)• Arthur Paul-Dubois-Taine (Honors)• Dylan Jude (Honors)

• Funding Sources: NSERC Discovery, NSERC Strategic, FQRNT Team Grants,NSERC Collaborative and Development, NSERC Engage, Bombardier Aerospace,Pratt&Whitney, Hydro Quebec.


Current Status of Aerodynamic Design OptimizationAerodynamic Design Optimization using Drag DecompositionAerodynamic Shape Optimization Through Drag DecompositionSensitivity-Based Sequential Sampling for Surrogate ModelsDesign for Predominantly Laminar Flow WingsConstrained Optimization of Multistage TurbomachineryDesign Optimization to Alleviate Gust Response and Flutter

Next Generation of AlgorithmsError-Norm Oriented Adaptation for High-Order MethodsCurrent Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil

Conclusion

Documents

Next Generation of Algorithms for Aerodynamic Design ...arrow.utias.utoronto.ca/ncss13/NCSA Presentations/Nadarajah... · Cross Validation (CV) More accurate than MSE Computationally