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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
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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)
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
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
3D NACA 0414 Wing, α=4 Deg -- Euler Solution (256x64x64) Pressure Distribution
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
3D NACA 848 Wing, α=4 Deg -- Euler Solution (256x64x64) Pressure Distribution
→ 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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationError-Norm Oriented Adaptation for High-Order Methods
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
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
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
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.
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
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)
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
airfoil3.aviMedia File (video/avi)
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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.
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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...
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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)
• 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?
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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 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?”
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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 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?”
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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 (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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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 (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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
Error-norm adjoint problem (J-S. Cagnone)
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
Error-norm adjoint problem (J-S. Cagnone)
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
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
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
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
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
Current Status of Goal-Oriented Mesh AdaptationError-norm adjoint problemResults: Linear advectionResults: Prandtl-Meyer expansionResults: Supersonic diamond airfoil
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
Current Status of Aerodynamic Design OptimizationNext Generation of Algorithms
Error-Norm Oriented Adaptation for High-Order MethodsConclusion
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.
Siva Nadarajah Next Generation of Algorithms for Aerodynamic Design Optimization: Current Status and Future Challengers
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