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HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF...
Transcript of HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF...
HIGH-FIDELITY AERO-STRUCTURAL DESIGNOPTIMIZATION OF A SUPERSONIC BUSINESS JET
Joaquim R. R. A. MartinsJuan J. Alonso
James J. Reuther
Department of Aeronautics and AstronauticsStanford University
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 1
Outline
• Introduction
– High-fidelity aircraft design optimization– The need for aero-structural sensitivities– Sensitivity analysis methods– Optimization problem statement
• Theory
– Adjoint sensitivity equations– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions and Future Work
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 2
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.
• Gradient-based optimization is the mostefficient and requires accurate sensitivityinformation.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 3
Aero-Structural Aircraft Design Optimization
• Aerodynamics traditionally has used ashape corresponding to the flying shapeof the wing, assuming that shape can bereproduced.
• Wing shape depends on aerodynamicsolution, so need to couple aerodynamicand structural analyses to obtain thesolution, specially for unusual designs.
• Want to optimize the structure as well,since there is a trade-off betweenaerodynamic performance and structuralweight:
Range ∝ L
Dln
(Wi
Wf
)
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 4
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.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 5
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
subject to CL = CLT
KS ≥ 0
xS ≥ xSmin
Lump stress constraints
gi = 1− σiV M
σyield≥ 0,
using the Kreisselmeier-Steinhauserfunction
KS (gi(x)) = −1ρ
ln
[∑
i
e−ρgi(x)
].
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 6
Methods for Sensitivity Analysis
• Finite-Difference: very popular; easy, but lacks robustness andaccuracy; run solver n times.
f ′(x) ≈ f(x + h)− f(x)h
+O(h)
• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver n times.
f ′(x) ≈ Im [f(x + ih)]h
+O(h2)
• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of n.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 7
Objective Function and Governing Equations
Want to minimize scalar objective function,
I = I(x, y),
which depends on:
• x: vector of design variables, e.g. structural plate thickness.
• y: state vector, e.g. structural displacements.
Physical system is modeled by a set of governing equations:
R (x, y (x)) = 0,
where:
• Same number of state and governing equations, nR
• nx design variables.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 8
Variational Equations
R =0
x
Iy
Total variation of the objective function:
δI =∂I
∂xδx +
∂I
∂yδy.
Variation of the governing equations,
δR =∂R
∂xδx +
∂R
∂yδy = 0.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 9
Adjoint Sensitivity Equations
Since the variation of the governing equations must be zero, we can add itto the total variation of the objective,
δI =∂I
∂xδx +
∂I
∂yδy + ψT
(∂R
∂xδx +
∂R
∂yδy
),
where ψ is a vector of arbitrary components known as adjoint variables.
Re-arrange terms,
δI =(
∂I
∂x+ ψT ∂R
∂x
)δx +
(∂I
∂y+ ψT ∂R
∂y
)δy.
If term in blue were zero, term in red would represent the total variation ofthe objective with respect to the design variables.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 10
Adjoint Sensitivity Equations
Since the adjoint variables are arbitrary, we can find a set such that,
[∂R
∂y
]T
︸ ︷︷ ︸(nR×nR)
ψ︸︷︷︸(nR×1)
= −∂I
∂y
T
︸ ︷︷ ︸(nR×1)
Adjoint valid for all design variables.
Now the total sensitivity of the objective is:
dI
dx︸︷︷︸(1×nR)
=∂I
∂x︸︷︷︸(1×nx)
+ ψT︸︷︷︸
(1×nR)
∂R
∂x︸︷︷︸(nR×nx)
The partial derivatives are inexpensive, since they don’t require the solutionof the governing equations.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 11
Aero-Structural Adjoint Equations
R =0
x
I
wA R =0
uS
Two coupled disciplines: Aerodynamics (RA) and Structures (RS).
R =[
RA
RS
], y =
[wu
]and ψ =
[ψA
ψS
]
Flow variables, w, five for each grid point.
Structural displacements, u, three for each structural node.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 12
Aero-Structural Adjoint Equations
∂RA
∂w
∂RA
∂u
∂RS
∂w
∂RS
∂u
T
[ψA
ψS
]= −
∂I
∂w
∂I
∂u
• ∂RA/∂w: a change in one of the flow variables affects only the residualsof its cell and the neighboring ones.
• ∂RA/∂u: wing deflections cause the mesh to warp, affecting theresiduals.
• ∂RS/∂w: since RS = Ku− f , this is equal to −∂f/∂w.
• ∂RS/∂u: equal to the stiffness matrix, K.
• ∂I/∂w: for CD, obtained from the integration of pressures; for KS, itszero.
• ∂I/∂u: for CD wing displacement changes the surface boundary overwhich drag is integrated; for KS, related to σ = Su.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 13
Lagged Aero-Structural Adjoint Equations
Since the factorization of the complete residual sensitivity matrix isimpractical, decouple the system and lag the adjoint variables,
[∂RA
∂w
]T
ψA = − ∂I
∂w︸ ︷︷ ︸Aerodynamic adjoint equations
−[∂RS
∂w
]T
ψ̃S
[∂RS
∂u
]T
ψS = −∂I
∂u︸ ︷︷ ︸Structural adjoint equations
−[∂RA
∂u
]T
ψ̃A.
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.
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Total Sensitivity
The aero-structural sensitivities of the drag coefficient with respect to wingshape perturbations are,
dI
dx=
∂I
∂x+ ψT
A
∂RA
∂x+ ψT
S
∂RS
∂x.
• ∂I/∂x: CD changes when the boundary over which the pressures areintegrated is perturbed; stresses change when nodes are moved.
• ∂RA/∂x: the shape perturbations affect the grid, which in turn changesthe residuals; structural variables have no effect on this.
• ∂RS/∂x: shape perturbations affect the structural equations, so this is∂K/∂x · u− ∂f/∂x.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 15
3D Aero-Structural Design Optimization Framework
• Aerodynamics: SYN107-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.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 16
Sensitivity of CD wrt Shape
1 2 3 4 5 6 7 8 9 10−8
−6
−4
−2
0
2
4
6
8
10x 10
−3
Shape variable, xj
d C
D /
d x A
Coupled adjoint Complex step Avg. rel. error = 3.5%
Student Version of MATLAB
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 17
Sensitivity of CD wrt Structural Thickness
11 12 13 14 15 16 17 18 19 20−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08d
CD /
d x S
Structural variable, xj
Coupled adjoint Complex step Avg. rel. error = 1.6%
Student Version of MATLAB
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 18
Sensitivity of KS wrt Shape
1 2 3 4 5 6 7 8 9 10−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Shape variable, xj
d C
D /
d x A
Coupled adjoint Complex step Avg. rel. error = 2.9%
Student Version of MATLAB
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 19
Sensitivity of KS wrt Structural Thickness
11 12 13 14 15 16 17 18 19 20−10
0
10
20
30
40
50
60
70
80
d C
D /
d x S
Structural variable, xj
Coupled adjoint Complex step Avg. rel. error = 1.6%
Student Version of MATLAB
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 20
Rigid Wing Aerodynamic Optimization Results
SYMBOL
SOURCE Rigid Baseline
Rigid, constrained, aero only design
ALPHA 3.440
4.113
CD 0.00743
0.00700
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AERODYNAMIC DESIGN
MACH = 1.500 , CL = 0.100
MCDONNELL DOUGLAS
Solution 1 Upper-Surface Isobars
( Contours at 0.05 Cp )
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 8.6% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 24.3% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 40.8% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25C
p
X / C 58.9% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 76.5% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 91.5% Span
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 21
Aeroelastic Wing Aerodynamic Optimization Results
SYMBOL
SOURCE Baseline Aero-elastic
Aero-elastic, Constrained, aero only design
ALPHA 3.330
4.095
CD 0.00741
0.00701
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AEROELASTIC DESIGN - OML ONLY
MACH = 1.500 , CL = 0.100
MCDONNELL DOUGLAS
Solution 1 Upper-Surface Isobars
( Contours at 0.05 Cp )
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 8.6% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 24.3% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 40.8% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25C
p
X / C 58.9% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 76.5% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 91.5% Span
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 22
Simplified Optimization Problem
minimize I = αCD + βW + γ max(0,−KS)2
subject to CL = CLT
xS ≥ xSmin
α = 104, β = 3.226× 10−3, γ = 103
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 23
Aero-Structural Optimization Convergence History
2 4 6 8 10 12
80
100
120
140
160
180
200
220
0 2 4 6 8 10 12
70
70.5
71
71.5
72
72.5
73
73.5
74
74.5
75
2 4 6 8 10 123000
3200
3400
3600
3800
4000
4200
4400
4600
Objective function
Weight (lbs)
Drag counts
Iteration number
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 24
Aero-Structural Optimization Results
SYMBOL
SOURCE Aero-elastic Baseline
Aero-elastic, Aero-structural design
ALPHA 3.330
3.689
CD 0.00741
0.00685
COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF FULL AERO-STRUCTURAL DESIGN
MACH = 1.500 , CL = 0.100
MCDONNELL DOUGLAS
Solution 1 Upper-Surface Isobars
( Contours at 0.05 Cp )
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 8.6% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 24.3% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 40.8% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25C
p
X / C 58.9% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 76.5% Span
0.2 0.4 0.6 0.8 1.0
-0.25
-0.15
-0.05
0.05
0.15
0.25
Cp
X / C 91.5% Span
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 25
Aero-Structural Optimization Results
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 26
Conclusions
• Presented a framework for high-fidelity aero-structural design.
• The sensitivities given by the aero-structural adjoint were extremelyaccurate when compared to the reference results.
• The coupled-adjoint method was shown to be extremely efficient, makingviable design optimization with respect to hundreds of design variables.
• Demonstrated the usefulness of the coupled-adjoint by optimizing asupersonic business jet configuration with aerodynamic and structuralvariables.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 27
Future Work
• Implement multiple load cases, each one with a KS adjoint.
• Use the flow solver in Navier-Stokes mode with a full-configuration mesh.
• Use a better structural model and a more established structural solver.
• Explore the design space for the supersonic business jet configurationand provide useful results to industry.
AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 28