Post on 31-Jul-2020
COMPLETE CONFIGURATION AERO-STRUCTURALOPTIMIZATION USING A COUPLED SENSITIVITY
ANALYSIS METHOD
Joaquim R. R. A. MartinsJuan J. Alonso
James J. Reuther
Department of Aeronautics and AstronauticsStanford University
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 1
Outline
• Introduction
– High-fidelity aircraft design optimization– The need for aero-structural sensitivities– Sensitivity analysis methods
• Theory
– Adjoint sensitivity equations– Lagged aero-structural adjoint equations
• Results
– Optimization problem statement– Aero-structural sensitivity validation– Optimization results
• Conclusions
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 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.
• Large number of design variables andcomplex models inccur a large cost.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 3
Aero-Structural Aircraft Design Optimization
• Aerodynamics and structures are coredisciplines in aircraft design and are verytightly coupled.
• For traditional designs, aerodynamicistsknow the spanload distributions that leadto the true optimum from experience andaccumulated data. What about unusualdesigns?
• Want to simultaneously optimize theaerodynamic shape and structure, sincethere is a trade-off between aerodynamicperformance and structural weight, e.g.,
Range ∝ L
Dln
(Wi
Wf
)
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 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.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 5
Methods for Sensitivity Analysis
• Finite-Difference: very popular; easy, but lacks robustness andaccuracy; run solver Nx times.
df
dxn≈ f(xn + h)− f(x)
h+O(h)
• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver Nx times.
df
dxn≈ Im [f(xn + ih)]
h+O(h2)
• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of Nx.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 6
Objective Function and Governing Equations
Want to minimize scalar objective function,
I = I(xn, yi),
which depends on:
• xn: vector of design variables, e.g. structural plate thickness.
• yi: state vector, e.g. flow variables.
Physical system is modeled by a set of governing equations:
Rk (xn, yi (xn)) = 0,
where:
• Same number of state and governing equations, i, k = 1, . . . , NR
• Nx design variables.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 7
Sensitivity Equations
� �
� � ��
Total sensitivity of the objective function:
dI
dxn=
∂I
∂xn+
∂I
∂yi
dyi
dxn.
Total sensitivity of the governing equations:
dRk
dxn=
∂Rk
∂xn+
∂Rk
∂yi
dyi
dxn= 0.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 8
Solving the Sensitivity Equations
Solve the total sensitivity of the governing equations
∂Rk
∂yi
dyi
dxn= −∂Rk
∂xn.
Substitute this result into the total sensitivity equation
dI
dxn=
∂I
∂xn− ∂I
∂yi
− dyi/ dxn︷ ︸︸ ︷[∂Rk
∂yi
]−1∂Rk
∂xn,
︸ ︷︷ ︸−Ψk
where Ψk is the adjoint vector.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 9
Adjoint Sensitivity Equations
Solve the adjoint equations
∂Rk
∂yiΨk = − ∂I
∂yi.
Adjoint vector is valid for all design variables.
Now the total sensitivity of the the function of interest I is:
dI
dxn=
∂I
∂xn+ Ψk
∂Rk
∂xn
The partial derivatives are inexpensive, since they don’t require the solutionof the governing equations.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 10
Aero-Structural Adjoint Equations
���
� � � � ��� � ����
�
Two coupled disciplines: Aerodynamics (Ak) and Structures (Sl).
Rk′ =[ Ak
Sl
], yi′ =
[wi
uj
], Ψk′ =
[ψk
φl
].
Flow variables, wi, five for each grid point.
Structural displacements, uj, three for each structural node.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 11
Aero-Structural Adjoint Equations
∂Ak∂wi
∂Ak∂uj
∂Sl∂wi
∂Sl∂uj
T [ψk
φl
]= −
∂I∂wi∂I∂uj
.
• ∂Ak/∂wi: a change in one of the flow variables affects only the residualsof its cell and the neighboring ones.
• ∂Ak/∂uj: wing deflections cause the mesh to warp, affecting theresiduals.
• ∂Sl/∂wi: since Sl = Kljuj − fl, this is equal to −∂fl/∂wi.
• ∂Sl/∂uj: equal to the stiffness matrix, Klj.
• ∂I/∂wi: for CD, obtained from the integration of pressures; for stresses,its zero.
• ∂I/∂uj: for CD, wing displacement changes the surface boundary overwhich drag is integrated; for stresses, related to σm = Smjuj.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 12
Lagged Aero-Structural Adjoint Equations
Since the factorization of the complete residual sensitivity matrix isimpractical, decouple the system and lag the adjoint variables,
∂Ak
∂wiψk = − ∂I
∂wi︸ ︷︷ ︸Aerodynamic adjoint
−∂Sl
∂wiφ̃l,
∂Sl
∂ujφl = − ∂I
∂uj︸ ︷︷ ︸Structural adjoint
−∂Ak
∂ujψ̃k,
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.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 13
Total Sensitivity
The aero-structural sensitivities of the drag coefficient with respect to wingshape perturbations are,
dI
dxn=
∂I
∂xn+ ψk
∂Ak
∂xn+ φl
∂Sl
∂xn.
• ∂I/∂xn: CD changes when the boundary over which the pressures areintegrated is perturbed; stresses change when nodes are moved.
• ∂Ak/∂xn: the shape perturbations affect the grid, which in turn changesthe residuals; structural variables have no effect on this term.
• Sl/∂xn: shape perturbations affect the structural equations, so this termis equal to ∂Klj/∂xnuj − ∂fl/∂xn.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 14
3D Aero-Structural Design Optimization Framework
• Aerodynamics: FLO107-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.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 15
Sensitivity of CD wrt Shape
1 2 3 4 5 6 7 8 9 10Design variable, n
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
dCD /
dxn
Complex step, fixed displacements
Coupled adjoint, fixed displacements
Complex step
Coupled adjoint
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 16
Sensitivity of CD wrt Structural Thickness
11 12 13 14 15 16 17 18 19 20Design variable, n
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
dCD /
dxn
Complex step
Coupled adjoint
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 17
Structural Stress Constraint Lumping
To perform structural optimization, we need the sensitivities of all thestresses in the finite-element model with respect to many design variables.
There is no method to calculate this matrix of sensitivities efficiently.
Therefore, lump stress constraints
gm = 1− σm
σyield≥ 0,
using the Kreisselmeier–Steinhauser function
KS (gm) = −1ρ
ln
(∑m
e−ρgm
),
where ρ controls how close the function is to the minimum of the stressconstraints.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 18
Sensitivity of KS wrt Shape
1 2 3 4 5 6 7 8 9 10Design variable, n
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
dKS
/ dx
n
Complex, fixed loads
Coupled adjoint, fixed loads
Complex step
Coupled adjoint
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 19
Sensitivity of KS wrt Structural Thickness
11 12 13 14 15 16 17 18 19 20Design variable, n
-10
0
10
20
30
40
50
60
70
80
dKS
/ dx
n
Complex, fixed loads
Coupled adjoint, fixed loads
Complex step
Coupled adjoint
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 20
Computational Cost vs. Number of Variables
0 400 800 1200 1600 2000Number of design variables (Nx)
0
400
800
1200
Nor
mal
ized
tim
e
Complex step
2.1
+ 0.
92 ·
Nx
Finite difference
1.0 + 0.38 · N x
Coupled adjoint
3.4 + 0.01 · Nx
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 21
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
where CD is that of the cruisecondition.
Subject to:
KS(σm) ≥ 0where KS is taken from a maneuvercondition.
With respect to: external shapeand internal structural sizes.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 22
Baseline Design
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 23
Aero-Structural Optimization Convergence History
0 10 20 30 40 50Major iteration number
3000
4000
5000
6000
7000
8000
9000W
eigh
t (lb
s)
Weight
60
65
70
75
80
Dra
g (c
ount
s)
Drag
-1.2
-1.1
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
KS
KS
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 24
Aero-Structural Optimization Results
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 25
Comparison with Sequential Optimization
CD
(counts)σmaxσyield
Weight(lbs)
Objective
All-at-once approachBaseline 73.95 0.87 9, 285Optimized 69.22 0.98 5, 546 87.12Sequential approachAerodynamic optimization
Baseline 74.04Optimized 69.92
Structural optimizationBaseline 0.89 9, 285Optimized 0.98 6, 567
Aero-structural analysis 69.95 0.99 6, 567 91.14
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 26
Conclusions
• Presented a framework for high-fidelity aero-structural design.
• The computation of sensitivities using the aero-structural adjoint isextremely accurate and efficient.
• Demonstrated the usefulness of the coupled adjoint by optimizing asupersonic business jet configuration with external shape and internalstructural variables.
9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 27