A COUPLED-ADJOINT METHOD FOR HIGH-FIDELITY AERO...
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A COUPLED-ADJOINT METHOD FORHIGH-FIDELITY AERO-STRUCTURAL
OPTIMIZATION
Joaquim Rafael Rost Avila Martins
Department of Aeronautics and AstronauticsStanford University
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Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Complex-step derivative approximation
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions
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High-Fidelity Aerodynamic Shape Optimization
• Start from a baseline geometry providedby a conceptual design tool.
• High-fidelity models required for transonicconfigurations where shocks are present,high-dimensionality required to smooththese shocks.
• Accurate models also required forcomplex supersonic configurations, subtleshape variations required to takeadvantage of favorable shock interference.
• Large numbers of design variables andhigh-fidelity models incur a large cost.
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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
)
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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.
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Introduction to Optimization
x1
x2
g
Iminimize I(x)
x ∈ Rn
subject to gm(x) ≥ 0, m = 1, 2, . . . , Ng
• I: objective function, output (e.g. structural weight).
• xn: vector of design variables, inputs (e.g. aerodynamic shape); boundscan be set on these variables.
• gm: vector of constraints (e.g. element von Mises stresses); in generalthese are nonlinear functions of the design variables.
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Optimization Methods
• Intuition: decreases with increasing dimensionality.
• Grid or random search: the cost of searching the designspace increases rapidly with the number of design variables.
• Genetic algorithms: good for discrete design variablesand very robust; but infeasible when using a large numberof design variables.
• Nonlinear simplex: simple and robust but inefficient formore than a few design variables.
x1
x2
g
I
• Gradient-based: the most efficient for a large number ofdesign variables; assumes the objective function is “well-behaved”.
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Gradient-Based Optimization: Design Cycle
x
Analysis
Sensitivity
Calculation
Optimizer
x = x+�x
• Analysis computes objective function andconstraints (e.g. aero-structural solver)
• Optimizer uses the sensitivity informationto search for the optimum solution(e.g. sequential quadratic programming)
• Sensitivity calculation is usually thebottleneck in the design cycle.
• Accuracy of the sensitivities is important,specially near the optimum.
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Sensitivity Analysis Methods
• Finite Differences: very popular; easy, but lacks robustness andaccuracy; run solver Nx times.
dfdxn≈ f(xn + h)− f(x)
h+O(h)
• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver Nx times.
dfdxn≈ Im [f(xn + ih)]
h+O(h2)
• Algorithmic/Automatic/Computational Differentiation: accurate;ease of implementation and cost varies.
• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of Nx.
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Research Contributions
• Complex-step derivative approximation
– Contributed new insights to the theory of this approximation, includingthe connection to algorithmic differentiation.
– Automated the implementation of the complex-step method for Fortrancodes.
– Demonstrated the value of this method for sensitivity validation.
• Coupled-adjoint method
– Developed the general formulation of the coupled-adjoint formultidisciplinary systems.
– Implemented this method in a high-fidelity aero-structural solver.– Demonstrated the usefulness of the resulting framework by performing
aero-structural optimization.
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The History of the Complex-Step Method
• Lyness & Moler, 1967:nth derivative approximation by integration in the complex plane
• Squire & Trapp, 1998:Simple formula for first derivative.
• Newman, Anderson & Whitfield, 1998:Applied to aero-structural code.
• Anderson, Whitfield & Nielsen, 1999:Applied to 3D Navier-Stokes with turbulence model.
• Martins et al., 2000, 2001:Automated Fortran and C/C++ implementation. 3D aero-structuralcode.
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Finite-Difference Derivative Approximations
From Taylor series expansion,
f(x+ h) = f(x) + hf ′(x) + h2f′′(x)2!
+ h3f′′′(x)3!
+ . . . .
Forward-difference approximation:
⇒ df(x)dx
=f(x+ h)− f(x)
h+O(h).
f(x) 1.234567890123484
f(x+ h) 1.234567890123456
∆f 0.000000000000028
x x+h
f(x)
f(x+h)
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Complex-Step Derivative Approximation
Can also be derived from a Taylor series expansion about x with a complexstep ih:
f(x+ ih) = f(x) + ihf ′(x)− h2f′′(x)2!− ih3f
′′′(x)3!
+ . . .
⇒ f ′(x) =Im [f(x+ ih)]
h+ h2f
′′′(x)3!
+ . . .
⇒ f ′(x) ≈ Im [f(x+ ih)]h
No subtraction! Second order approximation.
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Simple Numerical Example
Step Size, h
Norm
alized E
rror
,
e
Complex-Step
Forward-Difference
Central-Difference
Estimate derivative atx = 1.5 of the function,
f(x) =ex√
sin3x+ cos3x
Relative error defined as:
ε =
∣
∣
∣f ′ − f ′ref∣
∣
∣
∣
∣
∣f ′ref
∣
∣
∣
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Algorithmic Differentiation vs. Complex Step
Look at a simple operation, e.g. f = x1x2,
Algorithmic Complex-Step∆x1 = 1 h1 = 10−20
∆x2 = 0 h2 = 0f = x1x2 f = (x1 + ih1)(x2 + ih2)∆f = x1∆x2 + x2∆x1 f = x1x2−h1h2 + i(x1h2 + x2h1)df/dx1 = ∆f df/dx1 = Im f/h
Complex-step method computes one extra term.
• Other functions are similar:
– Superfluous calculations are made.– For h ≤ x× 10−20 they vanish but still affect speed.
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Implementation Procedure
• Cookbook procedure for any programming language:
– Substitute all real type variable declarations with complexdeclarations.
– Define all functions and operators that are not defined for complexarguments.
– A complex-step can then be added to the desired variable and thederivative can be estimated by f ′ ≈ Im[f(x+ ih)]/h.
• Fortran 77: write new subroutines, substitute some of the intrinsicfunction calls by the subroutine names, e.g. abs by c abs. But ... needto know variable types in original code.
• Fortran 90: can overload intrinsic functions and operators, includingcomparison operators. Compiler knows variable types and chooses correctversion of the function or operator.
• C/C++: also uses function and operator overloading.
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Fortran Implementation
• complexify.f90: a module that defines additional functions andoperators for complex arguments.
• Complexify.py: Python script that makes necessary changes to sourcecode, e.g., type declarations.
• Features:
– Compatible with many platforms and compilers.– Supports MPI based parallel implementations.– Resolves some of the input and output issues.
• Application to aero-structural framework: 130,000 lines of code in 191subroutines, processed in 42 seconds.
• Tools available at: http://aero-comlab.stanford.edu/jmartins/
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Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Complex-step derivative approximation
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions
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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.
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Sensitivity Equations
xn
Rk = 0
yiI
Total sensitivity of the objective function:
dIdxn
=∂I
∂xn+∂I
∂yi
dyidxn
.
Total sensitivity of the governing equations:
dRkdxn
=∂Rk∂xn
+∂Rk∂yi
dyidxn
= 0.
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Solving the Sensitivity Equations
Solve the total sensitivity of the governing equations
∂Rk∂yi
dyidxn
= −∂Rk∂xn
.
Substitute this result into the total sensitivity equation
dIdxn
=∂I
∂xn− ∂I
∂yi
− dyi/ dxn︷ ︸︸ ︷
[
∂Rk∂yi
]−1∂Rk∂xn
,
︸ ︷︷ ︸
−Ψk
where Ψk is the adjoint vector.
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Adjoint Sensitivity Equations
Solve the adjoint equations
∂Rk∂yi
Ψk = − ∂I∂yi
.
Adjoint vector is valid for all design variables.
Now the total sensitivity of the the function of interest I is:
dIdxn
=∂I
∂xn+ Ψk
∂Rk∂xn
The partial derivatives are inexpensive, since they don’t require the solutionof the governing equations.
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Aero-Structural Adjoint Equations
xn
Ak = 0 Sl = 0
wi
uj
I
Two coupled disciplines: Aerodynamics (Ak) and Structures (Sl).
Rk′ =[
AkSl
]
, yi′ =[
wiuj
]
, Ψk′ =[
ψkφl
]
.
Flow variables, wi, five for each grid point.
Structural displacements, uj, three for each structural node.
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Aero-Structural Adjoint Equations
∂Ak∂wi
∂Ak∂uj
∂Sl∂wi
∂Sl∂uj
T[
ψkφl
]
= −
∂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.
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Lagged Aero-Structural Adjoint Equations
Since the factorization of the complete residual sensitivity matrix isimpractical, decouple the system and lag the adjoint variables,
∂Ak∂wi
ψk = − ∂I
∂wi︸ ︷︷ ︸
Aerodynamic adjoint
−∂Sl∂wi
φl,
∂Sl∂uj
φl = − ∂I∂uj
︸ ︷︷ ︸
Structural adjoint
−∂Ak∂uj
ψk,
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,
dIdxn
=∂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.
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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.
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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
Avg. rel. error = 3.5%
Complex step, fixed displacements
Coupled adjoint, fixed displacements
Complex step
Coupled adjoint
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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
Avg. rel. error = 1.6%
Complex step
Coupled adjoint
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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− σmσyield
≥ 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.
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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
Avg. rel. error = 2.9%
Complex, fixed loads
Coupled adjoint, fixed loads
Complex step
Coupled adjoint
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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
Avg. rel. error = 1.6%
Complex, fixed loads
Coupled adjoint, fixed loads
Complex step
Coupled adjoint
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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 N
x
Finite difference
1.0 + 0.38 N x
Coupled adjoint
3.4 + 0.01 Nx
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Computational Cost Breakdown
2:40:64
1:20
0:60
< 0:001
0:01Nx
@Ak
@wi
k = �@I
@wi
�@Sl
@wi
~�l
@Sl
@uj�l = �
@I
@uj�@Ak
@uj~ k
dI
dxn= @I
@xn+ k
@Ak
@xn+ �l
@Sl@xn
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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.
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Baseline Design
CD = 0.007395
Weight = 9,285 lbs
0.5 1.4 0.0 1.0
Von Mises stresses (maneuver)Surface density (cruise)
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Design Variables
0.5 1.4
9 bumps along fuselage axis
10 skin thickness groups
Total of 97 design variables
10 Hicks-Henne bumps
Twist
6 de�ning airfoilsLE camber
TE camber
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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
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Aero-Structural Optimization Results
0.5 1.4 0.0 1.0
Surface density (cruise) Von Mises stress (maneuver)
CD = 0.006922
Weight = 5,546 lbs
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Comparison with Sequential Optimization
CD(counts)
σmaxσyield
Weight(lbs)
Objective
All-at-once approachBaseline 73.95 0.87 9, 285 103.90Optimized 69.22 0.98 5, 546 87.11Sequential 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 91.13
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Conclusions
• Shed a new light on the theory behind the complex-step method.
• Showed the connection between complex-step and algorithmicdifferentiation theories.
• The complex-step method is excellent for validation of more sophisticatedgradient calculation methods, like the coupled-adjoint.
Step Size, h
No
rm
alize
d E
rro
r
,e
Complex-Step
Forward-Difference
Central-Difference
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
Avg. rel. error = 3.5%
Complex step, fixed displacements
Coupled adjoint, fixed displacements
Complex step
Coupled adjoint
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Conclusions
• Developed the general formulation for a coupled-adjoint method formultidisciplinary systems.
• Applied this method to a high-fidelity aero-structural solver.
• Showed that the computation of sensitivities using the aero-structuraladjoint is extremely accurate and efficient.
• Demonstrated the usefulness of the coupled adjoint by optimizing asupersonic business jet configuration.
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 N
x
Finite difference
1.0 + 0.38 N x
Coupled adjoint
3.4 + 0.01 Nx
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Long Term Vision
Continue work on a large-scale MDO framework for aircraft design
ConceptualDesign
DetailedDesign
PreliminaryDesign
CentralDatabase
CAD
Discretization
Multi-DisciplinaryAnalysis
Optimizer
Aerodynamics
Structures
Propulsion
Mission
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Acknowledgments
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Acknowledgments
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Acknowledgments
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CFD and OML grids
v
u
CFD mesh point
OML point
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Displacement Transfer
u1;2;3
u4;5;6
u7;8;9
�ri
rigid link
associated point
OML point
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Mesh Perturbation
Baseline
1
2, 3
4
Perturbed
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Load Transfer
v
u
CFD mesh point
OML point
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Aero-Structural Iteration
CFD
CSM
Load
tran
sfer
Dis
plac
emen
t tra
nsfe
ru(0)
= 0
w(0)= w1
u(1)
1
2
35
4
w(N)
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