Post on 30-Jun-2020
HIGH-FIDELITY AERO-STRUCTURAL DESIGN OFCOMPLETE AIRCRAFT CONFIGURATIONS
Juan J. AlonsoDepartment of Aeronautics & Astronautics
Stanford Universityjjalonso@stanford.edu
DLR - Institute for Aerodynamics and Flow TechnologyBraunschweig, December 16, 2002
DLR Lecture, December 2002 1
Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions & future work
DLR Lecture, December 2002 2
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.
DLR Lecture, December 2002 3
Quiet Supersonic Platform (QSP) Program
• Range = 6,000 nmi
• Cruise Mach No. = 2.4
• TOGW = 100,000 lbs
• Payload = 20,000 lbs
• Initial Overpressure =0.3 psf
• Swing-wing concept
Gulfstream Aerospace Corporation QSJ Configuration
DLR Lecture, December 2002 4
Quiet Supersonic Platform (QSP) Program
• Is this combination of requirements achievable? Can we actually do this?This is a set of conflicting requirements: the airplane may not “close”.
• Classical sonic boom minimization theory says that ∆p ≡ W
L32.
• What is the necessary aircraft length? Can we achieve this with ourtarget TOGW?
• At Stanford, we have decided to focus on
– Using aerodynamic shape optimization to take advantage of thenonlinear interactions between shock waves and expansions andproduce shaped booms.
– Using Multidisciplinary Design Optimization (MDO) methods tominimize the weight of the airframe.
DLR Lecture, December 2002 5
Quiet Supersonic Platform (QSP) Program
Ground Plane
Mid Field
CFD Far Field
Near Field
Method for Computing Ground Boom
Signatures
Symmetry Plane View of CFD Computation
DLR Lecture, December 2002 6
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
)
DLR Lecture, December 2002 7
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.
DLR Lecture, December 2002 8
Introduction to Optimization
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� minimize 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.
DLR Lecture, December 2002 9
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.
• Evolutionary/Genetic algorithms: good for discretedesign variables and very robust; are they feasible whenusing a large number of design variables?
• Nonlinear simplex: simple and robust but inefficient formore than a few design variables.
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• Gradient-based: the most efficient for a large numberof design variables; assumes the objective function is“well-behaved”. Convergence only guaranteed to a localminimum.
DLR Lecture, December 2002 10
Gradient-Based Optimization: Design Cycle
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• 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, particularlyfor large dimensional design spaces.
• Accuracy of the sensitivities is important,specially near the optimum.
DLR Lecture, December 2002 11
Sensitivity Analysis Methods
• Finite Differences: 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).
• 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.
DLR Lecture, December 2002 12
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)
DLR Lecture, December 2002 13
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.
DLR Lecture, December 2002 14
Simple Numerical Example
Step Size, h
Norm
aliz
ed E
rror,
e
Complex-StepForward-DifferenceCentral-Difference
Estimate derivative atx = 1.5 of the function,
f(x) =ex
√sin3x + cos3x
.
Relative error defined as:
ε =
∣∣∣f ′ − f ′ref
∣∣∣∣∣∣f ′ref
∣∣∣.
DLR Lecture, December 2002 15
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.
DLR Lecture, December 2002 16
Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions & future work
DLR Lecture, December 2002 17
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.
DLR Lecture, December 2002 18
Sensitivity Equations
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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.
DLR Lecture, December 2002 19
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.
DLR Lecture, December 2002 20
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.
DLR Lecture, December 2002 21
Aero-Structural Adjoint Equations
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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 (Euler).
Structural displacements, uj, six for each structural node.
DLR Lecture, December 2002 22
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,it is zero.
• ∂I/∂uj: for CD, wing displacement changes the surface boundary overwhich drag is integrated; for stresses, related to σm = Smjuj.
DLR Lecture, December 2002 23
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.
DLR Lecture, December 2002 24
Total Sensitivity
The aero-structural sensitivities of the drag coefficient and stresses withrespect to wing shape 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 thickness variables have no effect on this term.
• Sl/∂xn: shape perturbations affect the structural equations, so this termis equal to ∂Klj/∂xnuj − ∂fl/∂xn.
DLR Lecture, December 2002 25
Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions & future work
DLR Lecture, December 2002 26
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.
DLR Lecture, December 2002 27
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
DLR Lecture, December 2002 28
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
DLR Lecture, December 2002 29
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 known 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.
DLR Lecture, December 2002 30
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
DLR Lecture, December 2002 31
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
DLR Lecture, December 2002 32
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
DLR Lecture, December 2002 33
Computational Cost Breakdown
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DLR Lecture, December 2002 34
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(gm) ≥ 0where KS is taken from a maneuvercondition.
With respect to: external shapeand internal structural sizes.
DLR Lecture, December 2002 35
Baseline Design
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DLR Lecture, December 2002 36
Design Variables
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DLR Lecture, December 2002 37
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
DLR Lecture, December 2002 38
Aero-Structural Optimization Results
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DLR Lecture, December 2002 39
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
DLR Lecture, December 2002 40
Outline
• Introduction
– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods
• Coupled-adjoint method
– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations
• Results
– Aero-structural sensitivity validation– Optimization results
• Conclusions & future work
DLR Lecture, December 2002 41
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
DLR Lecture, December 2002 42
Long Term Vision
Continue work on a large-scale MDO framework for aircraft and spacecraftdesign
ConceptualDesign
DetailedDesign
PreliminaryDesign
CentralDatabase
CAD
Discretization
Multi-DisciplinaryAnalysis
Optimizer
Aerodynamics
Structures
Propulsion
Mission
DLR Lecture, December 2002 43
Direct Interfaces to CAD
• Make use of existing CADpackages to control geometrymanipulation.
• Use CAD packages forstructural definition as well.
• Preferrably, CAD-independentimplementation.
• Master model for aircraftshape and structure geometryis defined parameterically.
• Aircraft model has over 5,000parameters that can becontrolled.
DLR Lecture, December 2002 44
Parallel/Distributed CAD Geometry Server
Arbitrary Network Interconnect (Ethernet)
SYN107−MB−AE SYN107−MB−AESYN107−MB−AE SYN107−MB−AE SYN107−MB−AE
SYN107−MB−AE SYN107−MB−AESYN107−MB−AE SYN107−MB−AE SYN107−MB−AE
SYN107−MB−AE SYN107−MB−AESYN107−MB−AE SYN107−MB−AE SYN107−MB−AE
CAD Model CAD Model CAD Model
PVM Session
Task 1
ProE
Task 2 Task N
AEROSURF MASTER PROGRAM
MPI COMMUNICATION LAYER FOR SOLVER AND OPTIMIZER
ProE
ProE
v_1, v_2, ..., v_N v_1, v_2, ... , v_N v_1, v_2, ... , v_N
Parallel Computer 1
Parallel Computer 2
• Completely distributed parallelsystem.
• CAD packages run on onecomputer, simulation/designon another.
• Geometry regeneration iscompletely parallel.
• PVM is used for distributedframework.
• Concept of Geometry Serverthat can be usedsimultaneously with multipleapplications.
DLR Lecture, December 2002 45
Future Work
A lot of work remains to be done for truly high-fidelity MDO environmentsto be realized: it is not simply a matter of development. In particular,
• Extension of coupled-adjoint procedure to multiple disciplines.
• Multilevel optimization strategies with variable fidelity approach.
• Combination of optimization strategies (evolutionary + gradient) forproblems with difficult design spaces.
• Algorithms for coordination in the design of multiple disciplines.
• Introduction of aeroelastic constraints (divergence, flutter) with bothlow- and high-fidelity methods into the design framework.
are some of the areas that we are currently pursuing.
See http://aero-comlab.stanford.edu for more details.
DLR Lecture, December 2002 46
DLR Lecture, December 2002 47
CFD and OML grids
v
u
CFD mesh point
OML point
DLR Lecture, December 2002 48
Displacement Transfer
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rigid link
associated point
OML point
DLR Lecture, December 2002 49
Mesh Perturbation
Baseline
1
2, 3
4
Perturbed
DLR Lecture, December 2002 50
Load Transfer
v
u
CFD mesh point
OML point
DLR Lecture, December 2002 51
Aero-Structural Iteration
CFD
CSM
Load tr
ansf
er
Dis
pla
cem
ent
tran
sfer
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1
2
35
4
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DLR Lecture, December 2002 52