Introductory Course to Design Optimization
Transcript of Introductory Course to Design Optimization
N. Gauger, Institute of Aerodynamics and Flow Technology
Nicolas Gaugerwith thanks to Joël Brezillon
Institute of Aerodynamics and Flow TechnologyDLR Braunschweig
(German Aerospace Center)
ERCOFTACIntroductory Course to
Design Optimization
April 1st-3rd, 2003
Adjoint Methods
N. Gauger, Institute of Aerodynamics and Flow Technology
Outline
� Requirements for Detailed Aerodynamic Shape Optimization� Tools in DLR’s Optimization Framework SPP� Motivation for Adjoint Approaches by some High-Lift-Case� The Dual or Adjoint Problem� Examples, Adjoint Euler Equations� Continuous Adjoint Approach / Implementation Aspects� Validation / Application of the Adjoint Approach� Continuous, Discrete and Hybrid Adjoint� Conclusion / Outlook
N. Gauger, Institute of Aerodynamics and Flow Technology
Requirements fordetailed design
• Compressible Navier-Stokes equations with models forturbulence and transition (at least Euler)
• Large number of design variables
• Complex geometries
• Physical and geometrical constraints
• Multi-point design• General optimization framework
Aerodynamic Shape Optimization
DeterministicOptimizationStrategies
Here local minima desired!
N. Gauger, Institute of Aerodynamics and Flow Technology
Starting GeometryStarting Geometry
Mesh generation Mesh generation• 2D C-mesh• MegaCads (Batch)
Optimization strategiesOptimization strategies
Parameter change
Evaluation ofcost functionEvaluation ofcost function
OutputOutput
• Gradient based - Finite differences - Adjoint
Geometry generationGeometry generation• B-spline• Bezier curves
Flow calculationFlow calculation• FLOWer
• Optimized configuration
• Flow field
Optimization ?
• Simplex• Simulated
annealing
• Free Form Deformation
• Centaur• Remesh
• TAU
Optimization Framework:Synaps Pointer Pro
N. Gauger, Institute of Aerodynamics and Flow Technology
MEGAFLOWMEGAFLOW
Block-structured capabilityBlock-structured capability
MegaCadsICEM Hexa
FLOWer
Unstructured capabilityUnstructured capability
Centaur
TAU
N. Gauger, Institute of Aerodynamics and Flow Technology
Structured Grid Generation Software MegaCads
� structuredmulti-block grids
� parametric system� basic functions for
geometry modeling(projection, intersection)
� script language forreplay capability
� ensures high quality grids� interactive and
batch functionality� well suited for
optimization loop
N. Gauger, Institute of Aerodynamics and Flow Technology
� accuracy� state-of-the-art turbulence models� finite volume discretization
on block-structured grids� central & upwind schemes
Reynolds-averaged Navier-Stokes Solver FLOWer
� performance� multigrid� implicit schemes for
time accurate flows� preconditioning for low speed flow� vectorization & parallelization
� flexibility� arbitrarily moving bodies� overset grids (Chimera)� deforming grids for coupling
with other disciplines� reliability
� comprehensive validation� production code in industry� quality assurance
N. Gauger, Institute of Aerodynamics and Flow Technology
Application to Aircraft in Cruise Flight
M = 0.75, � = 0.980
Re = 3x106
FLOWer, Navier-Stokes
block structured grid: - 45 blocks, - 3.8 million points
aerodynamic coefficients surface pressure distribution
N. Gauger, Institute of Aerodynamics and Flow Technology
Numerical Optimization of 2D High-Lift Devices
Application� drag optimization for 3-element airfoil� take-off configuration
(M�=0.2, Re=3.52x106)
Cost function:� minimum drag with constant lift
and constraint pitching momentDesign parameters (12)
� element position & deflection� element-size variations
parametric grid generationMegaCads software
N. Gauger, Institute of Aerodynamics and Flow Technology
Numerical Optimization of 2D High-Lift Devices
Application� drag optimization for 3-element airfoil� take-off configuration
(M�=0.2, Re=3.52x106)
Cost function:� minimum drag with constant lift
and constraint pitching momentDesign parameters (12)
� element position & deflection� element-size variations
parametric grid generationMegaCads software
0 1 2 3 4 5 66.50
7.00
7.50
8.00
8.50
9.00
optimization cycles
Fobj(X) = 100�CD
Testcase:DRA NHLP L1T2
CL = const. = 3,77Re = 3,56�106
-Cm � -Cm,start
optimized:�CD = -20,815 %
design parameter:element deflections
Ma�
= 0,2
cutout geometries
N. Gauger, Institute of Aerodynamics and Flow Technology
Numerical Optimization of 2D High-Lift Devices
Application� drag optimization for 3-element airfoil� take-off configuration
(M�=0.2, Re=3.52x106)
Cost function:� minimum drag with constant lift
and constraint pitching momentDesign parameters (12)
� element position & deflection� element-size variations
parametric grid generationMegaCads software
0 1 2 3 4 5 66.50
7.00
7.50
8.00
8.50
9.00
optimization cycles
Fobj(X) = 100�CD
Testcase:DRA NHLP L1T2
CL = const. = 3,77Re = 3,56�106
-Cm � -Cm,start
optimized:�CD = -20,815 %
design parameter:element deflections
Ma�
= 0,2
cutout geometries
pressure distribution
test case optimized
initial configuration, �=20.160
optimal configuration, �=17.930
N. Gauger, Institute of Aerodynamics and Flow Technology
Numerical Optimization of 2D High-Lift Devices
Application� drag optimization for 3-element airfoil� take-off configuration
(M�=0.2, Re=3.52x106)
Cost function:� minimum drag with constant lift
and constraint pitching momentDesign parameters (12)
� element position & deflection� element-size variations
parametric grid generationMegaCads software
0 1 2 3 4 5 66.50
7.00
7.50
8.00
8.50
9.00
optimization cycles
Fobj(X) = 100�CD
Testcase:DRA NHLP L1T2
CL = const. = 3,77Re = 3,56�106
-Cm � -Cm,start
optimized:�CD = -20,815 %
design parameter:element deflections
Ma�
= 0,2
cutout geometries
pressure distribution
test case optimized
initial configuration, �=20.160
optimal configuration, �=17.930
Computational effort: 350 CPU hrs, NEC-SX4, 1 proc.
~ 50 hrs, 4 procs NEC SX5
N. Gauger, Institute of Aerodynamics and Flow Technology
Finite Differences
• Finite Differences n design variables requiren+1 flow calculations
metric sensitivity � pressure variation � aerodynamic sensitivity
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variation of i-th design variable
i-th component of cost function´s gradient
n--loop
N. Gauger, Institute of Aerodynamics and Flow Technology
Motivation• detailed design optimization requires Navier-Stokes (at least Euler) computations
� each flow computation suffers from high computational costs
� deterministic optimization strategies should be preferred
(gradient based: steepest descent, conjugate gradient, QNTR, SQP, ... )
• for detailed design optimization a large number of design variables required
• Finite Differences n design variables requiren+1 flow calculations
• Adjoint Approach n design variables require1 flow and 1 adjoint flow calculation
independence of number ofdesign variables
high accuracy
Adjoint Method for Aerodynamic Shape Optimization
N. Gauger, Institute of Aerodynamics and Flow Technology
N. Gauger, Institute of Aerodynamics and Flow Technology
N. Gauger, Institute of Aerodynamics and Flow Technology
N. Gauger, Institute of Aerodynamics and Flow Technology
N. Gauger, Institute of Aerodynamics and Flow Technology
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drag, lift and pitching moment:
Governing Equations, Aerodynamic Coefficients
N. Gauger, Institute of Aerodynamics and Flow Technology
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adjoint formulation of cost function’s gradient
Continuous Adjoint Approach
�: vector of adjoint variables
N. Gauger, Institute of Aerodynamics and Flow Technology
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Continuous Adjoint Approach
pitching moment
lift
N. Gauger, Institute of Aerodynamics and Flow Technology
FLOWer Adjoint
Implementation aspects• adjoint flow equations and boundary conditions first derived
then discretized � continuous adjoint
• use of FLOWer infrastructure (e.g. multi-block capability ...)
• flux and boundary treatment modified for Euler and Navier-Stokes
Current status• Euler
- cost functions: drag, lift, moment and combinations- highly validated for 2D / 3D multi-block applications
• Navier-Stokes
- first verification results
N. Gauger, Institute of Aerodynamics and Flow Technology
n-th Iteration
d�/d
t,d�
1/dt
CL
CD
1000 2000 3000 4000 5000
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
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FLOWer MAIN FLOWer ADJOINT
FLOWer ADJOINT (Euler)
RAE2822(192x32)M
�=0.73, � = 2.0�
Drag Reduction
-12.4408 -9.55489 -6.66898 -3.78306 -0.897145 1.98877 4.87468 7.7606
�1
N. Gauger, Institute of Aerodynamics and Flow Technology
n-th Design Variable
-�C
m
0 10 20 30 40 50-5
-4
-3
-2
-1
0
1
2
AdjointFinite Differences
n-th Design Variable
-�C
L
0 10 20 30 40 50-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
AdjointFinite Differences
n-th Design Variable
-�C
D
0 10 20 30 40 50-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
AdjointFinite Differences
RAE2822M
�=0.73, � = 2.0�
50 design variables(B-spline)
Validation of Euler Adjointadjoint gradientvs. finite differences‘ gradient
drag
lift
moment
finite differences:51 calls of FLOWer MAINadjoint approach:1 call of FLOWer MAIN3 calls of FLOWer ADJOINT
N. Gauger, Institute of Aerodynamics and Flow Technology
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
Validation of adjoint gradient based optimization
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.00°
Constraints
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Deformation of camberline(20 Hicks-Henne functions)
Optimizer
N. Gauger, Institute of Aerodynamics and Flow Technology
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
Validation of adjoint gradient based optimization
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.00°
Constraints
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Deformation of camberline(20 Hicks-Henne functions)
Optimizer
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
N. Gauger, Institute of Aerodynamics and Flow Technology
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
Validation of adjoint gradient based optimization
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.00°
Constraints
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Deformation of camberline(20 Hicks-Henne functions)
Optimizer
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
N. Gauger, Institute of Aerodynamics and Flow Technology
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
Validation of adjoint gradient based optimization
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.00°
Constraints
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Deformation of camberline(20 Hicks-Henne functions)
Optimizer
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
N. Gauger, Institute of Aerodynamics and Flow Technology
� Steepest Descent
� Conjugate Gradient
� Quasi Newton Trust Region
Validation of adjoint gradient based optimization
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.00°
Constraints
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Deformation of camberline(20 Hicks-Henne functions)
Optimizer
N. Gauger, Institute of Aerodynamics and Flow Technology
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.0°
Constraints
� Lift, pitching moment and angle of attack held constant
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Constraints handled byfeasible direction
� Deformation of camberline
Multi-constraint airfoil optimization RAE2822
N. Gauger, Institute of Aerodynamics and Flow Technology
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.0°
Constraints
� Lift, pitching moment and angle of attack held constant
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Constraints handled byfeasible direction
� Deformation of camberline
Multi-constraint airfoil optimization RAE2822
N. Gauger, Institute of Aerodynamics and Flow Technology
Objective function
� Drag reduction for RAE 2822 airfoil
� M� =0.73, �=2.0°
Constraints
� Lift, pitching moment and angle of attack held constant
� Constant thickness
Approach
� FLOWer Euler Adjoint
� Constraints handled byfeasible direction
� Deformation of camberline
Multi-constraint airfoil optimization RAE2822
surface pressure distribution
N. Gauger, Institute of Aerodynamics and Flow Technology
Objective function
� Reduction of drag in 2 design points
Design points
� 1 : M�=0.734, CL = 0.80 , � = 2.8�, Re=6.5x106, xtrans=3%, W1=2
� 2 : M�=0.754, CL = 0.74 , � = 2.8�, Re=6.2x106, xtrans=3%, W2=1
Constraints
� No lift decrease, no change in angle of incidence
� Variation in pitching moment less than 2% in each point
� Maximal thickness constant and at 5% chord more than 96% of initial
� Leading edge radius more than 90% of initial
� Trailing edge angle more than 80% of initial
Multipoint airfoil optimization RAE2822
),(2
1iid
ii MCWI ��
�
�
N. Gauger, Institute of Aerodynamics and Flow Technology
Parameterization� 20 design variables changing camberline, Hicks-Henne functions
Optimization strategy� Constrained SQP� Navier-Stokes solver FLOWer, Baldwin/Lomax turbulence model� Gradients provided by FLOWer Adjoint, based on Euler equations
Results
Pt � Mi Clt Cdt (.10-4) Cl Cdt (.10-4) ��Cd/Cdt�Cl/Clt �Cm/Cmt
1 2.8 0.734 0.811 197.1 0.811 135.5 -31.2% 0% +1.6%
2 2.8 0.754 0.806 300.8 0.828 215.0 -27.4% +2.7% +2.0%
Multipoint airfoil optimization RAE2822
N. Gauger, Institute of Aerodynamics and Flow Technology
1. design point 2. design point
shape geometry
Multipoint airfoil optimization RAE2822
N. Gauger, Institute of Aerodynamics and Flow Technology
volume formulation (Jameson et al.)
surface formulation (Gauger)
Formulations of Adjoint Gradients
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e.g. )sin,(cos)( ��refpDT CCCk �
�
High accuracy butunpractical for 3D multi-block!
Way out:
N. Gauger, Institute of Aerodynamics and Flow Technology
n-th design variable
-gra
d(C
L)
0 10 20 30 40 50-16
-14
-12
-10
-8
-6
-4
-2
0
2
Adjoint (Volume)Adjoint (Surface)
n-th design variable
-gra
d(C
m)
0 10 20 30 40 50-5
-4
-3
-2
-1
0
1
2
Adjoint (Volume)Adjoint (Surface)
n-th design variable
-gra
d(C
D)
0 10 20 30 40 50-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Adjoint (Volume)Adjoint (Surface)
RAE2822M
�=0.73, � = 2.0�
50 design variables(B-spline)
Validation of Euler Adjoint
adjoint gradient volume formulationvs. surface formulation (Gauger)
drag
lift
moment
N. Gauger, Institute of Aerodynamics and Flow Technology
Wing Section
-�C
D
0 10 20 30-0.0006
-0.0005
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001 Adjoint (surf)Finite Differences
Wing Section
-�C
m
0 10 20 30-0.003
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
Adjoint (surf)Finite Differences
Wing Section
-�C
L
0 10 20 30-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001 Adjoint (surf)Finite Differences
ONERA M6 Wing(129x33x49)M
�=0.84, � = 3.1294�,
CLt = 0.3
32 design variables(spanwise twist)
adjoint gradient (surface formulation (Gauger) )vs. finite differences‘ gradient
Validation of Euler Adjoint
drag
lift moment
X
Y
Z
-4.59903 -3.09198 -1.58493 -0.0778884 1.42916 2.9362 4.44325 5.95029�1
Accuracy !
N. Gauger, Institute of Aerodynamics and Flow Technology
Drag reduction at constant lift� Mach number = 2.0� lift coefficient = 0.12
Design variables� fuselage contraction: 10 parameters� angle of attack: 1 parameter
Geometric constraints� minimum fuselage radius
Approach� FLOWer / FLOWer Euler Adjoint� keep lift constant by adjusting angle of attack� structured multi-block grid (MegaCads),
5 blocks, 200.000 grid points
Optimization of the body for a SCT configuration
N. Gauger, Institute of Aerodynamics and Flow Technology
10 sections controlled by Bezier nodesParameterizationResults
Optimization of the body for a SCT configuration
N. Gauger, Institute of Aerodynamics and Flow Technology
10 sections controlled by Bezier nodesParameterizationResults
Optimization of the body for a SCT configuration
�CD=3.5 %
Compared to FD:• 45 FLOWer calls saved
N. Gauger, Institute of Aerodynamics and Flow Technology
10 sections controlled by Bezier nodesParameterizationResults
Optimization of the body for a SCT configuration
Fuselage radius
N. Gauger, Institute of Aerodynamics and Flow Technology
Geometric constraints� minimum wing thickness distribution
along the spanwise directionApproach
� FLOWer / FLOWer Euler Adjoint� deforming mesh approach during optimization
Drag reduction at constant lift� Mach number = 2.0� lift coefficient = 0.1207
Design variables� twist deformation: 10 parameters� camberline (8 sections):80 parameters� thickness (8 sections) : 32 parameters� angle of attack: 1 parameter . 123 parameters
Optimization of the SCT´s wing
CamberlineThickness
N. Gauger, Institute of Aerodynamics and Flow Technology
Geometric constraints� minimum wing thickness distribution
along the spanwise directionApproach
� FLOWer / FLOWer Euler Adjoint� deforming mesh approach during optimization
Drag reduction at constant lift� Mach number = 2.0� lift coefficient = 0.1207
Design variables� twist deformation: 10 parameters� camberline (8 sections):80 parameters� thickness (8 sections) : 32 parameters� angle of attack: 1 parameter . 123 parameters
Optimization of the SCT´s wing
CamberlineThickness
Results
�CD=12.8 %
N. Gauger, Institute of Aerodynamics and Flow Technology
Geometric constraints� minimum wing thickness distribution
along the spanwise directionApproach
� FLOWer / FLOWer Euler Adjoint� deforming mesh approach during optimization
Drag reduction at constant lift� Mach number = 2.0� lift coefficient = 0.1207
Design variables� twist deformation: 10 parameters� camberline (8 sections):80 parameters� thickness (8 sections) : 32 parameters� angle of attack: 1 parameter . 123 parameters
Optimization of the SCT´s wing
CamberlineThickness
Results
�CD=12.8 %
Compared to FD:• 602 FLOWer calls saved
N. Gauger, Institute of Aerodynamics and Flow Technology
Optimization of the wing
BaselineOptimized
Optimization of the SCT´s wing
N. Gauger, Institute of Aerodynamics and Flow Technology
� The optimized wing set with the previous optimized body
Optimized Wing Combined with Optimized Body
13.2 % drag count decreased
N. Gauger, Institute of Aerodynamics and Flow Technology
cycle
d�/d
t,d�
1/dt
100 200 300 400
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
FLOWer ADJOINT
FLOWer MAIN
Navier-Stokes Adjoint
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e.g. drag reduction:
Adj. flux:= adjoint Euler flux + adj. viscous flux (mean flow)
(frozen turbulent viscosity)
Baldwin Lomax
Drag reduction flat plate
First verification results
Adjoint solververified against hand calculations!
N. Gauger, Institute of Aerodynamics and Flow Technology
Navier-Stokes Adjoint
,sin3 �� ��
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e.g. drag reduction:
Adj. flux:= adjoint Euler flux + adj. viscous flux (mean flow)
(frozen turbulent viscosity)
o
Ma2
73.0105.6Re 6
�
�
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Baldwin Lomax
Drag reduction RAE2822
First verification results
N. Gauger, Institute of Aerodynamics and Flow Technology
-0.38972 -0.237459 -0.0851982 0.0670627
�1
Navier-Stokes Adjoint
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Adjoint wall boundary condition
e.g. drag reduction:
Adj. flux:= adjoint Euler flux + adj. viscous flux (mean flow)
(frozen turbulent viscosity)
Baldwin Lomax
Drag reduction RAE2822
First verification results
N. Gauger, Institute of Aerodynamics and Flow Technology
• Continuous Adjoint - optimize then discretize - hand coded adjoint solver - time consuming in implementation - efficient in run and memory
• Discrete Adjoint / Algorithmic Differentiation (AD) - discretize then optimize - more or less automated generation of adjoint solver - for CFD restricted to FORTRAN (source to source) - memory effort increases (way out e.g. check-pointing)
• Hybrid Adjoint - use source to source AD tools - optimize differentiated code - merge “continuous and discrete” routines
Different Adjoint Approaches
N. Gauger, Institute of Aerodynamics and Flow Technology
• Adjoint approaches are essential for detailed aerodynamic design !• FLOWer ADJOINT / Continuous Adjoint Approach
- is very efficient - delivers exact gradients - handles multi-constraints (aerodynamic as well as geometric) - handles multipoint design problems - handles complex 3D multi-block configurations - highly validated for Euler
• Future work � MEGADESIGN - make Navier-Stokes adjoint as robust / validated as Euler adjoint - build up several adjoint turbulence models (use AD here, hybrid continuous/discrete adjoint approach) - implement adjoint solver for unstructured solver TAU (AD?) - one shot optimization (in collaboration with Uni Trier) - ...
Conclusion / Outlook