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


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


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!


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


MEGAFLOWMEGAFLOW

Block-structured capabilityBlock-structured capability

MegaCadsICEM Hexa

FLOWer

Unstructured capabilityUnstructured capability

Centaur

TAU


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


� 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


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


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




(M�=0.2, Re=3.52x106)





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




(M�=0.2, Re=3.52x106)





0 1 2 3 4 5 66.50

7.00

7.50

8.00

8.50

9.00

optimization cycles

Fobj(X) = 100�CD


CL = const. = 3,77Re = 3,56�106

-Cm � -Cm,start



Ma�

= 0,2

cutout geometries

pressure distribution

test case optimized

initial configuration, �=20.160

optimal configuration, �=17.930




(M�=0.2, Re=3.52x106)





0 1 2 3 4 5 66.50

7.00

7.50

8.00

8.50

9.00

optimization cycles

Fobj(X) = 100�CD


CL = const. = 3,77Re = 3,56�106

-Cm � -Cm,start



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


Finite Differences

• Finite Differences n design variables requiren+1 flow calculations

metric sensitivity � pressure variation � aerodynamic sensitivity

��

�

CrefD p

CpMC �

�� 2

2 dlnn yx )sincos( ��

dlnnCC y

Cxp

ref

)sincos(1�� ,

variation of i-th design variable

i-th component of cost function´s gradient

n--loop


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






0��

��

�

��

�

�

yg

xf

tw

)21()1( 2vEp �

��

��

��

�

pMppCp 2

)(2�

� ��

Cyxp

refD dlnnC

CC )sincos(1

��

� ��

Cxyp

refL dlnnC

CC )sincos(1

��

� ��

Cmxmyp

refm dlyynxxnC

CC ))()((1

2

compressible Euler equations

with

��

�

�

��

�

�

��

uHuv

puu

f

�

�

�

�

2

��

�

�

��

�

�

��

vHpv

vuv

g

�

�

�

�

2

��

�

�

��

�

�

�

Evu

w

�

�

�

�

, ,

perfect gas

dimensionless pressure

drag, lift and pitching moment:

Governing Equations, Aerodynamic Coefficients


0��

��

��

�

�

�

�

��

��

�

�

�

�

�

ywg

xwf

t

TT��

0,..., �� yx , 0�w� on far field

)(32 Idnn yx �� on wall

adjoint Euler equations

boundary conditions

� ��

C

IKdlxypI )()( 32 ��

� ��

D

TT dAgxfygxfy )()( ��

adjoint formulation of cost function’s gradient

Continuous Adjoint Approach

�: vector of adjoint variables


)sincos(2)( 2 ��

yxref

D nnCpM

Cd ��

��

dlnnCC

CK yC

xpref

D )sincos(1)( ��

)sincos(2)( 2 ��

xyref

L nnCpM

Cd ��

��

dlnnCC

CK xC

ypref

L )sincos(1)( ��

))()((2

)( 22 mxmyref

m yynxxnCpM

Cd ��

��

dlyynxxnCC

CKC

mxmypref

m ))()((1)( 2 � ��

drag

Continuous Adjoint Approach

pitching moment

lift


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-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

0.02��/�t (1-Block)CL (1-Block)CD (1-Block)��/�t (2-Block)CL (2-Block)CD (2-Block)��1/�t (1-Block)��1/�t (2-Block)

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-th Design Variable

-�C

m

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

AdjointFinite Differences


-�C

L

0 10 20 30 40 50-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2



-�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


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


� 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






Objective function


� M� =0.73, �=2.00°

Constraints


Approach



Optimizer









Objective function


� M� =0.73, �=2.00°

Constraints


Approach



Optimizer









Objective function


� M� =0.73, �=2.00°

Constraints


Approach



Optimizer









Objective function


� M� =0.73, �=2.00°

Constraints


Approach



Optimizer


Objective function


� M� =0.73, �=2.0°

Constraints

� Lift, pitching moment and angle of attack held constant


Approach


� Constraints handled byfeasible direction

� Deformation of camberline

Multi-constraint airfoil optimization RAE2822


Objective function


� M� =0.73, �=2.0°

Constraints



Approach






Objective function


� M� =0.73, �=2.0°

Constraints



Approach





surface pressure distribution


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 ��

�

�


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%



1. design point 2. design point

shape geometry



volume formulation (Jameson et al.)

surface formulation (Gauger)

Formulations of Adjoint Gradients

� ��

C

IKdlxypI )()( 32 ��

� ��

D

TT dAgxfygxfy )()( ��

)()( IKdlvnunwIC

yxTH� ��

wTH = ( �, �u, �v, �H )surface formulation (Weinerfelt)

� ��

Cyx

TH dlynxnvwIkI )())()((div ��

�

�

e.g. )sin,(cos)( ��refpDT CCCk �

�

High accuracy butunpractical for 3D multi-block!

Way out:


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)


-gra

d(C

m)

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2



-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


RAE2822M

�=0.73, � = 2.0�

50 design variables(B-spline)

Validation of Euler Adjoint

adjoint gradient volume formulationvs. surface formulation (Gauger)

drag

lift

moment


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 !


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


10 sections controlled by Bezier nodesParameterizationResults





�CD=3.5 %

Compared to FD:• 45 FLOWer calls saved




Fuselage radius


Geometric constraints� minimum wing thickness distribution

along the spanwise directionApproach

� FLOWer / FLOWer Euler Adjoint� deforming mesh approach during optimization


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








CamberlineThickness

Results

�CD=12.8 %








CamberlineThickness

Results

�CD=12.8 %

Compared to FD:• 602 FLOWer calls saved


Optimization of the wing

BaselineOptimized



� The optimized wing set with the previous optimized body

Optimized Wing Combined with Optimized Body

13.2 % drag count decreased


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

,sin3 ��

055

��

��

� nn

�

,cos2 ��

o

Ma0

734.0105.6Re 6

�

�

��

�

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 flat plate

First verification results

Adjoint solververified against hand calculations!



,sin3 ��

055

��

��

� nn

�

,cos2 ��





o

Ma2

73.0105.6Re 6

�

�

��

�

Baldwin Lomax

Drag reduction RAE2822



-0.38972 -0.237459 -0.0851982 0.0670627

�1


,sin3 ��

055

��

��

� nn

�

,cos2 ��

o

Ma2

73.0105.6Re 6

�

�

��

�





Baldwin Lomax

Drag reduction RAE2822



• 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


• 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

Documents

Introductory Course to Design Optimization