Adjoint Approaches in Aerodynamic Shape …gauger/gauger_VKI...1 N. Gauger et al. Intro to Optimization and MDO, VKI, March 6-10,2006 Adjoint Approaches in Aerodynamic Shape Optimization

1N. Gauger et al.Intro to Optimization and MDO, VKI, March 6-10,2006

Adjoint Approaches in Aerodynamic Shape Optimization and MDO Context I/II

Nicolas Gauger 1), 2)

1) DLR BraunschweigInstitute of Aerodynamics and Flow Technology

Numerical Methods Branch2) Humboldt University Berlin

Department of Mathematics

Introduction to Optimization and Multidisciplinary DesignVKI, March 6-10, 2006

http://www.mathematik.hu-berlin.de/~gauger


CollaboratorsWith contributions to this lecture:

• DLR: N. Kroll, J. Brezillon, A. Fazzolari,

R. Dwight, M. Widhalm

• HU Berlin: A. Griewank, J. Riehme

• Fastopt: R. Giering, Th. Kaminski

• TU Dresden: A. Walther, C. Moldenhauer

• Uni Trier: V. Schulz, S. HazraUniversity of Trier

FastOpt


Content of lecture

Why adjoint approaches?

What is an adjoint approach?

Continuous and discrete adjoint approaches / solvers

Validation and Application in 2D and 3D, Euler and Navier-Stokes

Algorithmic / Automated Differentiation (AD)

Coupled aero-structure adjoint approach

Validation and application in MDO context

One shot approaches


Requirements on CFD• high level of physical modeling

– compressible flow– transonic flow– laminar - turbulent flow – high Reynolds numbers (60 million)– large flow regions with flow separation – steady / unsteady flows

• complex geometries• short turn around time

Use of CFD in Aerodynamic Aircraft Design


Consequencessolution of 3D compressible Reynolds averaged Navier-Stokes equations turbulence models based on transport equations (2 – 6 eqn)models for predicting laminar-turbulent transition flexible grid generation techniques with high level of automation(block structured grids, overset grids, unstructured/hybrid grids)link to CAD-systemsefficient algorithms (multigrid, grid adaptation, parallel algorithms...)large scale computations ( ~ 10 - 25 million grid points)…

Use of CFD in Aerodynamic Aircraft Design


MEGAFLOW Software

Structured RANS solver FLOWer

block-structured grids moderate complex configurationsfast algorithms (unsteady flows)design optionadjoint option

Unstructured RANS solver TAU

hybrid grids very complex configurationsgrid adaptation fully parallel softwareadjoint option


Physical model→ 3D compressible Navier-Stokes equations→ arbitrarily moving bodies→ steady and time accurate flows→ state-of-the-art turbulence models (RSM)

Reynolds-Averaged Navier-Stokes Solver FLOWer

Numerical algorithms→ 2nd order finite volume discretization

(cell centered & cell vertex option)→ central and upwind schemes→ multigrid→ implicit treatment of turbulence equations→ implicit schemes for time accurate flows→ preconditioning for low speed flow→ vectorization & parallelization→ adjoint solver

Grid strategy→ block-structured grids→ discontinuous block boundaries→ overset grids (Chimera)→ deforming grids


Physical model→ 3D compressible Navier-Stokes equations→ arbitrarily moving bodies→ steady and time accurate flows→ state-of-the-art turbulence models

Reynolds-Averaged Navier-Stokes Solver TAU

Numerical algorithms→ 2nd order finite volume discretization

based on dual grid approach→ central and upwind schemes→ multigrid based on agglomeration → implicit schemes for time accurate flows→ preconditioning for low speed flow→ optimized for cash and vector processors→ MPI parallelization

Grid strategy→ unstructured/hybrid grids→ semi-structured sublayers→ overset grids (Chimera)→ deforming grids → grid adaptation (refinement, de-refinement)


Dual grid approach• solver independent of

cell types of primary grid• efficient edge-based data structure• agglomeration of dual cells

for coarser meshes (multigrid)

Hybrid Navier-Stokes Solver TAU

primary grid fine dual grid

dual grid, 2nd level dual grid, 3rd level

primary griddual grid


Hybrid Navier-Stokes Solver TAU

Local Mesh Adaptation• local grid refinement and de-refinement

depending flow solution • reduction of total number of grid points• efficient simulation of complex flow

phenomena

Overlapping grid technique• efficient approach for simulation

of complex configurations withmovable control surfaces (maneuvering aircraft)

• separate grids for movable surfaces

• parallel implementation

wing flap


• M∞=0.85, Re=32.5x106

• coupled CFD/structural analysis for wing deformation at α ≈ 1.5°• FLOWer, kω turbulence model, fully turbulent

ValidationHiReTT Wing/Body Configuration

3.5 million grid points


• M∞=0.85, Re=32.5x106

• coupled CFD/structural analysis for wing deformation at α ≈ 1.5°• FLOWer, kω turbulence model, fully turbulent

ValidationHiReTT Wing/Body Configuration

3.5 million grid points


Requirementscomplex configurations

compressible Navier-Stokes equationswith accurate models for turbulence and transition

validated and efficient CFD codes

multi-point design, multi-objective optimization, MDO

large number of design variables

physical and geometrical constraintsmeshing & mesh deformation techniques ensuring grid qualityefficient optimization algorithms

automatic framework

parameterization based on CAD model

Aerodynamic Shape Optimization


Requirementscomplex configurations

compressible Navier-Stokes equationswith accurate models for turbulence and transition

validated and efficient CFD codes

multi-point design, multi-objective optimization, MDO

large number of design variables

physical and geometrical constraintsmeshing & mesh deformation techniques ensuring grid qualityefficient optimization algorithms

automatic framework

parameterization based on CAD model


⇓

⇒ Sensitivity baseddeterministic optimizationstrategies !!!

⇒


Parametrizedairfoil

Design space

I cost

T

niiPII

,...,1

,......,=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=∇−

δδ

“ “ :

Line search

Search direction



0=∂∂

+∂∂

+∂∂

yg

xf

tw

∞∞

∞−=

pMppCp 2

)(2γ

∫ +=C

yxpref

D dlnnCC

C )sincos(1 αα

∫ −=C

xypref

L dlnnCC

C )sincos(1 αα

∫ −−−=C

mxmypref

m dlyynxxnCC

C ))()((12

Compressible 2D Euler-Equations

while

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+=

uHuv

puu

f

ρρρ

ρ2

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+=

vHpv

vuv

g

ρρ

ρρ

2

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

Evu

w

ρρρρ

, ,

Dimensionless pressure

Drag, lift, pitching moment coefficients

Pressure (ideal gas)

)21()1( 2vEp r

−−= ργ

Governing Equations and Aerodynamic Coefficients


• Finite Differences n design variables requiren+1 flow calculations

Metric sensitivities → pressure variation → aerodynamic sensitivity

∫∞∞

=Cref

D pCpM

C δγ

δ 22 dlnn yx )sincos( αα +

dlnnCC y

Cxp

ref

)sincos(1 αδαδ∫ ++ ,

i-th component of cost function‘s gradient

i-loopi=1,...,n

Variation of i-th design variable

Finite Differences


High number of design variables

• Finite Differences n design variables require n+1 flow calculations

• Adjoint Approach n design variables require 1 flow and1 adjoint flow calculation

Independent of number of design variables

High accuracy

Motivation of Adjoint Approach





Convection Eq.


• Continuous Adjoint- optimize then discretize- hand coded adjoint solvers- time consuming in implementation- efficient in run and memory

• Discrete Adjoint / Algorithmic Differentiation (AD)- discretize then optimize- hand coding of adjoint solvers or …- … more or less automated generation- memory effort increases (way out e.g. check-pointing)

Different adjoint approaches


How to get the gradient using adjoint theory

Let the optimization problem be stated as

and with the governing equations

with W the flow variables, X the mesh and D the design variables.

The goal here is to determine the derivatives of I with respect to D

We define the Lagrangian which is identical to I and its derivatives with respect to the design variables D

( ) 0,, =DXWR

( ),,, min D

DXWI

RIL TΛ+=


The derivatives of L with respect to the design variables D are:

( ) ( )( )Λ+= DXWRDXWIdDd

dDdL T ,,,,



( ) ( )( )

⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

+∂∂

Λ+⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

+∂∂

=

Λ+=

DR

dDdX

XR

dDdW

WRT

DI

dDdX

XI

dDdW

WI

DXWRDXWIdDd

dDdL T

,,,,




( ) ( )( )

⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

+⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

+⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

=

⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

+∂∂

Λ+⎭⎬⎫

⎩⎨⎧

∂∂

+∂∂

+∂∂

=

Λ+=

DRT

DI

dDdX

XRT

XI

dDdW

WRT

WI

DR

dDdX

XR

dDdW

WRT

DI

dDdX

XI

dDdW

WI

DXWRDXWIdDd

dDdL T

,,,,




=0

}

}}

The expensive component can be canceled by solving the adjoint

The derivatives of L with respect to D are:

equation

dDdW

WRT

WI

DRT

DI

dDdX

XRT

XI

dDdL

⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

+⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

+⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

=

Variations w. r. t. the flow variables

expensive to evaluate

Partial variations according to the design variables

relatively inexpensive

Metric sensitivities

relatively inexpensive with finite differences



After solving the adjoint equation,

the derivatives of L with respect to D are evaluated according to

⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

+⎭⎬⎫

⎩⎨⎧

∂∂

Λ+∂∂

=DRT

DI

dDdX

XRT

XI

dDdL

0=∂∂

Λ+∂∂

WRT

WI










0=∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

−∂

∂−

ywg

xwf

t

TT ψψψAdjointEuler-Equations:

Boundary conditions:

∫ ++−−=C

IKdlxypI )()( 32 ξξ δψδψδ

∫ +−+−−D

TT dAgxfygxfy )()( ξξηηηξ δδψδδψ

Adjoint volume formulation of cost function’s gradient:

Ψ: Vector of adjoint variables

)(32 Idnn yx −=+ ψψ

0=wδ,0,..., =ηξ δδ yxWall:Farfield:

Continuous Adjoint Approach


)sincos(2)( 2 ααγ yx

refD nn

CpMCd +=

∞∞

dlnnCC

CK yC

xpref

D )sincos(1)( αδαδ∫ +=

)sincos(2)( 2 ααγ xy

refL nn

CpMCd −=

∞∞

dlnnCC

CK xC

ypref

L )sincos(1)( αδαδ∫ −=

))()((2

)(22 mxmyref

m yynxxnCpM

Cd −−−=∞∞γ

dlyynxxnCC

CKC

mxmypref

m ))()((1)( 2 ∫ −−−= δ

Drag

Pitching moment

Lift

Continuous Adjoint Approach


Continuous adjoint• Euler implemented in FLOWer & TAU• surface formulation for gradient evaluation• one shot method (FLOWer)• coupled aero-structure adjoint (FLOWer) • Navier-Stokes (frozen μ) implemented

in FLOWer, robustness problems

Discrete adjoint• implemented in TAU • Euler & RANS with several turbulence

models• currently high memory requirements• experience with automatic differentiation

(FLOWer and TAUijk) moment

pressure drag

TAU-Code

comparison of gradients (airfoil, inviscid)

Adjoint solvers


Continuous adjoint• Euler implemented in FLOWer & TAU• surface formulation for gradient evaluation• one shot method (FLOWer)• coupled aero-structure adjoint (FLOWer) • Navier-Stokes (frozen μ) implemented

in FLOWer, robustness problems

Discrete adjoint• implemented in TAU • Euler & RANS with several turbulence

models• currently high memory requirements• experience with automatic differentiation

(FLOWer and TAUijk) moment

pressure drag

comparison of gradients (airfoil, inviscid)

TAU-Code

comparison of gradients (3-airfoil, viscous)

TAU-Code

Adjoint solvers


flowsolution

Rae2822M = 0.734α = 2.0˚

drag optimization

adjointsolution

3v multigrid

3v multigrid

Continuous adjoint Euler solver TAU

Runge-Kutta versus LUSGS


Continuous adjoint solver FLOWer

Adjoint solver on block-structured grids

• continuous adjoint approach• implemented in FLOWer• cost functions: lift, drag & moment

and combinations • adjoint solver based on multigrid• Euler & Navier-Stokes (frozen μ)

convergence history, FLOWer

-12.4408 -9.55489 -6.66898 -3.78306 -0.897145 1.98877 4.87468 7.7606

ψ1


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, α = 2.0°50 design variables(B-spline)

Validation of continuous adjoint solver in FLOWerAdjoint approach vs. finite differences‘ gradient

drag

lift

moment

finite differences: 51 calls of FLOWer MAINadjoint approach:1 call of FLOWer MAIN3 calls of FLOWer ADJOINT


Validation of adjoint gradient based optimization

Objective function

4 Drag reduction for RAE 2822 airfoil

4 M∞ =0.73, α=2.00°

Constraints

4 Constant thickness

Approach

4 FLOWer Euler Adjoint

4 Deformation of camberline(20 Hicks-Henne functions)

Optimizer

4 Steepest Descent

4 Conjugate Gradient

4 Quasi Newton Trust Region


Validation of adjoint gradient based optimization

Objective function


4 M∞ =0.73, α=2.00°

Constraints


Approach


4 Deformation of camberline(20 Hicks-Henne functions)

Optimizer

4 Steepest Descent

4 Conjugate Gradient

4 Quasi Newton Trust Region


Content of lecture

Why adjoint approaches?

What is an adjoint approach?

Continuous and discrete adjoint approaches / solvers

Validation and Application in 2D and 3D, Euler and Navier-Stokes

Algorithmic / Automated Differentiation (AD)

Coupled aero-structure adjoint approach

Validation and application in MDO context

One shot approaches

Adjoint Approaches in Aerodynamic Shape Optimization and MDO Context II


Orthogonalprojection

ii i

DTi

D bb

CbCb ∑=

∇+−∇=

2

123

03 =ba Ti , 2,1=i

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−=

=

∑=

+++ . 2,1

,

12

111

11

lbbabab

ab

i

l

i i

lTi

ll

},,{},,{ 321 DmL CCCaaa −∇∇∇=

)()( )(kLL XCrC ≈

0)()()(

)(

)(

)(

)( =∇=k

kT

LXk

kL

rrC

drXdC

k

},,{ 321 bbb

Schmidt - orthogonalization

:

In direction b3 the drag is reduced while thelift and pitching moment are held constant

it holdsIn direction r(k) the drag is reduced whilethe lift is held constant

.

X(k)

LC∇ DC∇−

r(k)

r

Treatment of Constraints


Orthogonalprojection

ii i

DTi

D bb

CbCb ∑=

∇+−∇=

2

123

03 =ba Ti , 2,1=i

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=−=

=

∑=

+++ . 2,1

,

12

111

11

lbbabab

ab

i

l

i i

lTi

ll

},,{},,{ 321 DmL CCCaaa −∇∇∇=

)()( )(kLL XCrC ≈

0)()()(

)(

)(

)(

)( =∇=k

kT

LXk

kL

rrC

drXdC

k

},,{ 321 bbb

Schmidt - orthogonalization

:

In direction b3 the drag is reduced while thelift and pitching moment are held constant

it holdsIn direction r(k) the drag is reduced whilethe lift is held constant

.

X(k)

LC∇ DC∇−

r(k)

r

Treatment of Constraints

A lot of other strategies andcommercial packages areavailable !!!


Objective function


4 M∞ =0.73, α=2.0°

Constraints

4 Lift, pitching moment and angle of attack held constant


Approach


4 Constraints handled byfeasible direction

4 Deformation of camberline

Multi-constraint airfoil optimization RAE2822


Objective function


4 M∞ =0.73, α=2.0°

Constraints

4 Lift, pitching moment and angle of attack held constant


Approach


4 Constraints handled byfeasible direction

4 Deformation of camberline

Multi-constraint airfoil optimization RAE2822

surface pressure distribution


Objective function

4 Reduction of drag in 2 design points

Design points

4 1 : M∞=0.734, CL = 0.80 , α = 2.8°, Re=6.5x106, xtrans=3%, W1=2

4 2 : M∞=0.754, CL = 0.74 , α = 2.8°, Re=6.2x106, xtrans=3%, W2=1

Constraints

4 No lift decrease, no change in angle of incidence

4 Variation in pitching moment less than 2% in each point

4 Maximal thickness constant and at 5% chord more than 96% of initial

4 Leading edge radius more than 90% of initial

4 Trailing edge angle more than 80% of initial

Multipoint airfoil optimization RAE2822

),(2

1iid

ii MCWI α∑

=

=


Parameterization4 20 design variables changing camberline, Hicks-Henne functions

Optimization strategy4 Constrained SQP

4 Navier-Stokes solver FLOWer, Baldwin/Lomax turbulence model

4 Gradients provided by FLOWer Adjoint, based on Euler equations

Results

Pt α Mi Clt Cdt (.10-4) Cl Cdt (.10-4) ΔCd/Cdt ΔCl/Clt ΔCm/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:

Surface formulations:

Adjoint gradient formulations

∫ ++−−=C

IKdlxypI )()( 32 ξξ δψδψδ

∫ +−+−−D

TT dAgxfygxfy )()( ξξηηηξ δδψδδψ

)()( IKdlvnunwIC

yxTH∫ ++−= δδψδ

wTH = ( ρ, ρu, ρv, ρH )

∫ +⋅+=C

yxTH dlynxnvwIkI )())()((div δδψδ rr

e.g. )sin,(cos)( ααrefpDT CCCk =r

High accuracy butunpractical for 3D multi-block!

Way out:

II.

I.

others …


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, α = 2.0°50 design variables(B-spline)

Adjoint gradient formulation: Volume formulation vs. surface formulation (I.)

drag

lift

moment


Objective function

4 drag reduction by constant lift

Design point

4 Mach number = 2.0

4 lift coefficient = 0.12

Constraints

4 fuselage incidence

4 minimum fuselage radius

4 wing planform unchanged

4 minimum wing thickness distribution in spanwise direction

Optimization of SCT Configuration (SCT – Supersonic Cruise Transporter)


Approach

4 FLOWer code in Euler mode with target lift option

4 Lift kept constant by adjusting angle of attack

4 FLOWer code in Euler adjoint mode

4 Adjoint gradient formulation: Surface formulation (II.)

4 Structured mono-block grid (MegaCads), 230.000 grid points

Optimization strategy

4 Quasi-Newton Method (BFGS algorithm)

Optimization of SCT Configuration


Fuselage

10 sections controlled by Bezier nodes

Design variables h fuselage: 10 parametersh twist deformation: 10 parametersh camberline (8 sections): 32 parametersh thickness (8 sections): 32 parametersh angle of attack: 1 parameter .

85 parameters



Camberline Thickness

Deformation in8 sections

Deformation in 8 sections


85 parameters




85 parametersThickness and camberline

Normalised airfoil



11 times faster than classical approach

14.6 Drag Counts

optimized geometry

baseline geometry



11 times faster than classical approach

14.6 Drag Counts

optimized geometry

baseline geometry



and Area RuleRadius of the fuselage in freestream direction



Wing section and pressure distribution

η=0.24

η=0.49

η=0.92



Validation of Discrete Adjoint Solver in TAU

viscous flow around RAE2822 airfoil, M=0.73, α=2.80, Re=6.5x106


Shape Optimization Based on Discrete AdjointObjective

• drag minimization for RAE 2822 airfoil atconstant lift, pitching moment and AoA

• projected steepest descent strategy • flow solver: viscous TAU-Code, SA model• adjoint solver: viscous discret TAU adjoint

20 design parameters

M∞ =0.73, α=2.80°, Re=6.5x106

Shape Optimization Based on Discrete Adjoint


• CD reduction at constant CL with varying angle-of-attack.

• Takeoff configuration, Re=14.7x106, Ma=0.1715, RANS+SAE

• Parameterized is only the “hidden” nose of the flap ~10 design vars.

• Exact adjoint gradients with Conjugate Gradient optimization.

• Drag reduction of 9 counts – lift unchanged.

Shape Optimization Based on Discrete Adjoint


Discrete Adjoint Solver

Advantage• exact discrete adjoint in TAU for most commonly used models and discretizations• solution via Krylov method requires 5% -10% of time needed the for flow solution

Problem• memory requirement for large scale

application efficient storage strategy(recalculation of terms)

TAU Main + Jacobianstorage

+ linear sol.storage

Memory (bytes) 25M 165M 290M

Factor increase x1.0 x6.6 x11.6

points in 1GB 2x106 300x103 170x103

Approximations of discrete adjoint• 1st order discretization (FOA)• assumption of constant coefficients

in the JST scheme (CCA)• gradients based on Euler solution• adjoint solution based on thin layer

viscous fluxes• assumption of constant eddy

viscosity




Advantage• exact discrete adjoint in TAU for most commonly used models and discretizations• solution via Krylov method requires 5% -10% of time needed the for flow solution

Problem• memory requirement for large scale

application efficient storage strategy(recalculation of terms)

TAU Main + Jacobianstorage

+ linear sol.storage

Memory (bytes) 25M 165M 290M

Factor increase x1.0 x6.6 x11.6

points in 1GB 2x106 300x103 170x103

Approximations of discrete adjoint• 1st order discretization (FOA)• assumption of constant coefficients

in the JST scheme (CCA)• gradients based on Euler solution• adjoint solution based on thin layer

viscous fluxes• assumption of constant eddy

viscosity



Algorithmic Differentiation (AD)

Work in progress and results

• ADFLOWer generated with TAF (3D Navier-Stokes,k-w), first verifications and validation

• Adjoint version of TAUij (2D Euler) + mesh deformationand parameterization with ADOL-C, validated versus finite differences and first applications

• First and second derivatives of a “FLOWer-Derivate”(2D Euler) + mesh deformation and parameterizationgenerated with TAPENADE, used for One Shot (Piggy Back)




Evaluation of Simple Example:

Simple Example



Forward Derived Evaluation Trace of Simple Example


idvdy

vi … Adjoint variable


Reverse Derived Evaluation Trace of Simple Example


FastOpt

Test configuration2d NACA12k-omega (Wilcox) turbulence modelcell-centred metric2 time steps on fine gridtarget sensitivity: d lift/ d alpha

StepsModifications of FLOWer code (TAF Directives, slight recoding, etc...)tangent-linear code (verification + useful per se small dimensional design problems) adjoint codeeifficient adjoint code

Major challengememory management (all variables in one big field 'variab')complicates detailed analysis and handling of deallocation

ADFLOWer by TAF


TAF CPUs Code lines solve rel CPU solve memoryNominal 166000 1.0 57tangent 293 268000 3.3 75adjoint 253 310000 6.3 489

Usually better for larger configurations

Ma = 0.734α = 2.8°Re = 6x10^6kw turbulence model

ADFLOWer FastOpt


Demonstrates convergence of discrete sensitivities including turbulence (tangent linear model)

Same sensitivity for Euler adjoint

Same sensitivity for Navier-Stokes adjoint once it runs for 2000 time steps


ADFLOWer FastOpt


Demonstrates convergence of discrete sensitivities including turbulence (tangent linear model)

Same sensitivity for Euler adjoint

Same sensitivity for Navier-Stokes adjoint once it runs for 2000 time steps


ADFLOWer FastOpt


Differentiate entire design chain

• Adjoint version of entire design chain by ADOL-C:TAUij (2D Euler) + mesh deformation + parameterization

• validated versus finite differences

design vector (P) → defgeo → difgeo → meshdefo → TAUij → CDxnew dx m

surface grid (static) grid (static)

Px

xdx

dxm

mC

dPdC new

new

DD

∂∂

⋅∂∂

⋅∂∂

⋅∂

∂=

)()(

Idx

xxxdx

new

oldnew

new

=∂

−∂=

∂∂ )()(

and

TAUij_AD meshdefo_AD defgeo_AD


• RAE2822Ma = 0.73α = 2.0°(mesh 161x33)

• Dessign variables:20 Hicks-Henne forcamberline deformation

• run time- primal: 2 minutes- adjoint: 16 minutes

• run time memory- primal: 8 MB- adjoint: 45 MB

Differentiate entire design chainValidation


Differentiate entire design chainApplication

RAE2822Ma = 0.73α = 2.0°(mesh 161x33)


• Continuous Adjoint- optimize then discretize- hand coded adjoint solvers- time consuming in implementation- efficient in run and memory

• Discrete Adjoint / Algorithmic Differentiation (AD)- discretize then optimize- hand coding of adjoint solvers or …- … more or less automated generation- 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


Coupled Aero-Structure Adjoint

Motivation

Wing deflection up to 7% of wing span!

Deflected aerodynamicoptimal shape can beworse than the initial …

Boeing 737Boeing 737--800 at ground and in cruise (Ma = 0.76)800 at ground and in cruise (Ma = 0.76)



AMP wing

15 design variables(shape bumping functions based on Bernstein polynomials)

Ma=0.78alpha=2.83

Drag reduction byconstant lift

Taking into accountstatic deformation

NASTRANshell/beam model126 nodes

FLOWer MAIN/ADJOINT15 design variablesMa=0.78alpha=2.83(300.000 cells)


A

TAD

S

TS

dR

dC

dR ψψ ~⎟

⎠⎞

⎜⎝⎛

∂∂

−∂

∂=⎟

⎠⎞

⎜⎝⎛

∂∂

Pd

dC

Pw

wC

PC

dPdC DDDD

∂∂

∂∂

+∂∂

∂∂

+∂

∂=

PR

PR

PC

dPdC ST

SAT

ADD

∂∂

−∂∂

−∂

∂= ψψ

S

TSD

A

TA

wR

wC

wR ψψ ~⎟

⎠⎞

⎜⎝⎛

∂∂

−∂

∂=⎟

⎠⎞

⎜⎝⎛

∂∂

0=AR

0=−= aKdRS

Structure:

Aerodynamics, e.g Euler Eqn.:

K: Symmetric stiffness matrixa: Aerodynamic forced: Displacement vectorP: Vector of Design variables


Adjoint Gradient:

Aero/Structure Adjoint System:

Conventional Gradient:

::

S

A

ψψ Aerodynamic Adjoint

Structure Adjoint~: Lagged ...



Pad

PK

PaKd

PR

KKd

aKddR

wa

waKd

wR

wC

PC

dC

PR

dR

S

TS

S

D

DD

AA

∂∂

−∂∂

=∂

−∂=

∂∂

==∂

−∂=

∂∂

∂∂

−=∂

−∂=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

)(

)(

)(

,

, : perturb shape by d,P → calculate change in CFD residual

: perturb shape by d,P → calculate change in drag coefficient

: treat → boundary condition∫ +∂∂

Cyx nn

wp )...sincos(... αα

: treat → boundary condition∫ ∂∂

C wp ......

… has been derived in the last lecture!



Finite Differences:Perturb the shape by each designvariable and converge the aero-elastic loop until stationary behavior

Coupled Aero-Structure Adjoint:Each 100 iterations the laggedis updated ...

Sψ~

AMP wing

Aψ


PR

PR

PC

dPdC ST

SAT

ADD

∂∂

−∂∂

−∂

∂= ψψ

Validation of Adjoint Gradient


15 design variablesMa=0.78alpha=2.83(300.000 cells)

AMP wing



PR

PR

PC

dPdC ST

SAT

ALL

∂∂

−∂∂

−∂

∂= ψψ

Validation of Adjoint Gradient


15 design variablesMa=0.78alpha=2.83(300.000 cells)

AMP wing




AMP wing

240 design variables(control points free form deformation)

Ma=0.78alpha=2.83

Drag reduction byconstant lift ΔCD= 24.9 %

ΔCL= 0.1%

feasible direction method



AMP wing


Ma=0.78alpha=2.83


baseline

optimized



Comparison of numerical effort:(PC Pentium IV, 2.6 GHz, 2GB RAM)

• Coupled adjoint: 15 days(11 gradient and 91 state evaluations)

• Finite differences: 227 days

AMP wing


Ma=0.78alpha=2.83



Aero-Structure MDO

FWW

CCR

D

L

−∝ ln

)1(0 ksWW λ+=

∑ −=

n

nks ))exp(ln(1

0

0

σσσβ

βλ=0.2 , σ0 =30.000 and β=40

Range R:

Weight W:

Fuel Weight F

Kreisselmeier-Steinhauser:


Aero-Structure MDO

FWW

CCR

D

L

−∝ ln

)1(0 ksWW λ+=

∑ −=

n

nks ))exp(ln(1

0

0

σσσβ

βλ=0.2 , σ0 =30.000 and β=40

Range R:

Weight W:

Fuel Weight F

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

+=

0

1

1ln

WFks

ksCC

D

L

λ

λ

Kreisselmeier-Steinhauser:PR

Pks

dPdks AT

∂∂

+∂∂

= ψ

pksnnn zyx ∂

∂−=++ 432 ψψψ

adjoint b.c.


AMP wing


Ma=0.78alpha=2.83

Range maximization byconstant lift

Aero-Structure MDO

ΔCD = -25 %

Δks = -10 %

ΔR = +37 %

feasible direction method


I∇

start geometryx0

ψ

x0

w

xn+1

k-loop

k-loop

Adjoint Based Optimization

min Ι (w,x)s.t. R(w,x)=0

optimizationstrategy

RANS solverR(wk,xn)=0

gradient

∫=∇V

mn

m dVxiI ))(,(w,)( δψ

Adjoint solverR*(w,ψk,xn)=0

dim x = M

n-loop n=1,…,N

m-loopm=1,…,M

All at once?


),(),(),,( xwRxwIxwL Tψψ −=

Simultaneous Pseudo-Time stepping- One Shot Approach -

0),(0),,(0),,(

==∇=∇

xwRxwLxwL

x

w

ψψ

(state equation)

(adjoint equation)

(design equation)

University of Trier

min Ι (w,x)

s.t. R(w,x)=0

dim x = M

( )( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡∇∇

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂∂∂

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ+Δ+Δ+

−

RLL

xRwRxRLLwRLL

xw

xxww

x

wT

xxxw

Twxww

1

0////

ψψψ

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−∇−∇−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ −

RLL

xRIxRB

Ixw

x

wT

1

0//0

00

ψ

KKT

Newton SQPmethod

inexact Newton rSQP method

simultaneous preconditionedpseudo time stepping


Lx∇

start geometryx0

ψk+1

x0

wk+1

xk+1

primal updatewk+1=wk-Δt·R(wk,xk)

gradient

∫ ++=∇V

mkkk

mx dVxlL ))(,,(w)( 11 δψ

k-loop dual update

ψk+1= ψk-Δt·R*(wk+1,ψk,xk)

}{ 111 LxRBLBtxx w

T

kxkkk ∇⎟

⎠⎞

⎜⎝⎛

∂∂

−∇Δ−= −−+

m-loopm=1,…,M

design update

Bk – BFGS updatesof reduced Hessian Lxx

Simultaneous Pseudo-Time stepping- One Shot Approach - University of Trier


Optimization problem• drag reduction for RAE 2822 • inviscid flow• M=0.73, a=20

Tools• FLOWer• FLOWer adjoint



Optimization at the cost of 4 flow simulations!



Optimal Design ScenarioPiggy–Back Approach

Problem: s.t.

where and are state and design variables

Available:

Code for and

Assumption:

and

Notation:

where the Lagrangian is formed w.r.t.

),(min uyf ,0),( =uycny ℜ∈ mu ℜ∈

),( uyf ),(),(),(1

uycuycy

yuyG−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−≈

)(, 1,2 mnCfG +ℜ∈ 1),( <≤∂∂ ρuyGy

,),(),(),,( yyLagrangianuyGyuyfuyyN +≡+≡0),(),( =−≡ yuyGuyc


Piggy–Back ApproachSingle-step-one-shot

→−=

→=

→=

−+

+

+

),(

),(

),(

,1

1

,1

1

kkkukkk

kkkyk

kkk

uyyNHuu

uyyNy

uyGy primal feasibility at

dual feasibility at

optimality at

*y

*y

*u

where reduced gradient

and is a suitable preconditioner

≈+= uuu fGyN

kH


Spectral AnalysisPiggy–Back Approach

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−=

∂∂

−−−

+++

uuTuuy

yuTyyy

uy

kkk

kkk

NHIGHNHNGNGG

uyyuyy

111

111

0

),,(),,(

1

2),(

)()(

],)([],)([)(

])1()(det[)(

−−−≡

∇≡

−+≡

IGGZ

IZNIZH

HHP

Ty

Tu

T

TTuy

T

λλ

λλλ

λλλ

has at as eigenvalues the roots of ),,( *** uyy

where

λ

Rows of span tangent space of],)([ IZ Tλ { }yuyG λ=),(


Contractivity in convex casePiggy–Back Approach

2/)1(101

−⇒−>⇔<

HHH

f

f

λλ i.e. positive definite

Numerical experience shows:

Reduced Hessian immediate blow-up

Projected Hessian full-step convergence⇒−≡ )1(HH

⇒≡ )1(HH


Transonic case: NACA 0012 at Ma = 0.8 with α = 2°

Cost function: glide ratio

“FLOWer-Derivate” (2D Euler) + mesh deformation +parameterization

First and second derivatives by AD tool TAPENADE

Geometric constraint: constant thickness

Camberline/Thickness decomposition,20 Hicks-Henne coefficients define camberline

Piggy–Back Approach - Application


min CD/CL (Inverse Glide Ratio)

Wing Shapes

Piggy–Back Approach - Results

NACA 0012

Ma = 0.8 with α = 2°


Thanks for your attention!

Adjoint Approaches in Aerodynamic Shape …gauger/gauger_VKI...1 N. Gauger et al. Intro to Optimization and MDO, VKI, March 6-10,2006 Adjoint Approaches in Aerodynamic Shape Optimization

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