Constrained Single-Step One-Shot Method with Applications ...

Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes

Constrained Single-Step One-Shot Method with Applications in Aerodynamics

Nicolas R. Gauger1),2)

1) German Aerospace Center (DLR) BraunschweigInstitute of Aerodynamics and Flow TechnologyNumerical Methods Branch (C2A2S2E)

2) Humboldt University BerlinDepartment of Mathematics


Collaborators

• HU Berlin: A. Griewank, A. Hamdi, E. Özkaya, A. Plocke

• DLR: C. Ilic

• Uni Paderborn: A. Walther

• Uni Trier: V. Schulz, S. Schmidt


Goal: s.t.

where and are the state and design variables.

Given fixed point iteration (e.g. pseudo-time

stepping) to solve PDE

Assumptions:

always invertible. IFT given s.t.

contractive:

),(min uyfu

,0),( =uycy u

),(1 uyGy kk =+

., 1,2CfG ∈1),(),( <≤= ρuyGuyG T

yy

Problem Statement

.0),( =uyc

⇒

G

yu !,∃yc

∂∂

.0),( =uyc


⎪⎩

⎪⎨

⎧

−==

=

−+

+

+

Tkkkukkk

Tkkkyk

kkk

uyyNBuuuyyNy

uyGyOS

),,(),,(

),()(

11

1

1

yyuyGuyfuyyL T)),((),(),,( −+=One-Shot approach

⎪⎩

⎪⎨

⎧

===−=

=−=

0),,(0),,(

0),(

Tuu

Tyy

y

uyyNLyuyyNL

yuyGL

primal update

dual update

design update

One-step one-shot (step k+1):

Stationary point:

yyuyyN T−=43421

),,(

Aims: Choose B such that: • Convergence of (OS).• Bounded retardation.

shifted Lagragian


Jacobian of the extended iteration:

Whenever we can define such that

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−=

∂∂

=−−−

+++

uuTuuy

yuTyyy

uy

uyykkk

kkk

NBIGBNBNGNGG

uyyuyyJ

111),,(

111*

0

),,(),,(

***

B

constJGy <

−

−

∗)(ˆ1)(1

ρρ

Bounded retardation

we have bounded retardation.


where

Eigenvalues of are the zeros of the equation

( ) .)(

,)()(1

1⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

−−

IGIG

NNNN

IIGGH uy

uuuy

yuyyTy

Tu

λλλ

*J

0))()1det(( =+− λλ HB

).1(21

−HB f

Necessary (but not sufficient) condition for contractivity:

Necessary condition for contractivity

0fTBB = and [Griewank, 2006]


Remark:

Deriving (sufficient) conditions on B for J* to have a

spectral radius smaller than 1 has proven difficult.

Instead, we look for descent on the augmented Lagrangian

,),,(2

),(2

:),,(22

4342144 344 2143421yyNyuyyNyuyGuyyL TT

ya −+−+−=

βα

where α > 0 and β > 0 .

Exact penalty function: La

primal residual Lagrangiandual residual


),,,( uyyMsLLL

au

ay

ay

−=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∇∇∇

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

− Tu

Ty

uyyNByuyyN

yuyGuyys

),,(),,(

),(),,(

1

The full gradient of La is given by

where

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−

=BNG

GIINIGI

MT

yuT

u

y

yyT

y

βαβ

βα

,0),(,0,),(

and .

Correspondence condition


Consequence (Correspondence condition):

There is a 1-1 correspondence between the stationary points

of La and the roots of s if

,0]))((det[ ≠−−−− yyyT

y NIGIGI βαβ

for which it is sufficient that

.1)1( 2yyNβραβ +>−

Correspondence condition

[Hamdi, Griewank, 2008]


.uuyuTyuu

Tu NNNGGB ++= βα

Theorem (Descent condition):is a descent direction for all large positive B

if and only if),

2())(

21)(

2())(

21( 1

yyT

yyyyT

yy NIGGINIGGI ββαβ ++−+>+− −

),,( uyys

.2

1)1( yyNβραβ +>−which is implied by

Descent condition

Theorem: A suitable B is given by:

Satisfied for with .yyNc =2)1(2,2ρ

αβ−

==c

c

[Hamdi, Griewank, 2008]


A suitable B is given by

.)1,2,1 >>=⇒= αβc

uyuT

yuTa

u NNyNGyGL +−+−=∇ )()( βα

.uuyuTyuu

Tu NNNGGB ++= βα

Descent for with .yyNc =2)1(2,2ρ

αβ−

==c

c

One-step one-shotAerodynamic shape design

Instead BFGS updates for the Hessian

(In practice choose

.)()(00

2

**

yuuTT

yuuT

B

uuyuTyuu

Tu

au NyNGyGNNNGGL

434213214444 34444 21→→

−+−+++=∇ βαβα

The gradientis evaluated by Algorithmic Differentiation (AD).


Transonic case: RAE 2822 at Ma = 0.73 with α = 2°

Cost function: drag (cd)

TAUij (2D Euler) + mesh deformation + parameterization

First and second derivatives by AD tool ADOL-C

Geometric constraint: constant thickness

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

One-step one-shotAerodynamic shape design


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

design vector (P) → defgeo → difgeo → meshdefo → flow solver → CD

xnew dx m

surface grid grid

Px

xdx

dxm

mC

dPdC new

new

DD

∂∂⋅

∂∂⋅

∂∂

⋅∂∂

=)(

)(Id

xxx

xdx

new

oldnew

new

=∂−∂

=∂∂ )()(and

TAUij_AD meshdefo_AD defgeo_AD

Automatic Differentiation of Entire Design Chain


One-step one-shotDrag reduction • RAE 2822, M = 0.73, α = 2.0°• inviscid flow, mesh 161x33 cells• 20 design variables (Hicks-Henne)• One-step one-shot

Flow Solver: TAUij• Compressible Euler• Explicit RK-4• Multigrid• Implicit residual smoothing


Primal compared to coupled iteration

Retardation-Factor = 4 [Özkaya, Gauger, 2008]


Update the penalty parameter in each one-shot step k:

hC

u

Du

∇∇

=0λ

0),1(1 >+=+ cchkk λλ,0 ↑⇒> λh

Penalty function for lift: 0),(!

arg, ≤−= hCCh LettL

Treatment of lift constraint by penalty multiplier method

A good starting value is:

Redefine objective function: hCf D λ+=

↓⇒< λ0h

),(..),(min arg, uyGyandCCtsuyC ettLLDu=≥

0;),(min →+ hhuyCDuλ


Constrained One-Step One-ShotDrag reduction by constant lift• RAE 2822, M = 0.73, α = 2.0°• inviscid flow, mesh 161x33 cells• 40 design variables (Hicks-Henne)• One-step one-shot

Flow Solver: TAUij• Compressible Euler• Explicit RK-4• Multigrid• Implicit residual smoothing


Retardation-Factor = 6

Primal compared to coupled iteration

[Gauger, Plocke, 2008]


History of Penalty Multiplier

[Gauger, Plocke, 2008]


Extension to Navier-Stokes (ELAN Code)

Flow Solver: ELAN (TU Berlin)• 3D Navier-Stokes (RANS)• incompressible with pressure

correction• multiblock• k-ω (Wilcox) turbulence

model (and others)• Fortran (20.000 lines)

AD Tool: TAPENADE (INRIA)• source to source• reverse for first derivatives• tangent on reverse for

second derivatives


Drag reduction with lift constraint • NACA 4412• Re = 1.000.000, α=5.1°• RANS• k-ω (Wilcox) turbulence model• 300 surface mesh points

Approaches for Optimization• one-shot method• entire design chain differentiated• gradient smoothing• penalty multiplier method

Extension to Navier-Stokes (ELAN Code)



Extension to Navier-Stokes (ELAN Code)Approaches for Optimization• one-shot method• entire design chain differentiated• gradient smoothing• penalty multiplier method



5% drag reduction

Extension to Navier-Stokes (ELAN Code)Approaches for Optimization• one-shot method• entire design chain differentiated• gradient smoothing• penalty multiplier method

[Özkaya, Gauger, 2009]


Thanks for your attention!

Constrained Single-Step One-Shot Method with Applications ...

Documents

Transcript of Constrained Single-Step One-Shot Method with Applications ...