constrained single-step one-shot method with applications
TRANSCRIPT
Slide 1
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
Slide 2
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Collaborators
• HU Berlin: A. Griewank, A. Hamdi, E. Özkaya, A. Plocke
• DLR: C. Ilic
• Uni Paderborn: A. Walther
• Uni Trier: V. Schulz, S. Schmidt
Slide 3
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
⎪⎩
⎪⎨
⎧
−==
=
−+
+
+
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
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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.
Slide 6
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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]
Slide 7
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
),,,( 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
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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]
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
.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]
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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).
Slide 12
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
Slide 13
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
• 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
Slide 14
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
Slide 15
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Primal compared to coupled iteration
Retardation-Factor = 4 [Özkaya, Gauger, 2008]
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
Slide 17
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
Slide 18
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Retardation-Factor = 6
Primal compared to coupled iteration
[Gauger, Plocke, 2008]
Slide 19
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
History of Penalty Multiplier
[Gauger, Plocke, 2008]
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Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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
Slide 21
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
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)
Slide 22
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Drag reduction with lift constraint • NACA 4412• Re = 1.000.000, α=5.1°• RANS• k-ω (Wilcox) turbulence model• 300 surface mesh points
Extension to Navier-Stokes (ELAN Code)Approaches for Optimization• one-shot method• entire design chain differentiated• gradient smoothing• penalty multiplier method
Slide 23
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Drag reduction with lift constraint • NACA 4412• Re = 1.000.000, α=5.1°• RANS• k-ω (Wilcox) turbulence model• 300 surface mesh points
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]
Slide 24
Trier, June 3-5, 2009 Nico GaugerPDE Constrained Optimization of Certain and Uncertain Processes
Thanks for your attention!