automated extension of fixed-point pde solvers for optimal
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Slide 1
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
Nico Gauger1),2), Andreas Griewank2)
1) DLR BraunschweigInstitute of Aerodynamics and Flow TechnologyNumerical Methods Branch
2) Humboldt University BerlinDepartment of Mathematics
Slide 2
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
CollaboratorsWith contributions to this talk:
• HU Berlin: J. Riehme
• Fastopt: R. Giering, Th. Kaminski
• TU Dresden: C. Moldenhauer, S. Schlenkrich,
A. Walther
FastOpt
Slide 3
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
OutlineMEGAFLOW Software (FLOWer, TAU)ADFLOWerAutomatic differentiation of entire design chainAll-at-Once / One-Shot / Piggy-BackFeasibility studyGoals
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
Slide 4
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
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
Slide 5
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
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 and tested
• First and second derivatives of a “FLOWer-Derivate”(2D Euler) + mesh deformation and parameterizationgenerated with TAPENADE, used for All-at-Once (Piggy-Back)
Slide 6
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
FastOptADFLOWer by TAF( )
Test configuration2d NACA0012k-omega (Wilcox) turbulence modelcell-centred metric2000 time steps on fine gridtarget sensitivity: d lift/ d alpha
StepsModifications of FLOWer code (TAF Directives, slight recoding, etc...)tangent-linear code (verification) adjoint codeefficient adjoint code
Major challengememory management (all variables in one big field 'variab')complicates detailed analysis and handling of deallocation
(Gauger,Giering,Kaminski)
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
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
(Gauger,Giering,Kaminski)
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Ma = 0.734α = 2.8°Re = 6x10^6kw turbulence model
ADFLOWer
(Gauger,Giering,Kaminski)
Demonstrates convergence of discrete sensitivities including turbulence
Same sensitivity for Navier-Stokes adjoint (Wilcox kw) and tangent linear model
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Demonstrates convergence of discrete sensitivities including turbulence
Same sensitivity for Navier-Stokes adjoint (Wilcox kw) and tangent linear model
ADFLOWer
(Gauger,Giering,Kaminski)
Ma = 0.734α = 2.8°Re = 6x10^6kw turbulence model
Slide 10
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
• Adjoint version of entire design chain by ADOL-C (TU-Dresden)• 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
(Gauger,Walther)
Automatic Differentiation of Entire Design Chain
Slide 11
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
• Run time (2000 fixed-point iterations)- primal: 2 minutes- adjoint: 16 minutes
• Tape size: 340 MB (reverse accumulation approach!)[Christianson in 94]
• Run time memory- primal: 8 MB- adjoint: 45 MB
Automatic Differentiation of TAUij (fixed-point solver)
(Schlenkrich,Walther)
Slide 12
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Automatic Differentiation of Entire Design Chain
Drag reduction • RAE 2822, M = 0.73, α = 2.0°• inviscid flow, mesh 161x33 cells• 20 design variables (Hicks-Henne)• steepest descent
(Gauger,Moldenhauer)
First Application / Validation:
Slide 13
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
∫=∇V
mn
m dVxiI ))(,(w,)( δψ
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
Adjoint solverR*(w,ψk,xn)=0
dim x = M
n-loop n=1,…,N
m-loopm=1,…,M
“All-at-Once”!
∫=∇V
mn
m dVxiI ))(,(w,)( δψ
Slide 14
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Activities on “All-at-Once”
• MEGADESIGN (BMWi Project) (Ends 2007):
Aerodynamic shape optimization using simultaneous pseudo-timestepping (published in JCP 2005: Hazra, Schulz, Brezillon, Gauger)
• Optimization with PDE (DFG-SPP 1253) (Starts this Year):
Multilevel parameterizations and fast multigrid methods for shape optimization (Gauger / Schulz)
Automated extension of fixed point PDE solvers with bounded retardation (Gauger / Griewank / Slawig)
Slide 15
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Problem: s.t.
where and are state and design variables
Available:
Code for and
Contractive fixed-point iteration:
and
Shifted Lagrangian:
Lagrangian is formed w.r.t.
),(min uyf ,0),( =uycny ℜ∈ mu ℜ∈
),( uyf ),(),(),(1
uycuycy
yuyG−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−≈
)(, 1,2 mnCfG +ℜ∈ 1),( <≤∂∂ ρuyGy
,),(),(),,( yyLagrangianuyfyuyGuyyN TT +≡+≡0),(),( =−≡ yuyGuyc
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
Slide 16
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−+
+
+
+
)()()(
)(
,,1
,,
,,
,
1
1
1
1
kkkukk
kkku
kkky
kky
k
k
k
k
uyyNHuuyyNuyyN
uyN
uuyystates
adjoint states
adjoint designs
designs
positive definite matrix preconditioner0>kH
Strategy: differentiate → iterate
“Piggy-back”: automated extension to coupled iteration:
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
(Griewank)
Slide 17
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Jacobian of the extended iteration:
Whenever we can define such that
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−=
∂∂
=−−−
+++
uuTuuy
yuTyyy
uy
kkk
kkk
NHIGHNHNGNGG
uyyuyyJ
1*
1*
1*
*
111*
0
),,(),,(
H
constGJ
GJ
yy
<≈−−
))(log())(log(
)(1)(1 **
ρρ
ρρ
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
(Griewank)
we have bounded retardation
Slide 18
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
where
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
−−
IGIG
NNNN
IIGGH uTy
uuuy
yuyyTy
Tu
11 )(,)()(
λλλ
0))()1det(( *
Eigenvalues of are the zeros of the equation*J
=+− λλ HH
)1/()(* λλ −> HH 1−≤λ
Necessary condition for contractivity:
for
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
(Griewank)
Slide 19
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
)1/()(* λλ −> HH 1−≤λ
Necessary condition for contractivity:
for
Usually not satisfied by reduced Hessian
Promising alternative: )1(−H
)1(H
[Griewank 05]
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
(Griewank)
Slide 20
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
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
Feasibility Study
(Gauger,Griewank,Riehme)
Slide 21
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
min CD/CL (Inverse Glide Ratio)
Wing Shapes
NACA 0012
Ma = 0.8 with α = 2°
Results
Simulation -Piggy-back -
original -optimized -
(Gauger,Griewank,Riehme)
Slide 22
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Globalization: enforce convergence by line search w.r.t.
augmented Lagrangian
With , the directional derivative is given by
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−−−−−−−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
−=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
∇∇∇
−−−u
y
yuu
yuyyy
uyyyT
u
y
u
y
T
u
y
y
NHyNyG
HNGNGIINGINGI
NHyNyG
NHyNyG
ppp
111 )2/()2/()2/()()2/()2/()2/()(
βαβββαβα
yG
yyNyuyyNyuyGuyyp Ty −+−+−=
2
2
2
2),,()2(),()2(:),,( βα
which is a definite quadratic form for contractive
and suitable weights ., ℜ∈βα
))(21( Tyyy GGG +=
(Griewank)
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
Slide 23
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
For suitable choices and the smallest eigenvalue of
must converge to a KKT point.
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−−−−−−−−
HNGNGIINGINGI
yuu
yuyyy
uyyy
)2/()2/()2/()()2/()2/()2/()(
βαβββαβα
[ ] [ ] [ ]),(),,(),(,)1(,, ,1
*,,,111 kkkukkkykkkkkkkkkk uyyNHuyyNuyGuyyuyy −+++ −+−= γγ
can be bounded away from zero. Then one can design a line
search procedure for determining such that the iterates
βα , H
kγ
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
(Griewank)
Slide 24
Oxford, June 1st, 2006 Nico Gauger and Andreas Griewank3rd European Workshop on Automatic Differentiation
Goals:Full fidelity optimization without model reductionAutomated adjoint generation with verifiable resultsBounded retardation of the original convergence rate Bounded increase of computational effort per stepBounded increase in overall storage requirementEfficient storage and linear algebra for the preconditioner of the combined iterationSelective evaluation or approximation of second derivativesStability with respect to the design parameterization
Automated extension of fixed-point PDE solversfor optimal design with bounded retardation
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