Continuous and Discrete adjoint methodologies within ESI CFD … · 2012. 4. 18. · Open...

Continuous and Discrete adjoint methodologies within ESI CFD solvers

Copyright © ESI Group, 2012. All rights reserved.Copyright © ESI Group, 2012. All rights reserved.

methodologies within ESI CFD solvers

Guillaume Pierrot

ESI Group

1

Rationale

Continuous Adjoint Solver within PAM-FLOW

Agenda

Copyright © ESI Group, 2012. All rights reserved.

Continuous Adjoint Solver within PAM-FLOW

Discrete Adjoint solver interfaced with CFD-ACE+

Morphing

Different problematics:

Rationale

global optimization


0.041.05

- 17 %

local improvement

- 2500%

V

One would like to improve the performance of some given design wrt some criterion: the “cost function”

may be lift, drag..

design parameters may be nodes coordinates, CAD parameters, level set position..

Rationale


position..

Standard numerical simulation provides a way to evaluate a design “a posteriori”

Optimal design tools automatically select the best (or at least a better) design wrt some given criterion

Different optimization methods:

non-gradient based (e.g. genetic algorithms): slow but may succed in finding a global optimum

gradient-based: quicker but stop as soon as a local optimum is found

Rationale


Different approaches for computing the gradient:

by Finite Differences PAM-OPT

by using the adjoint state PAM-FLOW Adjoint Solver, i-adjoint

Rationale


by using the adjoint state PAM-FLOW Adjoint Solver, i-adjoint

Both have their pro and cons

By Finite Differences:

Flexible, black box tool: very generic, no knowledge on the underlying application solver is needed

But requires a number of runs proportional to the number of design parameters

Rationale


parameters

Not sustainable when it tends to be large (e.g. free shape optimization)

In pratic, used together with a surrogate model (for CPU savings), which introduces further approximation and complexity

−−−≈∇

+++

n

nnnIIIIII

Iδβδβδβ

βδβββδβββδβββ ..,,,

21

222111

CFD run 1 CFD run 2 CFD run n

By using the adjoint state:

Less generic: does require some knowledge of the underlying application

Rationale


Less generic: does require some knowledge of the underlying application code

More complicated requires a dedicated tool, the so-called « adjoint solver »

But requires only one primal+one adjoint run cost is independant on the number of design parameters

Well suited for shape optimization

2 different approaches for computing the adjoint state:

Linearize/Dualize then Discretize Continuous Adjoint Method

Rationale


Linearize/Dualize then Discretize Continuous Adjoint Method

Discretize then Linearize/Dualize Discrete Adjoint Method

ESI adjoint solutions:

Continuous adjoint solver embedded into PAM-FLOW (vertex-centered FV)

Rationale


Continuous adjoint solver embedded into PAM-FLOW (vertex-centered FV) (2006)

Discrete adjoint library interfaced with CFD-ACE+ (cell-centered FV,multiphysics) (2012)

Incompressible Navier-Stokes:

( ) =∂+∇−⋅∇ pvvv 0η αα

Continous adjoint solver

symmetrysymmetry

inle

tin

let

groundground

ou

tlet

ou

tletwallwall


( )

( )

Γ==⋅∂=⋅

Γ=∂−

ΓΓ=

Γ=

=⋅∇

=∂+∇−⋅∇

symn

outn

groundwall

in

tvnv

Fvnp

v

Vv

v

pvvv

on 2100

on

on 0

on

0

0

,,; α

η

η

α

ααα

U

mass & momentum mass & momentum

conservationconservation

boundary conditionsboundary conditions

Optimization set-up:

( ) ( ) ∫∫


volumicvolumic cost cost

functionfunction

surfacic cost surfacic cost

functionfunction


( ) ( )

+

∂−=∂=

+=Ω ∫∫Ω∂Ω

bcs equations S-N

and with

..

,,,,),,,,(!

ts

vpvS

nnviSpvjSpvI

αβαβ

αβ

αβ

αβ

αβαβ

ααβ

ααβ

αβ

α

ηδτ

ττ

KKT theory:

( )

( ) ( ) ( )

=Ω αβ

αβ

ααβ

αβ

α τ dcbbbbqwSpvn

D

t

NDNl ,,,,,,,,,,,,


flow eqsflow eqs

bcsbcs

constitutive constitutive

relationsrelations


( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( )∫

∫∫∫

∫∫∫

∫

Ω

ΩΓΓ

ΓΓ∪ΓΓΩ

Ω

∂−+

∂−−+⋅+⋅−

+−+⋅−+⋅∇−

+⋅∇+Ω

αβ

αβ

αβ

αβ

αβαβ

αβ

αα

αα

α

β

αβ

αααβ

αβ

α

ηδττ

δτ

ττ

dvS

cvpbnvbtn

bVvbnFvq

wvvqwSpvI

symsym

groundwallinout

N

D

t

N

DinN

U

,,,,,,

KKT theory:

( )( ) 0=

∂

Ω∂αβ

αβ

α

αβ

αβ

ααβ

αβ

α τ

dcbbbbqw

dcbbbbwqSpvn

D

t

NND

n

D

t

NND

,,,,,,,

,,,,,,,,,,,,lFlow equations + BCsFlow equations + BCs



( )ββDNND

Adjoint equations + BCsAdjoint equations + BCs

Shape derivative (0 at Shape derivative (0 at

optimum)optimum)

( )( ) 0=

∂

Ω∂αβ

αβ

α

αβ

αβ

ααβ

αβ

α

τ

τ

,,,

,,,,,,,,,,,,

Spv

dcbbbbwqSpvn

D

t

NNDl

( )Ω∂

Ω∂ αβ

αβ

ααβ

αβ

α τ dcbbbbwqSpvn

D

t

NND ,,,,,,,,,,,,l

Adjoint equations:


( ) ∂

−

∂

∂=∇+∇⋅∇−∇⋅−∂−jj

pvvvvvααββ η


( )

∂

∂=⋅∇

∂

∂−

∂

∂∂=∇+∇⋅∇−∇⋅−∂−

∗

∗∗∗

p

jv

v

j

S

jpvvvvv

ααβ

βααβ

αβ η *

cost function dependant cost function dependant

source termssource terms

Adjoint bcs:


( )

ΓΓΓ⋅∂

∂−=∗ groundwallin

n

iv on

τ αUU

cost function dependant cost function dependant



( )

( ) ( )

( ) ( )

Γ=⋅

∂

∂−⋅

∂

∂−=⋅∂⋅

⋅∂

∂−=⋅

Γ

∂

∂+

∂

∂+⋅+⋅=∂−

ΓΓΓ⋅∂

−=

∗∗

∗∗∗∗

∗

sym

α

β

α

β

n

out

α

β

α

β

n

groundwallin

tnS

jt

v

itvn

n

inv

nS

j

v

ivnvnvvvnp

nv

on 21

on

on

,,; αητ

η

τ

ααα

α

αUU

Shape derivative:


( ) ( )∫Γ

∗ ⋅

∂

∂+∇⋅∇−

Ω∂

∂=

Ω∂

∂

wall

nS

jSvv

Iϕη

αβ

αβ

ααl


keep the flow / change the shapekeep the flow / change the shape keep the shape / change the flowkeep the shape / change the flow

Application to Aero Force:


( ) ( ) ( )∫∫ΓΓ

⋅⋅=⋅∂−⋅=

wallwall

dndvdnpI n βατη

( ) =∇+∇⋅∇−∇⋅−∂− ∗∗∗ 0pvvvvv *

ααβα

β η

cost functioncost function


( )

=⋅∇

=∇+∇⋅∇−∇⋅−∂−

∗

∗∗∗

0

0

v

pvvvvv *α η

( ) ( )∫Γ

∗ ⋅∇⋅∇−=Ω∂

∂

wall

nvv ϕη ααladjoint systemadjoint system

adjoint bcsadjoint bcs

shape derivativeshape derivative

source termsource term

( ) ( )

( )

Γ==⋅∂=⋅

Γ⋅+⋅=∂−

ΓΓ=

Γ−=

∗∗

∗∗∗∗

∗

∗

symn

outn

groundin

wall

tvnv

vnvnvvvnp

v

dv

on 2100

on

on 0

on

,,; αη

ηα

U

Pressure Drop:


( ) ( ) ( ) ( )∫∫∫ΓΓΓΩ

⋅⋅+⋅−=∂−⋅+−=∇+=

walloutin

vnnvvvnvvpvI n βαα τηη 22 2222

U

( ) =∇+∇⋅∇−∇⋅−∂− ∗∗∗ 0pvvvvv *

ααβα

β η

cost functioncost function


( )

=⋅∇

=∇+∇⋅∇−∇⋅−∂−

∗

∗∗∗

0

0

v

pvvvvv *α η

( ) ( )∫Γ

∗ ⋅∇⋅∇−=Ω∂

∂

wall

nvv ϕη ααladjoint systemadjoint system

adjoint bcsadjoint bcs

shape derivativeshape derivative

( ) ( )

( ) ( )

Γ−

⋅−−

⋅+⋅=∂−

Γ−=

Γ=

∗∗∗∗

∗

∗

out

n

in

wall

F

nvvn v

vnvnvvvnp

Vv

v

on

2

on

on 0

2

η


Some complication: turbulent viscosity


( )

=

+=αβη

ηηη

Sgt

t

Smagorinsky modelSmagorinsky model

new variablesnew variables


KKT: ( )

( )

( ) ∫Ω

−+

Ω

=Ω

µη

τ

µητ

αβ

αβ

αβ

ααβ

αβ

α

αβ

αβ

ααβ

αβ

α

t

n

D

t

NDN

t

n

D

t

NDN

Sg

dcbbbbqwSpv

dcbbbbqwSpv

0 ,,,,,,,,,,,,

,,,,,,,,,,,,,,

l

l

standard lagrangianstandard lagrangian

turbulent turbulent

lagrangianlagrangian

new variablesnew variables

Modified adjoint equations:


( ) ∂

−

∂

∂=

∂

∂−∇+∇⋅∇−∇⋅−∂−jjg

pvvvvv ηη ααββ


( )

∂

∂=⋅∇

∂

∂+∇∇=

∂

∂−

∂

∂∂=

∂

∂∂−∇+∇⋅∇−∇⋅−∂−

∗

∗∗

∗∗∗∗

p

jv

jvv

v

j

S

j

S

gpvvvvv

tηη

ηη

αα

ααβ

βαβ

βααβ

αβ

*

new termnew term

volumetric cost function is allowed to volumetric cost function is allowed to

depend on turbulent viscositydepend on turbulent viscosity

Modified adjoint bcs:


( )

ΓΓ∂

−=i

v on U

new termsnew terms


( )

( ) ( )

( ) ( )

Γ=

∂

∂⋅+⋅

∂

∂−⋅

∂

∂−=⋅∂⋅

⋅∂

∂−=⋅

Γ

∂

∂+

∂

∂+

∂

∂+⋅+⋅=∂−

ΓΓ⋅∂

∂−=

∗∗∗

∗∗∗∗∗

∗

sym

α

β

α

β

n

out

α

β

α

β

n

wallin

nS

gttn

S

jt

v

itvn

n

inv

nS

gn

S

j

v

ivnvnvvvnp

n

iv

on 21

on

on

,,; αηητ

ηη

τ

β

αβ

αααα

α

β

αβ

αU

Modified shape derivative:


( ) ( )∫ ⋅

∂∂

+∂

+∇⋅∇−∂

=∂

ngjj

SvvI

ϕη ααα~l

new termnew term


( ) ( )∫Γ

∗ ⋅

∂

∂

∂

∂+

∂

∂+∇⋅∇−

Ω∂

∂=

Ω∂

∂

wall

nS

gj

S

jSvv

I

t

ϕη

ηαβ

αβ

αβ

αα~l

αβ

αβηη

S

gS

∂

∂+=~

linearized linearized

turbulent turbulent

viscosityviscosity

More complications..

turbulence models (k-epsilon,Spalart-Allmaras..)

law of the wall



natural convection (+Temperature equation)

MHD

Chemistry ..


( )( )

= 0β

β

,

,!

uR

uI

PhysicsPhysics

exact derivation

u

Iu

u

RT

∂

∂−=

∂

∂ ∗

Dual PhysicsDual Physics

Co

ntin

uo

us a

djo

int s

olv

er

Pri

mal so

lver


dis

cre

tizati

on

consistency?

Co

ntin

uo

us a

djo

int s

olv

er

Pri

mal so

lver

CFD-ACE+Open FOAMFluent

( ) 0, =βhh uR

Discrete PhysicsDiscrete Physics

hh

T

u

Iu

u

Rh

∂

∂−=

∂

∂ ∗

Discrete dualDiscrete dual

dis

cre

tizatio

n

( )

=⋅∇

∇=∇⋅∇+∗

∗∗

0

2

v

pvvv µε )(*

=⋅∇

∇=∆−⊗⋅∇

0v

pvvv µNavierNavier--

StokesStokes

Pros:

Relatively easy-to-implement

Independant of the numerical scheme of the application code

CPU and memory efficient

Continuous adjoint solver


Cons:

Gradient inconsistency: the computed adjoint state is not the adjoint state of the computed physical field

Requires by hand differentiation of the underlying physical model may be tricky (turbulence models..)

High cost maintenance: any model addendum in the primal solver requires a specific development effort counterpart in the adjoint solver

Enriched with converters in 2010 so that it can accomodate results of alternative CFD codes

the CFD may be run using OpenFOAM or Star-CCM+ and the adjoint using PAM-FLOW



Only tet mesh

Integrated within ESI Visual process environment

Both academic and industrial proof of value

Drag reduction: 13% after 1 optimization cycleDrag reduction: 13% after 1 optimization cycle



Blue: Original surfaceBlue: Original surfaceMagenta: optimized surfaceMagenta: optimized surface

Initial design

Optimized design

Drag coefficient

0.342

0.298

Predicted drag coefficient:

Cd = 0.342



-30%

Optimization set-up:

Discrete adjoint solver

mesh nodes mesh nodes

coordinatescoordinates


( )( )

= 0uxRts

uxI

,..

,!

KKT theory: ( ) ( ) ( )uxRuuxIuuxT

,),(,,∗∗ +=l

coordinatescoordinates

DOF vector: (v1,v2,v3,p)DOF vector: (v1,v2,v3,p)

KKT theory:


( ) ( )0=

∂

+∂

=∂ ∗

∗

uuxRuxIuux

T,),(,,l

discrete

adjoint


( ) ( )0=

∂

∂+

∂

∂=

∂

∂ ∗u

u

uxR

u

uxI

u

uux ,),(,,ladjoint equation

( ) ( ) ∗∗

∂

∂+

∂

∂=

∂

∂u

x

uxR

x

uxI

x

uuxT

,),(,,l sensitivity

vector

Discrete vs. Continuous shape derivative:

*u

x

R

x

I

x

TT

∂

∂+

∂

∂=

∂

∂l


dis

cre

ted

iscre

te


xxx ∂∂∂

( ) ( )∫Γ

∗ ⋅

∂

∂−∇⋅∇−

Ω∂

∂=

Ω∂

∂

wall

nS

jSvv

Iϕη

αβ

αβ

ααl

dis

cre

ted

iscre

te

co

ntin

uo

us

co

ntin

uo

us

hh00 BUT:BUT:Ω∂

∂≠

∂

∂ I

x

IT

Indeed:


T

x

I

∂

∂

Ω∂

∂I Lie Lie

derivativederivative

Sensitivity Sensitivity

to the node to the node

positionspositions


keep the flow / change the shapekeep the flow / change the shape advect the flow as shape changes advect the flow as shape changes

They are connected: ( ) ( )UU

Iu

u

II

x

I

wall

T

∇⋅

∂

∂−∇⋅

∂

∂−

Ω∂

∂→

∂

∂ϕϕ

Similarily:


( ) ( ) ( )

∇⋅

∂

∂−

∇⋅

∂

∂−

Ω∂

∂→

∂

∂ ∗∗

∗ UuU

Ruu

u

RuRu

x

RT

wall

TT

ϕϕ ,,, *

( )

∇⋅

∂

∂+

∂

∂−

Ω∂

∂→

∂

∂ ∗U

U

Iu

U

RI

x

TT

ϕ,l


Hence:( )

( ) ( )

∇⋅

∂

∂+

∂

∂−

Ω∂

∂+

∇⋅

∂

+

∂

−Ω∂

→∂

∗∗

uu

Iu

u

RuR

UU

uUx

TT

wallwall

ϕ

ϕ

,,

,

Finally:

00

00

( )

∇⋅

∂

∂+

∂

∂−

Ω∂

∂≈

∂

∂ ∗U

U

Iu

U

RI

x

T

wall

T

wall

ϕ,l

3 choices for computing the shape derivative:

*u

x

R

x

I

x

TT

∂

∂+

∂

∂=

∂

∂l


dis

cre

ted

iscre

te

exactexact


xxx ∂∂∂

( ) ( )∫Γ

∗ ⋅

∂

∂−∇⋅∇−

Ω∂

∂=

Ω∂

∂

wall

nS

jSvv

Iϕη

αβ

αβ

ααl

dis

cre

ted

iscre

teco

ntin

uo

us

co

ntin

uo

us

( )

∇⋅

∂

∂+

∂

∂−

Ω∂

∂≈

∂

∂ ∗U

U

Iu

U

RI

x

T

wall

T

wall

ϕ,l

hyb

rid

hyb

rid

valid at valid at convergenceconvergence

good (?) good (?) compromise if compromise if one can not one can not easily move easily move the nodes the nodes

The adjoint matrix:


333

222

111

321

321

321

v

R

v

R

v

Rv

R

v

R

v

Rv

R

v

R

v

R

vvv

vvv

vvv

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

3

2

1

v

Rv

Rv

R

p

p

p

∂

∂∂

∂∂

∂

p

R

p

R

p

Rvvv

∂

∂

∂

∂

∂

∂ 321

p

Rp

∂

∂

AA BB

B’B’ CC

==


Typical block stencil:

Huge, (not so) sparse, unsymmetric, undefinite matrix

Hard to solve (saddle-point system)

Use an optimal preconditioner (DDM, Multigrid, Subdomain deflation)


Or a segregated-like approach: ( ) n

TT

nnx

u

R

u

IxxP

∂

∂−

∂

∂=−+1


If matrix has to be assembled, out of coring may be needed

uu ∂∂

( )( )

+

−′

−′

−=

−

−−

−−

I

AD

IB

I

CBDB

I

I

BDIP A

A

A

0

00

0

0

0

1

11

11

( )ADA DIAGα=


( )( )

= 0β

β

,

,!

uR

uI

PhysicsPhysics

exact derivation

u

Iu

u

RT

∂

∂−=

∂

∂ ∗

Dual PhysicsDual PhysicsP

rim

al so

lver


dis

cre

tizati

on


Pri

mal so

lver

CFD-ACE+

( ) 0, =βhh uR

Discrete PhysicsDiscrete Physics

hh

T

u

Iu

u

Rh

∂

∂−=

∂

∂ ∗

Discrete dualDiscrete dual

co

nsis

ten

cy?

( )

=⋅∇

∇=∇⋅∇+∗

∗∗

0

2

v

pvvv µε )(*

exact derivation

Pros:

Gradient consistency: the computed adjoint state is the real adjoint of the computed physical field



Cons:

Depends on the very numerical scheme of the application code

How to build the discrete adjoint operator?

Different approaches for building the discrete adjoint operator:

By hand



Automatic (Algorithmic) Differentiation

By hand:

Requires an exhaustive knowledge of the primal solver



tedious and error prone!

High maintenance cost

Each application solver adjoint derivation has to be adressed independantly

By Algorithmic Differentiation:

The source code istelf is differentiated

2 modes: direct / reverse

2 approaches: source transformation / operator overloading



Pros & cons:

Low cost maintenance: the code is differentiated once and for all (no further effort for accomodating newly developped models in the primal solver)

But each application solver adjoint derivation has to be adressed mostly independantly

May turn to be tedious and time consuming depending on the code structure and programming language

Very invasive: requires full access to the source code

Severe CPU efficiency and memory consumption challenges

Academic validation against FD


Dynamic library interfaced with CFD-ACE+


Any mesh topology

Limitations:

Not yet parallelized

Enriched with converters so that it can accomodate results of alternative CFD codes

the CFD may be run using OpenFOAM or Star-CCM+ and the adjoint using PAM-FLOW


A word on rigorous validation:

-step_factor*Computed_Gradient


Relative error :=100* ABS(Expected_Improvement-Effective_Improvement)/Effective_Improvement

Expected Improvement := step_factor*Computed_Gradient_Norm**2

Effective Improvement := I_oldshape-I_newshape

I_oldshapeI_newshape


Airfoil Tet

Airfoil Hex

Aifoil Hex

S-Bend (Viscart)

Ahmed Body (Viscart)


Physics Laminar Laminar Frozen Turbulent

Frozen Turbulent

Frozen Turbulent

Relative error (%)

0,07 0,14 0,37 5,15 0,11

At first bound interior node displacement to sruface nodes one thanks to harmonic mapping (ALE-like):

Minimizes mesh isotropic distortion

Morphing


ΓΩ∂=

Γ=

=∆−

wall

wallsurf

x

xx

x

\ on 0

on

0

δ

δδ

δ

surfxKx δδ =int

Not good for boundary layers (anisotropic mesh)

Use rigidification instead :

Morphing


( )( )

( )( )

∑

∑

−∈

−∈

∈−

−

=

1

1

1

1

nblNj

ij

nblNj

jij

nbli

i

i

xx

xxx

x

I

I

/

/

)(

δ

δwallwall

ii

j1j1

j2j2

Then combine Laplace and rigidification:

Morphing

=∆− x 0δ


( )( )

( )( )

∑

∑

−∈

−∈

≤∈−

−

=

1

1

1

1

nblNj

ij

nblNj

jij

Nnbli

i

i

xx

xxx

x

I

I

/

/

)(

δ

δ

ΓΩ∂=

Γ=

=∆−

wall

NNbl

x

xx

x

\

)(

on 0

on

0

δ

δδ

δ

NΓ

For boundary nodes, apply LSQ morphing:

Sample the surface nodes

Apply LSQ operator (exact for polynomial up to

Morphing

( ) ( )j

s

i

surf yx δδ →


Apply LSQ operator (exact for polynomial up to desired degree)

∑∈

Φ=Sj

j

j

ssurf xyxx )()( δδ

( )

−==Φ ∑

∈≤≤

Si

iinii xxWJx2

1)()(minarg)( λλ ε

k

nodei

kiki pxpxp ∀=∑ )()(λ subject to

Finally, adapt the shape derivative accordingly via chain rule:

:

Morphing


∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

s

surf

surfsurf

in

ins y

x

xx

x

xy

lll

Do not forget this step, otherwise the gradient is wrong !

:

Morphing option available both within PAM-FLOW Continuous Adjoint Solver and ACE+ Discrete one

Morphing


:

Limited to small displacements

:

Additionaly, the tool provides the value of the maximal step factor so that all volume remain positive

:

Continuous and Discrete adjoint methodologies within ESI CFD … · 2012. 4. 18. · Open...

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Transcript of Continuous and Discrete adjoint methodologies within ESI CFD … · 2012. 4. 18. · Open...