Inverse method for pyrolysable and ablative materials with optimal control formulation

IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse

Inverse method for pyrolysable and ablative materials with

optimal control formulation S.Alestra1, J.Collinet2, and F.Dubois3

[email protected] [email protected]

2EADS ASTRIUM STLes Mureaux, France

1EADS Innovation WorksToulouse, FRANCE

3Conservatoire National des Arts et Métiers

Paris, France

[email protected]

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Astrium Space Transportation


Atmospheric re-entry missions Reentry Aerothermodynamics Team at EADS ASTRIUM-ST

Multiphysics : aerodynamics, aerothermodynamics, plasma

design and sizing of the Thermal Protection System (TPS) of the aerospace vehicles

the identification of heat fluxes is of great industrial interest

ARD

Industrial problem

Huygens probe(on Titan)

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Internal energy balance

Pyrolysis:Arrhenius equation

Surface energy balance

Surface Recession

xhm

tdTAhF

xT

xtTC gg

T

Tgp

01

TBn

v

c

v

eAt

1

ConductionAblation

PyrolysisRadiationConvection

)(

)()()(

2

144

0

xThhHm

hhHmTThh

wrvc

wrcgrwwr

hydrchemmechc sssms

)(0 t ),( txT

xm

tg

Decompositionand mass balance

General equations of direct probleminput data: heat flux output data: temperature

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Evaluate the heat fluxes from temperature measurements

on thermal protection with ablation and pyrolysisInverse Heat problem is hard !! : see theory, diffusion aspect, ..

« Monopyro » 1D numerical tool, EADS ASTRIUM ST

The inverse method

(t), t in [O,T]p(t)= (t) ?

T in [0,T]

Pyrolysis gas

Material

Pyrolysed

PyrolysisZone

VirginM

aterial

Material Pyrolysable Ablative

Radiation Flux

Convection Flux

Heat Flux ofPyrolysis gas

Ablation heat flux

Radaition Flux

Blocking Flux

Radiation Flux

Convection Flux

FibresResinCoke

Pyrolytic Carbon

p(t)= (t) ?

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Direct Problem

vector of temperature T and ablation s, functions of time t and position x.

=> system of coupled nonlinear time domain evolution differential equations:

The other variables described above are hidden in the formulation of FSystem is rewritten in reduced variables

etsx )(1

etsxttxsTxT

pWFdtdW

f ,,,00)0,()0,(

,

0

)(

),(tstxT

W

1,0 ,t

s

e

T1 T2 T3

(t)X (t)

p(t)= (t) Heat Flux

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Direct Discrete scheme

Heat Flux N~=2000K number of one-dimensional grid points (~100), N number of time (N~=2000) iterations in the numerical scheme

The equation is written at time (n+1) :

Linearization at time n forward time discrete linearized Euler scheme, with initial condition vanishing: stability

Nnw

pwftww n

nn

00

,

0

11

Nnw

wwpwdfpwftww nnnn

nn

00

,,

0

11

nnK

nnn sTTTw ,,,, 21

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Cost Function

time domain unknown heat flux convection coefficient

Quadratic error or cost function j(p)

Measured temperatureComputed temperatureTo minimize this quantity, by optimization algorithm we need the derivatives of J(p), with respect to p.

The inverse method

N

n

nm

nm

Wiables

N tTpwpwJpJ1

2

var

1 )(),...,()(

Nppp ,...,1

nmnmT

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Adjoint SystemAdjoint variable : dual multiplyer ofLagrangian L + calculus of variations

Cancel the variations of L with respect to Direct system, forward in time, initial vanishing condition Cancel the variations of L with respect to w Adjoint system, backward in time, final vanishing condition

The inverse method

1

0

11

2/1

1

2

varint

2/12/1

var

11

,,,

,...,,,...,,,...,,,

N

n

nnnnnn

nN

n

nnm

iablesadjo

N

wiables

N

pparameter

N

wwpwdfpwftwwtT

wwppLwpL

00

2,,

2/1

22/1122/112/12/1

nN

tTwwpwfdpwdft

N

nm

nm

nnnnnntnn

2/1n nw

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Gradient computation

With this particular choice of , the gradient of the cost function is simply obtained by :

Variations L function of p discrete gradients

apply an iterative inverse procedure minimizing J(p) to estimate the unknown parameter optimal function

The inverse method

pL

pJJ

1

0

12/1 ,N

n

nnnnn wwwpdfw

pf

pJ

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Gradient computationThe inverse method

0

,*

Tft

ttpTWf

t obs

dtppWf

pL

pj t

,

Adjoint State

Final Condition

Gradients

00

),(

tW

pWftWDirect State

Initial condition

time

time

W = (T,s)

df / dp =

complex, non linear

df / dW = complex, non linear, tables

Measurements

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OptimizationThe inverse method

2

2

dpjd

j(p*)

p*j (p)

j(p*)

p*j (p)

)( pj

Direct problemT(p)

Optimization p+ p

1) Steepest Descent to explore2) Quasi Newton to finish convergence

(T(p)-)**2

p0,

P optimal

)( pj

dpdj

Gradient

2

2

dpjd

Approximation ofHessian

Direct + Adjoint systemCan be computed by Automatic Differentiation tool

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Automatic Differentiation TAPENADE, INRIA Sophia Antipolis, France

The inverse method

Program (Fortran) : sequence of elementary arithmetic operations

Derivatives can be computed automatically

If the code is modified, it is more easy to compute new adjoints and new gradients

Input• function f• cost function J(p)

Output functions f’gradient dJ/dp

AD Tool

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Automatic Differentiation

),,,,,,,(1 ptwwwwfww knj

kni

nj

ni

ni

ni

ni

kni

knj

k jkn

i

knj

kni

nj

ni

kjn

ini w

Jw

ptwwwwf

),,,,,,,(1

N

n

nm

nm

Wiables

N tTpwpwJJ1

2

var

1 )(),...,(

The inverse method

Direct problem instruction

Cost Function

Adjoint system instructions : differentiation in reverse mode, with push, pop

Gradient computed by reverse mode

1

0

12/1 ,N

n

nnnnn wwwpdfw

pf

pJ

time

time

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Virgin material, low heat flux)Pseudo measurements very well rebuilt (RMS<1K)Automatic Differentiation (AD) sucessful

Some applications

Heat Flux without AD Heat Flux with AD

Gradient

Quasi Newton

Cost Function

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Carbon/Resin with ablation, pyrolysis & pseudo measurements Results OK with pyrolysis and ablation (without and with AD) Results OK with 2% noise on pseudo measurement Tichonov regularization to stabilize the solution

Some applications

Convection (noise without regularization)

Convection (noise with regularization)

Cost Function

GradientQuasi Newton

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ARD1998 on Ariane Flight 503First use of the inverse method for « in-flight »

rebuilding during ARD post-flight analysis (1999)Last improvements of the method OK

Some applications

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Plasma Torch Facility

Material to be tested

Nozzle

Ablation compensation

Fluxmeters

Measurements:• Laser (ablation)

• Pyrometer (surface temperature)

TC1

TC2

TC3

TC4

TC5

TC6

TC7

TC8

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Plasma torch: First results ONLY ONE SENSOR USED

Influence of sensor used Ablation restitution

Influence of sensor used Temperature at thermocouple

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Missing sensors Temperature at Thermocouple

Missing sensors convection coefficient restitution

Influence of sensor used Heat Flux restitution

Influence of sensor used Temperature at surface

ONLY ONE SENSOR USED

SEVERAL SENSORS USED AT THE SAME TIME

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Conclusion & perspectivesConclusion:

Inverse method sucessfully implemented First tests of Automatic Differentiation promising Validation OK for pseudo-measurements (with or without noise) Good results obtained on hard cases

Perspectives: Theoretical aspects (observability, identificability) to be analyzed Improve robustness of the method (initial guess,uncertainties on

noise, regularization) test on industrial re-entry problems Improve automatic differentiation version for hard cases

Documents

Inverse method for pyrolysable and ablative materials with optimal control formulation