First experiments of Automatic Differentiation on some ... EuroAd... · Tapenade on recent MONOPYRO...

transcript

09/12/2010 1

ALESTRA Stéphane, SRITHAMMAVANH Vassili

EADS INNOVATION WORKS

First experiments of AutomaticDifferentiation on some inverse problemsin aerospace applications.

11th European Workshop on Automatic Differentiation

Cranfield (UK) 09/12/10

S.Alestra / V.Srithammavanh

EADS INNOVATION WORKS

• EADS : European Aeronautic Defence and Space company

Civil and military aircraft

Satellites and space rockets

Helicopters

Communication systems

• EADS IW : INNOVATION WORKS (Research Center)

EADS Technical innovation potential

Link between EADS Business Units and Academics laboratories

S.Alestra / V.Srithammavanh09/12/2010

EADS IW Applied Mathematics & Simulation Research Team

Historically, competencies in Inverse methods and Optimal Control

Then,develop progressively competencies in Optimization

I Identification of aerodynamic coefficients

E.Leibenguth, A.Charpe, V.Srithammavanh, S.Alestra

• Context : Trajectory control of Atmospheric re-entry probe

• Necessary to identify accurately the aerodynamic behaviour of theprobe

• Frame of the study

Trajectory measurements on ground with probe’s shot by acannon

Determine the aerodynamic coefficients

09/12/2010 3S.Alestra / V.Srithammavanh

State vector

Direct problem : to model the probe movement

• Aerodynamic forces

DRAG LIFT

• 2D Dynamics equation solved by RUNGE-KUTTA (RK4)

Dynamic pressure Reference area

• Aerodynamic moment

mqmLD CCCC ,,,

depending on

Aerodynamic coefficients

Angle of attack

Velocity

Position

Anglular velocity

Inverse problem : coefficients identification

• Problem : reconstruct the aerodynamic coefficient from measurements of angle of attack and velocity

• Parameter: iimqmaNAii MCmCCCMp ,,,,,

• Constrained minimization

•TO COMPUTE Gradient

subject to :

tables

Dimension of parameters = around 1000=4*50*50

Use adjoint techniques

• Manual

• AD

)(XFpX p

f : norm

Numerical resolution usingAutomatic Differentiation (AD)

and Optimization• Gradient computation

Adjoint / Gradients obtained manually by Optimal control

AD : allows to obtain automatically this adjoint of time from the Fortran 77 trajectory code

TAPENADE mode REVERSE

• OPTIMIZATION

Quasi Newton / GRG / SQP

insystemadjointbackwardobtainto0

)(,)(),,(Lagrangian

XFXXJXpL

Storing all X(t) and use (t) to compute gradients

Adjoint equation

Direct equation

t=0 t=T

Checkpointing : Store intermediate snapshot ofX and regenerate X at the same time than

ADJOINT

XFpX p

)(, XFpp

GRADIENTS

GRADIENTSADJOINT

DIRECT

Gradients

Cost of Gradients computation

To analyze with INRIA Sophia

DIRECT

Numerical results

• Angle of attack measurements / simulation • Identification of Aerodynamic moment

• Good results :

• Aerodynamic coefficients obtained consistent with trajectory physics

• BUT Necessity of efficient / powerful large scale optimizers (number parameters > 5000 )• Investigations with Edimburgh University, Cerfacs, …

machalpha

Restitution in(t)

Long time

Intermediate time

Industrial context :

Atmospheric re-entry missions

Design and sizing of the Thermal Protection System (TPS)

Heat fluxes identification : very important industrial interest

ARD Huygens probe (on Titan)

II Inverse method for non linear ablative thermicsS.Alestra, J.Collinet (EADS), F.Dubois (CNAM)

40 Th AIAA Thermophysics, Seattle (June 08)International Journal of Engineering Systems Modelling and Simulation (IJESMS) 2009

Identify heat fluxes from temperature measurements ?

(t)p(t)= (t) ?

THE INVERSE PROBLEM

Heat fluxes

Temperature sensor

Thermal Protection System

Ablation & Pyrolyses

Direct Problem

Temperatures

Ablation

etsx )(1

etsxtt

f ,,,0

0)0,()0,(

1,0 ,t

T1 T2 T3

09/12/2010 p11

(function of time t and position X)

• Remark : use of reduced variables

• State vector

• Model

• « Monopyro » : one dimensional numerical tool (implement manual adjoint computation)

p :Heat FLUX

Direct Discrete scheme

K grid points, N time iterations in the numerical scheme

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

Assumption of MONOPYRO :Linearization at time n direct system, forward intime, stability

wwpwdfpwft

ww nnnnnn

ww nnn

nnn sTTTw ,,,, 21

Cost Function

time domain unknown heat flux convection coefficient

Quadratic error or cost function j(p)

Measured temperature Computed temperature

we need the derivatives of J(p), with respect to p.

p is large scale input parameter = 2000 Need adjoint reverse mode

Wiables

N tTpwpwJpJ1

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

Nppp ,...,1

Adjoint System

Adjoint variable : dual multiplyer of

Lagrangian L + calculus of variations

Cancel the variations of L with respect to Direct system, forward in time

Cancel the variations of L with respect to w Adjoint system, backward in time

varint

2/12/1

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

nnnnnn

iablesadjo

wiables

pparameter

wwpwdfpwft

wwppLwpL

22/1122/112/12/1

tTwwpwfdpwdft

nnnnnntnn

2/1nnw

Gradient computation (manually)

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

Variations L function of p discrete gradients

Test using AD to compute automatically adjoint state with TAPENADE (INRIASophia Antipolis)

12/1 ,N

nnnnn wwwp

Gradient : test with Automatic Differentiation Tapenade

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

ptwwwwf

),,,,,,,(1

Wiables

N tTpwpwJJ1

1 )(),...,(

Direct problem instruction

Cost Function

Differentiation in reverse mode, with push, pop

Gradient computed by reverse mode

12/1 ,N

nnnnn wwwp

09/12/2010 p16

Heat flux identificationCarbon/Resin with ablation, pyrolysis

Results OK with pyrolysis and ablation – Good correlation beetwen ADand manual adjoint computation

Cost Function

Convection restitution

ARD Heat flux identification

First use of the inverse method for ARD post-flight analysis

Good correlation between measurement and AD restitution

Recent evolutions of MONOPYRO direct code (2009/2010)

• Add of Temperature, ablation, mass flow to state vector

• Multi layers

• Multi sensors

• Adapative grid in space & time

• At each time, non linear equation to solve F(U,p,t)=0

• Newton method with Fkinsol library

• Complexity of the Fortran code : common, interpolation tables switch, staticarray declaration, many imbricated routines

Tapenade tested for faisability study on this more complex and industrialproblem

Tapenade on recent MONOPYRO direct code (2009/2010)

Tangent mode (linearization) is OK, but hard to obtain !

Complexity of the code, duplication of arrays with differentiation

CPU and memory constraints

First results are promising but costly (number of parameters)

Adjoint mode is under development

• Use reverse differentiation + mathematic adjoint algorithms

• Problem of storing and memory stack at backward sweep

TAPENADE has given good faisability results but still work to do !!

• First promising applications of AD (TAPENADE) to 2 aerospace applications

• Collaboration with INRIA SOPHIA about TAPENADE

• Interest in AD Hessian calculation to improve optimization

• Interest to know best practices and methodological guidelines in AD

• Extension to Fortran 95 and C language and check

• Checkpointing use to optimize adjoint backward calculations and performances

• Future possible applications inside EADS : sensitivity analysis, shape optimization,…

Conclusion / Perspectives

THANK YOU FOR YOUR ATTENTION !!!!

First experiments of Automatic Differentiation on some ... EuroAd... · Tapenade on recent MONOPYRO...

Documents