Inverse method for pyrolysable and ablative materials with optimal control formulation

1. 1EADS Innovation Works Toulouse, FRANCE 2EADS ASTRIUM ST Les Mureaux, France 3Conservatoire National des Arts et Métiers Paris, France stephane.alestra@eads.net jean.collinet@astrium.eads.net fdubois@cnam.fr Inverse method for pyrolysable and ablative materials with optimal control formulation S.Alestra1, J.Collinet2, and F.Dubois3

2. Industrial problem • 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 Huygens probe (on Titan) ARD

3. General equations of direct problem input data: heat flux output data: temperature Internal energy balance Pyrolysis: Arrhenius equation Decomposition and mass balance Surface Recession Surface energy balance

4. Convection Flux Radiation Flux Radaition Flux Blocking Flux Pyrolysis gas Heat Flux of Ablation heat flux Pyrolysed Material Material Pyrolysis gas Ablative Pyrolysis Zone Pyrolysable Virgin Material Fibres Resin Coke Pyrolytic Carbon Radiation Flux Convection Flux The inverse method « Monopyro » 1D numerical tool, EADS ASTRIUM ST • Evaluate the heat fluxes from temperature measurements on thermal protection with ablation and pyrolysis Inverse Heat problem is hard !! : see theory, diffusion aspect, .. q (t), t in [O,T] p(t)=a0 (t) ? T in [0,T] p(t)=a0 (t) ?

5. T2 T3 T1 s q (t) X (t) e p(t)=a0 (t) Heat Flux 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 F • System is rewritten in reduced variables

6. Direct Discrete scheme • Heat Flux N~=2000 • K 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

7. The inverse method Cost Function • time domain unknown heat flux convection coefficient • Quadratic error or cost function j(p) • Measured temperature • Computed temperature • To minimize this quantity, by optimization algorithm  we need the derivatives of J(p), with respect to p.

8. The inverse method Adjoint System • Adjoint variable : dual multiplyer of • Lagrangian L + calculus of variations • Cancel the variations of dL with respect to dj Direct system, forward in time, initial vanishing condition • Cancel the variations of dL with respect to dw  Adjoint system, backward in time, final vanishing condition

9. The inverse method Gradient computation • With this particular choice of j, the gradient of the cost function is simply obtained by : • Variations dL function of dp  discrete gradients • apply an iterative inverse procedure minimizingJ(p)to estimate the unknown parameter optimal function

10. The inverse method Gradient computation Direct State Initial condition W = (T,s) time df / dW = complex, non linear, tables time Adjoint State Final Condition Measurements df / dp = complex, non linear j Gradients

11. The inverse method Optimization p0, q Direct problem T(p) (T(p)-q)**2 P optimal Gradient Approximation of Hessian Optimization p+ Dp 1) Steepest Descent to explore 2) Quasi Newton to finish convergence Direct + Adjoint system Can be computed by Automatic Differentiation tool

12. The inverse method Automatic Differentiation TAPENADE, INRIA Sophia Antipolis, France • 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 Output functions f’ gradient dJ/dp • Input • function f • cost function J(p) AD Tool

13. The inverse method Automatic Differentiation • Direct problem instruction • Cost Function • Adjoint system instructions : differentiation in reverse mode, with push, pop • Gradient computed by reverse mode time time

14. Some applications Virgin material, low heat flux) • Pseudo measurements very well rebuilt (RMS<1K) • Automatic Differentiation (AD) sucessful Heat Flux with AD Heat Flux without AD Cost Function Gradient Quasi Newton

15. Some applications 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 Cost Function Convection (noise without regularization) Convection (noise with regularization) Gradient Quasi Newton

16. Some applications ARD • 1998 on Ariane Flight 503 • First use of the inverse method for « in-flight » rebuilding during ARD post-flight analysis (1999) • Last improvements of the method OK

17. Plasma Torch Facility • Measurements: • Laser (ablation) • Pyrometer (surface temperature) Fluxmeters Nozzle TC1 TC2 TC3 TC4 TC5 TC6 TC7 TC8 Material to be tested Ablation compensation

18. Plasma torch: First results ONLY ONE SENSOR USED Influence of sensor used  Temperature at thermocouple Influence of sensor used  Ablation restitution

19. Influence of sensor used  Temperature at surface Influence of sensor used  Heat Flux restitution ONLY ONE SENSOR USED Missing sensors  convection coefficient restitution Missing sensors  Temperature at Thermocouple SEVERAL SENSORS USED AT THE SAME TIME

20. Conclusion & perspectives • Conclusion: • 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