Calibration and validation of computer models for radiative shock experiment

Transcription

1 Calibration and validation of computer models for radiative shock experiment Jean Giorla 1, J. Garnier 2, E. Falize 1,3, B. Loupias 1, C. Busschaert 1,3, M. Koenig 4, A. Ravasio 4, C. Michaut 3 1 CEA/DAM/DIF, Arpajon, France 2 Laboratoire J-L Lions, Université Paris VII, France 3 LUTH, Observatoire de Paris, Meudon, France 4 LULI, Palaiseau, France contact: jean.giorla@cea.fr 1

2 Radiative shocks in astrophysics & laboratory Polars are close binaries containing a magnetic white dwarf accreting material from a secondary star. The matter is channelled toward the magnetic pole, hits the surface, a radiative shock propagates through the accretion column. Warner B., Cataclysmic variable stars (Cambridge Astrophysic Series,1995) 2

3 POLAR project aim to mimic the accretion column Polars are close binaries containing a magnetic white dwarf accreting material from a secondary star. The matter is channelled toward the magnetic pole, hits the surface, a radiative shock propagates through the accretion column. obstacle The plasma is channelled by the tube The plasma hits the obstacle A radiative shock is produced Secondary star Accretion column White dwarf Falize E. et al. Astrophys. Space Science 336 (2011) 3

4 POLAR campaign on LULI facility E. Falize et al. HEDP, 8 (2012) Laser CH sheet Tube vacuum Ti or Al sheet z Expanding plasma 8 experiments available ( 4 Ti & 4 Al ) Quartz collision time time The aim of this study is to calibrate the simulation parameters from these 8 experiments and to quantify the uncertainty on the prediction of the collision time of a new experiment. 4

5 Outline Sources of uncertainty Experimental errors Modeling & input uncertainties Numerical uncertainties Quantification of the uncertainties using a Bayesian framework 5

6 Sources of uncertainty ASME V&V dx : errors on experimental inputs Experimental errors Truth T: collision time t coll d D Simulation model d model d input Modeling assumptions Simulation inputs d num Numerical solutions Experimental data, D Simulation result, S Each error d is a realization of a random variable (or field) e with known or unknown PDF. 6

7 Experimental errors Laser pulse 1.5 ns J Variations in the driving conditions: laser energy U = 15% duration U = 10% shape (modes 1-4) U = 15% z Expanding plasma Variations in the physical hardware: sheet densities U = 1 to 10% sheet thicknesses U = 10 to 20% tube length U = 20% collision time t coll 8-12ns time Uncertainty on collision time measurement: U = 250 ps (~2-3%) 7

8 Total data uncertainty is about U 3 ns. Total uncertainty on collision time is obtained by propagating input uncertainties with a Monte-Carlo method. Data uncertainties are huge in these preliminary experiments, mainly due to poor target characterizations. 8

9 Modeling uncertainty For a given laser energy E, the Truth is the simulation model + a discrepancy: Truth (E) = S model (E) + e mod (E) e mod is a Gaussian process with standard deviation s mod Input uncertainties Thermal model : diffusion with an electronic flux limiter f elec geometric mean between f elec * F theoretical and computed flux T e. The laser energy E is measured outside the chamber, not on the target E target = E * K laser K laser comes from the non-perfect transmission into the last lenses. => The simulation output S is a function of E, K laser and f elec : S ( K laser * E, f elec ) 9

10 void Ti Numerical uncertainties CH plasma blow-off hydro shock Heated matter laser laser absorption 10

11 void Ti Numerical uncertainties CH plasma blow-off hydro shock Heated matter laser laser absorption The mesh is converged for the hydro shock and the blow-off t coll error ~ -20 ps Grid Convergence Index ~ 30 ps negligible compared to data uncertainty 11

12 void Ti plasma blow-off Numerical uncertainties density profile n c CH ray-trace laser absorption laser The mesh is converged for the hydro shock and the blow-off t coll error ~ -20 ps Grid Convergence Index ~ 30 ps The laser energy is absorbed along raytraces until the critical density n c. The ray-trace is deflected according to the density gradient at n c. This gradient is very sharp, and needs a negligible compared to data uncertainty very fine mesh. Important numerical error 12

13 ratio K num ±GCI (factor of safety Fs=1.25) Numerical errors in the absorbed laser energy E abs Errors estimated on a simplified problem (only CH and X-ray diffusion scheme). The grid was halved 7 times (maximum reasonable run time) The convergence is very low with an apparent order p << 1 The ratio K num = E converged_mesh abs / E reference_mesh abs mainly depends on f elec 0,9 0,7 0,5 0,05 0,07 0,09 0,11 0,13 flux limitor f elec The ratio K num will be applied to the input energy: S converged (E) S reference ( {K num +dk num }*E ) dk num is a realization of the Gaussian process e num of standard deviation s num = GCI/2 13

14 Final statistical model: collision time prediction for a new E Truth Simulation model uncertainty T (E) = S ( {K num +e num } * K laser * E, f elec ) + e mod (E) laser energy Correction terms due to non converged mesh flux limitor transmission trough last lenses e mod & e num are Gaussian processes with standard deviations s mod & s num and correlation length L. 14

15 Unknown parameters T (E) = S ( {K num +e num } * K laser * E, f elec ) + e mod (E) e mod & e num are Gaussian processes with standard deviations s mod & s num and correlation length L. 15

16 Bayesian inference We infer K Laser & s mod from the data D [1,2] and we tune f elec & L in order to maximize the posterior likelihood [3] using Bayes theorem of conditional probabilities posterior likelihood * prior P post (k Laser, s mod D, f elec,l) = P(D k Laser, s mod, f elec,l) * p(k Laser, s mod ) / k (f elec & L) maximizes the posterior likelihood k (integral of the numerator) [1] Kennedy, O Hagan, J.R. Statist. Soc. B (2001) [2] Higdon et al, SIAM J. Sci. Comput. (2004) [3] Han, Santner, Rawlinson, Technometrics (2009) 16

17 Algorithm: double loop & monotonic realizations as the collision time is a monotonic function of laser energy E j = 1, J f elec (j) & L (j) given Outer loop tuning Inner loop calibration k = 1, K sampling of K laser (k) & s mod (k) realization T(E i ) at points {E i } i=1,i monotonic realization? no post(k) = 0 yes post(k) = likelihood(k) * prior(k) total likelihood = S k post(k) 17

18 The thermal flux limiter that maximizes the posterior likelihood is f elec = Optimal correlation length is L = 150 J 18

19 Posterior probability P post (k Laser, s mod ) s mod marginal PDF (UA) marginal PDF (UA) posterior PDF (UA) k Laser mean = 0.74 ns s mod (ns) mean = 0.87 k Laser 19

20 Predictive uncertainty on collision time CH/Titanium targets CH/Aluminum targets Only 100 monotonic realizations plotted 20

21 Summary The uncertainty on the collision time is inferred from the data using a Bayesian approach. The PDFs we obtained are relatively vague due to large input uncertainties in these preliminary experiments. Future work Application to 2012 experiments whose target metrology and laser diagnostics were greatly improved. Development of adaptive mesh techniques for the laser absorption. 21