THE ROLE OF CORRELATIONS IN MONTE CARLO CRITICALITY SIMULATIONS

Transcription

1 THE ROLE OF CORRELATIONS IN MONTE CARLO CRITICALITY SIMULATIONS ANDREA ZOIA CEA/SACLAY Séminaire MANON Andrea ZOIA DEN/DANS/DM2S/SERMA/LTSD March 22nd 2016 APRILE 13, 2016 PAGE 1

2 OUTLINE q Power iteragon with Monte Carlo q The impact of correlagons in crigcality simulagons q Entropy and convergence q SpaGal moments and correlagons q RelaGon to neutron clustering theory q PerspecGves CEA March 22nd 2016 PAGE 2

3 THE CRITICAL BOLTZMANN EQUATION We would like to determine q the fundamental mode ϕ 1 q and the associated fundamental eigenvalue k 1 of Critical Boltzmann equation for the neutron flux ϕ Net disappearance operator L Creation (fission) operator F CEA March 22nd 2016 PAGE 3

4 POWER ITERATION A generalized eigenvalue equation Power iteration algorithm: q Guess solution q Iterate Hypothesis Convergence CEA March 22nd 2016 PAGE 4

5 MONTE CARLO APPROACH: CRITICALITY SIMULATION 1 st gen. 2 nd gen. 3 rd gen. 4 th gen. 5 th gen. Source Fission chain t Source 1 st gen. 2 nd gen. 3 rd gen. N 0 particles Fission generations CEA March 22nd 2016 PAGE 5

6 POWER ITERATION: THE STANDARD (?) TOOL Source 1 st gen. G th gen. (G+1) th gen. M th gen. N 0 particles Convergence to the fundamental mode ϕ 1 (statistical equilibrium) Stationarity: sample ϕ 1 = <ϕ 1 (g)> Hypothesis: (I)ID replicas What about correlations? CEA March 22nd 2016 PAGE 6

7 A TOY MODEL OF A NUCLEAR REACTOR Neutrons in a box Scattering Capture Fission Descendants per fission Reflecting boundary conditions Assumption: the reactor is critical Expected fundamental mode ϕ 1 : spatially uniform over the box CEA March 22nd 2016 PAGE 7

8 IMPACT OF SYSTEM SIZE L ON POWER ITERATION Generations Delta-like source at the center of the box Initial number of neutrons per generation N = 10 4 CEA March 22nd 2016 PAGE 8

9 IMPACT OF SYSTEM SIZE L ON POWER ITERATION Neutrons per generation N = 10 4 Neutron clustering CEA March 22nd 2016 PAGE 9

10 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Pure diffusion (ideal gas) Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 10

11 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 11

12 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 12

13 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Diffusion + branching + capture (critical gas) Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 13

14 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 14

15 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Mixing time Initial condition: N 0 particles with uniform density Clustering Capture Fission Diffusion CEA March 22nd 2016 PAGE 15

16 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Mixing time Renewal time? Initial condition: N 0 particles with uniform density Clustering Capture Fission Diffusion CEA March 22nd 2016 PAGE 16

17 IMPACT OF POPULATION SIZE N ON POWER ITERATION Generations Delta-like source at the center of the box System size L = 400 cm CEA March 22nd 2016 PAGE 17

18 IMPACT OF POPULATION SIZE N ON POWER ITERATION N System size L = 400 cm Neutron clustering CEA March 22nd 2016 PAGE 18

19 THE ENTROPY FUNCTION: CONVERGENCE ANALYSIS Shannon entropy: m m Generations to convergence: CEA March 22nd 2016 PAGE 19

20 THE EFFECTS OF CLUSTERING ON THE ENTROPY FUNCTION Shannon entropy: Theoretical expected value for independent replicas Measured value Impact of correlations between generations CEA March 22nd 2016 PAGE 20

21 ANALYSIS OF SPATIAL MOMENTS: THE CENTER OF MASS CEA March 22nd 2016 PAGE 21

22 A STATISTICAL MECHANICS DESCRIPTION Neutrons as a collection of N stochastic particles: {x 1, x 2, x i, x N } A remarkable identity for the spatial moments: Square COM Mean square displacement: Mean square pair distance: CEA March 22nd 2016 PAGE 22

23 A STATISTICAL MECHANICS DESCRIPTION Neutrons as a collection of N stochastic particles: {x 1, x 2, x i, x N } A remarkable identity for the spatial moments: Average particle density ψ Square COM Mean square displacement: Pair correlation function h Mean square pair distance: CEA March 22nd 2016 PAGE 23

24 RECIPROCITY OF RANDOM WALKS Forward time flow Backward time flow Measure in z z t Source Measure in z z t Source CEA March 22nd 2016 PAGE 24

25 THE AVERAGE NEUTRON DENSITY z Measurement z t Source z 0 CEA March 22nd 2016 PAGE 25

26 THE PAIR CORRELATION FUNCTION Correlated measurements z 1 z 2 z 1 à z z 2 à z z à z 0 z t Source z 0 CEA March 22nd 2016 PAGE 26

27 THE HOMOGENEOUS CUBE REACTOR Average neutron density: Pair correlation function: System size L Single dimensionless parameter Population size N Migration area M 2 CEA March 22nd 2016 PAGE 27

28 POWER ITERATION AS A FUNCTION OF χ Uniform initial CEA March 22nd 2016 PAGE 28 condition

29 POWER ITERATION AS A FUNCTION OF χ Uniform initial CEA March 22nd 2016 PAGE 29 condition

30 THE HOMOGENEOUS CUBE REACTOR: SPATIAL MOMENTS Mean square displacement Mean square pair distance CEA March 22nd 2016 PAGE 30

31 THE HOMOGENEOUS CUBE REACTOR: SPATIAL MOMENTS Mean square displacement Mean square pair distance Square COM Fluctuations of COM CEA March 22nd 2016 PAGE 31

32 SPATIAL MOMENTS: STATISTICAL ANALYSIS Theory MC Theory MC CEA March 22nd 2016 PAGE 32

33 CONCLUSIONS q StaGsGcal mechanics approach to power iteragon q Neutron clustering can be suppressed by acgng on χ q Applicability to real- world systems? CEA March 22nd 2016 PAGE 33

34 Thanks for your axengon q E. Dumonteil et al., Annals of Nuclear Energy 63, (2014). q A. Zoia, E. Dumonteil, A. Mazzolo, C. de Mulatier, A. Rosso, Phys. Rev. E 90, (2014). q C. de Mulatier, E. Dumonteil, A. Rosso, A. Zoia, J. Stat. Mech. P08021 (2015). q B. Houchmandzadeh, E. Dumonteil, A. Mazzolo, A. Zoia, Phys. Rev. E 92, (2015). q M. Nowak et al., Monte Carlo power iteration: entropy and spatial correlations, submitted. CEA March 22nd 2016 PAGE 34

35 PAGE 35 CEA 10 AVRIL 2012 APRILE 13, 2016 Commissariat à l énergie atomique et aux énergies alternagves Centre de Saclay Gif- sur- YveXe Cedex T. +33 (0) Secr :+33 (0) Etablissement public à caractère industriel et commercial RCS Paris B DEN/DANS DM2S SERMA