MONTE CARLO POWER ITERATION: ENTROPY AND SPATIAL CORRELATIONS

Transcription

1 MONTE CARLO POWER ITERATION: ENTROPY AND SPATIAL CORRELATIONS ANDREA ZOIA, M. NOWAK (CEA/SACLAY) E. DUMONTEIL, A. ONILLON (IRSN) J. MIAO, B. FORGET, K. S. SMITH (MIT) NEA EGAMCT meeting Andrea ZOIA DEN/DANS/DM2S/SERMA/LTSD July 6th JUILLET 2016 PAGE 1

2 OUTLINE Power iteration with Monte Carlo The impact of correlations in criticality simulations Entropy and convergence Spatial moments and correlations Relation to neutron clustering theory Perspectives CEA July 6th 2016 PAGE 2

3 THE CRITICAL BOLTZMANN EQUATION We would like to determine the fundamental mode j 1 and the associated fundamental eigenvalue k 1 of Critical Boltzmann equation for the neutron flux j Net disappearance operator L Creation (fission) operator F CEA July 6th 2016 PAGE 3

4 POWER ITERATION A generalized eigenvalue equation Power iteration algorithm: Guess solution Iterate Hypothesis Convergence CEA July 6th 2016 PAGE 4

5 Source MONTE CARLO APPROACH: CRITICALITY SIMULATION 1 st gen. 2 nd gen. 3 rd gen. 4 th gen. 5 th gen. Fission chain t Source 1 st gen. 2 nd gen. 3 rd gen. N 0 particles Fission generations CEA July 6th 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 j 1 (statistical equilibrium) Stationarity: sample j 1 = <j 1 (g)> Hypothesis: (I)ID replicas What about correlations? CEA July 6th 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 j 1 : spatially uniform over the box CEA July 6th 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 July 6th 2016 PAGE 8

9 IMPACT OF SYSTEM SIZE L ON POWER ITERATION Neutrons per generation N = 10 4 Neutron clustering CEA July 6th 2016 PAGE 9

10 IMPACT OF POPULATION SIZE N ON POWER ITERATION Generations Delta-like source at the center of the box System size L = 400 cm CEA July 6th 2016 PAGE 10

11 System size L = 400 cm IMPACT OF POPULATION SIZE N ON POWER ITERATION N Neutron clustering CEA July 6th 2016 PAGE 11

12 THE ENTROPY FUNCTION: CONVERGENCE ANALYSIS Shannon entropy: m m Generations to convergence: CEA July 6th 2016 PAGE 12

13 THE EFFECTS OF CLUSTERING ON THE ENTROPY FUNCTION Shannon entropy: Theoretical expected value for independent replicas Measured value Impact of correlations between generations CEA July 6th 2016 PAGE 13

14 ANALYSIS OF SPATIAL MOMENTS: THE CENTER OF MASS CEA July 6th 2016 PAGE 14

15 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 July 6th 2016 PAGE 15

16 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 y Square COM Mean square displacement: Pair correlation function h Mean square pair distance: CEA July 6th 2016 PAGE 16

17 RECIPROCITY OF RANDOM WALKS Forward time flow Backward time flow Measure in z z t Source Measure in z z t Source CEA July 6th 2016 PAGE 17

18 THE AVERAGE NEUTRON DENSITY z Measurement z t Source z 0 CEA July 6th 2016 PAGE 18

19 THE PAIR CORRELATION FUNCTION Correlated measurements z 1 z 2 z 1 z z 2 z z z 0 z t Source z 0 CEA July 6th 2016 PAGE 19

20 THE HOMOGENEOUS CUBE REACTOR Average neutron density: Pair correlation function: System size L Single dimensionless parameter Population size N Migration area M 2 CEA July 6th 2016 PAGE 20

21 POWER ITERATION AS A FUNCTION OF c Uniform initial condition CEA July 6th 2016 PAGE 21

22 POWER ITERATION AS A FUNCTION OF c Uniform initial condition CEA July 6th 2016 PAGE 22

23 THE HOMOGENEOUS CUBE REACTOR: SPATIAL MOMENTS Mean square displacement Mean square pair distance CEA July 6th 2016 PAGE 23

24 THE HOMOGENEOUS CUBE REACTOR: SPATIAL MOMENTS Mean square displacement Mean square pair distance Square COM Fluctuations of COM CEA July 6th 2016 PAGE 24

25 SPATIAL MOMENTS: STATISTICAL ANALYSIS Theory MC Theory MC CEA July 6th 2016 PAGE 25

26 THE HOOGENBOOM-MARTIN PWR BENCHMARK Spatial moments Power iteration CEA July 6th 2016 PAGE 26

27 CONCLUSIONS Statistical mechanics approach to power iteration Neutron clustering can be suppressed by acting on c Applicability to real-world (heterogeneous) systems? CEA July 6th 2016 PAGE 27

28 Thanks for your attention E. Dumonteil et al., Annals of Nuclear Energy 63, (2014). A. Zoia, E. Dumonteil, A. Mazzolo, C. de Mulatier, A. Rosso, Phys. Rev. E 90, (2014). C. de Mulatier, E. Dumonteil, A. Rosso, A. Zoia, J. Stat. Mech. P08021 (2015). B. Houchmandzadeh, E. Dumonteil, A. Mazzolo, A. Zoia, Phys. Rev. E 92, (2015). M. Nowak et al., Annals of Nuclear Energy 94, 856 (2016). CEA July 6th 2016 PAGE 28

29 PAGE 29 CEA 10 AVRIL JUILLET 2016 Commissariat à l énergie atomique et aux énergies alternatives Centre de Saclay Gif-sur-Yvette Cedex T. +33 (0) Secr :+33 (0) Etablissement public à caractère industriel et commercial RCS Paris B DEN/DANS DM2S SERMA

30 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Pure diffusion (ideal gas) Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 30

31 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 31

32 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 32

33 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Diffusion + branching + capture (critical gas) Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 33

34 SPATIAL BEHAVIOUR OF THE NEUTRON DENSITY Fluctuations: Initial condition: N 0 particles with uniform density CEA March 22nd 2016 PAGE 34

35 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 35

36 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 36

37 A STATISTICAL MECHANICS DESCRIPTION Neutrons as a collection of N particles Spatial moments: Mean square displacement: Mean square pair distance: Center of mass: CEA April 11th 2016 PAGE 37