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Nuclear Engineering and Technology 49 (2017) 1157e1164
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Original Article
Clustering and traveling waves in the Monte Carlo criticality simulation of decoupled and conﬁned media
Eric Dumonteil a, *, Giovanni Bruna a, Fausto Malvagi b, Anthony Onillon a, Yann Richet a
a Institut de Radioprotection et de Sûrete Nucleaire, IRSN/PSN-EXP/SNC, 92262, Fontenay-aux-Roses, France
b Den-Service D’etudes des Reacteurs et de Mathematiques Appliquees (SERMA), CEA, Universite Paris-Saclay, F-91191, Gif-sur-Yvette, France
a r t i c l e i n f o a b s t r a c t
Article history: Received 3 June 2017
Received in revised form 19 July 2017
Accepted 26 July 2017 Available online 3 August 2017
Keywords: Monte Carlo Criticality Biases Clustering Traveling waves
1. Introduction
The Monte Carlo criticality simulation of decoupled systems, as for instance in large reactor cores, has been a challenging issue for a long time. In particular, due to limited computer time resources, the number of neutrons simulated per generation is still many order of magnitudes below realistic statistics, even during the start-up phases of reactors. This limited number of neutrons triggers a strong clustering effect of the neutron population that affects Monte Carlo tallies. Below a certain threshold, not only is the variance affected but also the estimation of the eigenvectors. In this paper we will build a time-dependent diffusion equation that takes into account both spatial correlations and population control (ﬁxed number of neutrons along generations). We will show that its solution obeys a traveling wave dynamic, and we will discuss the mechanism that explains this biasing of local tallies whenever leakage boundary conditions are applied to the system.
© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
correlations affecting some branching processes evolving in inﬁnite
media in dimension 1 or 2 were diverging in amplitudes, while in
Monte Carlo neutron transport codes [1,2] are often used as a reference tool by the nuclear industry, as the approximations on which they rely to solve the Boltzmann equation in ﬁssile media (the so-called critical Boltzmann equation [3]) are extremely sparse. Their growing use in the past few decades is strongly correlated to the increase of computer resources and now ranges from nuclear fuel cycle studies to criticality safety assessment and reactor physics simulations. However, in this last application, and especially in the case of large reactor cores or loosely coupled systems [4,5], a strong undersampling effect biases the estimates of the variance of ﬂux-based quantities [6e10]. Worse, in a work inspired by recent developments in population ecology [11e14], Dumonteil, Mazzolo and Zoia have shown that non-Poisson spatial ﬂuctuations were caused by a neutron clustering phenomenon [15e18]: even for intermediate or high numbers of simulated neutrons, those ﬂuctuations can make it hard to estimate ﬂux-based standard deviations. The very ﬁrst description of this mechanism typical of birthedeath processes dates back to the 1980s, when Cox and Griffeath [19] showed that the spatial
* Corresponding author.
E-mail address: eric.dumonteil@irsn.fr (E. Dumonteil).
dimension 3 those spatial correlations would saturate. Later on, while being translated from the ﬁeld of probabilities to the ﬁelds of population ecology (describing spatial patterns appearing in water column plankton [11] or describing bacterial growth in Petri dishes [13,14]), neutron transport [15,16], and statistical mechanics [17,18], this process was referred to as clustering or Brownian bugs, whether it be for ﬁnite or inﬁnite media in any dimension and with different kinds of boundary conditions. In the present paper, we will show that space-dependent biases observed while simulating the neutron transport in decoupled systems ﬁnd their origin in these spatial correlations, when leakage boundary conditions are employed. Section 2 will discuss the phenomenology of these biases on a commercial reactor benchmark, and on a simpliﬁed model grasping its main characteristics [the mass-preserved one-dimensional (1D) binary branching Brownian motion on a segment with Dirichlet boundary conditions]. In Section 3 we will build a functional equation modeling the simpliﬁed case, based on a generalized Fisher equation with time-dependent coefﬁcients that accounts for population control and which incorporates spatial correlations. In Section 4 we will rely on an asymptotic analysis to establish a deep connection between traveling waves proper to quadratic terms in the neutronic ﬁeld equation and clustering. In
particular, wewill show that the neutron clusters triggera traveling
http://dx.doi.org/10.1016/j.net.2017.07.011
1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1158 E. Dumonteil et al. / Nuclear Engineering and Technology 49 (2017) 1157e1164
wave dynamic on the ﬂux causing the bias on local tallies. Some numerical solutions of this equation retrieved under simplifying hypotheses will be compared to the numerical ﬁndings of the ﬁrst section. Conclusions will be drawn in the ﬁnal section.
2. Biases associated to the Monte Carlo simulation of large reactor cores
2.1. Commercial reactor critical benchmark
The Expert Group on Advanced Monte Carlo Techniques belongs to the Working Party on Nuclear Criticality Safety of the OECD Nu-clear Energy Agency. Its aim is, amongst others, to guide Monte Carlo criticality practitioners through ﬁnding their ways in deﬁning the most appropriate simulation parameters, so as to minimize biases in the Monte Carlo estimate of different local quantities or in the estimate of their variances. This group, as well as recent work, has pointed out strong bias in both the estimate of the ﬂux and its variance, which depends on the spatial position of the tally volume [20,21]. This bias isproneto develop inparticular forlooselycoupled systems. Thus, a benchmark named R1 is currently under study, which proposes to tally the ﬂux in different radial zones of a critical commercial reactor [22]. This reactor has been simulated with the MORET 5.B.2 Monte Carlo code [23], exploiting a quarter symmetry. Axiallyaveraged ﬂuxesare presented ontheleftpartofFig.1 andthe associated ”apparent” 1-s error bars are provided by the left plot of Fig. 2 (these error bars are calculated by the Monte Carlo code using the central limit theorem). As expected, the highest uncertainties are located in low ﬂux regions, where neutrons leak out of the core. Surprisingly enough, though, the ”true” error bars given by the right plot of Fig. 2 exhibit spatial patterns: these errors seem to be big near the leaking boundaries of the reactor core but are also close to the reﬂecting boundaries and at the center of the core. Such nontrivial spatial patterns are even more strikingon the right plot of Fig.1, where the undersampling bias is estimated and is shown to be bigger at the center of the core, and close to the leaking boundaries. In particular, the ﬂux is overestimated near the leaking edges and is underestimated at the center. Also, the amplitude of this bias, be it positive or negative, seems to be inversely proportional to the number of neutrons per generation, as revealed by Fig. 3 In the following parts of this paper, we will try to model this very last
phenomenon, also reported by many authors and papers.
2.2. Mass preserving branching Brownian motion on a 1D conﬁned medium with Dirichlet boundary conditions
In order to explain these observations, different capabilities of the MORET 5.B.2 Monte Carlo code were successively disabled (simpliﬁed geometry, one group cross-sections, etc.) to grasp the phenomenology discussed in the present paper with the simplest model. In this respect it appeared that a mass-preserving binary branching Brownian motion [19,24,25] on a segment (of half length L arbitrarily set to 20) with Dirichlet (leakage) boundary conditions allowed to observe precisely an underestimation of the ﬂux in the central region while reproducing an overestimation of the ﬂux close to the boundaries. It is worth noting that such a modeling is more suited to Monte Carlo dynamic simulations (which make use of an algorithmic loop over time steps instead of a loop over neutron generations): for that reason a methodology to model generation-based simulations directly, instead of time-dependent problems, was proposed in a very recent work [26]. However, for the particular phenomenon described in this paper no signiﬁcant differences were observed between criticality and dynamic simu-lations: continuous time processes will be used as a means to draw conclusions that apply to both processes. The mass-preserving mechanism used in the simulations is fully described [27,28,18]. It is based on a combination between splitting and Russian roulette techniques: each time a particle is captured by a physics process, another is picked randomly and split, while each time a ﬁssion occurs, a randomly picked particle is Russian rouletted. The diffu-sion coefﬁcient D was set to 1, while the binary process was such that the capture cross-section g was equal to the two-daughter particles ﬁssion cross-section b, and both were set to 0.1. Typical realizations of such a process are provided in Fig. 4. As expected, the top plot of this ﬁgure highlights a strong particle clustering mechanism [16], and reveals that, after a short time, only one cluster remains [17]. Interestingly enough, though, looking at this process on a large time window (bottom plot), a qualitative view of the problem under consideration emerges: when only one cluster of particles strikes one of the boundaries while wandering around, the constraint on the overall mass Nof our mass-preserving process refrains the particle cluster from leaking out of the system, the splitting rate increases dramatically until the cluster is ”reﬂected” to the other side of the system. Therefore the Dirichlet boundary conditions cannot be properly taken into account. When the system
is not prone to trigger a clustering effect (i.e., for coupled
Fig. 1. MORET 5.B.2 simulation of the R1 OECD/NEA benchmark: axially averaged ﬂuxes with 104 active cycles of 104 neutrons (left plot). Ratio of the axially averaged ﬂuxes between a simulation with 106 active cycles of 102 neutrons and a simulation with 102 active cycles of 106 neutrons (right plot).
E. Dumonteil et al. / Nuclear Engineering and Technology 49 (2017) 1157e1164 1159
Fig. 2. MORET 5.B.2 simulation of the R1 benchmark with 104 active cycles of 104 neutrons: 1-s error bars (”apparent errors”) on the axially averaged ﬂuxes (left plot) and 1-s error bars (”true errors” estimated by independent simulations) on the axially averaged ﬂuxes with 104 active cycles of 104 neutrons (right plot).
conﬁgurations, with dominant ratio sensitively less than 1), the splitting mechanism that compensates leakages picks particles
according to a converged eigenvector and the bias disappears. Fig. 5
vtf ¼ Dv2f þ ðb gÞf þ lf2; (2)
sums up these discussions: the typical cosine shape of the mean ﬂux solution of the one-groupBoltzmann critical equation for a slab geometry is progressively distorted following a ﬂat distribution at the center with a strong decay near the extremities, when the number of simulated particles gets smaller. For a small number of particles ðN ¼ 50Þ, the cluster density proﬁle has been numerically
calculated by averaging 〈xðtÞ 〈xðtÞ〉〉t over Dt ¼ 1000 (a.u.), where 〈xðtÞ〉 is the average over all realizations of the process at a given
time. The typical resulting proﬁle of the clusters given in Fig. 6 exhibits exponentially decaying tails with a cutoff at ±40, corre-sponding to the maximum distance between two particles. This typical proﬁle has a spatial extension proportional to the number of particles: when this extension is smaller than the typical size of the system, the clusters wander around randomly so as to produce the ﬂat structures for the mean ﬂux discussed above.
3. The time-dependent generalized Fisher equation with population control
In this section we will build a stochastic model for our process (mass-preserved binary branching Brownian motion on a segment) and discuss some of its analytical solutions, obtained under simplifying hypotheses, in the case of Dirichlet or Neumann
boundary conditions.
where the coefﬁcient l is a saturation rate in front of f2. Indeed, depending on the sign of l, this saturation rate can be seen as imposingeithera maximal threshold value for the ﬂux (for negative l) above which negative counter-reaction will take place, or a minimal threshold value for the ﬂux (for positive l), below which positive counter-reaction will develop. To give a concrete example, in population ecology b and g can be interpreted, respectively, as the birth and death rates of a given species, whereas l might be interpreted as a saturation due to the competition for resources between individuals of this species. The squared ﬂux term arises from the fact that local interactions between individuals are combinatorial: the survival of an individual in x depends on the number of individuals in the vicinity of x. Therefore the death probability has to be adjusted by a saturation term proportional to
the number of pairs of individuals NðxÞ in ½x;x þ dx (where
NðxÞ ¼ fðxÞdx is the number of individuals in ½x;x þ dx), hence being quadratic in the f-ﬁeld.
When the competition term depends on the local density of individuals, the clustering effect described in Section 1 cannot be
neglected anymore as it affects the density itself. Therefore the
3.1. Generalized Fisher equation in the context of population ecology
The key ingredient used as a starting point of our modeling is the Fisherequation [29,30], also known inpopulation biologyas the spatial logistic equation, or in theoretical physics as the KPP (Kol-mogorovePetrovskiiePiskunov) equation. To build this equation, one starts with the diffusion equation
vtf ¼ Dvxf þ ðb gÞf; (1)
where f is a density, D is the diffusion coefﬁcient, b is a repro-duction rate, and g is a disappearance rate. The Fisher equation is a diffusion equationwith a supplementary quadratic term in the ﬂux. It takes the following form
1 In the following we will adopt a convention where the functions variables are
implicit in order to lighten the equations. In this case, f ¼ fðx;tÞ.
Fig. 3. MORET 5.B.2 simulation of the R1 benchmark with 104 active cycles: axially integrated ﬂux in each pin-cell as a function of the number of neutrons per cycle. The ﬂuxes are divided by reference ﬂuxes obtained by simulating 106 neutrons per cycle. Fits of ﬂux plots for each pin-cell indicate that the bias is inversely proportional to the
number of neutrons per cycle.
1160 E. Dumonteil et al. / Nuclear Engineering and Technology 49 (2017) 1157e1164
Fig. 4. x positions (x-axis) of the particles versus time t (y-axis) for two realizations of a mass-preserved binary branching Brownian motion on a segment (between 20 and 20) with Dirichlet boundary conditions and with N ¼ 50 particles. Top plot: ﬁrst realization observed between t ¼ 0 and t ¼ 300, bottom plot: second realization observed between t ¼ 0 and t ¼ 7250.
Fig. 5. Normalized particle densitydobtained by averaging over Dt ¼ 1000 (a.u.)dfor a mass-preserved binary branching Brownian motion on a segment (between -20 and 20) with Dirichlet boundary conditions and with N particles. Black curve: N ¼ 1000, blue curve: N ¼ 100, red curve: N ¼ 10, green curve: N ¼ 5.
Poisson spatial distribution of individuals used to build Eq. (2) is no longer true and one has to resort to a generalized Fisher equation (see the pioneer work of Birch and Young [31] for the full devel-
opment). The generalized Fisher equation
Fig. 6. Normalized cluster densitydobtained by averaging over Dt ¼ 1000 (a.u.)dfor a mass-preserved binary branching Brownian motion on a segment (between -20 and 20) with Dirichlet boundary conditions and with N ¼ 50 particles.
kernel nðrÞthat deﬁnes the spatial scale over which the competition occurs.
Z
vtf ¼ Dvxf þ ðb gÞf þ l dy nðjx yjÞGðx;y;tÞ; (3)
3.2. Adaptation of the generalized Fisher equation to reactor
physics: time-dependent coefﬁcients and population control
makes use of the pair correlation function G(x,y,t) which is deﬁned such as Gðx;y;tÞdxdy is the expected number of pair of individuals
with one individual in x and the other in y, and of the competition
The critical Boltzmann equation is commonly used in reactor physics to model the neutron transport in ﬁssile media [3], together
with its approximate form, the critical diffusion Eq. (1). In neutron
E. Dumonteil et al. / Nuclear Engineering and Technology 49 (2017) 1157e1164 1161
transport, the bf and gf reaction rates now represent ﬁssion and capture events, and are therefore respectively proportional to the ﬁssion and capture cross-sections. However, whether it be for reactor physics itself or for the simulation of reactor physics with Monte Carlo neutron transport codes, feedback effects arising from local/global constraints on the neutron population are left aside by
the transport/diffusion equation. These feedback effects arise, for
each generation. It is consequently proportional to the number of captures (g) and to the number of leakages DRL dx vxfðx;tÞ
and is smaller when productions by ﬁssion (b) are important. Finally, it is more convenient to use the normalized and centered
pair correlation function gðx;y;tÞ deﬁned as
example, from:
spatial and temporal control rod adjustments of the power
Gðx;y;tÞ fðx;tÞfðy;tÞ
fðx;tÞfðy;tÞ
(6)
during the operation of reactorcores. Those adjustments aim for instance at preventing local power excursions.
the so-called ”weight-watching” mechanisms (Russian roulette
Upon injection in Eq. (3) we get our equation for the dynamic of a branching Brownian motion in a conﬁned medium with popu-
lation control
and splitting) which are used at the end of each cycle (or each time step) of Monte Carlo criticality (or dynamic) simulations to adjust the number of simulated neutrons, so as to prevent any divergence or disappearance of the global population. This mechanism is also referred to as the ”renormalization” method
in this context.
vtfðx;tÞ ¼ Dvxfðx;tÞ þ ðb gÞfðx;tÞ 1 ZL
þlðtÞ@1 þ dy gðx;y;tÞfðy;tÞAfðx;tÞ;
L
(7)
In both casesdreactor physics or its Monte Carlo simu-lationdthe fundamental ingredient lacking in the transport equa-tion is the introduction of a feedback reaction rate term that.
depends on the local/global neutron population. This term de-scribes the rate at which local adjustments of the neutron density occur and relates the ﬁssion/capture probabilities of a neutron along its path to the local/global neutron densities seen by this neutron. In particular, it shall be sensitive to spatial correlations. The structure of this term is therefore precisely the one discussed in Section 3.1.: l dy nðjx yjÞGðx;y;tÞ; and
depends on time. As mentioned at the beginning of this section, the local adjustments per unit of time depend on time/genera-tion and the population number is constantly readjusted. In pretty much the same manneras in aworkof Newman et al. [32] we can therefore adjust the population by using a time-dependent adjustment coefﬁcient lðtÞ.
In the following, and without loss of generality, we will keep in mind the Monte Carlo criticality simulation of a 1D branching
Brownian with keff <1: at each cycle (or similarly for any time), the number of neutrons produced by ﬁssion is smaller than the number of neutrons at the beginning of the cycle and the l coefﬁcient is therefore positive. It represents the splitting mechanism which pro-
duces neutrons to compensate for leakages and absorptions. In this case,thecompetitionkernelnðrÞcanbesimpliﬁedbecausethespatial range onwhich thepopulationcontrol isapplied doesnot dependon thedistancebetweenneutrons:itisa globalconstraintontheoverall population and will therefore be set to 1 in the following. Also, the population-controlcriteriacanbemadeexplicitbyintegratingEq.(3)
over the positions between L and L which gives
together with its limit condition fð±L;tÞ ¼ 0 and with lðtÞ given by Eq. (4). We also recall here that gðx;y;tÞ is the normalized and centered pair correlation function of our system.
4. Clustering and traveling waves in bounded domains
The rather intricate form of this generalized Fisher equation with population control does not allow for direct solving as the pair correlation function has not been made explicit. However, sur-prisinglyenough, it is provided in a recent work of De Mulatier et al. [18], where the authors were able to provide an amenable form of the correlation function for branching Brownian motion with population control on bounded domains. In this section we will therefore use an asymptotic analysis of Eq. (7), and we will show that its solution, in the case where the neutron population is small and whenever leakage boundary conditions are used, differs from the ”deterministic” one, allowing to understand the cause and the structure of the bias observed in Section 2.
4.1. Large population size
Whenever the neutron population N is very large, we have gðx;y;tÞ/0 as the spatial correlations arising from the branching process vanish (see for instance [18,31]. Hence, the pair correlation function G becomes separable in x and y and we have Gðx;y;tÞ/fðx;tÞfðy;tÞ so that dy Gðx;y;tÞ/fðx;tÞ and dx dy Gðx;y;tÞ/1. The stationary limit of Eq. (7) takes in this
case the very simple form
0 1 L
v2f B dx v2fðxÞCf ¼ 0: (8)
Z L
bþ g D dx v2fðx;tÞ
lðtÞ ¼ Z L Z L L ; (4) dx dy Gðx;y;tÞ
L L
where we have used the following normalization relation
ZL
dx fðx;tÞ ¼ 1: (5)
L
L
Noticing that RL dx vxfðxÞ ¼ vxfðxÞjx¼±L, this equation sim-pliﬁes to
vxf vxfðxÞjx¼±Lf ¼ 0: (9)
In the case of reﬂecting boundary conditions at each side of the domain, we have to use the Neumann boundary conditions
vxfjx¼±L ¼ 0. Eq. (9) is therefore trivially veriﬁed, ensuring that the Monte Carlo criticality renormalization in this case is unbiased.
In the case of absorbing boundary conditions at each side of the
domain, we have to use the Dirichlet boundary conditions given by
In our example with keff <1, this positive coefﬁcient is inter-preted as the splitting rate that compensates the neutron loss at
fðx;tÞjx¼±L ¼ 0. Surprisingly enough, this boundary condition applied on a positive and symmetric function ensures that the
1162 E. Dumonteil et al. / Nuclear Engineering and Technology 49 (2017) 1157e1164
coefﬁcient in front of the term linear in f is strictly positive. To go further, as a guess function it is possible to test the cosine solution of the diffusion approximation of the stationary Boltzmann critical equation with leakages. This test function can be written as
fðxÞ ¼ A cos 2 L , where A ¼ 4L is deﬁned upon normalization of
the density function fðxÞ. This implies that vxfðxÞjx¼±L ¼ 4L2. As a consequence, we have
þL þL
dy Gðx;y;tÞ ¼ fðx;tÞ g∞ðjrjÞdrfðx;tÞ: (14)
L L
This hypothesis on the structure of the spatial correlation function is relevant for strongly decoupled systems which typical dimensions are order of magnitudes bigger than the typical spatial
correlation length. The lastexpression can then be transformed into
2
vxf þ 4L2 f ¼ 0; (10)
þL
dy Gðx;y;tÞ ¼ g∞fðx;tÞ2; (15)
ensuring that Eq. (9) is veriﬁed. The solution of this equation is indeed a cosine, ensuring also that the Monte Carlo criticality renormalization in this case is unbiased. It is noticeable that none of the coefﬁcients of this equation depend on the values of b, g, or D, and that there is always a solution to this problem, unlike for the critical diffusion equation where a criticality condition linking the geometry and the compositions has to be met. However, this can be understood if we keep in mind that this is precisely the purpose of the renormalization: whatever the parameters characterizing the system, the simulation converges to an unbiased estimate of
fðxÞ.