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Nuclear Engineering and Technology 49 (2017) 1240e1249
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Original Article
A lumped parameter method of characteristics approach and multigroup kernels applied to the subgroup self-shielding calculation in MPACT
Shane Stimpson a, *, Yuxuan Liu b, Benjamin Collins a, Kevin Clarno a
a Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN 37831, USA
b Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 1901 Cooley Building, 2355 Bonisteel Boulevard, Ann Arbor,
MI 48109, USA
a r t i c l e i n f o a b s t r a c t
Article history: Received 5 June 2017 Accepted 7 July 2017
Available online 17 July 2017
Keywords: CASL
Lumped Parameter M&C 2017
MOC MPACT
Subgroup Self-Shielding
1. Introduction
An essential component of the neutron transport solver is the resonance self-shielding calculation used to determine equivalence cross sections. The neutron transport code, MPACT, is currently using the sub-group self-shielding method, in which the method of characteristics (MOC) is used to solve purely absorbing ﬁxed-source problems. Recent efforts incorporating multigroup kernels to the MOC solvers in MPACT have reduced runtime by roughly 2. Applying the same concepts for self-shielding and devel-oping a novel lumped parameter approach to MOC, substantial improvements have also been made to the self-shielding computational efﬁciency without sacriﬁcing any accuracy. These new multigroup and lumped parameter capabilities have been demonstrated on two test cases: (1) a single lattice with quarter symmetry known as VERA (Virtual Environment for Reactor Applications) Progression Problem 2a and (2) a two-dimensional quarter-core slice known as Problem 5a-2D. From these cases, self-shielding computational time was reduced by roughly 3e4, with a corresponding 15e20% increase in overall memory burden. An azimuthal angle sensitivity study also shows that only half as many angles are needed, yielding an additional speedup of 2. In total, the improvements yield roughly a 7e8 speedup. Given these performance beneﬁts, these approaches have been adopted as the default in MPACT.
© 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/).
in the code has been improved [7e12]. This paper presents some of
the improvements made to the self-shielding calculation.
The Consortium for Advanced Simulation of Light Water Re-actors (CASL) [1] aims to provide high-ﬁdelity simulations of nu-clear reactor core physics. To accomplish this, CASL is developing the Virtual Environment for Reactor Applications (VERA) [2], which consists of a collection of physics codes and multiphysics coupling
drivers. The MPACT code [3] is the primary deterministic neutron
The resonance self-shielding calculation is a critical part of the neutron transport solve used to obtain accurate cross sections and ﬂux solutions. MPACT is currently using the subgroup method [13] to accomplish this. Recent efforts to improve the efﬁciency of the MOC solvers, which are the workhorse for radial transport in
MPACT, have yielded efﬁcient multigroup kernels that loop over
transport solver in VERA, predominantly using the two- several energy groups rather than one group at a time [9,14]. This
dimensional/one-dimensional (2D/1D) method [4,5] to solve 3D transport problems. In this approach, the 2D method of charac-teristics (MOC) is used for each plane to solve the radial transport problem, and 1D pin-wise nodal methods are used axially [6]. Considerable attention has recently been focused on improving the computational performance of MPACT, and almost every sequence
* Corresponding author.
E-mail address: stimpsonsg@ornl.gov (S. Stimpson).
approach is consistent with the MOC kernels in CASMO (K. Smith, personal communication, 2015) and OpenMOC [15]. These kernels have sped up the MOC sweeping time by roughly 2 during the eigenvalue calculation. The subgroup calculation typically requires a substantial amount of time, and it had not been reevaluated to take advantage of these new kernels. The ﬁrst improvement addressed herein is the integration of multigroup kernel concepts into the subgroup calculation, which is then used as the basis for further
extensions.
http://dx.doi.org/10.1016/j.net.2017.07.006
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/).
S. Stimpson et al. / Nuclear Engineering and Technology 49 (2017) 1240e1249 1241
The next improvement discussed is what is being termed lumped parameter MOC. Because the subgroup calculation in MPACT is solving purely absorbing ﬁxed-source problems (FSPs), as will be discussed later, multiple sweeps are performed only to update the boundary angular ﬂuxes. Because it is solving purely absorbing FSPs, the sweep procedure can be condensed to allow for instantaneous propagation of the ﬂux across a spatial domain without the need to sweep along all segments in a ray. This re-
quires an initial sweep to determine the lumped parameter values,
where
0 1 Ngrp X Ngrp X
qgðrÞ ¼ ðrÞfg0 ðrÞ þ ðrÞfg0 ðrÞ :
eff g0¼1 f;g0 g0¼1; s0;g0/g
(1b)
The scalar ﬂux ðfgÞ is obtained by integrating the angular ﬂux ð4gÞ over all angles (Eq. 1c):
which are ray-dependent. Then, a number of fast sweeps can be performed to resolve the boundary angular ﬂuxes. Once the boundary angular ﬂuxes are considered to be converged, a ﬁnal
sweep is completed to tally the scalar ﬂux, which is then used to
fg0 ðrÞ ¼ Zp4g0 r;U0dU0: (1c)
0
update the cross sections for the eigenvalue calculation that follows.
Section 2 presents the underlying theory, including back-ground on MOC and details of the general subgroup method. Once the foundation is set, the initial improvements to incorpo-rate multigroup kernels are shown, along with corresponding algorithms. A batching scheme is also presented to help minimize the memory burden incurred by solving all pseudogroups concurrently, which is the combination of resonant groups, cat-egories, and levels. Then the lumped parameter MOC is derived, expanding on the algorithms to highlight the new calculation scheme. Section 3 showcases the test cases used to demonstrate these new capabilities. First, a single lattice test case known as VERA Progression Problem 2a is shown. The previous subgroup scheme is compared with the new approach with multigroup kernels and lumped parameter MOC for various numbers of batches. Next, a 2D quarter-core pressurized water reactor prob-lem known as VERA Progression Problem 5a-2D [16] is shown. It provides an interesting comparison with the lattice problem, as radial decomposition is used to solve it. Finally, the sensitivity of the number of azimuthal angles used to solve the subgroup calculation is explored, providing some insights into additional performance gains.
Although the results presented in this work focus on steady-state problems without feedback, it is worth noting that the self-shielding calculation is even more important to problems with
feedback. As the eigenvalue calculation is being executed, the
Here, Sx;g is total, ﬁssion, or scattering cross section; cg is the ﬁssion spectrum; keff is the system eigenvalue, r represents the spatial vector; U represents the angular vector consisting of both
azimuthal and polar angles; g denotes the neutron energy group index; and qg is the g-group source term that contains both the ﬁssion and scattering terms. Whereas some applications use a linear source representation spatially [20,21], most use a ﬂat source approximation, as is used in this work. The MOC equations in this article focus on ﬂat source 2D radial applications with isotropic scattering kernels. MOC is applied to this problem by introducing a characteristic direction and casting a 1D version of Eq. (1) along the problem. By assuming the spatial dependence of the total cross section/source is ﬂat within each region, the angular ﬂux can be found at any point s along the ray:
4 ðs;UÞ ¼ 4inðUÞeSt;gs þ 1 eSt;gs : (2)
t;g
By using the discrete ordinates approximation and applying spatial discretization, Eq. (2) now becomes Eq. (3), where a denotes the angle index and r denotes the spatial region index:
4 ðsÞ ¼ 4in eSt;g;rs þ qg;r 1 eSt;g;rs : (3) t;g;r
From this, the equations for the outgoing angular ﬂux (Eq. 4a) at the end of the ray (la,r) and the average angular ﬂux along the ray
(Eq. 4b) are determined:
simulation will iterate between the neutronics and thermal hy-draulic calculations every outer iteration. If the change in temper-
ature is large enough, the self-shielding calculation will be called
4out ¼ 4inaeSt;g;rla;r þ St;g;r 1 eSt;g;rla;r ; (4a)
again to update the equivalence cross sections. This process can result in several subgroup self-shielding calculations; if the time for this sequence is large, it can be a substantial burden on the overall calculation. The approaches presented here do not affect the number of times self-shielding will be instantiated in such a calculation, but they do address the efﬁciency of each instantiation, and the same speedups observed here apply to calculations with feedback.
2. Governing theory
2.1. Method of characteristics
MOC is a widely used deterministic method of solving the multigroup Boltzmann neutron transport equation (Eq. 1)
[5,9,15,17e19]:
out in
4 ¼ g;a g;a þ g;r : (4b)
t;g;r a;r t;g;r
When tracing along each ray, the outgoing angular ﬂux ð4outÞ is
then used as the incoming angular ﬂux ð4g;aÞ for the next segment in the ray. If the segment terminates along a parallel domain
boundary, it will be sent to the neighboring process to be used in the next iteration. The average angular ﬂux ð4g;aÞ is used to tally the scalar ﬂux in each region ðfg;rÞ, which is used to update the source term in Eq. (3).
2.2. Subgroup self-shielding
MPACT is currently using the subgroup self-shielding method, in which the detailed cross section behavior of each
coarse energy group is replaced by its probability density rep-
U,V4gðr;UÞ þ XðrÞ4gðr;UÞ ¼ qgðrÞ; t;g
resentation that preserves certain integrals. There are two (1a) groups of methodologies for determining the subgroup proba-
bility tables: (1) the physical probability table and (2) the
1242 S. Stimpson et al. / Nuclear Engineering and Technology 49 (2017) 1240e1249
mathematical probability table [22]. MPACT uses the physical probability table, in which the resonance integral tables are converted into a set of subgroup levels and weights by preser-ving effective cross sections over a range of background cross sections. The effective cross section is evaluated by these sub-group levels sx;g;l and weights wx;g;l as:
sx;gyPlsx;g;lfg;lwx;g;l ; (5) l g;l x;g;l
where fg;l is the subgroup level-dependent ﬂux. Assuming that subgroup parameters have been determined, application of the subgroup method involves obtaining the level-dependent ﬂux (or
the equivalence cross section that can be converted from ﬂux) from
"X X # ,V4gðr; Þ þ ðrÞ þ g ðrÞ 4gðr; Þ
a;g p
¼ 1 lg XðrÞDug : (9) p
The corresponding subgroup FSP can be written as: 2 3
U,V4g;c;lðr;UÞ þ 4 X ðrÞ þ lg XðrÞ54g;c;lðr;UÞ a;g;c;l p
¼ 1 lg XðrÞDug: (10) p
an FSP.
To obtain the subgroup FSP, we start with the continuous-
energy slowing-down equation in terms of lethargy:
Eq. (10) must be solved for every resonance group g, resonance category c [10], and subgroup level l, resulting in signiﬁcant
computing time for subgroup calculation. Note that Eq. (9) is a
U,V4ðr;U;uÞ þ Stðr;uÞ4ðr;U;uÞ
X Zu X u0u
¼ ðr;u Þfðr;u Þ du ; (6)
iso uεiso s;iso iso
purely absorbing problem that is achieved by the approximations
discussed previously. The isotopes in the problem are generally divided into a setof resonantcategories. Forexample, 238Umay be in one category, 235U and other heavy metals in another, natural zir-
conium isotopes in a third, and all others in a fourth category. A
typical set of resonance categories used for most pressurized water
where 4ðr;U;uÞ and fðr;u0Þ are the neutron angular and scalar ﬂuxes, respectively. Ss;iso is the macroscopic cross section of reac-
2
tion channel x for isotope iso. aiso ¼ Aiso1 and εiso ¼ ln aiso
are the maximum fraction of energy loss and maximum lethargy gain per neutron scattered off isotope iso, respectively. Eq. (6) is deﬁned in the resolved resonance energy range (roughly from 1 eV to 25 keV), where the ﬁssion source term is neglected and isotropic scattering in a laboratory frame of reference is assumed [23]. The scattering source can be further simpliﬁed by applying the inter-mediate resonance approximation [24],
U,V4ðr;U;uÞ þ Stðr;uÞ4ðr;U;uÞ
¼ 1 lSpðrÞ þ ð1 lÞSsðr;uÞfðr;uÞ ; (7)
where lSpðrÞ ¼ PlisoSp;isoðrÞ and ð1 lÞSsðr;uÞ ¼ Pð1 lisoÞ iso iso
Ss;isoðr;uÞ. Integrating Eq. (7) over energy groups yields the multigroup FSP,
2 3 U,V4gðr;UÞþSt;gðrÞ4gðr;UÞ¼ 1 1lg4XðrÞþXðrÞ5fgðrÞ
RS;g p
X
þ g ðrÞ ug:
p
(8)
Note that Ss;g is written as the sum of resonance scattering SRS,g and potential scattering Sp. MPACT uses heterogeneous pin cell conﬁgurations to generate resonance integral table and subgroup parameters. As long as a consistent form of FSP is used to obtain the subgroup parameters and to perform the subgroup FSP calculation,
another two approximations can be made to further simplify the
reactor applications in MPACT is shown in Table 1. This approach is commonly used in self-shielding methods to minimize the total number of transport calculations required to obtain accurate results. Once the ﬂux solution for each resonant group, category, and sublevel are obtained, the cross sections can be updated. The details of the cross section updates will be left to other resources [3,13] as the focus of this article is to focus on obtaining faster ﬂux solutions
that feed into the cross section updates.
2.3. Multigroup kernels for subgroup
The preexisting subgroup calculation in MPACT consists of several loops: (1) over resonant groups, (2) over resonant cate-gories, and (3) over sublevels [3]. These are typically solved one at a time, but the multigroup kernels allow for some or all to be solved concurrently. The calculation scheme for the three-loop approach is shown in Fig.1. As can be seen, inside these three loops, there is an iteration loop in which transport sweeps for each resonant group, category, and level are performed. Here, Sp is the potential cross section, St is the total cross section, and Seq is the equivalence cross section.
To take advantage of the multigroup kernels that have been implemented into MPACT [14], the scheme must be slightly restructured. For the purposes of this work, a single combination of group/category/level is considered to be a pseudogroup [26]. The numberof pseudogroups for the entire subgroup calculationwill be the product of the number of resonant groups, the average number of subgroup categories per group, and the number of subgroup levels. In theory, the number of categories can vary from group to group, although this is not the case for the current libraries avail-able to MPACT. In the 47-group library [27] used in this work, there are 17 resonant groups, four categories, and four levels yielding 272
Table 1
A typical set of resonance categories.
FSP without sacriﬁcing the accuracy of the effective cross section [25], i.e., in the scattering term, eliminating the resonance scat-
tering lgSRS;g and assuming an isotropic ﬂux 4gðr;UÞ¼4pfgðrÞ. Applying the two approximations and moving the term containing
ﬂux from the right side to the left gives:
Category
1 2 3
4
Isotopes
U-238
U-235, other actinides, and important ﬁssion products Clad isotopes
Poison isotopes (AIC, Gd, Hf, etc.)
S. Stimpson et al. / Nuclear Engineering and Technology 49 (2017) 1240e1249 1243
Fig. 1. Pseudocode for preexisting subgroup scheme with separate loops over resonant group, category, and level.
pseudogroups. Based on this concept, a transport kernel could be constructed to sweep over all pseudogroups concurrently, effec-tively vectorizing the three loops of the original algorithm. How-ever, the sources, cross sections, scalar ﬂuxes, and angular ﬂuxes must be stored for each pseudogroup up front, whereas in the previous scheme, storage of only one groupat a time was necessary. Fig. 2 shows the pseudocode for the refactored scheme, taking advantage of the multigroup kernel concept. When Fig. 2 is compared with Fig. 1, it can be seen that the three loops over resonant group, subgroup category, and subgroup level are condensed into a single loop over pseudogroups. What is not seen is the actual multigroup MOC kernel, inwhich the loop over groups/ pseudogroups is moved to the innermost loop, which is being called on line 6.
As might be expected, the memory required to store source and ﬂux data for 272 pseudogroups can be signiﬁcant. One way to mitigate memory concerns while allowing the scheme to make use of themultigroupkernelsistodividethepseudogroupsintobatches.
Here, the 272 pseudogroups are decomposed into Nbatch batches, dividingasevenlyaspossible.Themainadvantageof themultigroup
kernels is that they eliminate the duplicate work of connecting the modular rays for the entire domain by moving the loop over groups to the innermost loop [14]. When using batches, some of this advantage is compromised, as the modular rays setup will still be
performed Nbatch times. However, this is still a signiﬁcant reduction comparedwith performing itforeach pseudogroup,asisthe casefor
the original approach. Fig. 3 shows the pseudocode for the batched approach, where each batch contains a starting and stopping pseu-dogroup index. The primary difference between Figs. 2 and 3 is that Fig. 3 includes an additional outer loop over the number of batches, and the bounds for the pseudogroup loops are updated to reﬂect the
lower and upper pseudogroup indexes for each batch.
It is worth noting that using multigroup kernels necessitates that all groups, at least within each batch, perform the same number of sweeps. To get an idea of how this might affect performance, Fig. 4 shows the number of iterations to reach a converged solution for the same 2D quarter-core slice shown in the results using the one-group sweeping approach, in which each pseudogroup is solved separately. FromFig.4, itcan beseenthatmostpseudogroups(about 66%) take ﬁve iterations, another 28% take four iterations, and only 6% take three iterations. Each batch will be limited by the pseu-dogroup that is slowest to converge, which in most cases will be ﬁve iterations. So there will be some additional overhead, as some pseudogroups will likely be more converged than necessary, although it will usually be only one to two additional iterations. In the next subsection, which pertains to the lumped parameter MOC approach, it will be seen that there are basically two full sweeps for each pseudogroup (ﬁrst and last) with several fast sweeps in be-tween the converge. In this lumped parameter scheme, the over-head incurred from the multigroup kernels is mitigated, as the additional iterations observed are just the fast iterations and most of the burden is on the ﬁrst and last sweeps.
2.4. Lumped parameter MOC
In addition to using multigroup kernels, a lumped parameter MOC approach has been applied to the subgroup self-shielding problem [26]. Because the self-shielding calculation is a purely absorbing FSP, and because multiple sweeps are performed only to update the boundary angular ﬂuxes, the sweep procedure can be condensed to allow for the instantaneous propagation of the ﬂux across a spatial domain without the need to sweep along all seg-ments in a ray as is typically done. This requires an initial sweep to
tabulate lumped parameter coefﬁcients for the angular ﬂux
Fig. 2. Pseudocode for subgroup scheme using the multigroup transport kernel, where groups, categories, and levels are vectorized as pseudogroups.
1244 S. Stimpson et al. / Nuclear Engineering and Technology 49 (2017) 1240e1249
Fig. 3. Pseudocode for subgroup scheme using the multigroup transport kernel and batching. This is very similar to Fig. 2, but with the introduction of an outer loop over batches.
Fig. 4. Number of iterations to converge each of the 272 pseudogroups for a two-dimensional quarter-core slice with 16 radial decomposition zones. 68% of the pseudogroups here require 5 iterations to converge, 28% need 4 iterations, and only 6% need 3 iterations.
propagation. Subsequent sweeps use the lumped parameters to instantly update the angular ﬂux, bypassing all calculations along
the ray. Once the boundary angular ﬂuxes are considered to be
same time, effectively two equations are needed for each angle and
ray (Eqs. 7a and 7b):
converged, an additional sweep is completed to tally the scalar ﬂux. Because the MOC kernels in MPACT sweep over two angles
traveling in opposite directions (forward and backward) at the
out;for in;for pg;a pg;a
(7a)