A lumped parameter method of characteristics approach and multigroup kernels applied to the subgroup self-shielding calculation in MPACT

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A lumped parameter method of characteristics approach and multigroup kernels applied to the subgroup self-shielding calculation in MPACT. The neutron transport code, MPACT, is currently using the subgroup 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 developing a novel lumped parameter approach to MOC, substantial improvements have also been made to the self-shielding computational efficiency without sacrificing any accuracy
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:
An essential component of the neutron transport solver is the resonance self-shielding calculation used to
Received 5 June 2017
Accepted 7 July 2017
Available online 17 July 2017
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 xed-source problems. Recent efforts incorporating multigroup kernels to the MOC solvers in
Keywords:
CASL
Lumped Parameter
M&C 2017
MOC
MPACT
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 efciency without sacricing 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
Subgroup 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 benets, 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
1. Introduction
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-
The resonance self-shielding calculation is a critical part of the
actors (CASL) [1] aims to provide high-delity simulations of nu-
neutron transport solve used to obtain accurate cross sections and
clear reactor core physics. To accomplish this, CASL is developing
ux solutions. MPACT is currently using the subgroup method [13]
the Virtual Environment for Reactor Applications (VERA) [2], which
to accomplish this. Recent efforts to improve the efciency of the
consists of a collection of physics codes and multiphysics coupling
MOC solvers, which are the workhorse for radial transport in
drivers. The MPACT code [3] is the primary deterministic neutron
MPACT, have yielded efcient 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
approach is consistent with the MOC kernels in CASMO (K. Smith,
transport problems. In this approach, the 2D method of charac-
personal communication, 2015) and OpenMOC [15]. These kernels
teristics (MOC) is used for each plane to solve the radial transport
have sped up the MOC sweeping time by roughly 2 during the
problem, and 1D pin-wise nodal methods are used axially [6].
eigenvalue calculation. The subgroup calculation typically requires
Considerable attention has recently been focused on improving the
a substantial amount of time, and it had not been reevaluated to
computational performance of MPACT, and almost every sequence
take advantage of these new kernels. The rst improvement
addressed herein is the integration of multigroup kernel concepts into
the subgroup calculation, which is then used as the basis for further
* Corresponding author.
extensions.
E-mail address: stimpsonsg@ornl.gov (S. Stimpson).
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/
X
X
c
ðrÞ
1
g
@
A
n
p
p
4
4
k
4
q
g
g
g
S
g;a
g;a
S
q
g;r
g;a g;
4  4
q
g;a
S s S
g;a
in
S. Stimpson et al. / Nuclear Engineering and Technology 49 (2017) 1240e1249
1241
The next improvement discussed is what is being termed
where
lumped parameter MOC. Because the subgroup calculation in
MPACT is solving purely absorbing xed-source problems (FSPs),
as will be discussed later, multiple sweeps are performed only to
update the boundary angular uxes. Because it is solving purely
0 1
Ngrp X Ngrp X
qgðrÞ ¼ ðrÞfg0 ðrÞ þ ðrÞfg0 ðrÞ :
eff g0¼1 f;g0 g0¼1; s0;g0/g
absorbing FSPs, the sweep procedure can be condensed to allow
(1b)
for instantaneous propagation of the ux 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,
The scalar ux ðfgÞ is obtained by integrating the angular ux
ð4gÞ over all angles (Eq. 1c):
which are ray-dependent. Then, a number of fast sweeps can be
performed to resolve the boundary angular uxes. Once the
boundary angular uxes are considered to be converged, a nal
fg0 ðrÞ ¼ Zp4g0 r;U0dU0:
(1c)
sweep is completed to tally the scalar ux, which is then used to
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-
Here, Sx;g is total, ssion, or scattering cross section; cg is the
ssion 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
ssion and scattering terms. Whereas some applications use a
linear source representation spatially [20,21], most use a at source
approximation, as is used in this work. The MOC equations in this
article focus on at 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 at within each region, the angular ux can be
found at any point s along the ray:
 
4 ðs;UÞ ¼ 4inðUÞeSt;gs þ 1  eSt;gs : (2)
t;g
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-
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
state problems without feedback, it is worth noting that the self-
From this, the equations for the outgoing angular ux (Eq. 4a) at
shielding calculation is even more important to problems with
the end of the ray (la,r) and the average angular ux along the ray
feedback. As the eigenvalue calculation is being executed, the
(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 efciency of each instantiation,
and the same speedups observed here apply to calculations with
feedback.
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 ux ð4outÞ is
then used as the incoming angular ux 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 ux ð4g;aÞ is used to tally the
2. Governing theory
scalar ux in each region ðfg;rÞ, which is used to update the source
term in Eq. (3).
2.1. Method of characteristics
2.2. Subgroup self-shielding
MOC is a widely used deterministic method of solving the
multigroup Boltzmann neutron transport equation (Eq. 1)
[5,9,15,17e19]:
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;
(1a)
resentation that preserves certain integrals. There are two
groups of methodologies for determining the subgroup proba-
t;g
bility tables: (1) the physical probability table and (2) the
l
U
U
U
P
f w
X
1
4 5
l
U
U
U
1
l
D
p
4
1 e
0 0 0
a
p
1 
4
 
 
1
þ1
A
iso
1
p
4
1
l
D
p
4
1
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
"X X #
,V4gðr; Þ þ ðrÞ þ g ðrÞ 4gðr; Þ
a;g p
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 ux. Assuming that
subgroup parameters have been determined, application of the
subgroup method involves obtaining the level-dependent ux (or
the equivalence cross section that can be converted from ux) from
¼ 4plg p ðrÞDug :
The corresponding subgroup FSP can be written as:
2 3
X X
,V4g;c;lðr; Þ þ ðrÞ þ g ðrÞ 4g;c;lðr; Þ
a;g;c;l p
X
¼ g ðrÞ ug:
p
(9)
(10)
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 signicant
computing time for subgroup calculation. Note that Eq. (9) is a
purely absorbing problem that is achieved by the approximations
U,V4ðr;U;uÞ þ Stðr;uÞ4ðr;U;uÞ
X Zu X u0u
¼ ðr;u Þfðr;u Þ du ;
iso uεiso s;iso iso
(6)
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
uxes, 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
dened in the resolved resonance energy range (roughly from 1 eV
to 25 keV), where the ssion source term is neglected and isotropic
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 ux 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 ux solutions
that feed into the cross section updates.
scattering in a laboratory frame of reference is assumed [23]. The
scattering source can be further simplied by applying the inter-
2.3. Multigroup kernels for subgroup
mediate resonance approximation [24],
The preexisting subgroup calculation in MPACT consists of
several loops: (1) over resonant groups, (2) over resonant cate-
U,V4ðr;U;uÞ þ Stðr;uÞ4ðr;U;uÞ
gories, and (3) over sublevels [3]. These are typically solved one at a
¼ 4plSpðrÞ 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,
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.
2 3
U,V4gðr;UÞþSt;gðrÞ4gðr;UÞ¼ 1 1lg4XðrÞþXðrÞ5fgðrÞ
RS;g p
X
þ g ðrÞ ug:
p
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
(8)
Note that Ss;g is written as the sum of resonance scattering SRS,g
and potential scattering Sp. MPACT uses heterogeneous pin cell
congurations to generate resonance integral table and 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
parameters. As long as a consistent form of FSP is used to obtain the
subgroup parameters and to perform the subgroup FSP calculation,
Table 1
another two approximations can be made to further simplify the
A typical set of resonance categories.
FSP without sacricing the accuracy of the effective cross section
Category
Isotopes
[25], i.e., in the scattering term, eliminating the resonance scat-
tering lgSRS;g and assuming an isotropic ux 4gðr;UÞ¼4pfgðrÞ.
Applying the two approximations and moving the term containing
ux from the right side to the left gives:
1
2
3
4
U-238
U-235, other actinides, and important ssion products
Clad isotopes
Poison isotopes (AIC, Gd, Hf, etc.)
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A lumped parameter method of characteristics approach and multigroup kernels applied to the subgroup self-shielding calculation in MPACT. The neutron transport code, MPACT, is currently using the subgroup 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 developing a novel lumped parameter approach to MOC, substantial improvements have also been made to the self-shielding computational efficiency without sacrificing any accuracy.

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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)

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