A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

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A multilevel in space and energy solver for multigroup diffusion eigenvalue problems. n this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step.
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Original Article
A multilevel in space and energy solver for multigroup diffusion
eigenvalue problems
Ben C. Yee*, Brendan Kochunas, Edward W. Larsen
Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 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:
In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving
Received 2 June 2017
Received in revised form
14 July 2017
Accepted 27 July 2017
Available online 5 August 2017
multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three
additional features: (1) a grey (one-group) diffusion equation used to efciently converge the ssion
source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs
required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In
MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by
Keywords:
Multigroup diffusion
Multilevel
Eigenvalue
performing work on lower-order equations with only one group and/or coarser spatial grids. Results
from several Fourier analyses and a one-dimensional test code are provided to verify the efciency of the
MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear
solver. These results highlight the potential efciency of the MSED method as a solver for multidi-
mensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future
work. Our ultimate goal is to implement the MSED method as an efcient solver for the two-
dimensional/three-dimensional coarse mesh nite difference diffusion system in the Michigan parallel
characteristics transport code. The work in this paper represents a necessary step towards that goal.
© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the
1. Introduction
from existing ideas (multigrid-in-space [3] and two-grid in energy
[4]) as well as new ideas (space-dependent Wielandt shift [5]). The
The multigroup diffusion eigenvalue problem is an approxima-
three primary components of MSED are: (1) a grey(one-group)
tion to the multigroup neutron transport eigenvalue problem that
diffusion equation, used to converge the eigenvalue and ssion
is widely used for reactor physics simulations. The solution is
source, (2) a space-dependent Wielandt shift, used to reduce the
frequently used to accelerate the source iteration procedure for
number of power iterations (PIs) required for convergence, and (3)
solving neutron transport problems via methods such as coarse
a multigrid-in-space solver, used to solve the xed-source grey and
mesh nite difference (CMFD) [1]. Although solving a diffusion
multigroup diffusion linear systems.
problem requires signicantly fewer computational resources than
The MSED method can be viewed as an extension of the CMFD
solving a transport problem, this cost is still not trivial. Many
method. In CMFD, the convergence of a higher-order (more un-
transport codes that use CMFD-like procedures (e.g., the Michigan
knowns) transport or nodal diffusion system is accelerated by
parallel characteristics transport (MPACT) code [2]) have a CMFD
leveraging a lower-order diffusion system. In MSED, this lower-
eigenvalue problem with hundreds of millions of unknowns, and
order diffusion system is itself accelerated by simpler diffusion
obtaining solutions to this problem constitutes a large portion of
equations with even fewer unknowns. Fig. 1A provides a visuali-
the computational effort.
zation of this hierarchy. Alternatively, the MSED method can be
In this work, we introduce a new multilevel in space and energy
viewed as an extension of the multigrid method to nonspatial
diffusion (MSED) method for solving the multigroup diffusion
variables. Fig. 1B provides a visualization of the changes in spatial
eigenvalue problem. This is a multicomponent method that draws
and energy grid sizes in the MSED iteration scheme. These two
gures are further explained in Sections 2 and 3.
Of the methods that have been developed for reactor physics
* Corresponding author.
simulations, the multilevel coarse mesh rebalance (MLCMR) and
E-mail addresses: bcyee@umich.edu (B.C.
Yee),
multilevel surface rebalance (MLSR) methods developed by van
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/
^
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1126
B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134
t
e
e
d
d
e
g
d
e
m
e
d
d
e
m
d
g m
e
d
g
e
g
e
g
g
d
(A)
(B)
Fig. 1. Each gure provides an overview of the MSED iteration procedure. (A) The hierarchy of the equations in MSED is shown, with a scale on the left describing the relative
complexity of these equations (i.e., the number of unknowns). The higher (red) dashed box encloses the equations used in the CMFD method, while the lower (green) dashed box
encloses the equations used in the MSED method. (B) An MSED iteration is broken up into four steps, and the changes in the energy and spatial grid sizes at each step are visualized.
Geemert and others [6e8] are the most similar to MSED. The
found that the MSED method with a grey equation already per-
MLCMR and MLSR methods are techniques for nodal diffusion
forms very well. Our Fourier analysis and numerical results indicate
problems, which leverage solutions from a series of lower-order
that the MSED method, as described in this paper, has a spectral
coarse-grid one-group diffusion equations in order to minimize
radius of ~0.1. Because of this low spectral radius and the fact that a
the number of iterations required on the full higher-order nodal
grey system is simpler (easier to implement and solve) than a two-
diffusion equations. In this sense, the approaches taken by the
group system, we have not yet been compelled to study the use of a
MSED method and the MLCMR/MLSR methods are similardboth
two-group system in the MSED method. Recent work by Cornejo
methods minimize the computational effort required by shifting
and Anistratov, however, has shown that additional energy grid(s)
iterations from higher-order equations to lower-order equations
between one group and G groups [12,13] can provide tangible im-
that are less costly to solve. The MSED method is also similar to
provements in the runtime, and it may be possible to use a similar
multilevel CMFD methods [1,9,10]. In these cited works, the
strategy to improve the MSED algorithm. In future work, we will
multigroup CMFD equations are themselves accelerated by two-
assess the potential benet of both introducing an extra two-group
group CMFD equations.
system to MSED and replacing the one-group system in MSED with
The MSED method draws from elements of both the rebalance
a two-group system.
and multilevel CMFD concepts, but there are several important
Another motivation for the development of the MSED method is
distinctions. First, the CMFD problem motivating the development
to reduce the reliance of MPACT on black-boxKrylov linear
of MSED is orders of magnitude larger (in terms of the number of
solvers. In recent years, many diffusion and CMFD codes have
unknowns and the number of processors used) than any of the
become increasingly reliant on Krylov methods for solving their
applications in the works referenced above. Second, unlike the
linear systems. These methods are generallyeasy to implement due
MLCMR/MLSR techniques, MSED operates on a diffusion/CMFD
to their availability in various linear algebra libraries such as PETSc
system in which the unknowns representing the neutron net cur-
[14]. They perform reasonably well when compared to other
rent have already been eliminateddthis simplies the process of
frequently used linear solvers such as SOR or GausseSeidel. How-
collapsing onto coarser spatial/energy grids. Third, whereas the
ever, many Krylov solvers (GMRES in particular) require a signi-
multilevel CMFD techniques use ux-weighted cross-sections and
low-order consistency factors(commonly denoted by D, and
cant amount of memory and may not be well-suited for high-
performance computing applications where memory is a limiting
sometimes described as a driftvector/term) to generateits group-
resource. Moreover, Krylov methods generally do not account for
collapsed equations, the MSED method uses both ux-weighted
the physics and structure of the problem being solved, and their
cross-sections and ux-weighted diffusion coefcients, and does
not need additional consistency factorslike the D in the multi-
convergence is typically slow for large problems unless a good
physics-based or problem-dependent preconditioner is applied to
level CMFD methods. In this sense, the collapse in energy in MSED
the system. In manycases, such a preconditioner may not be known
is similar to that of the MLCMR/MLSR methods. Another example in
and, even if one exists, constructing the preconditioner and
which the diffusion coefcients are group-collapsed via ux-
applying it to the linear system may require a signicant compu-
weighting can be found in Schunert et al. [11]; there, the SN FEM-
tational effort. The approach taken by MSED is fundamentally
discretized transport equations are accelerated by coarse-group
different from those taken by the Krylov methods. Whereas Krylov
FEM-discretized diffusion equations. However, the coarsening in
methods are applicable to general linear systems, MSED is opti-
space in MSED differs from all of the aforementioned exam-
mized only for multigroup diffusion/CMFD eigenvalue problems.
plesdthe spatial variable in MSED is collapsed using a standard
MSED leverages our knowledge of the physics and structure of the
multigrid approach in which coarse-grid equations are error
multigroup diffusion problem and is designed to exploit the unique
equations rather than approximations to the original system.
features of this problem.
Moreover, this collapse in MSED is performed on both the grey and
In the following sections, we provide an overview of the theory
the multigroup equations, as illustrated by Fig. 1B.
for the three components of MSED, describe the full algorithm, and
Lastly, we note that, like the MLCMR/MLSR methods, MSED uses
present results from our Fourier analysis and one-dimensional (1D)
a one-group (grey) low-order equation rather than a two-group
test code. This paper should be viewed as an initial report for the
equation. Although the referenced multilevel CMFD methods all
development of the MSED method, and the work presented in this
use a two-group structure as their coarsest energy grid, we have
paper is a necessary initial step towards our ultimate goal of
f  f f  f
2 2 2 2
1
  D
D
D
D
D
x
x
x
2 2
j
1 1
X
2 2
G
ðlÞ
  D f
D þ S
D
D
D
x
x
x
2 2
j
1 1
2 2
X
X
n
S f
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B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134
1127
implementing MSED as a solver for the multidimensional CMFD
problem in MPACT. As this development progresses and as we learn
more about the performance of the MSED method, it is likely that
changes will be made to the algorithm so that it performs optimally
in parallel on both two-dimensional (2D) and three-dimensional
(3D) CMFD problems.
" ðlþ1Þ ðlþ1Þ ðlþ1Þ ðlþ1Þ #
jþ1;g j;g j;g j1;g
jþ1;g j1;g
jþ2 j2
   
lþ1 G lþ1
þSt;j;gfj;g  Ss0;j;g0 /gfj;g0
g0¼1
(4a)
2. Theory
The 1D multigroup diffusion equation (the blue box in Fig. 1A),
discretized using a second-order nite-difference scheme, is given
¼ lðlÞcj;g X nSf;j;g0 fj;g0 ;
g0¼1
by:
" #
1 fjþ1;g  fj;g fj;g  fj1;g
jþ1;g j1;g t;j;g j;g
jþ j
G G (1)
 Ss0;j;g0/gfj;g0 ¼ lcj;g nSf;j;g0 fj;g0
g0¼1 g0¼1
Here, j is the spatial index, g is the group index, G is the number
PG P ðlÞ
ðlþ1Þ ðlÞ g0¼1 j f;j;g0 j;g0
PG P ðlþ1Þ
g0¼1 j f;g0 j;g0
    
fðlþ1Þ ¼ f lþ2  fj;g 2 :
(4b)
(4c)
of groups, Djþ½,g is the diffusion coefcient, St,j,g is the total cross-
section, Ss0,j,g/g is the differential scattering cross-section, nSf,j,g
is the neutron multiplicity times the ssion cross-section, cj,g is the
ssion spectrum, fj,g is the multigroup scalar ux (spatially aver-
aged over cell j), Dxj is the width of cell j, and
Here, l is the iteration index for PI.
Algorithm 1: A standard PI step
Input: fj,g, l(l)
Dxjþ2 2Dxj þ Dxjþ1:
(2)
Result: f(lþ1), l(lþ1)
1. Update the scalar ux using Eq. (4a).
Eq. (1) is often represented using matrix or operator notation as
2. Update the eigenvalue using Eq. (4b).
M f ¼ lF f:
(3)
3. Renormalize the scalar ux using Eq. (4c).
In this paper, a double underscore denotes a matrix while a
single underscore denotes a column vector.
Although a CMFD system differs slightly from Eq. (1) due to the
presence of correction factors from the transport system, we can
account for these correction factors with a minor mod-
icationdthis is discussed briey in Section 2.2. The remainder of
the theory presented in this paper can be applied to both diffusion
and CMFD systems with no other modications.
In the following subsections, three modications are made to
Algorithm 1 in order to establish the MSED method illustrated in
Fig. 1. Each subsection describes a modication, the motivation for
that modication, and its effect on accelerating the PI scheme. In
Section 3, we combine these components (Algorithms 2-4) to
create Algorithm 5, which describes the full MSED iteration scheme
depicted in Fig. 1.
For simplicity and brevity, we only present the theory for the
MSED method in one dimension for the specic nite-difference
2.2. Grey diffusion equation
spatial discretization in Eq. (1). This spatial discretization is used
by most CMFD systems, including the one in MPACT. The general-
ization of the theory to two and three dimensions is straightfor-
ward; we would only need to add spatial indices to the subscripts
and additional leakage terms. Moreover, the geometric multigrid
method described in Section 2.2 is still valid in higher dimensions,
provided that the spatial grid is Cartesian. The generalization of the
MSED method to other spatial discretizations is less straightfor-
ward but should be possible if one carefully modies the denitions
of the grey diffusion coefcients (Eq. (6d)) and the spatial inter-
polation and restriction operators of the multigrid solver (Eqs.
(15)).
The remainder of this section consists of four subsections, each
describing one algorithm that is a component of the MSED method.
The rst subsection describes the standard PI method for solving
Eq. (1). Subsequent subsections describe how each component of
MSED is used to improve standard PI.
The rst and perhaps most important component of the MSED
method is the grey (one-group) diffusion equation (the box with
the solid red background in Fig.1A). We note that the term greyis
not typically used in reactor physics. The history of this term stems
from the study of the radiative transfer equations, and in particular,
this term is frequently used when describing the numerical simu-
lation of these equations for astrophysics and atmospheric appli-
cations. Despite its origins, we feel that it is appropriate to use the
term greyin this paper, as the development of our grey diffusion
equation was strongly motivated by our knowledge of similar
concepts in radiative transfer. When numerically simulating radi-
ative transfer, a grey diffusion or transport equation is often used to
accelerate the convergence of the more complex, multifrequency
(i.e., energy-dependent or multigroup) radiative transfer problems
through some scale-bridging Algorithm [15,16]. In our work, the
purpose of the grey diffusion equation is more or less the same; we
use the grey diffusion system to accelerate the convergence of the
more complex multigroup diffusion system.
2.1. Power iteration
To derive the grey diffusion equation, we sum Eq. (1) over the
groups g. The collapse is straightforward for all of the terms except
The standard PI scheme for solving diffusion eigenvalue prob-
the leakage term. Performing the collapse yields the following grey
lems can be dened as follows:
equation:
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A multilevel in space and energy solver for multigroup diffusion eigenvalue problems. n this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step..

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Nuclear Engineering and Technology 49 (2017) 1125e1134 Contents lists available at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article A multilevel in space and energy solver for multigroup diffusion eigenvalue problems Ben C. Yee*, Brendan Kochunas, Edward W. Larsen Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 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 2 June 2017 Received in revised form 14 July 2017 Accepted 27 July 2017 Available online 5 August 2017 Keywords: Multigroup diffusion Multilevel Eigenvalue 1. Introduction In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidi-mensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal. © 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/). from existing ideas (multigrid-in-space [3] and two-grid in energy [4]) as well as new ideas (space-dependent Wielandt shift [5]). The The multigroup diffusion eigenvalue problem is an approxima-tion to the multigroup neutron transport eigenvalue problem that is widely used for reactor physics simulations. The solution is frequently used to accelerate the source iteration procedure for solving neutron transport problems via methods such as coarse mesh finite difference (CMFD) [1]. Although solving a diffusion problem requires significantly fewer computational resources than solving a transport problem, this cost is still not trivial. Many transport codes that use CMFD-like procedures (e.g., the Michigan parallel characteristics transport (MPACT) code [2]) have a CMFD eigenvalue problem with hundreds of millions of unknowns, and obtaining solutions to this problem constitutes a large portion of the computational effort. In this work, we introduce a new multilevel in space and energy diffusion (MSED) method for solving the multigroup diffusion eigenvalue problem. This is a multicomponent method that draws * Corresponding author. three primary components of MSED are: (1) a “grey” (one-group) diffusion equation, used to converge the eigenvalue and fission source, (2) a space-dependent Wielandt shift, used to reduce the number of power iterations (PIs) required for convergence, and (3) a multigrid-in-space solver, used to solve the fixed-source grey and multigroup diffusion linear systems. The MSED method can be viewed as an extension of the CMFD method. In CMFD, the convergence of a higher-order (more un-knowns) transport or nodal diffusion system is accelerated by leveraging a lower-order diffusion system. In MSED, this lower-order diffusion system is itself accelerated by simpler diffusion equations with even fewer unknowns. Fig. 1A provides a visuali-zation of this hierarchy. Alternatively, the MSED method can be viewed as an extension of the multigrid method to nonspatial variables. Fig. 1B provides a visualization of the changes in spatial and energy grid sizes in the MSED iteration scheme. These two figures are further explained in Sections 2 and 3. Of the methods that have been developed for reactor physics simulations, the multilevel coarse mesh rebalance (MLCMR) and E-mail addresses: bcyee@umich.edu (B.C. Yee), bkochuna@umich.edu multilevel surface rebalance (MLSR) methods developed by van (B. Kochunas), edlarsen@umich.edu (E.W. Larsen). http://dx.doi.org/10.1016/j.net.2017.07.014 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/). 1126 B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134 t e d e d e d e m d g m e g d m d e e g g d e g g d e (A) (B) Fig. 1. Each figure provides an overview of the MSED iteration procedure. (A) The hierarchy of the equations in MSED is shown, with a scale on the left describing the relative complexity of these equations (i.e., the number of unknowns). The higher (red) dashed box encloses the equations used in the CMFD method, while the lower (green) dashed box encloses the equations used in the MSED method. (B) An MSED iteration is broken up into four steps, and the changes in the energy and spatial grid sizes at each step are visualized. Geemert and others [6e8] are the most similar to MSED. The MLCMR and MLSR methods are techniques for nodal diffusion problems, which leverage solutions from a series of lower-order coarse-grid one-group diffusion equations in order to minimize the number of iterations required on the full higher-order nodal diffusion equations. In this sense, the approaches taken by the MSED method and the MLCMR/MLSR methods are similardboth methods minimize the computational effort required by shifting iterations from higher-order equations to lower-order equations that are less costly to solve. The MSED method is also similar to multilevel CMFD methods [1,9,10]. In these cited works, the multigroup CMFD equations are themselves accelerated by two-group CMFD equations. The MSED method draws from elements of both the rebalance and multilevel CMFD concepts, but there are several important distinctions. First, the CMFD problem motivating the development of MSED is orders of magnitude larger (in terms of the number of unknowns and the number of processors used) than any of the applications in the works referenced above. Second, unlike the MLCMR/MLSR techniques, MSED operates on a diffusion/CMFD system in which the unknowns representing the neutron net cur-rent have already been eliminateddthis simplifies the process of collapsing onto coarser spatial/energy grids. Third, whereas the multilevel CMFD techniques use flux-weighted cross-sections and low-order “consistency factors” (commonly denoted by D, and sometimes described as a “drift”vector/term) to generateits group-collapsed equations, the MSED method uses both flux-weighted cross-sections and flux-weighted diffusion coefficients, and does not need additional “consistency factors” like the D in the multi-level CMFD methods. In this sense, the collapse in energy in MSED is similar to that of the MLCMR/MLSR methods. Another example in which the diffusion coefficients are group-collapsed via flux-weighting can be found in Schunert et al. [11]; there, the SN FEM-discretized transport equations are accelerated by coarse-group FEM-discretized diffusion equations. However, the coarsening in space in MSED differs from all of the aforementioned exam-plesdthe spatial variable in MSED is collapsed using a standard multigrid approach in which coarse-grid equations are error equations rather than approximations to the original system. Moreover, this collapse in MSED is performed on both the grey and the multigroup equations, as illustrated by Fig. 1B. Lastly, we note that, like the MLCMR/MLSR methods, MSED uses a one-group (grey) low-order equation rather than a two-group equation. Although the referenced multilevel CMFD methods all use a two-group structure as their coarsest energy grid, we have found that the MSED method with a grey equation already per-forms very well. Our Fourier analysis and numerical results indicate that the MSED method, as described in this paper, has a spectral radius of ~0.1. Because of this low spectral radius and the fact that a grey system is simpler (easier to implement and solve) than a two-group system, we have not yet been compelled to study the use of a two-group system in the MSED method. Recent work by Cornejo and Anistratov, however, has shown that additional energy grid(s) between one group and G groups [12,13] can provide tangible im-provements in the runtime, and it may be possible to use a similar strategy to improve the MSED algorithm. In future work, we will assess the potential benefit of both introducing an extra two-group system to MSED and replacing the one-group system in MSED with a two-group system. Another motivation for the development of the MSED method is to reduce the reliance of MPACT on “black-box” Krylov linear solvers. In recent years, many diffusion and CMFD codes have become increasingly reliant on Krylov methods for solving their linear systems. These methods are generallyeasy to implement due to their availability in various linear algebra libraries such as PETSc [14]. They perform reasonably well when compared to other frequently used linear solvers such as SOR or GausseSeidel. How-ever, many Krylov solvers (GMRES in particular) require a signifi-cant amount of memory and may not be well-suited for high-performance computing applications where memory is a limiting resource. Moreover, Krylov methods generally do not account for the physics and structure of the problem being solved, and their convergence is typically slow for large problems unless a good physics-based or problem-dependent preconditioner is applied to the system. In manycases, such a preconditioner may not be known and, even if one exists, constructing the preconditioner and applying it to the linear system may require a significant compu-tational effort. The approach taken by MSED is fundamentally different from those taken by the Krylov methods. Whereas Krylov methods are applicable to general linear systems, MSED is opti-mized only for multigroup diffusion/CMFD eigenvalue problems. MSED leverages our knowledge of the physics and structure of the multigroup diffusion problem and is designed to exploit the unique features of this problem. In the following sections, we provide an overview of the theory for the three components of MSED, describe the full algorithm, and present results from our Fourier analysis and one-dimensional (1D) test code. This paper should be viewed as an initial report for the development of the MSED method, and the work presented in this paper is a necessary initial step towards our ultimate goal of B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134 1127 implementing MSED as a solver for the multidimensional CMFD problem in MPACT. As this development progresses and as we learn more about the performance of the MSED method, it is likely that changes will be made to the algorithm so that it performs optimally in parallel on both two-dimensional (2D) and three-dimensional (3D) CMFD problems. 2. Theory The 1D multigroup diffusion equation (the blue box in Fig. 1A), discretized using a second-order finite-difference scheme, is given by: " # 1 fjþ1;g fj;g fj;g fj1;g jþ1;g j1;g t;j;g j;g jþ j G G (1) Ss0;j;g0/gfj;g0 ¼ lcj;g nSf;j;g0 fj;g0 g0¼1 g0¼1 Here, j is the spatial index, g is the group index, G is the number of groups, Djþ½,g is the diffusion coefficient, St,j,g is the total cross-section, Ss0,j,g/g is the differential scattering cross-section, nSf,j,g is the neutron multiplicity times the fission cross-section, cj,g is the fission spectrum, fj,g is the multigroup scalar flux (spatially aver-aged over cell j), Dxj is the width of cell j, and " ðlþ2Þ ðlþ2Þ ðlþ2Þ ðlþ2Þ # jþ1;g j;g j;g j1;g jþ1;g j1;g jþ2 j2 lþ1 G lþ1 þSt;j;gfj;g Ss0;j;g0 /gfj;g0 g0¼1 ðlÞ X ðlÞ j;g f;j;g0 j;g0 g0¼1 PG P ðlÞ ðlþ1Þ ðlÞ g0¼1 j f;j;g0 j;g0 PG P ðlþ1Þ g0¼1 j f;g0 j;g0 fðlþ1Þ ¼ f lþ2 fj;g 2 : Here, l is the iteration index for PI. Algorithm 1: A standard PI step Input: fj,g, l(l) (4a) (4b) (4c) Dxjþ2 ≡ 2Dxj þ Dxjþ1: (2) Result: f(lþ1), l(lþ1) 1. Update the scalar flux using Eq. (4a). Eq. (1) is often represented using matrix or operator notation as 2. Update the eigenvalue using Eq. (4b). M f ¼ lF f: (3) 3. Renormalize the scalar flux using Eq. (4c). In this paper, a double underscore denotes a matrix while a single underscore denotes a column vector. Although a CMFD system differs slightly from Eq. (1) due to the presence of correction factors from the transport system, we can account for these correction factors with a minor mod-ificationdthis is discussed briefly in Section 2.2. The remainder of the theory presented in this paper can be applied to both diffusion and CMFD systems with no other modifications. For simplicity and brevity, we only present the theory for the MSED method in one dimension for the specific finite-difference spatial discretization in Eq. (1). This spatial discretization is used by most CMFD systems, including the one in MPACT. The general-ization of the theory to two and three dimensions is straightfor-ward; we would only need to add spatial indices to the subscripts and additional leakage terms. Moreover, the geometric multigrid method described in Section 2.2 is still valid in higher dimensions, provided that the spatial grid is Cartesian. The generalization of the MSED method to other spatial discretizations is less straightfor-ward but should be possible if one carefully modifies the definitions of the grey diffusion coefficients (Eq. (6d)) and the spatial inter-polation and restriction operators of the multigrid solver (Eqs. (15)). The remainder of this section consists of four subsections, each describing one algorithm that is a component of the MSED method. The first subsection describes the standard PI method for solving Eq. (1). Subsequent subsections describe how each component of MSED is used to improve standard PI. 2.1. Power iteration The standard PI scheme for solving diffusion eigenvalue prob- lems can be defined as follows: In the following subsections, three modifications are made to Algorithm 1 in order to establish the MSED method illustrated in Fig. 1. Each subsection describes a modification, the motivation for that modification, and its effect on accelerating the PI scheme. In Section 3, we combine these components (Algorithms 2-4) to create Algorithm 5, which describes the full MSED iteration scheme depicted in Fig. 1. 2.2. Grey diffusion equation The first and perhaps most important component of the MSED method is the grey (one-group) diffusion equation (the box with the solid red background in Fig.1A). We note that the term “grey” is not typically used in reactor physics. The history of this term stems from the study of the radiative transfer equations, and in particular, this term is frequently used when describing the numerical simu-lation of these equations for astrophysics and atmospheric appli-cations. Despite its origins, we feel that it is appropriate to use the term “grey” in this paper, as the development of our grey diffusion equation was strongly motivated by our knowledge of similar concepts in radiative transfer. When numerically simulating radi-ative transfer, a grey diffusion or transport equation is often used to accelerate the convergence of the more complex, multifrequency (i.e., energy-dependent or multigroup) radiative transfer problems through some scale-bridging Algorithm [15,16]. In our work, the purpose of the grey diffusion equation is more or less the same; we use the grey diffusion system to accelerate the convergence of the more complex multigroup diffusion system. To derive the grey diffusion equation, we sum Eq. (1) over the groups g. The collapse is straightforward for all of the terms except the leakage term. Performing the collapse yields the following grey equation: 1128 B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134 "D E# "D E# jþ1;jþ1 j1;j1 jþ1 j1 j jþ1 j j1 "D E D E * +# * + jþ2;j j2;j a;j j f;j j j jþ2 j j2 In this equation, the grey scalar flux and the bracketed grey quantities are given by: naturallya greyquantity.This termisthe samefor both thegreyand multigroup systems upon convergence. Algorithm 2 can still be viewed as a PI scheme on the multigroup system, but it has one important modification: at the beginning of each multigroup PI, we create a grey eigenvalue problem (Step 1), solve the grey eigenvalue problem (Step 2), and use the resulting grey solution to improve the working estimate of the eigenvalue, eigenvector, and fission source (Step 3). Then, in Step 4, a standard PI step is performed on the multigroup diffusion system. Step 4 is a Fj ≡Xfj;g; (6a) g¼1 Sa;j≡ 1 XSa;j;gfj;g; (6b) j g¼1 nSf;j≡ 1 XnSf;j;gfj;g; (6c) j g¼1 Dj1;j2 ≡ 1 XDj1;gfj2;g: (6d) 2 g¼1 Here, Sa,j,g is the absorption cross-section, defined by X Sa;j;g ¼ St;j;g Ss0;j;g/g0 : (7) g0¼1 PI in the sense that we are solving a fixed-source problem, in which the source is determined by the current estimate of the eigenvector, in order to obtain an improved estimate of the eigenvector. Because we normalize the eigenvector and update the eigenvalue in the grey system, these steps are not necessary on the multigroup sys-tem and are not explicitly included in Algorithm 2. (Nonetheless, we have found that performing an additional update of the eigenvalue after Step 4 of Algorithm 2 can provide a marginal improvement in the convergence of the eigenvalue.) With the grey diffusion equation providing an efficient means of converging the fission source and eigenvalue, iterations on the multigroup system (Step 4 in Algorithm 2) are only needed to converge the energy dependence of the scalar flux (and, conse-quently, the scattering source). As a result, the required number of multigroup PIs is one to two orders of magnitude smaller than that of standard PI. Although Algorithm 2 requires PIs on the grey diffusion system in addition to PIs on the multigroup diffusion system, these new grey iterations are significantly cheaper than iterations on the larger multigroup system. In short, the incorpo-ration of the grey diffusion system makes it possible to shift the bulk of the work in the iteration scheme from the multigroup The careful collapse of the diffusion coefficient via Eq. (6d) is necessary to ensure that the grey diffusion coefficients are positive and well-defined when (Fjþ1Fj)/0. If a different spatial dis-cretization is used, the collapse of the diffusion coefficients may require modification to ensure consistency and positivity. To ac-count for CMFD correction factors (D), a minor change can be made to the definition of the collapsed diffusion coefficient in Eq. (6d). Algorithm2brieflydescribeshowthegreydiffusionequationcan be incorporated into the standard PI scheme. By using a grey equa-tion, Algorithm 2 is able to efficiently converge the eigenvalue and fission source. The basic reason for this efficiency is that the right side of Eq. (1) is separable in the indices g and g0da mathematical representation of the assumption that the energy of a neutron born from fission is independent of the energy of the incoming neutron that induced the fission. Because of this property, the fission source in a multigroup diffusion eigenvalue problem ( g0¼1nSf,j,g0fj,g0) is Algorithm 2: A PI step accelerated by a grey diffusion equation. Input: fj,g, l(l) Result: f(lþ1), l(lþ1) 1. Compute F(l,0) and the grey cross-sections from fj,g using Eqs. (6). 2. Perform M PIs (Algorithm 1 with G¼1) on the grey diffusion eigenvalue problem (Eq. (5)) to obtain the normalized grey scalar flux Fðl;MÞ and its corresponding eigenvalue l(l,M). M can be either a fixed number or the number of PIs required to converge the grey diffusion solution to a certain tolerance. 3. Update the multigroup scalar flux and eigenvalue: lþ1 ðl;MÞ fj;g ¼ fj;g Fðl;0Þ ; (8a) lðlþ1Þ ¼ lðl;MÞ (8b) 4. Consider Eq. (4a), except with l(lþ1) instead of l(l) and with all of the iteration indices on f incremented by 1/2. Solve this equation to obtain fðlþ1Þ. diffusion system to the smaller grey system. In our work, we have also explored the use of an alternate grey diffusion equation, derived from multiplying Eq. (1) bya space- and group-dependent function fj,g and then summing over g. (In Eq. (5), fj,g ¼1.) A Fourier analysis of this alternate grey diffusion equation indicates that choosing fj,g to be an estimate of the multigroup adjointeigenfunctionleadstoaniterationschemewithanimproved spectral radius and a reduced likelihood of instability (e.g., fj,g could be the infinite-medium adjoint eigenfunction in each spatial cell). However, we have not yet encountered any physically relevant ex-amples for which the MSED method is unstable with fj,g ¼1 or for whichtheconvergencerateissignificantlyimprovedbysomechoice offj,gs1.Assuch,wehavenotfoundareasontojustifytheadditional cost of computing fj,g. Nonetheless, this alternate grey diffusion equationshouldbeconsideredifoneencountersaprobleminwhich the use of the standard grey diffusion equation results in instability. 2.3. Space-dependent Wielandt shift The second component of the MSED method is the space-dependent Wielandt shift (SDWS) [5]. The spectral radius of the PI scheme is determined by the dominance ratiodthe ratio of the second largest eigenvalue (in magnitude) to the largest eigenvalue. When problems are optically thick (e.g., realistic reactor problems), this dominance ratio approaches 1 and the PI converges slowly. In the previous section, we noted that the number of multigroup PIs required to solve Eq. (1) may be significantly reduced by leveraging the solution of the grey diffusion system. However, obtaining this grey solution still requires computational effort and, if PI is used, many iterations may be required to converge the grey diffusion system. Thus, even when the grey diffusion equation is used, a technique for reducing the dominance ratio of diffusion systems is still desired. Before we introduce the space-dependent Wielandt shift, we first describe the standard Wielandt shift (WS). The Wielandt shift B.C. Yee et al. / Nuclear Engineering and Technology 49 (2017) 1125e1134 1129 is a well-established acceleration technique for the PI scheme which shifts the eigenvalue spectrum of a diffusion system by some estimate of the true eigenvalue, l', in order to reduce the domi-nance ratio of the system [17]. The PI scheme with WS is described by the following equations: 0ðlÞ n 0ðlÞ n ðlÞoo PLEPS;j P ∞;j In Eq. (11), l0(l) is the shift used in the PARCS code (Eq. (10)) and l∞,j is the infinite-medium eigenvalue in spatial cell j, defined by the following equation: 2 ðlþ1Þ ðlþ1Þ ðlþ1Þ ðlþ1Þ3 6 jþ1;g j;g j;g j1;g7 jþ1;g j1;g jþ2 j2 lþ1 G 0ðlÞ lþ1 (9a) j;g s0;j;g0/g j j;g f;j;g0 j;g0 t;j;g g0¼1 h ðlÞ 0ðlÞi G ðlÞ j j;g f;j;g0 j;g0 g0¼1 PG P 0ðlÞ ðlþ1Þ h ðlÞ 0ðlÞi ðlÞ lðlþ1Þ ¼ g0¼1 j f;j;g0 j j;g0 1 j j;g0 ; (9b) g0¼1 j Sf;g0 fj;g0 2 fj;g 1Þ ¼ flþ11fj;g 1: (9c) X St;j;gf∞;j;g Ss0;j;g0/gf∞;j;g0 G g0¼1 (12) ¼ l∞;jcj;g nSf;j;g0 f∞;j;g0 : g0¼1 Eq. (12) can be rewritten as a GG linear system using matrix/ vector notation as follows: 2 3 6St;j Ss0;j 7f∞;j ¼ l∞;jcjnSf;jf∞;j: (13) Here, St;j is a GG diagonal matrix, Ss0;j is a GG matrix, f∞;j; cj, and nSf;j are length-G column vectors, and the superscript T represents the transpose operator. The structure of the fission source allows us to deduce that there is only one non-zero eigen-value solution to this infinite-medium problem, which can be computed directly using the following expression: Eqs. (9) and Algorithm 3 describe the Wielandt shift technique for both multigroup and grey diffusion systems. Although l' does not have space dependence in the case of a standard Wielandt shift, we have included spatial indices on l' so that Eqs. (9) can be used to > 2 31 >1 l∞;j ¼ >nSf;j4St;j Ss0;j 5 cj; : (14) describe both SDWS and standard WS. Algorithm 3: A PI step accelerated by (space-dependent) Wielandt shift Input: f(l), l(l) Result: f(lþ1), l(lþ1) 1. Compute the Wielandt shift parameter l0(l). 2. Update the scalar flux using Eq. (9a). 3. Update the eigenvalue using Eq. (9b). 4. Renormalize the scalar flux using Eq. (9c). Typically, one determines the Wielandt shift from the most recent iterates of the eigenvalue. In the 3D multigroup diffusion code PARCS (Purdue advanced reactor core simulator) [18], the Wielandt shift is an iteration-dependent quantity that depends on the two most recent estimates of the eigenvalue: From Eq. (14), we see that a drawback of the PLEPS shift is that it requires the computation of l∞,j in each spatial cell. Each of these computations requires solving a GG linear system. This is a trivial burden for the grey diffusion system, but not for the multigroup diffusion system. Moreover, a drawback of both space-dependent and standard Wielandt shifts is that they can significantly increase the condition number of the diffusion system to which they are applied. There is a trade-off that must be considered when Wielandt shifts are useddalthoughtheWielandtshiftreducesthenumberofPIsrequired, it also significantly increases the number of linear solver iterations required per PI unless a very effective preconditioner is applied. This trade-offisbeneficialforthegreydiffusionsystemsincethenumberof PIsrequiredcanbereducedbyonetotwoordersofmagnitude.Onthe multigroup system, however, the number of PIs required has already been significantly reduced from leveraging the grey diffusion system. At best, the application of a Wielandt shift on the multigroup system l0ðlÞ≡ maxflðlÞ c1jlðlÞ lðl1Þj c0;lmin g: (10) could save a few iterations; this would not offset the extra multigroup linearsolveriterationsrequiredandthecostofcomputingl∞,j.Because of this, we onlyapplya Wielandt shifttothe greysystemin MSED and Here, c1 and c0 are user-defined constants (with typical values of 10 and 0.02, respectively) while lmin is chosen such that it is physically impossible for lto be less than lmin (typically, lminz1/3). This type of shift works well if one has a good estimate of the true eigenvalue and the eigenvalue does not change significantly from iteration to iteration (i.e., when we are near convergence). The recently developed space-dependent Wielandt shift im-proves upon the PARCS shift by providing a physically-motivated shift that is more effective at the beginning of the iteration scheme, when one does not necessarily have a good estimate of the true eigenvalue [5]. Briefly, SDWS is a class of shift techniques in which the shift l' in Eq. (9a) is allowed to vary in space. Several variants of SDWS are described by Yee et al. [5], but, in this sum-mary, we focus on the PARCS local eigenvalue positive source

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