Improvements and Applications of the Uniform Fission Site Method
in Monte Carlo
MASSACHUSETTS WNT l'UTE
OF TECHNOLOGY
By
Jessica Lynn Hunter
OCT 29 2014
B.S., Nuclear Engineering, 2011
Rensselaer Polytechnic Institute
LIBRARIES
SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCE AND ENGINEERING IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SEPTEMBER 2014
2014 Massachusetts Institute of Technology. All rights reserved.
Signature redacted
Author:
t
(]
Certified by:
Jessica Lynn Hunter
Department of Nuclear Science and Engineering
Signature rn Sd acted
August 20, 2014
Kord S. Smith, Ph.D.
KEPCO Professor of the Practice of Nuclear Science and Engineering
Certified by:
Signature redacted
Thesis Supervisor
Benoit Forget, Ph.D.
V
Accepted by:
Associate Professor of Nuclear Science and Engineering
Signature
redacted__Reader
Mujid S. Kazimi, Ph.D.
TEPCO Professor of Nuclear Engineering
Chairman, Department Committee on Graduate Students
Improvements and Applications of the Uniform Fission Site Method
in Monte Carlo
By
Jessica Lynn Hunter
Submitted to the Department of Nuclear Science and Engineering on August
Partial Fulfillment of the Requirements for the Degree of
Master of Science in Nuclear Science and Engineering
2 0 th,
2014, in
Abstract
Monte Carlo methods for reactor analysis have been in development with the eventual goal of
full-core analysis. To attain results with reasonable uncertainties, large computational resources
are needed. Variance reduction methods have been developed in order to reduce the
computational resources required to obtain results in a practical amount of time. This work seeks
to expand research in the Uniform Fission Site (UFS) method, a variance reduction technique
recently developed that causes uniformity in uncertainty distributions by forcing uniformity in
source distributions. This work aims to both improve the method as well as investigate its use
with a source acceleration method, Coarse Mesh Finite Difference (CMFD) acceleration. Both
techniques have been implemented into OpenMC, a continuous energy Monte Carlo code.
The UFS method uses weights to alter the number of neutrons born at a fission site. It
operates on a superimposed mesh, in which each mesh cell contains a different weight. These
weights use an estimate of the source fraction and fuel volume fraction within the cell to produce
uniformity. In current implementations, the fuel volumes are assumed to be dispersed equally
over all mesh cells. This work aims to provide an estimate of the fuel volume fraction in each
cell in order to improve the accuracy of the method for irregular geometries. The new fuel
volume approximation method is tested on a toy problem and on a model of the Advanced Test
Reactor, a core with highly irregular geometry. Figures of merit were calculated for a basic
Monte Carlo simulation, a simulation with the standard UFS implementation, and the new UFS
method with estimated volume fractions. With the toy problem, the new method showed
significant improvement and had the highest figure of merit. In the case of the ATR, the long run
time for the approximation lowered the figure of merit. Both problems demonstrated that the use
of the standard UFS implementation on an irregular geometry produced higher uncertainties than
not using the method at all. The UFS method, when used with the estimated volume fractions,
behaved as expected and produced uniform uncertainty distributions.
The investigation of the use of the UFS method with CMFD acceleration was conducted
using the 3-D BEAVRS benchmark. Results showed that keeping CMFD acceleration on during
active batches maintained a stationary source and reduced the variance for assembly results. The
UFS method stacked on this, reducing the maximum relative uncertainties. The UFS method had
variable results with different tallies, but no interference between the two methods was observed.
Thesis Supervisor: Kord S. Smith
Title: Professor of the Practice of Nuclear Science and Engineering
Thesis Supervisor: Benoit Forget
Title: Associate Professor of Nuclear Science and Engineering
2
ACKNOWLEDGEMENTS
This research was performed under appointment to the Rickover Fellowship Program in Nuclear
Engineering sponsored by Naval Reactors Division of the U.S. Department of Energy.
I would like to extend thanks to my thesis co-advisor, Professor Kord Smith. Without his infinite
patience over the last three years I would not have been able to complete this work. His
knowledge and experience in the field of light water reactor design is inspirational.
I would also like to thank my other thesis co-advisor, Professor Ben Forget. His insight and
direction in the area of Monte Carlo analysis has been crucial to the progress of this work.
I would also like to express my deepest gratitude to my Rickover fellowship mentor and advisor,
Dr. Thomas Sutton at Knolls Atomic Power Laboratory. His depth of knowledge and inquisitive
nature were invaluable, his suggestions were useful, and his humor indispensable.
Without Paul Romano this research would not be possible. His authorship of the OpenMC code,
as well as his willingness (and timeliness) to assist whenever and wherever a bug arose were well
appreciated.
I would also
like to particularly thank Bryan Herman, whose
saint-like patience and
understanding has allowed me to incorporate his research into this work. His friendship and
informal guidance have instructed me just as much (if not more) than my coursework, and for
that I am eternally grateful.
I would like to thank my family and close friends, Lulu Li, Lindsay O'Brien, and Aaron Ennis
for their support. Without their friendship I would not have been able to absorb all that MIT had
to offer. Lastly, I would like to thank my best friend and the love of my life, Zachary Hoagland,
for sticking with me despite everything, and for providing objective advice when it was needed
most.
3
Contents
1 Introduction............................................................................................................................10
1.1 Current Research in Reactor Methods..............................................................................10
1.2 OpenM C and the M IT BEAVRS Benchmark............................................................... 11
1.3 Monte Carlo simulations..................................................................................................13
1.3.1 Fission Source Generations in Monte Carlo .............................................................. 15
1.3.2 Statistical Uncertainty in M onte Carlo .................................................................. 16
1.3.3 Distributions of Uncertainty in Monte Carlo and the UFS method.............................18
1.4 Thesis Objectives ............................................................................................................ 20
2 UFS Theory and Background ................................................................................................. 21
2.1 Introduction.....................................................................................................................21
2.2 Altering the Neutron Distribution .................................................................................... 21
2.3 Approximating the Redistribution Factor.........................................................................22
2.4 UFS in Current Monte Carlo Codes ................................................................................. 23
2.5 Proposed Improvements..................................................................................................25
3 Volume Fraction Approximation............................................................................................26
3.1 Description of Method.....................................................................................................26
3.2 Testing Implementation ...................................................................................................
28
3.3 Advanced Test Reactor....................................................................................................31
3.4 Results.............................................................................................................................32
4 UFS method and CMFD Acceleration....................................................................................39
4.1 Introduction.....................................................................................................................39
4.2 CMFD Theory and Background.......................................................................................39
4.3 Results.............................................................................................................................44
4.3.1 Source Convergence ................................................................................................. 45
4.3.2 Uncertainty Distributions .......................................................................................... 49
5 Conclusions............................................................................................................................57
4
5.1 Improvement through approximated fuel volume fractions .............................................. 57
5.2 CMFD acceleration and the UFS method.........................................................................58
5.3 Future Work .................................................................................................................... 59
5
LIST OF FIGURES
Figure 1.1: BEAVRS Benchmark. Radial structure and enrichment loading pattern for cycle 1. Red,
yellow, and blue indicate 1.6, 2.4, and 3.1 w/o U235 regions respectively.................................. 12
Figure 1.2: BEAVRS Benchmark. Left: Axial cross section cut at row 8. Right: Axial planes used in the
model, excluding partial control rod insertion planes ................................................................
13
Figure 1.3: Monte Carlo neutron transport algorithm ............................................................................. 14
Figure 1.4: The convergence of the source distribution for 20 million particles per fission source
generation for the 3-D BEAVRS model.........................................................................................16
Figure 1.5:Axial core averaged tally data using the BEAVRS 3-D OpenMC model........................... 18
Figure 1.6: Tally assembly data from the BEAVRS 3-D OpenMC model...............................................19
Figure 2.1: Core averaged axial distribution of uncertainties at the 95% Confidence level for BEAVRS 3D in OpenMC. ..............................................................................................................................
23
Figure 2.2: Distributions of 95% Relative Confidence Intervals for BEAVRS 3-D, with and without UFS.
a. binned distribution, b. cumulative distribution .....................................................................
24
Figure 2.3: Shannon entropy for BEAVRS 3-D with and without UFS, at 20 million particles per batch. 24
Figure 2.4: A midplane XY slice of the ATR core........................................................................... 25
Figure 3.1: Algorithm for OpenMC run mode for estimating volume fractions..................................27
Figure 3.2: a. (Left) Visualization of toy problem. Fuel cylinder 10cm high with a diameter of 10cm.
Water with a height of 15 cm and diameter of 22 cm. b. (Right) UFS mesh dimensions (black grid)
show n overlaying geom etry....................................................................................................... 28
Figure 3.3: Core-averaged data from toy problem. a. (Left) Axial normalize source. b. (Right) Axial
relative standard deviations .....................................................................................................
30
Figure 3.4: a. (Lower right) an XY slice of the ATR core. b. (Upper right) Serpentine fuel elements
surrounding a flux trap..................................................................................................................32
Figure 3.5: Normalized radial source distribution of ATR.................................................................. 33
Figure 3.6: Core-averaged data for the ATR model with 10 axial units............................................. 33
Figure 3.7: Core averaged data for the ATR model. a.(Left) Normalized source means for various cases.
b. (Right) 95% Confidence intervals for the mean data on the left............................................. 34
Figure 3.8: a. (Left) 95% Relative confidence intervals for each region binned. b. (Right) Cumulative
fractions of regions binned by relative confidence interval......................................................... 34
Figure 3.9: Uncertainty distributions for a single seed of the ATR OpenMC model. a. (Left) Binned
distribution of relative standard deviations. b. (Right) Cumulative distribution of relative standard
dev iations......................................................................................................................................37
6
Figure 3.10: Core-averaged axial uncertainty distributions for the ATR OpenMC model. a. (Left) Coreaveraged axial normalized source mean distributions. b. (Right) Core-averaged axial relative
standard deviation distributions ...............................................................................................
38
Figure 4.1: Flowchart from [7] showing the algorithm for the acceleration method in the MC framework.
.....................................................................................................................................................
40
Figure 4.2: Source convergence for BEAVRS 3-D OpenMC model.................................................. 45
Figure 4.3: (blowup of Figure 4.2) Shannon entropy during deviation from inactive to active batches. ... 46
Figure 4.4: Right tail of Shannon entropy for seed 2 ......................................................................... 47
Figure 4.5: Right tail of Shannon entropy for seed 3 ......................................................................... 47
Figure 4.6: Right tail of Shannon entropy for seed 4 .......................................................................... 48
Figure 4.7: Right tail of Shannon entropy for seed 5 ......................................................................... 48
Figure 4.8: a. (Left) Mean and uncertainty data for an assembly tally with 24 axial nodes for a standard
OpenMC calculation. b. (Right) Data for an assembly tally with 24 axial nodes with CMFD
acceleration used during inactive batches ................................................................................
49
Figure 4.9: a. (Upper left) UFS turned on during all active batches. b. (Upper right) CMFD and UFS, with
CMFD on only during active batches. c. (Lower left) CMFD on during all active batches. d. (Lower
right) CMFD and UFS method on during all active batches. All plots are based on an assembly mesh
tally w ith 24 axial nodes................................................................................................................50
Figure 4.10: a. (Upper) Regions binned according to 95% confidence intervals. b. (Lower) Fraction of
regions show n cum ulatively ..................................................................................................... 51
Figure 4.11: a. (Upper) Binned fractional regions for a pin tally with 100 axial nodes. b. (Lower)
Cumulative Fraction regions for a pin tally with 100 axial nodes .............................................. 52
Figure 4.12: Core averaged axial data from an assembly mesh with 24 axial nodes. a. (Left) Normalized
source for all cases. b. (Right) 95% Confidence intervals for the data on the left ....................... 54
Figure 4.13: Core averaged axial data from a pin mesh with 100 axial nodes. a. (Left) Normalized source
for all cases. b. (Right) 95% Confidence intervals for the data on the left..................................
55
Figure 4.14: Axially integrated radial normalized source distribution for the BEAVRS 3-D OpenMC
mod el............................................................................................................................................56
7
LIST OF TABLES
Table 3.1: Percent error in volume fraction estimates as a function of UFS mesh cell density............. 29
Table 3.2: Run times (wall clock) for volume estimate calculations ................................................
30
Table 3.3: Figures of merit for toy problem with times scaled to 64 processors ................................ 31
Table 3.4: Figures of merit for various cases for the ATR OpenMC model ....................................... 35
Table 3.5: Maximum and average uncertainties for the 5 independent runs of the ATR OpenMC model. 36
8
ACRONYMS
ATR Advanced Test Reactor
BEAVRS Benchmark for Evaluating and Validating of Reactor Simulations
CMFD Coarse Mesh Finite Difference
HZP Hot Zero Power
LWR Light Water Reactor
MC Monte Carlo
NDA Nonlinear Diffusion Acceleration
PWR Pressurized Water Reactor
UFS Uniform Fission Site
9
1 INTRODUCTION
1.1 CURRENT RESEARCH IN REACTOR METHODS
Current research for reactor core design and analyses mainly fall into two categories;
deterministic
calculations
and
stochastic
simulations
(or,
Monte
Carlo
simulations).
Deterministic calculations are numerical solutions to the transport equation that require a
discretization of time, energy, angular, and spatial variables. Often the geometry of a core is
simplified in order to run deterministic calculations with reasonable efficiency. Design and
analytical tools used for production today rely on deterministic methods. [1]
Monte Carlo simulations have been around for hundreds of years, dating back to the
approximation of pi using Buffon's Needle simulation in 1777 [1]. The beginning of its nuclear
application is often credited to Enrico Fermi, who used statistical sampling in the early 1930s to
predict the slowing down of neutrons, 15 years before the work of Stan Ulam and John von
Neumann [2]. The formulation resurged shortly after digital computers arrived. Computers could
be used to generate pseudorandom numbers and perform the summations needed to make Monte
Carlo methods feasible. The term "Monte Carlo" was coined during nuclear bomb research in the
late 40s and the method became popular shortly after [1]. Between then and the last decade or so
Monte Carlo methods had been developed for nuclear applications and core analysis, but were
not widely used to tackle steady-state full-core simulations. Several Monte Carlo codes are in use
today, along with OpenMC. The most well-known among them are MCNP [3], SERPENT [4],
and KENO-VI [5].
Monte Carlo methods are stochastic simulations of particle behavior using neutrons to
estimate some desired quantity. They are stochastic in that they rely on random interaction
probabilities in order to simulate the life of a neutron from birth (through fission or fixed source)
through death (absorption or escape of problem boundaries). As a neutron travels and interacts in
the core, random probability sampling is used to determine which interaction event will occur.
These events are tracked and the average behavior provides estimates of the quantity of interest.
Monte Carlo simulations, in contrast to deterministic methods, provide a continuous treatment of
time, energy, direction, and space. This allows for a nearly exact treatment of energy, geometry,
10
and physics of the simulation, removing discretization errors that would be present in
deterministic methods. [6]
Monte Carlo is not without its drawbacks, however. The process of simulating a sample
of neutrons introduces statistical error. This error can be very large and is combatted by
increasing the number of neutrons simulated and the time to run the simulations. For this reason
Monte Carlo implementations are often run in parallel on multiple processors which reduce the
wall time significantly. Running full core Monte Carlo models requires enormous computational
resources to achieve reasonable turn-around for design applications.
For steady-state eigenvalue calculations the spatial distribution of the fission source must
be known before the quantities of interest can be tallied. This means that the simulation begins
with some initial guess or distribution, and must slowly resolve the source distribution through
simulation of many generations of neutrons. Only after the fission source distribution is
stationary can the quantities of interest be tallied. There are two disadvantages introduced by this
process. First, source convergence can take many neutron histories, which increases the
simulation time [7]. Computational resource requirements are the main disadvantage to using
Monte Carlo methods for production-level work. Second, this fission source iteration introduces
cycle correlation, which causes an under prediction of uncertainty [8]. This cycle correlation has
recently been suggested to be the results of particle clustering, in which particles cluster together
as a result of the asymmetry between neutrons dying uniformly over the core but being born at
previous fission sites [9].
Two approaches to solving some of these problems are variance reduction and
acceleration methods. The goal of this thesis is to increase the viability of Monte Carlo methods
for full-core applications by investigating alternate strategies in both of these areas.
1.2 OPENMC AND THE MIT BEAVRS BENCHMARK
This thesis uses the Open Monte Carlo (OpenMC) code as its main analysis tool. OpenMC is a
continuous-energy Monte Carlo code developed at MIT starting in 2011. It has recently been
made available to the public domain and has an active development team. The purpose of
OpenMC development was to provide a high-performance computing platform for developing
new algorithms. As such it has an advantage of being written in contemporary FORTRAN and is
11
open-source, encouraging collaboration between developers at many institutions. OpenMC was
chosen for this analysis because of its availability and its proficiency in parallel calculations. [10]
A large portion of this thesis uses the MIT Benchmark for Evaluating and Validating of
Reactor Simulations (BEAVRS) [11]. The benchmark was developed in 2013 to provide a
detailed benchmark to validate high-fidelity full-core modeling capabilities. Unlike previous
benchmarks, BEAVRS is a detailed full-core model LWR with 2 cycles of measured reactor
data. It consists of geometry and materials for a Westinghouse 4-Loop PWR with 193
assemblies. Figure 1.1 shows a radial cross section of the core taken from benchmark
specifications.
Core Barrel
Pressure Vessel
Neutron Shield Panel
t
7F
I
Figure 1.1: BEAVRS Benchmark. Radial structure and enrichment loading pattern for cycle 1. Red, yellow,
and blue indicate 1.6, 2.4, and 3.1 w/o U235 regions respectively.
Each assembly consists of a 17 x 17 array of pins having one of three different enrichments.
Guide
tube
positions
sometimes
contain
one
12
of several
different
burnable
absorber
configurations. Figure 1.2 shows the axial cross section and axial planes used in the model. More
details can be found in the online specification [11].
Elevation (cm)
455.444
ifi
i
435.444
426.617
423.272
421.223
416.720
412.529
405.713
401.767
365.864
360.149
313.667
307.952
261.470
255.755
209.273
203.558
157.076
151.361
104.879
99.1640
45.0790
42.0700
41.0870
it1l fi1
1P
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Description
Highest Extent
Top of Upper Nozzle
Bottom of Upper Nozzle
lbp of Fuel Rod
Bottom of Top End Plug
Grid 8 Top
Grid 8 Bottom
Control Rod Step 228
Top of Active Fuel
Grid 7 Top
Grid 7 Bottom
Grid 6 Tp
Grid 6 Bottom
Grid 5 Top
Grid
5 Bottom
Grid
4 Tp
Grid 4 Bottom
Grid 3 Top
Grid 3 Bottom
Grid 2 Top
Grid 2 Bottom
Control Rod Step 0
Grid
1 Top
Bot. of Burnable Absorbers
Grid I Bottom
Bottom of Active Fuel
Bottom of Fuel Rod
Bottom of Support Plate
Lowest Extent
Figure 1.2: BEAVRS Benchmark. Left: Axial cross section cut at row 8. Right: Axial planes used in the
model, excluding partial control rod insertion planes.
For the analyses in this thesis, a 3-D OpenMC model of this benchmark model was used. The
operating condition simulated was the beginning of Cycle 1 at Hot Zero Power (HZP). The
benchmark has been tested using both OpenMC as well as the code MC21 (Monte Carlo for the
2 1st
Century), the in-house code at Knolls Atomic Power Laboratory. The benchmark is available
in open literature [12], [11].
1.3 MONTE CARLO SIMULATIONS
The basic algorithm of a Monte Carlo code is a construction of loops around the life of
an individual neutron. Figure 1.3 shows the basic process in a flowchart. At the lowest level is
the progression of a single neutron. The neutron is initialized at a fission site and then moves in
some direction. The distance and direction it moves is randomly sampled from a free flight
probability distribution determined by the total macroscopic cross section of the material the
neutron is transported through. At the new location, if it has not escaped the problem boundary,
13
the neutron undergoes a collision. Random sampling is used to determine with which isotope it
will collide and what type of collision occurs (fission, absorption, scattering, etc.). The new
direction is calculated via collision physics and the process is repeated until the particle is
absorbed or leaked. If it has collided in a fissile material, its probability of creating fission
neutrons is sampled. The number of neutrons born from this site is calculated and their location
and other pertinent information are stored for the next fission source generation, the next level in
the loop structure.
Neutron histories are collected into what are known as generations. The next generation
of neutrons is selected from the daughter neutrons that resulted from the previous generation's
fissions. Source sites are randomly sampled so that the number of neutrons simulated per
generation remains constant.
13e,
gin
PHej
Yes
Initiailize Fission
trhien
i
K Nreurn Per
Freea
i=
I. r
Irir
*r
i+1
Yesj =j +
I
SoinCe
G eneratton
k< K?
Yes k= k+
1
j
Begin Particle k
Yes
Partide
ahsorbed?
No
Particle
Collision
Ca[lulate path
escaped?
Physics
and move
Figure 1.3: Monte Carlo neutron transport algorithm
14
At the highest level are batches, which is a collection of fission generations in which the
quantities of interest are tallied. During each batch, the quantities of interest (reaction rates,
fluxes, surface currents) are estimated, along with its variance.
1.3.1 Fission Source Generations in Monte Carlo
For Monte Carlo criticality calculations, the fission source distribution is unknown. The
simulation begins with a guess or some other user-determined distribution and the shape of the
fission source is slowly estimated over the course of many fission generations. This shape must
be stationary before the user begins tallying quantities of interest in order to estimate unbiased
results. Batches run before the fission shape is stationary are termed inactive batches and are user
determined since codes usually have no way of determining automatically when the source is
converged. Tallies do not begin until the active batches. Since no tallying takes place during
inactive batches they are often considered "wasted". Research efforts in shortening this process
(acceleration methods) of convergence are discussed at length in this thesis.
Estimating the source convergence before running the simulation is guesswork. The
common work-around is to overestimate the number of inactive batches needed for convergence.
Several diagnostic tools have been developed over the years to determine source convergence.
Among these are several variations on entropy [13] [14], which is described below, as well as a
new method that measures the degree of particle clustering [9].
Shannon entropy is a concept from information theory that has been recently used as a
diagnostic tool for source convergence [13] and is available in OpenMC. It provides a single
number to characterize a distribution as opposed to examining 2-D or 3-D arrays of data on a
batch by batch basis. A mesh is superimposed over all fissionable regions and fission sites from
each batch are tallied within some user-selected mesh. This discretized source estimate is then
used to calculate the Shannon entropy value:
N
Hsrc = -
IP - log
2
(PI)
(1.1)
J=1
where N is the number of mesh cells and P is the number of source sites in the J-th cell . This
value is calculated for each batch and it can be plotted to demonstrate source stationarity in a
simple line plot. Below is a plot of the Shannon entropy for the BEAVRS 3-D benchmark. This
15
simulation was run with 20 million particles per fission generation, and one fission generation
per batch.
12.18
12.16
12.14
12.12
12.1
12.08
12.06
12.04
12.02
12
0
50
100
150
200
250
Batches
Figure 1.4: The convergence of the source distribution for 20 million particles per fission source generation
for the 3-D BEAVRS model.
The value of Shannon entropy seems to have converged shortly after 150 batches. Based on
Figure 1.4 it would be ideal to begin tallying anywhere after batch 160. In the BEAVRS
simulations presented in this thesis, active tallies begin after batch 200.
1.3.2 Statistical Uncertainty in Monte Carlo
Monte Carlo is a methodology used to estimate population means from sample means. It is based
on two well-known statistical concepts: the law of large numbers and the central limit theorem.
The expected value of x, or the population mean is defined as
b
(Z) =
z(x)f(x) dx
(1.2)
a
where f(x) is the probability distribution function, x is a random variable, and a and b are
bounds [15]. The sample mean of a function of a random variable x is defined as
N
_1
z =
z(xi)
(1.3)
i=1
16
I
where Nis a finite number of histories. The law of large numbers states that
im z = (z)
(1.4)
as long as the mean exists and the variance is bounded [1]. This implies that with increasing
histories (N) the sample mean will approach the expected value. With a large enough value of N,
a Monte Carlo simulation will produce accurate results.
This begs the question of how many histories N are required to obtain accurate results.
The central limit theorem provides the answer. For z obtained by samples from a distribution
with mean (z) and standard deviation a(z),
lim Prob
N-+
{if- (Z)
c(z)/vW
U2
1
= --
< A
eT
du
(1.5)
-_
This implies that with a very large number of independent random variables any distribution will
have a mean that is normally distributed, or, z is asymptotically distributed as a normal
distribution, with mean y = (z) and standard deviation a(z)/V7V. The central limit theorem also
upholds the law of large numbers. As A approaches 0, the right side of Eq. (1.5) approaches 0,
showing that as Napproaches o the sample mean z approaches the true mean (z) [1].
These two concepts make Monte Carlo methods feasible; the law of large numbers says
that an estimate of the true value can be found with a large enough number of histories, while the
central limit theorem determines how well that estimate corresponds to the true value. Because
of the normal distribution it is possible to find the variance since the sample variance can be used
to estimate the population variance. Now, using the sample variance, it is possible to construct
confidence intervals, which are intervals of values in which there is a certain probability that the
true value is contained within the interval [16]. The sample variance is defined as:
N
s2(z) = N-
[Z(Xi) - Z]2
1
(1.6)
i=1
By separating the summation it is possible to simplify the sample variance into a more practical
form:
s 2 (z) =
N
(z 2 -I2)
(1.7)
The sample standard deviation, s(z), is determined by collecting Z2 and 12 during simulation.
17
The central limit theorem also states that the uncertainty in the estimated expected value
is proportional to 1//N. In order to decrease the uncertainty in Monte Carlo results by a factor
of two, the number of particles, or events, must be quadrupled.
It is important to note that these equations are valid only when tally realizations are
independent of one another. The fission generations described earlier are correlated from
generation to generation, causing an under prediction of variance [8]. This effect on the variance
has been shown to be decreased by running multiple fission generations per batch [8]. This issue
is given a thorough treatment elsewhere [7] and suggests that for accurate variances, running
several simulations with different number seeds is an alternative solution.
1.3.3 Distributions of Uncertainty in Monte Carlo and the UFS method
Due to the statistical methods described in the previous section, Monte Carlo uncertainty
distributions tend to follow an inverse relationship to the distribution of neutron tracks. More
events occur where there are more neutrons, meaning that regions of the core with low neutron
populations will have higher uncertainties. A clear example of this is given
below in Figure 1.5,
where the top and bottom of the core have much higher relative uncertainties than the center.
1
1
1
1
/
t
1
1
CI1
11
oCg
-
z
N
10
20
90
40
ai
60
70
a
90
1o
Pin Axial Unit
Figure 1.5:Axial core averaged tally data using the BEAVRS 3-D OpenMC model.
18
The data from this plot is derived from a nu-fission tally on a pin mesh using 1 billion histories
with the OpenMC model of the BEAVRS benchmark. It is integrated radially to clearly show the
axial distribution of source and its relative uncertainty distribution. This relationship can also be
demonstrated by plotting the mean data of the source distribution versus its 95% relative
confidence interval, as is shown below in Figure 1.6.
0035
+
Assembly tally data
Exponential fit curve
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".
~
2
.
>
gthe
*
-.
Mean
Figure 1.6: Tally assembly data from the BEAVRS 3-D OpenMC model.
Data in Figure 1.6 is
from an assembly tally
on the same data as Figure 1.5. A fitted line is added
to the plot in order to clearly demonstrate the trend in the data. The fitted line demonstrates that
the lower the source, the higher the uncertainty. This plot shows the data for the entire volume of
the core (24 axial nodes).
These plots demonstrate the need for variance reduction methods in Monte Carlo. The
design of a reactor relies on the prediction of behavior at all locations in the core. Shielding
design in particular suffers from higher uncertainties present along the outer regions of a core.
There are a large variety of methods for variance reduction for Monte Carlo applications. A
majority focus on increasing the number of events in the specific direction, area, or tally that is
of interest. The Uniform Fission Site Method (UFS) is a variance reduction method that alters the
uncertainty distribution by altering the neutron source distribution. Instead of a localized effect
19
like most variance reduction methods, UFS affects all fissionable regions within the core. The
UFS method causes uniformity in the neutron source distribution with the effect of lowering
relative uncertainty distributions in low population regions. The method saves computational
effort by spending less time tracking neutrons in regions where the uncertainty is well known.
The method was introduced in MC21, and since its debut has had minor investigations into
optimization of several of its parameters [17].
1.4 THESIS OBJECTIVES
The purpose of this thesis is to forward research in variance reduction methods by investigating
improvements to the Uniform Fission Site Method. The method, introduced recently in 2012, has
not yet had widespread use. This work aims to make the UFS method more applicable and robust
for future use. The UFS applications to date have been restricted to regular lattices or geometry
in which fuel volumes are constant across a super-imposed mesh. Though these applications can
be very accurate for regular geometries, the method may be improved by removing the
assumption of equal fuel volumes and replacing it with an approximation. In chapter 2 the theory
of UFS is described, as well as a proposal to improve the method. Chapter 3 is devoted to
implementing, testing, and analyzing the new UFS approximation.
Quite separate from this goal is an investigation into whether the UFS method can be
successfully combined with an acceleration method, the Coarse Mesh Finite Difference (CMFD)
method that has been recently implemented into OpenMC. Chapter 4 is devoted to this analysis.
20
2 UFS THEORY AND BACKGROUND
2.1 INTRODUCTION
The Uniform Fission Site (UFS) method is a variance reduction method introduced by Kelly,
Sutton, et al. [8]. It is a variance reduction method that focuses on the reduction of uncertainties
in regions of the core with fewer collisions, and therefore higher relative statistical uncertainties.
This technique is a fission source weighting method that is applied during the simulation with the
objective of creating a uniform distribution of uncertainties. The new source distribution
indirectly lowers the uncertainties in low collision areas of the core.
The key mechanism of the UFS method is to start neutrons uniformly over the core while
maintaining an unbiased solution. By starting histories everywhere in the core, the probability of
collisions
in typically low collision areas are increased, directly affecting statistics. The
uniformity of starting neutrons also implies the inverse; less neutrons are started in high-collision
areas and its relative uncertainty is increased. The UFS method is particularly useful in
applications in which low-collision areas are the regions of interest, or in any region in which a
uniform uncertainty distribution is desired over many regions.
2.2 ALTERING THE NEUTRON DISTRIBUTION
Starting neutrons uniformly over the core is achieved by biasing the number of neutrons created
during a generation according to a user-defined Cartesian mesh. In OpenMC, the equation used
to determine the number of neutrons created after a collision at any site is:
n
= Wneutron
k
1 vaf
eff Ut
(2.1)
where keff is the average eigenvalue over active batches, vaf is the microscopic neutron
production cross section and
at is the
microscopic total cross section, and wneutron is the weight
of the neutron causing the fission event.
21
Since the goal is to start neutrons uniformly in a volume, the UFS method modifies this equation
with a redistribution factor on a chosen mesh. Every time a neutron undergoes fission in cell k,
the number of neutrons created becomes:
1
UFS
= Wneutron k
=
~eff
v f vk
(2.2k
Ut Sk(2)
where Vk, the fraction of the total fissionable material volume is contained cell k and Sk is the
fraction of the fission source in cell k. This redistribution factor ensures that neutrons are created
uniformly per unit fuel volume. The new neutrons are randomly selected from the array
containing neutrons for the next generation. To maintain an unbiased solution each neutron is
given a starting weight modified by the inverse to the redistribution factor,
Wstart
= Wo*
(Sk/Vk).
2.3 APPROXIMATING THE REDISTRIBUTION FACTOR
Each mesh cell, given that the values of Sk and Vk are correct, will produce neutrons evenly over
the core. The fraction of fission source located in cell k,
Sk,
is easily approximated. Before
beginning a new generation the source array is sorted according to the UFS mesh and normalized
into a new array of
Sk
values. The volume fraction, Vk , however, is simply approximated as
equal in every mesh cell. There are several reasons for this. First, core designs often employs
patterns and lattices in such a way that a Cartesian mesh would contain equal amounts of fuel.
Pin and assembly lattices which are loaded equally are the best example of this. Second, this
assumption is simple; it cuts the time and effort required to find the fuel volumes in every cell.
In its standard implementation the value of Vk is reduced to 1/N where Nis the number of mesh
cells containing fissionable material. In regular geometry this assumption holds well; as long as
the amount of fuel in each cell is zero or equal, the value of Vk is accurate. In cells that have no
fissionable material, the weighting factor is never used, and the 1/N weighting over each cell
that does contain fissionable material is eventually normalized. In irregular geometry, however,
Vk is inaccurate and as a result the starting neutrons are not uniform over the fissionable
geometry.
22
2.4 UFS IN CURRENT MONTE CARLO CODES
In Kelly, Sutton, et al, the UFS method was used on the NEA Monte Carlo performance
benchmark with MC21 [8]. The method was used to achieve the 95/95 goal as proposed by
Smith [18]. The 95/95 goal was to attain errors less than 1% at the 95% confidence level for 95%
of regions [8]. The results achieved in the regular geometry of the NEA benchmark showed a
remarkable reduction in the relative uncertainties at the edges of the core while maintaining an
unbiased solution. The power distribution of the NEA Monte Carlo performance benchmark is
very non-uniform both radially and axially which contributed to the success of the UFS method
in their results. Later research on improving the method involved examining the effects of the
size of each mesh cell as well as modifying the weight windows [17]. MC21 uses survival
biasing, a method of variance reduction in which neutrons are given weights and are killed or
split according to weight windows. In MC21 the UFS method caused greater variability in the
weights, and thus the previous weight windows reduced its efficiency. This was solved by
biasing the weight windows in proportion to the redistribution factor.
Weight windows are not currently available in OpenMC, and survival biasing is not a
default option, so no alterations to the basic method were completed beyond the theory already
presented in this chapter. In order to demonstrate the effectiveness of the UFS method in
OpenMC, results using 5 independent runs of the BEAVRS 3-D OpenMC model are summarized
in the figures below.
-non-UFS
UFS
Pin Axial Unit
Figure 2.1: Core averaged axial distribution of uncertainties at the 95% Confidence
OpenMC.
23
level for BEAVRS 3-D in
Figure 2.1 shows the relative uncertainty distribution for a nu-fission tally on a pin mesh with
100 axial nodes. The UFS method for this data was generated on an assembly mesh with 24 axial
nodes. Non-uniformity on the either end of the distribution is due to the fact that leakage tends to
reduce the number of tracks near the core edges. The method's efficiency in reducing the
maximum relative uncertainties can also be shown in binned and cumulative distributions. Figure
2.2a and Figure 2.2b show all of the data from the nu-fission tally on a pin mesh with 100 axial
nodes. The data is sorted into bins to demonstrate the shift in relative uncertainties.
0----ASE
-BASE
007f--JUFS
UFS
07
005
0
0 601
004
0
V
00
002
004
006
0o
01
95% Relative Confidence
012
0 14
016
018
0
Interval
002
004
006
006
01
95% Relative Confidence
0 12
014
16
0
Interval
Figure 2.2: Distributions of 95% Relative Confidence Intervals for BEAVRS 3-D, with and without UFS. a.
binned distribution, b. cumulative distribution.
These shifts demonstrate that the UFS method is properly executed in OpenMC. It should be
noted here as well that the UFS has no significant impact on the fission source convergence.
Figure 2.3 shows typical Shannon entropy for a single run, with and without UFS.
12.18
12.16
Base
12.14
12.12
12.1
12.08
12.06
12.04
12.02
12
0
50
100
150
200
250
Batches
Figure 2.3: Shannon entropy for BEAVRS 3-D with and without UFS, at 20 million particles per batch.
24
18
2.5 PROPOSED IMPROVEMENTS
One of the primary goals of this work is to improve the UFS method by adding an approximation
of fuel volume fractions for irregular geometries. Shown below is an example of irregular
geometry, an XY plot of the OpenMC model of the Advanced Test Reactor.
Figure 2.4: A midplane XY slice of the ATR core.
It is clear that employing a regular Cartesian mesh over this geometry would not produce equal
fuel volume fractions in any cell, barring the simplest of meshes. It is proposed, then, to create an
additional simulation that would approximate the amount of fuel in each cell. This would allow
for a full array of 1 k values, much like the source fraction array, to be referenced when using the
redistribution factor. The goal of this is to create a more robust implementation of the UFS
method and expand its applications to non-regular geometry. The investigation will explore
whether or not the approximation for Vk is worth the additional effort by examining results and
figures of merit.
25
3 VOLUME FRACTION APPROXIMATION
3.1 DESCRIPTION OF METHOD
In order to estimate the value of vk for a given cell the total fuel volume in the core must be
known, as well as the amount of fuel in cell k. In simple regular geometries these volumes may
be known by the user, but may be infeasible to hand calculate the fractions in each cell and enter
them by hand into the simulation. In irregular geometry, although the total fuel volume may be
known, sorting it into a Cartesian mesh is nightmarish. Thus, an automated method to
approximate volumes is necessary.
There are a couple solutions to approximating volumes that use simulations, which are
described later in this work. For this study, a method that was easy to implement and simple to
debug was chosen. Some of the current plotting routines in OpenMC provide a possible solution.
In plotting routines, a false particle is sent to the location corresponding to each pixel of the
image. At that location the geometry is searched to determine the material assigned. The pixel is
then assigned a color based on the material. Instead of sending particles to every specific
location, the time spent in simulation can be controlled by generating random particles, or
locations within the geometry. Using the locations, a similar routine can be used to find the
material. Once the material is found, whether or not it is fissionable is easily determined from a
flag in the materials file specified by the user as "fissionable". Once it is fissionable, it is sorted
according to the UFS mesh to be later normalized into fractions. This process is outlined in
Figure 3.1.
This simulation was implemented as a run mode into OpenMC. This implies that a user,
without running the entire Monte Carlo simulation, can run this simple simulation and generate
an output XML file containing the volume fractions for each cell. Once that XML file is
generated, it can be used multiple times to provide a fuel volume fraction array during a normal
UFS run. Users have the options of defining the number of sites generated as well as the UFS
mesh dimensions and boundaries. Since this simulation may require a very large amount of sites,
the routine was parallelized with MPI.
26
Command
line run
mode
Total
Number
o1
porticles P in
ML input
randomN
locaonN
Gen erate
Fmnd cell barsed
on
Isi>P
IocaItIon
res4
VolumeFraction =
Bins/Sum(Bins)
X
Find
material
based of] cell
iCite ?5tumt'e
)
ic'1n
I 1 t lt
F ind "fissionable"
flag;basedl on
materiMa
-+
Is it
fissionable?
NV
Add 1 to proper
Volume Fraction Bin
Figure 3.1: Algorithm for OpenMC run mode for estimating volume fractions.
As with any Monte Carlo simulation, the approximation of fuel volume fractions improves with
increasing amount of sites. This presents a possible pitfall of this method; a user will not know
when the approximation is good enough to use with the UFS method, since the actual fuel
volume fractions are unknown. Very complicated geometries, especially ones with thin fuel
plates like the ATR model, may need an extremely high number of sites. The advantage to this,
however, is that once the XML file is generated it will never need to be generated again for the
same model. It will only need to be generated if the user decides to change dimensions or
boundaries of the UFS mesh. An important part of this analysis is to examine whether or not the
benefits of using fuel volume fractions is worth the extra time spent generating them.
27
3.2 TESTING IMPLEMENTATION
In order to prove that this new run mode of OpenMC was accurate, a toy problem was used in
which the exact fuel volume fractions were known. The model consisted of a 5.0 cm radius
cylinder of fuel surrounded by water on all sides. The height of the cylinder was 10.0 cm, with
2.5 centimeters of water on the top and bottom and water extending 6.0 cm radially. The fuel
consisted of U-235 at 24 g/cc, a fictional density contrived to approximately achieve criticality.
Vacuum boundary conditions were used on all outer surfaces.
Fuel
Water
Half-filled UFS cell
Figure 3.2: a. (Left) Visualization of toy problem. Fuel cylinder 10cm high with a diameter of 10cm. Water
with a height of 15 cm and diameter of 22 cm. b. (Right) UFS mesh dimensions (black grid) shown overlaying
geometry.
The UFS mesh was laid over this model in such a way that axially two cells would be half-filled
with fuel, while radially all cells would have equal amounts of fuel. This mesh, a 2 by 2 by 9
mesh covering a 10cm x 10cm x 15 cm volume, allows for simple hand calculations of the
volumes in each cell which can be checked with the simulation output. The volume fractions
were estimated using different numbers of randomly generated sites to get a feel for how many
sites were needed for a specific accuracy. The table below demonstrates the estimates of fuel
volume fractions for various UFS cell site densities compared to the true value. The deviation
from the true volume fractions for each position is presented as percent error. The maximum
percent error is presented for each run on the bottom row of Table 3.1.
28
Table 3.1: Percent error in volume fraction estimates as a function of UFS mesh cell density.
Site Density (sites/cell)
4.17E+04
4.17E+05
6.19
6.39
5.42
0.29
3.41
0.00
0.10
4.70
1.15
5.26
2.83
0.07
4.27
0.67
2.02
2.47
3.29
3.82
0.34
0.77
0.10
0.84
0.62
0.24
0.17
0.19
0.07
0.36
8.14
2.p4 -
0.53
0.31
(2,2)
8.64
3.34
0.29
0.26
(2,1)
17.33
2.50
11.69
9.26
1.46
1.06
1.90
1.51
0.10
0.41
0.24
0.05
0.17
0.14
0.43
0.19
2.02
0.14
I.T
15.89
9.26
1.03
2.78
3.55
0.84
0.77
2.88
0.94
0.26
0.48
0.29
0.98
0.05
0.43
0.34
0.02
0.17
0.10
0,26
1.90
1.85
3.31
0.96
0.58
4.72
0.48
0.05
0.02
0.34
0.00
0.34
0.62
1.01
1.15
0.24
0.17
0.05
0.62'
0.14
0.53
0.05
0.00
0.00
0.00
0.14
0.05
0.29
0.05
0.05
0.00
0.19
0.05
4.70
2.35
0.62
0.29
0.05
X, Y
(2,1
3
(1,2)
(1,1)
4
5
(2,2)
(2,1)
(1,2)
(1,2)
(1,1)
(2,2)
6
(1,2)
(1,1)
(2,2)
7
2
4.75
(1,1)
(2,2)
1.03
(1,2)
(1,1)
(2,2)
(2,1)""
Maximum % Error
4. 17E+07
4.17E+08
0.48
0.58
0.05
0.05
0.00
.00
0.62
0.00
0.00
0.00
0.0
0.02
0.02
0.07
0.17
0.00
0.07
0.00
0.02
0.02
0.07
0.07
0.00
0.00
0.02
0.00
0.00
0.05
0.02
0.00
0.00
0.02
0.00
0.00
0.02
0.05
0.00
0.05
0.00
0.07
12.65
0.89
9.07
14.11
10.08
5.42
24.77
0.10
.,35
2.98
(2,1)
(1,2)
(2,1)
8
4.17E+06
% Error
(1,2)
(1,1)
(2,2)
,
Z
2
4.17E+03
k
4
;
026
0.05
0.02
0.02
$_
0.00
0.02
r
0.00
As the number of sites increase, the error in the volume fraction estimate generally decreases.
Some volume fraction errors increase due to the stochastic nature of sampling; sites are randomly
generated in some region more than others in any given simulation.
These results suggest that to achieve fractions with uncertainties less than 1%, a site
density of approximately 4 million sites per UFS mesh cell is required. These values, along with
some prior knowledge of the size of a model, hint at the number of sites that should be used for a
29
desired volume accuracy. Each simulation was run in parallel on 16 processors. The total time to
run each simulation is recorded below in Table 3.2.
Table 3.2: Run times (wall clock) for volume estimate calculations.
Total Sites
Site Density (sites/cell)
Time (s)
1.50E+10
4.17E+03
242.4
1.50E+09
4.17E+04
25.6
1.50E+08
4.17E+05
3.6
1.50E+07
4.17E+06
1.4
1.50E+06
4.17E+07
1.2
1.50E+05
4.17E+08
1.2
The XML file containing the estimates resulting from a density of 4.17E6 sites per cell
was then fed into an OpenMC simulation tallying the fission source and flux. Below are the axial
results without UFS, with equal volumes (marked as UFS standard), and the UFS with provided
volume fractions.
001
12
0 009
--
-BASE
UFS standard
UFS wVolume Approximation
--
o008
1-
.co~
0 000
u4
Uoa
z
A000
b m
4 --
000
oC
0 .2
-BASE
-- UFS standard
-UFS
w/Volume Approximation
4
a
1
12
14
16
12
Axial Unit
14
16
Axial Unit
Figure 3.3: Core-averaged data from toy problem. a. (Left) Axial normalize source. b. (Right) Axial relative
standard deviations.
Figure 3.3b shows the relative standard deviations for a tally on the source, whose means are
shown in Figure 3.3a. The toy problem has the lowest maximum uncertainty when the UFS
method is used with the estimated volume fractions. Figure 3.3b also demonstrates that the
standard UFS method may not produce the desired results if the fuel volumes are incorrect; the
30
maximum relative uncertainty for the standard UFS method is higher than not using the method
at all. Table 3.3 below shows the figures of merit for the above cases. The figure of merit is the
reciprocal of the time used to run the simulation multiplied by the square of the maximum
relative uncertainty. The time used is the time spent in active batches. Two columns are provided
in Table 3.3 under the UFS method with fuel volume approximations. The first includes the time
spent estimating the fuel volume fractions, and the second does not. This is to demonstrate how
the figure of merit is changed by the volume approximation time if a faster method is developed.
The data from Figure 3.3 was produced from running 64 processors; in order for consistency the
run time of the volume approximation was scaled appropriately.
Table 3.3: Figures of merit for toy problem with times scaled to 64 processors.
BASE
UFS standard UFS w/Vol. Approx.
Time in active batches (s)
6.73
6.69
6.33
6.33
Time approximating volumes (s)
N/A
N/A
0.90
N/A
Maximum Relative Uncertainty
0.0086
0.0094
0.0059
0.0059
2010
1692
3975
4540
Figure of Merit
The maximum relative uncertainties in Table 3.3 are taken from the entire 3-D data array of
uncertainties; they are not core-averaged as in Figure 3.3b. The UFS method with volume
approximation gives the highest figure of merit, largely due to short time required to run the
volume fraction simulation. Fractions with higher accuracy took much longer to achieve without
much gain in accuracy, and decreased the benefit of using the method. The toy problem
demonstrated that the method was implemented correctly and that there are benefits in estimating
the fuel volumes. In order to prove that this will work on a large scale and to test the limits of
the improvement, a larger, more complex model is required.
3.3 ADVANCED TEST REACTOR
The Advanced Test Reactor (ATR) was built at the Idaho National Laboratory from 1961 to
1965. It began full power operation in 1969, and since then has been used to study the effects of
radiation on reactor fuel and structural materials. The core is well known for its serpentine fuel
arrangement, shown in Figure 3.3a below.
31
The core contains 40 fuel elements that wrap around 9 flux traps. Each fuel element
consists of 19 concentric fuel plates, as shown in Figure 3.3b. The fuel is set into a beryllium
block, with rotating control cylinders around the fuel as well as control rods in the center.
Detailed geometry has been published in the "International Handbook of Evaluated Criticality
Safety Benchmark Experiments" [19]. The data from the benchmark was used to construct an
OpenMC model of the core.
Concentric fuel plates
Flux trap
Rotating Control
Cylinder
r
d
Withdrawn Control
Rods
Figure 3.4: a. (Lower right) an XY slice of the ATR core. b. (Upper right) Serpentine fuel elements
surrounding a flux trap.
3.4 RESULTS
The ATR model was run with 1 million particles per batch for 50 active batches and 150 inactive
batches. Five individual simulations were run for each case, and statistics were performed using
the independent runs due to the cycle correlation from fission source generations. For this study
tally results were based on an overlaid Cartesian mesh. This was for simplicity; often the
quantities of interest with the ATR model are data within flux traps or fuel elements. A single
32
tally, nu-fission, was run with a 60 by 60 radial mesh with 10 axial nodes. The UFS mesh was 10
by 10 with 5 axial nodes. An axially integrated radial distribution of the source is shown below.
.1.6
12
0.8
0.6
0.4
02
0
Figure 3.5: Normalized radial source distribution of ATR.
The radial plot shows a definite tilt in the model, which is carried through in the following
results. Figure 3.6 below shows the core-averaged axial distribution of the source means and
95% confidence intervals. The right tail also demonstrates a tilt in the source which is due to the
withdrawn control rods shown in Figure 3.4.
2.
1
n
nr.I
0
U
UA
04=
u
I
t
U
I
I
V
2
3
4
5
6
7
8
9
1
"
'Ix
ATR Axial Unit
Figure 3.6: Core-averaged data for the ATR model with 10 axial units.
The ATR geometry was run in the volume approximation run mode using a site density of 4
million sites per UFS mesh cell. The simulation took a total of 92.7 seconds on 120 processors. It
is apparent, from Figure 3.5, that most of this volume is non-fuel, which makes the brute-force
33
method of generating random sites across the entire mesh inefficient. The core-averaged axial
results are shown in the figure below.
16
-BASE
-UFS
4
0 04
-
-UFS
standard
w/Volume Approxumation
v
0)
004
u
0030
03
.Z
9
0t
00%3
96
0034
04
-- UFS w/oluncAAproxitnalionn
0 032'
S
Axial Unit
3
5
7
a
9
0
Axial Unit
Figure 3.7: Core averaged data for the ATR model. a.(Left) Normalized source means for various cases. b.
(Right) 95% Confidence intervals for the mean data on the left.
The 95% confidence widths for the source tally indicate that the use of the standard UFS method
on irregular geometry is detrimental to its purpose. Using the wrong volumes, as also
demonstrated in the toy problem, will not affect the mean data but may push the uncertainties in
the wrong direction. This issue is not mentioned in previous investigations but may be an issue if
use of the method becomes more widespread. It is also clear from the uncertainty distributions
that although using the volume approximation doesn't bring much benefit for this model, it still
operates as expected; the maximum relative uncertainties are reduced and the uncertainty
distribution is more uniform.
UFS
i
standard
UFS wrbolunie forxition 1
sote
7a
~00
aI'
rj
004a00
4c).0
-'-UIS
95% Relative Confidence
Interval
standard
95% Relal ive Confidence
Interval
Figure 3.8: a. (Left) 95% Relative confidence intervals for each region binned. b. (Right) Cumulative
fractions of regions binned by relative confidence interval.
34
The data from the 3-D tally results are summarized in Figure 3.8a and Figure 3.8b, in which each
region is binned according to its 95% confidence interval. This confirms that the use of the UFS
method in highly irregular geometry produces unfavorable results. It also suggests, however, that
there is not much to be gained by using the UFS method with the ATR model, even with
approximated volumes. The slight shift in uncertainties is an improvement,
although the
maximum relative 95% confidence interval, shown below in Table 3.4: Figures of merit for various
cases for the ATR OpenMC model. has increased. This lack of improvement in the figure of merit is
exacerbated by the time required to generate the volume fractions.
Table 3.4: Figures of merit for various cases for the ATR OpenMC model.
BASE
UFS standard
Average time in active batches (s)
690
687
696
696
Time approximating volumes (s)
N/A
N/A
92.7
N/A
Maximum Relative Uncertainty
0.215
0.2128
0.2237
0.2237
Figure of Merit for Maximum
0.0314
0.0321
0.0253
0.0287
Average Relative Uncertainty
0.0348
0.0360
0.0347
0.0347
Figure of Merit for Average
1.197
1.123
1.053
1.194
UFS w/Vol. Approx.
The average relative uncertainty for the new method has decreased, although the additional time
required still makes the figure of merit lower. A possible solution to this is running volume
approximations with fewer sites to cut down the time, but the radial source was already very
uniform in the ATR model. Perhaps another model, irregular but with a very non-uniform power
distribution, would capitalize on the possible advantages of using the volume approximation. The
toy problem was very non-uniform and saw much better improvements, which leads to the
conclusion that the method of approximating volumes may still be viable with other irregular
geometry, or regular geometry with different fuel loads. It should also be noted that in this case
the time is calculated such that the time to run the volume approximation would occur each time;
in reality this file would only be generated once, and with multiple runs the effect of this
additional time would become less and less.
Another way to view this data, however, is to look at the individual runs. Five
independent runs for each case show more hopeful results. The statistics for these runs are based
35
on the 50 active batches for each run, and therefore only relative standard deviations are
reported. Table 3.5 shows a summary of each individual seed. The first row of UFS with volume
approximation includes the entire time spent calculating volumes, while the second row shows
the figures of merit when the extra time is not taken into account.
Table 3.5: Maximum and average uncertainties for the 5 independent runs of the ATR OpenMC model.
1
2
3
4
5
Case
Total Time (s)
BASE
Max Rel. a
FOM-MAX
Avg Rel. 6 FOM-AVG
691
0.090
0.179
0.0118
10.391
UFS standard
686
0.081
0.225
0.0135
7.999
UFS w/Vol. App.
789
0.070
0.256
0.0117
9.256
UFS w/Vol. App.
697
0.070
0.290
0.0117
10.488
BASE
685
0.079
0.233
0.0118
10.482
UFS standard
683
0.071
0.290
0.0134
8.158
UFS w/Vol. App.
785
0.069
0.265
0.0116
9.470
UFS w/Vol. App.
692
0.069
0.300
0.0116
10.738
BASE
689
0.111
0.118
0.0118
10.427
UFS standard
689
0.070
0.298
0.0134
8.081
UFS w/Vol. App.
792
0.063
0.316
0.0117
9.227
UFS w/Vol. App.
699
0.063
0.358
0.0117
10.451
BASE
691
0.079
0.231
0.0118
10.388
UFS standard
690
0.123
0.096
0.0134
8.068
UFS w/Vol. App.
788
0.067
0.284
0.0117
9.271
UFS w/Vol. App.
695
0.067
0.321
0.0117
10.507
BASE
692
0.094
0.162
0.0118
10.375
UFS standard
688
0.081
0.220
0.0134
8.095
UFS w/Vol. App.
789
0.068
0.272
0.0117
9.261
UFS w/Vol. App.
696
0.068
0.308
0.0117
10.494
In this table the figures of merit for the maximum relative standard deviation and the average
relative standard deviation are labeled as FOM-MAX and FOM-AVG, respectively. The lowest
36
relative uncertainties and the highest figures of merit are emphasized in boldface
for each seed
for the first three rows, but the bottom row is not included since it assumes zero calculation
time.
The new method of using approximated volumes consistently produced the
lowest
maximum and average uncertainty, although the extra time required reduced the
figures of merit.
For four out of five seeds, however, the new method had the highest FOM-MAX.
As was
observed earlier, using the standard UFS method increases the average relative
standard
deviation. Below in Figure 3.9 and are the uncertainty distributions for a single seed.
All 5 seeds
show nearly identical behavior.
-BASE
-UFS
standard
-tUFS w/Volume Approximation
012
09
c on
01
04
0207
00
a02
00
001
002
Dos
--BASE
004
00
o06
Relative Standard Deviation
00
01
-U
-UFS
002
003
S
004
standard
wNolume Approximation
006
Relative Standard Deviation
Figure 3.9: Uncertainty distributions for a single seed of the ATR OpenMC model. a.
(Left) Binned
distribution of relative standard deviations. b. (Right) Cumulative distribution of relative
standard
deviations.
Figure 3.9 and Figure 3.10 demonstrate the effectiveness of the new approximation
method in
individual seeds, as well as the poor performance of the standard UFS method.
37
00.
b
002
-BASE
-UFS
standard
-- UFS w/Volume Approximation
1.4
0.018
c
0
1.2
ui
'0
0.01
061
04
02
-BASE
-UFS
-UFS
2
3
4
5
0.01
standard
w/Volume Approximation
6
7
8
o.aia
9
2
Axial Unit
4
x
6
7
8
9
Axial Unit
Figure 3.10: Core-averaged axial uncertainty distributions for the ATR OpenMC model. a. (Left) Coreaveraged axial normalized source mean distributions. b. (Right) Core-averaged axial relative standard
deviation distributions.
Based on these results, it is clear that the new method will produce the highest figures of merit if
the time to produce the volume fraction estimates was reduced. These calculations were
performed with the assumption that the volume fraction estimates would be produced before
every run. If this file were only produced once, and used many times, the figures of merit for the
UFS method with approximation would be the largest.
38
1J
4 UFS METHOD AND CMFD ACCELERATION
4.1 INTRODUCTION
This chapter covers the other main goal of this work, to investigate the effects of using a
combination of UFS and CMFD. The aim of this study is to explore the robustness of the UFS
method and make a recommendation on the use of both methods simultaneously. CMFD is a
source acceleration method, though it has been shown to have an effect on variance through its
impact on cycle correlation. While some studies have noted that CMFD acceleration can reduce
the effects of cycle correlation [20], a later study has shown that for consecutive batches cycle
correlation is reduced, but the correlation between batches that are lagged has slightly increased
[21].
It has been predicted that neither algorithm should interfere with the other, but the effects
of using them together have not been researched, other than inclusion of results from the MC21
analysis of the BEAVRS benchmark [12]. This study showed that using UFS alone did flatten
the relative error, but using CMFD with UFS did not show appreciable changes in either the
source convergence or the relative error distributions. It is unclear whether the results had CMFD
turned off during active batches or whether or not the UFS method was used throughout the
entire calculation. Similar plots will be shown to compare to these results using OpenMC with
the 3-D version of the BEAVRS benchmark. The goal of this work is to examine the degree of
effectiveness of using both methods in concert on a highly-detailed 3-D model.
Since this work presents no new research in CMFD alone, only a brief treatment will be
given its theory and implementation in OpenMC, as they are thoroughly covered in the thesis of
Herman [7]. The theory presented here is to provide a framework to understand the results of this
research.
4.2 CMFD THEORY AND BACKGROUND
In the last 5 years, methods combining deterministic and stochastic methods have garnered
considerable research efforts for Monte Carlo applications. Nonlinear Diffusion Acceleration
(NDA) methods have evolved from simple few-group 1-D and 2-D problems to full core LWR
models . CMFD, in particular, has recently been the method used in the latest evolution ofNDA
39
efforts to study realistic LWR models. The work of Lee applied
CMFD acceleration to 1-D, 2-D,
and 3-D problems using a multigroup Monte Carlo code [22] , [23]. Herman's
work builds on
Lee's by implementing CMFD acceleration in a continuous-energy Monte
Carlo code, OpenMC
[7].
CMFD is an NDA method that uses second order multigroup diffusion equations
on a
coarse spatial mesh. The intent of CMFD acceleration is to produce a better
estimate of the
fission source distribution by using use Monte Carlo tallies to estimate CMFD parameters.
Based
on the result from the CMFD calculation, the Monte Carlo source is then altered by
weighting
the source neutron distribution. If the CMFD source is accurate, then the new
Monte Carlo
source should converge with fewer fission generations. As a convenient byproduct,
the CMFD
method can also provide adjoint flux distributions, dominance ratios, and higher flux
harmonics.
Batch i
tally
ND A
Batch i + 1
uNDno
RUR
"tally
Calculate
D_
ES& DC
MOdifV
Solv
Calculate Equivalence
Figure 4.1: Flowchart from
171
NDA
MC SOUrce
N DA eqs.
showing the algorithm for the acceleration method in the MC framework.
The CMFD acceleration process first requires the neutron balance equations. Balance
can be
checked by calculation of macroscopic cross sections and diffusion coefficients, which
are also
needed for the succeeding steps. The neutron balance formulation, as provided in [7]:
40
uE(xy,z) (
(-
+ + 9mn~nM'
n~'n mn
(~m,nmnAnA'
AV
G
=
Z
n
vsls
m,n1
(4.1)
)
nm
h=1
G
(Vf f 1,m~n
',m,nA 1Armn
)
1+k
eff
The first two terms are surface area-integrated net current over respective surfaces (l
z, m, n)
with surface normal in direction u in energy group g. The second row is the volume integrated
total reaction rate over group g. The third row is a summation over all groups of the volumeintegrated scattering production rate of neutrons that begin with energy in group h and leave the
reaction in group g. keff is the core multiplication factor, and the rest of the term in the fourth
row is the volume-integrated fission production rate of neutrons that fission in group h and leave
in group g.
The quantities in angled brackets represent scalars from Monte Carlo tallies which can be
used to verify that the balance equation is satisfied. The total, scattering production, and fission
production macroscopic cross sections can be calculated from the same tallies:
ti mni
-
-
(d
h-kg
_
(Vssimn
vsIsl,mn
h-+g
AmAn)
(l
m,n1
(5-hnAU AV
1
(l
(4.2)
AauvmAD
-
-
Vf Ifimn
m,nA1
mXf
1
-(Vfl,m~nh9D
(m-nAh
A
)
2
tl,mn
(4.3)
w
1 Omn1
m,nl
uL vAw
AmAn
(4.4)
Conserving neutron balance also requires preservation of leakage rates, which are represented by
diffusion coefficients. The diffusion coefficients used in this implementation are derived for a
coarse energy transport reaction rate:
41
~nIO~
Dm9
uvp
w
where
w9
9
vw(4.6)
9
9
ti,m,n~lm,n'A m'~n)
=
v
-
(VSZSi~mn mnA1 Lm'n )(46
The Coarse Mesh Finite Difference method is a second order finite discretization of the
multigroup diffusion equations which result in a system of linear equations to solve. The method
involves cell-to-cell coupling, given by Eq. 4.7, and cell-to-boundary coupling, given by Eq. 4.8.
These equations relate the current to the flux.
I- g
1
D9
Tm,n
2
1,m,n
Au
+
1 ,m,n 1
'Au
l,m,n 1 1
259
1
1-
-
11,m,n
l,m,nJ
l,m,n
4D9m~
1+ u'9i
I~m~
(4
(4.7)
flug
m,n
a
The
m )
m ,n
D9
+
1-
~1+,m,n
(4.8)
u.9
~i2,m,n
refers to the left or right, front or back, top or bottom surface in any direction. The albedo
, is defined as the ratio of incoming to outgoing partial current on any surface:
u ,g
f T,n
,mn
J~"+
1 ,m,n
_
-},
(4.9)
Eq. 4.7 and 4.8 can be rewritten with a linear coupling term D and a nonlinear equivalence term
D. The nonlinear equivalence parameter is necessary and is used to force a solution consistent
with a higher order transport solution:
i
Z,m,n
S
,m,n(
*1,m,n T O&m,n) + Dl n(g
42
a
1,m,n
+
fm,n)
(4.10)
-
I1mn
lmn
-
,m,n +
(4.11)
I,mn
,m,n
The nonlinear equivalence parameter is unknown, and is dependent on the flux updated on the
next derivation. These equations, substituted into the neutron balance equation, form a system of
linear equations, for which one cell is shown:
1
u
[(j
u"9
-
-g.
1-1,m,n
1-1/2,m,n
1
1-2,m,n
+~~~
-Du'
l-Z,m,n
1+ 2 ,m,n
l+Z,m,n
/
+
1.m9
(D+
] Eg (g
l+1,mn
1+2,m,n
1+U,m,n
G
1)(4.12)
tl,m,n
l,m,n
G
N'
-
h-*g
=h~
k
lm,n 1ss,m,n
h=1
1v
if
h-4g
-
uE(x,y,z)
m,n
efh=1
The eigenvalue problem is solved using a system of linear equations, producing the CMFD
solution. The derivation of the system of linear equations is left out for brevity, but can be found
in [7] and [6]. When the solution to the linear system of equations is obtained,
1
MD = kFk
the multigroup fluxes
(D,
,
(4.13)
and implicitly the source distribution, are known. This vector will
provide a more accurate source than standard Monte Carlo after a fission generation. After the
solution is calculated, the source distribution is modified to reflect it. A probability mass function
is generated from the solution:
V-+g
h
(4.14)
mXh=1 f fl,m,n 1 m,n n
A
PG~mn
9
h
_
The probability mass function describes how likely it is for a neutron to be born in a particular
cell and energy group. To get an estimate of the expected number of neutrons, Eq. 4.8 is
43
multiplied by the number of source neutrons. It is then compared to the Monte Carlo estimated
source, described by ws, to produce weight adjustment factors.
g
fm,n
_
Npj"m~
-
;
s E (g,1, m, n)
(4.15)
The neutrons are then assigned a new weight based on their previous weight and the adjustment
factors.
ws = ws x fmmn ;
s E (g, 1, m, n)
(4.16)
Implementation of this method has already been described in [7], and in the following section
CMFD results will match Herman's results.
4.3 RESULTS
For this section 5 separate independent simulations of the BEAVRS 3-D OpenMC model were
run. Statistics were performed using the means of this data to construct 95% confidence widths.
As with previous BEAVRS data shown in this work, the models were run for 250 batches at 20
million neutrons per batch, and a single fission generation per batch. The data presented is the
tally result of 50 active batches, or 1 billion histories. Two tallies are presented in this work: nufission on a pin mesh and nu-fission on an assembly mesh, with axial nodes of 24 and 100,
respectively. This section will focus on the results of the assembly tallies due to correlation
effects. Several cases were run for this study:
1) BASE, in which a standard Monte Carlo simulation was run with no special options
2) UFS, in which the Uniform Fission Site method alone was turned on during all batches
3) CMFD, in which CMFD acceleration was turned on during inactive batches only
4) CMFD with UFS, in which CMFD acceleration is on during inactive batches only and
the UFS method is present during all batches
5) CMFD, in which CMFD acceleration is on during all batches
6) CMFD with UFS, in which both options are on for all batches
The motivation for turning off CMFD acceleration during active batches is the possible time it
would save, since CMFD acceleration is intended for source convergence during inactive
batches. Since the purpose of the UFS method is to improve tally uncertainties, however, CMFD
44
acceleration was turned on during active batches to gain insight into the effect of CMFD
acceleration on the ability of the UFS method to alter the confidence interval distribution. The
standard UFS method, which assumes equal fuel volumes in every UFS mesh cell, was used in
this study. This was optimal since the UFS mesh was chosen so that each cell contained an
assembly.
4.3.1 Source Convergence
Presented below are the Shannon entropies for the cases discussed above. A single run of every
case is presented, and all independent runs are represented below in Figure 4.3-4.7.
12.
-Base
-- CMFD (CMFD off during active)
-
12.15
UFS
-CMFD
and UFS (CMFD off during active)
-CMFD
(all batches)
--
CMFD and UFS (all batches)
12.1
2
12.05
12
}
11.95
0
50
100
150
200
250
Batches
Figure 4.2: Source convergence for BEAVRS 3-D OpenMC model.
As expected, the UFS method has no impact on the source convergence. The CMFD method
begins at batch 5 and flushes at batch 10 in order to remove the bias in diffusion parameters from
the initial source guess. The source almost immediately converges once CMFD tallies are
flushed, requiring 100 fewer batches than standard Monte Carlo. CMFD acceleration with the
UFS method turned on shows no significant changes.
45
12.014
-Base
12.013
CMFD (CMFD off during active)
UFS
-CMFD
-
12.012
and UFS (CMFD off during active)
CMFD (all batches)
CMFD and UFS (all batches)
12.011
12.01
12.009
12.008
150
160
170
180
190
200
210
220
230
240
250
Batches
Figure 4.3: (blowup of Figure 4.2) Shannon entropy during deviation from inactive to active batches.
A look at the tight tail of Figure 4.2 yields interesting behavior. A black dotted line has been
added to Figure 4.3 to show where the active batches begin. From batches 150 to 250, the
standard and UFS methods are not yet stationary. This suggests that for those methods, the
source was not completely converged. For cases with CMFD acceleration the source has reached
stationarity. In the cases where CMFD acceleration is turned off during active batches, however,
the source begins to drift from its previously steady state, as the source distribution is once more
controlled by Monte Carlo fission generations. This suggests that some level of convergence is
lost once the CMFD method is turned off. This behavior has been noted before, in the work of
Wolters [20], which suggested that the method be kept on in order to reap the benefits of a "more
converged" source. These results agree that to maintain the level of stationarity reached before
active batches the CMFD method should remain on. When CMFD acceleration is used with the
UFS method, the source still maintains similar stationarity. The rest of the independent runs are
provided in Figure 4.4-Figure 4.7 below.
46
12.014
-Base
-CMFD
12.013
(CMFD off during active)
UFS
12.012
-CMFD
and UFS (CMFD off during active)
-CMFD
(all batches)
-- CMFD and UFS (all batches)
12.011
12.010
12.009
12.008
150
160
170
180
190
200
210
220
230
240
250
Batches
Figure 4.4: Right tail of Shannon entropy for seed 2.
12.014
-Base
-CMFD
12.013
(CMFD off during active)
-UFS
12.012
-CMFD
and UFS (CMFD off during active)
-CMFD
(all batches)
-- CMFD and UFS (all batches)
12.011
12.01
12.009
12.008
150
160
170
180
190
200
210
220
Batches
Figure 4.5: Right tail of Shannon entropy for seed 3.
47
230
240
250
12.014
Base
--
-CMFD
12.013
(CMFD off during active)
-UFS
-CMFD
and UFS (CMFD off during active)
CMFD (all batches)
12.012
- CMFD and UFS (all batches)
12.011
12.010
12.009
12.008
150
160
170
180
190
200
210
220
230
Batches
240
250
Figure 4.6: Right tail of Shannon entropy for seed 4.
12.014
-Base
-CMFD
(CMFD off during active)
-UFS
12.013
_CMFD and UFS (CMFD off during active)
12.012
-CMFD
(all batches)
-CMFD
and UFS (all batches)
i 12.011
w
12.010
12.009
12.008
150
160
170
180
190
200
210
220
Batches
Figure 4.7: Right tail of Shannon entropy for seed 5.
48
230
240
250
4.3.2 UncertaintyDistributions
Several methods of visualizing the same data are presented in order to illustrate specific points.
The entire distribution, represented as mean versus 95% confidence interval, for each case is
provided in the figures below.
BASE
CMFD (off during active batches)
0.035
0,03
-
Assembly tally data
--
Exponential fit curve
*
Z
a-a
- Assembly tally data
Exponential fit curve
00Q25
*0
U
-
-0,03
002
a
0.025
";"
'
4
U0015
u~
0.01
0o005
S"
Me
i
2
2.
0
05
Mean
.
2
25
Mean
Figure 4.8: a. (Left) Mean and uncertainty data for an assembly tally with 24 axial nodes for a standard
OpenMC calculation. b. (Right) Data for an assembly tally with 24 axial nodes with CMFD acceleration used
during inactive batches.
Figure 4.8a shows the data for a standard simulation; this sets a baseline for comparison in the
next plots. Figure 4.8b is shown here to demonstrate that turning off CMFD during active
batches has no significant effect on the overall uncertainty distribution. The data is plotted with
an exponentially fit curve in order to facilitate trend observations in data.
49
UFS
0.035
CMFD
0.035
and UFS (CMFD off during active batches)
- Assembly tally data
0.03
-Exponential
fit curve
* 00
25
Assembly tally data
fit curve
-Exponential
03
0
00
25
2
Mean
o0
CMFD (on during active batches)
02
+
0 03
*
Mean
-
0.0051
-
0015
-
''
,
0
+
on
05
00
-Assembly tally data
-- Exponential fit curve
t7
0350
0.005
002
*
CMFD
during active batches C
1
CMFD and UFS (CMFD on during active batches)
0035
-Assembly tally data
Exponential fit curve
15
awe
UFS
2
m1
0015
(CMFD on during
,,
.
a.
UFS
r
active batches g)
0.5
25
Mean
CF
...
15
2
Mean
turnedon during all active batches. aby right
axial
MD
aUy with2
nodes.
The goal of the UFS method is uniformity in uncertainty distributions. In these plots, that would
be demonstrated by a reduction in vertical spread of the data. This behavior is not demonstrated
when comparing Figure 4.8a with Figure 4.9a as well as Figure 4.9c and Figure 4.9d. The UFS
method decreases the maximum relative uncertainties for both cases, but seems to have increased
the spread in uncertainties overall. This may be due to the lack of convergence in the source
propagating through to UFS weights. Having CMIFD acceleration turned on during inactive
batches helps the UFS method slightly in decreasing uncertainty overall, which is demonstrated
in Figure 4.9a versus Figure 4.9b. Since CMFD acceleration is turned off during the active
tallying in this case, it can be concluded that the improvement between these two figures is the
50
25
result of the converged source in Figure 4.9b due to CMFD acceleration during the inactive
batches.
The most notable results occur when CMFD acceleration is turned on during active
batches. Figure 4.9c demonstrates that leaving CMFD acceleration on results in a tighter, lower
relative uncertainty distribution. The UFS method aids in bringing down the maximum
uncertainties in Figure 4.9d. The ability of CMFD during active batches to bring down the
uncertainty provides ample motivation to use both UFS and CMFD as variance reduction
methods. This assembly data is summarized in Figure 4.10, which shows the binned and
cumulative fraction of regions versus 95% confidence interval.
009
-BASE
009
--
CvIFD
(inactive only)
CMFD (on during active)
UFS
0.07
-
CMFD+UFS (CMFD inactive only)
CMFD+UFS (CMFD during active)
-
a0 0.06
0 04
Z 00-
0 02
--
001
00
0
0.01
0.015
95% Relative Confidence Interval
00
020
.
00.0.4.1.1.0
*
073
03--BASE
~
-- CMFD (inactive only)
(on during active)
02---CMFD
-UFS
01--CMFD+UFS
00.
-0015
001
95% Relative Confidence
Figure 4.10:
(CMFD inactive only)
CMFD+UFS (CMFD) during active)
0002
Interval
a. (Upper) Regions binned according to 95% confidence intervals. b. (Lower) Fraction of regions
shown cumulatively.
51
Figure 4.10 emphasizes the unexpected trend in the data; the UFS method, although reducing the
maximum relative uncertainties, seems to have increased the spread of uncertainty in the data
due to the non-converged source. The pin uncertainty distributions, shown below in Figure 4.11,
display different trends.
-BASE
-CMFD
(inactive only)
- CMFD (on during active)
-UFS
-- CMFD+UFS (CMFD inactive only)
- CMFD+UFS (CMFD during active)
006
*c 0040
00s
'0
-
001
0
002
004
0.06
00
0.1
0.12
014
0.16
0.16
95% Relative Confidence Interval
09
[
008
06
R!
0.7
0
S06
05
Li, 04
00
03
-BASE
-CMFD
(inactive only)
- CFD (on during active)
-UFS
- CMFD+UFS (CMFD inactive only)
-CMFD+UFS
(CMFD during active)
0.1
a
0
002
0.04
0.06
000
0.1
95% Relative Confidence
012
Interval
014
0 16
0
18
Figure 4.11: a. (Upper) Binned fractional regions for a pin tally with 100 axial nodes. b. (Lower) Cumulative
Fraction regions for a pin tally with 100 axial nodes.
For Figure 4.11, the difference between the data is hard to distinguish; the BASE data lies on top
of the CMFD only data, and the UFS only data lies on top of the CMFD and UFS data. The
differences between the pin data and the assembly can in part be explained by the difference in
size of the regions. As shown by Yamamoto in [24], smaller tally regions are less affected by
cycle correlation, such that a smaller region will have variances that are closer to the true
52
variance. This CMFD method has been shown to have an effect on cycle correlation, causing an
effect on the variance [21], [20]. In this case CMFD is acting on an assembly level however, and
in relation to the pin errors this reduction would not be visible. This explains why there is
seemingly no difference between the pin variance distributions with CMFD on and off. For the
pin tallies, there is little cycle correlation to begin with, so nothing is altered in the variance
distribution by using the CMFD method. The other notable result of the pin data is that the UFS
method has reduced the maximum relative 95% confidence intervals, made the distribution more
uniform, and has shifted the average relative uncertainty higher. This behavior was observed in
chapter 3 when the standard UFS method was used with irregular geometry.
The assembly data and pin data have noticeably different behavior with regards to the
behavior of the UFS method. In the pin data, the method behaves as expected. The higher peak
and lowered tail ends implying uniformity, with the right tail end showing reduced maximum
uncertainty. The assembly data, however, does not show this behavior. Its peak is shifted much
more towards higher uncertainties, and the peak is lower, showing the uncertainties to be less
uniform. Although the behavior of the assembly UFS only data may be explained by a source
that is not quite converged, it still behaves a bit worse than expected even when the source has
been converged by CMFD (see Figure 4.9c versus Figure 4.9d). The only difference between this
data is tally region size, which suggests that this difference may be related to the difference
between the UFS mesh cell size compared to the tally mesh cell size. The CMFD and UFS
method both operated on an assembly sized mesh for this study, having the same number of axial
nodes. A small study has been previously done on the effects of absolute UFS cell size on the
maximum relative uncertainty [17]. In that study, only the maximum relative uncertainty was
observed, not the trends from the entire uncertainty distribution. It was also not intended to study
the tally mesh size in relation to the UFS mesh size, although it did show trends that the larger
the UFS mesh cell (which happened to be much larger than the tally mesh size), the higher the
maximum relative uncertainty. The conclusions of the study were that if the mesh is too large to
adequately capture the spatial variation of the source, the method will not be as effective. This
study is not helpful in this case, and perhaps more research should be conducted using a UFS
mesh the exact size of the tally mesh.
The figures below show the core-averaged axial distributions for each set of data. Figure
4.12b shows the slight gain in beginning the calculation with a source more converged than the
53
standard base simulation, and a larger gain in keeping the CMFD method on during active
batches. For the axial distributions, UFS is shown to always flatten the distribution and bring
down the maximum uncertainties, but gives a higher average relative uncertainty when used
without CMFD.
)
-BASE
-CMFD (imactiv oly
-CMFD
d
(ona
-UFS
-CND+UFS
-CMFD*UFS
active)
iCMFD inactive only
fCMFD dff* active)
Assembly Axial Unit
-BASE
UFS
1
1--~CM DUFS (CI&D
-CMUFD-FS
CMFD
Asemty Axial
inactnve onlyt
&uinadive)
Unit
Figure 4.12: Core averaged axial data from an assembly mesh with 24 axial nodes, a. (Left) Normalized
source for all cases. b. (Right) 95% Confidence intervals for the data on the left.
The core-averaged data shows the same trends as the 3-D distributions, with the
exception that axially the UFS distribution behaves more as expected, producing more uniform
distributions. This implies that the radial distribution may be having more of an impact on the
UFS method. More research into the behavior of UFS on uniform distributions may be helpful,
54
as this core has a non-uniform axial distribution but a reasonably uniform radial distribution,
shown in Figure 4.14.
1.6
1.2
-
1.
08
06
-BASE
CMFD (inactive only)
04
02
-
CMFD (on during active)
---
UFS
-
00
10
30
0
40
CMFD+UFS (CMFD inactive only)
-- CMFD+UFS (CMFD during active)
60
50
0
-
Z
90
80
100
Pin Axial Unit
-BASE
-CMFD
0-14-
(inactive only)
CMFD (on during active)
-UFS
-
012
(CMFD during active)
-
0.
CMFD+UFS (CMFD inactive only)
-CMFD+UFS
U
0
10
30
30
50
40
1
70
8)
90
100
Pin Axial Unit
Figure 4.13: Core averaged axial data from a pin mesh with 100 axial nodes. a. (Left) Normalized source for
all cases. b. (Right) 95% Confidence intervals for the data on the left.
55
Axiall} Inteonate Radial Source Disribution TASE)
1.2
Figure 4.14: Axially integrated radial normalized source distribution for the BEAVRS 3-D OpenMC
56
model.
5 CONCLUSIONS
The main goal of this work was to expand research in the variance reduction technique known as
the Uniform Fission Site method, both by improving the method as well as examining its
behavior when combined with a source acceleration method, CMFD acceleration.
5.1 IMPROVEMENT THROUGH APPROXIMATED FUEL VOLUME FRACTIONS
Previous implementations of the UFS method assume equal fuel volume fractions in all cells of a
super-imposed mesh on fissionable geometry. The assumption proves false in cases of irregular
non-Cartesian geometry as well as geometries in which the mesh does not perfectly enclose
fissionable regions.
In this thesis a method of approximating fuel volume fractions was
implemented into OpenMC with the intent of replacing the previous assumption of equal fuel
volumes. The method, a Monte Carlo simulation in which the volumes were approximated
through integrating random sites in fissionable geometry, was demonstrated to work well on a
toy problem. The results of the toy problem, a simple Uranium cylinder surrounded by water,
showed that using an assumption of equal fuel volumes pushed the maximum relative
uncertainty higher, while the new method reduced the maximum relative uncertainty while
producing the most uniform uncertainty distribution. The new method produced the highest
figure of merit.
The new method was then used on the ATR OpenMC model, which was highly irregular
with complex fuel. Five separate independent Monte Carlo runs were used to construct 95%
confidence interval distributions in order to bypass effects of cycle correlation. In those results,
the new method was successful at bringing down the average relative uncertainty and causing
uniformity in the distributions, but produced a higher maximum uncertainty and the lowest figure
of merit. The five separate runs were then examined individually, and the average and maximum
uncertainties and accompanying figures of merit were calculated. In each of the five seeds, the
new method of using approximated volumes consistently produced the lowest maximum and
average relative uncertainties. The standard UFS method also reduced the maximum relative
uncertainty but increased the average relative uncertainty. The highest figures of merit for the
reduction of maximum relative uncertainty were produced by the new method for four out of five
57
seeds. The extra time required to generate the volume fraction file, however, reduced the figures
of merit for the average relative uncertainties. Once the fact that the file only needs to be
produced once and can be used again and again is taken into account, the UFS method using
approximated volumes is the most successful.
5.2 CMFD ACCELERATION AND THE UFS METHOD
Another goal of this thesis was an investigation of the use of CMFD acceleration and the
UFS method in combination. CMFD acceleration, an NDA method previously implemented into
OpenMC, was turned on at the same time as the UFS method in order to see if their benefits
would stack. Cases with the various methods turned on and off were presented with regards to
two goals: source convergence and variance reduction. Two CMFD cases were presented, one in
which CMFD acceleration was turned on only during inactive batches, and one in which it was
turned on during all batches. Data was presented for two tallies, one on a pin mesh and the other
on an assembly mesh. Five independent runs of the BEAVRS 3-D OpenMC model were used to
construct 95% confidence intervals.
Source convergence was much improved by all CMFD cases, and stacking UFS on top
did not have any effect on source stationarity. The two CMFD cases, however, presented
different behaviors when switching from inactive to active batches. For the case in which CMFD
was turned off during active batches, the source distribution began to drift from its previous level
of stationarity. When CMFD was on during all batches, however, the source distribution
remained stationary.
The two tallies presented different results in regard to variance reduction. When the
CMFD method was used on the assembly mesh it acted as a variance reduction method, reducing
the uncertainty of the entire distribution, and much more so when the method remained on
through all batches. On top of this behavior, the UFS method reduced the maximum relative
uncertainties. Strange behavior was observed in the assembly UFS results, however. When UFS
was turned on for the assembly results, without CMFD, it produced data with uncertainties with
a higher spread, the opposite of its desired result. The only difference between the two tallies is
the size of the tally region. Possible causes may be the unexplored effects of UFS on a tally in
which the mesh cell sizes are equal and the unexplored effect of UFS on a more uniform radial
distribution. The core averaged axial distributions for the assembly results showed more
58
expected results, with UFS flattening the axial uncertainty distribution and reducing the
maximum relative uncertainties.
The pin data, however, showed different behavior. The pin mesh data showed that CMFD
had no visible effect on the variance distributions, and the UFS method decreased the maximum
error and produced more uniform uncertainty distributions. The lack of difference between the
CMFD distribution and the Monte Carlo distribution is explained by the size of the regions, as
well as the difference in the magnitude of uncertainties. As tally regions become smaller there is
less cycle correlation from the fission source distribution and the difference between the
"apparent" variance and "true variance" becomes less and less. The difference in magnitude of
error between the assembly data and pin data make it clear that any effects CMFD may have
would not be visible.
Much is gained from using the CMFD acceleration method during all batches. Not only does
the source become stationary many batches sooner, but it also stabilizes the stationarity of the
source which indirectly lowers the uncertainties. On the assembly level, the CMFD acceleration
method also reduced the uncertainty. Thus this method is recommended no matter what size of
the tally regions, and whether or not multiple seeds are used.
When using the UFS method with a radially non-uniform power distribution, the UFS
method successfully stacks with this effect. In both assembly and pin tally, the CMFD method
and UFS method had no effects on each other; they simply provided separate effects on the same
source distribution.
5.3 FUTURE WORK
Work in the area of the UFS method is not complete. More work can be done in altering or
providing correct volume fractions. An alternate method of approximating volumes is an
integrated track method, in which particles travel stream through the geometry and calculate the
path lengths through fissionable geometry. This may be a more efficient use of particles in the
sense that a single particle may be able to stream through multiple regions before absorption or
escape. A disadvantage of this, however, is that the particles must actually collide and be tracked
through geometry as in a normal simulation with tallies. This could occur during an actual Monte
Carlo simulation or as a separate run mode. The efficiency could be improved, however, by
sending particles on fixed tracks through the geometry, sampling only path lengths in each
59
region, not unlike the Method of Characteristics, in which individual tracks are summed and
multiplied by the spacing to calculate the volume [25].
Another alternative method of producing volume fractions is to remove the random location
aspect. Instead of randomly generating site locations anywhere within the mesh, generate a site
method in regular intervals across the geometry as done in the current OpenMC plotting routine.
This would allow for greater control of volume accuracy, and the user could control exactly how
many sites are generated in each UFS mesh cell. This would guarantee always-increasing volume
accuracy with increased site locations.
It is clear from this work that more research in the UFS method is necessary. What makes
UFS ineffective requires exploration so that users do not accidentally create worse uncertainty
distributions for themselves. The UFS method has great worth for non-uniform applications, but
the limits of its usefulness require further investigation. Additionally, more work is required in
evaluating the cycle auto-correlation coefficients when using CMFD to help justify some of the
observed results.
60
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