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 t I I I 3l i. s u{ 4 N 4 i' it .,tc t it - t(i i llillIl 1111 9hJ11 1111 fill 37.8790 j Il 1111 iHl l 1111111MAIM 36.0070 35.1600 20.0000 0.X0000 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 0.03 *s. 0025 0.01p :on rder to / eay .... *lower* es , 0005 - Fir 0 . e th uei msta the re t Ts pt show 1.6 Tat 0.5 in h hd e "f fo th e seydt firm the BEVR 3-D OpnM 1 ~ ~ mode "" 15 ". ~ 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 BIBLIOGRAPHY [1] William L. Dunn and J. Kenneth Shutis, Exploring Monte CarloMethods. Burlington, MA: Elsevier, 2012. [2] N. Metropolis, The Beginning of the Monte Carlo Methos Los Alamos Science Special Issue, 1987. [3] Arthur R. Forster et al., MCNP Version 5 5th Topical Meeting on IndustrialRadiationand RadioisotopeMeasurement Applcations, 2004. [4] E. Fridman, "Serpent Monte-Carlo Code: An Advanced Tool for Few-Group Cross Section Generation," ATW-International JournalFor Nuclear Power, vol. 58, no. 3, pp. 156-157, March 2013. [5] D. F. Hollenbach, L. M. Petrie, and N. F. 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