# Comparison of Variance Reduction Techniques for the Simulation of a Nuclear Reactor

Arief Rahman Hakim

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Arief Rahman Hakim. Comparison of Variance Reduction Techniques for the Simulation of a Nuclear Reactor. Engineering Sciences [physics]. 2020. irsn-04738980

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Submitted on 16 Oct 2024

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# Projet de Fin d'Études (PFE)

Spécialité : Conception des Installations Nucléaires

Année scolaire : 2020

## Comparison of Variance Reduction Techniques for the Simulation of a Nuclear Reactor

NON CONFIDENTIEL

**Auteur :**
Arief Rahman HAKIM

**Promotion :**
17/09/2020

**Tuteur ENSTA ParisTech :**
Kim PHAM

**Tuteur organisme d'accueil :**
Mariya BROVCHENKO (IRSN)
Odile PETIT (CEA)

Stage effectué du 01/04/2020 au 30/09/2020

NOM de l'organisme d'accueil : Institut de Radioprotection et de Sûreté Nucléaire

Adresse : 31 avenue de la Division Leclerc, 92260 Fontenay-aux-Roses, France
**Abstract**

In the context of aging French PWRs, it is important to be able to rely on calculation codes to determine the quantities of interest such as neutron fluence on reactor structures. Monte Carlo simulation is a stochastic method to solve particle transport problems by simulating particles from their life to their death. However, for a deep penetration problem, a natural Monte Carlo simulation would be inefficient to obtain quantities of interest due to the low probability of occurrences. Variance Reduction Technique (VRT) can be used to increase the efficiency of the Monte Carlo simulation. In this study, Adaptive Multilevel Splitting (AMS), a newly developed VRT in TRIPOLI-4 code, will be used to estimate the neutron flux on the reactor structures. AMS parameters were optimized to maximize the efficiency of the Monte Carlo simulation. AMS results then benchmarked against the previous study conducted with other VRTs in MCNP available at the beginning of the internship. Based on the benchmark, the neutron flux obtained from AMS simulations showed an agreement with the previous study. The performance of the Adaptive Multilevel Splitting is proven able to obtain non-biased results with satisfactory standard deviation and simulation time.

**Keywords:**

Monte Carlo simulation, Variance Reduction Technique, Adaptive Multilevel Splitting, TRIPOLI-4, MCNP.
Acknowledgements

I would like to thank the EIT-InnoEnergy for sponsoring my study at European Master's in Nuclear Energy. I am grateful to be part of the EMINE program. Special thanks to Quentin Mihatsch, EMINE Coordinator, who continuously assists us at any conditions. Furthermore, I would like to extend my gratitude to Anne-Lise Gloanec (coordinator of M2 - Master in Nuclear Energy at INSTN) and Kim Pham (coordinator of M2 - Nuclear Plant Design at ENSTA Paris Tech). Thanks for all the help during our master journey on the second year of the EMINE program.

This thesis would not have been possible without the help, support and patience of my supervisors Mariya Brovchenko (IRSN) and Odile Petit (CEA). I am truly grateful for their detailed, critical and constructive feedbacks throughout the process of the thesis. I am honored to be supervised by Mariya and Odile. I admire their dedication as my supervisors and mentors. A special thanks to Mariya who made it possible to start the internship at IRSN despite the ongoing COVID-19 situation.
# Table of Contents

<table><tr><td>Abstract</td><td>2</td></tr><tr><td>Acknowledgements</td><td>3</td></tr><tr><td>Table of Contents</td><td>4</td></tr><tr><td>List of Figures</td><td>5</td></tr><tr><td>Chapter 1 – Introduction</td><td>6</td></tr><tr><td>I.1. Background</td><td>6</td></tr><tr><td>I.2. Objective</td><td>6</td></tr><tr><td>I.3. Structure of the Report</td><td>6</td></tr><tr><td>Chapter 2 – Theories</td><td>7</td></tr><tr><td>II.1. Monte Carlo Simulation</td><td>7</td></tr><tr><td>II.2. Variance Reduction Techniques</td><td>7</td></tr><tr><td>II.2.1. Motivation</td><td>7</td></tr><tr><td>II.2.2. Figure of Merit</td><td>8</td></tr><tr><td>II.2.3. Adaptive Multilevel Splitting</td><td>9</td></tr><tr><td>Chapter 3 – Methodology</td><td>12</td></tr><tr><td>III.1. Model Description</td><td>12</td></tr><tr><td>III.2. TRIPOLI-4 Model</td><td>13</td></tr><tr><td>III.3. Workflow of the Study</td><td>14</td></tr><tr><td>Chapter 4 – Results and Discussion</td><td>16</td></tr><tr><td>IV.1. AMS Parameters</td><td>16</td></tr><tr><td>IV.1.1. Splitting rate</td><td>16</td></tr><tr><td>IV.1.2. Resample Collisions</td><td>18</td></tr><tr><td>IV.1.3. Target(s) Choice for the Stopping Criterion</td><td>19</td></tr><tr><td>IV.2. AMS Using Spatial Dependence Only Importance Function</td><td>20</td></tr><tr><td>IV.3. AMS Using Pre-calculated INIPOND Importance Map</td><td>24</td></tr><tr><td>IV.4. AMS Using the Importance Map Derived From ADVANTG</td><td>27</td></tr><tr><td>Chapter 5 – Conclusions</td><td>29</td></tr><tr><td>Reference</td><td>30</td></tr><tr><td>Appendix</td><td>31</td></tr><tr><td>A.1. Simulation Results</td><td>31</td></tr><tr><td>A.2. Comparison of the FOM in an Analog Monte Carlo Simulation</td><td>33</td></tr><tr><td>A.3. Verification and Validation of the TRIPOLI-4 Input File</td><td>35</td></tr><tr><td>A.4. Simulation Parameters</td><td>37</td></tr></table>
List of Figures

<table><tr><td>Figure 1</td><td>The flowchart of the AMS fundamental algorithm</td><td>10</td></tr><tr><td>Figure 2</td><td>The illustration of how AMS algorithm works in step 1 (1), step 2 (2), step 3 (3), and step 4 (4). Dashed-lines denote the suppressed particles</td><td>11</td></tr><tr><td>Figure 3</td><td>The neutrons paths to reach the detectors in the air in the modified Tihange-I model. Circles represent the sphere detectors defined in the model</td><td>12</td></tr><tr><td>Figure 4</td><td>The workflow of the study</td><td>14</td></tr><tr><td>Figure 5</td><td>Total clock time vs splitting rate in AMS simulation using line-wise importance function</td><td>16</td></tr><tr><td>Figure 6</td><td>The evolution of detector 4004's relative standard deviation</td><td>17</td></tr><tr><td>Figure 7</td><td>The evolution of detector 4006's relative standard deviation when the resample collisions feature is on (blue) and off (black)</td><td>18</td></tr><tr><td>Figure 8</td><td>The YZ axis plot of particles collisions sites when 7 detectors (1) and only detector 4001 (2) were used as targets choice in an AMS line-wise simulation</td><td>19</td></tr><tr><td>Figure 9</td><td>The collisions sites plot on YZ axis of AMS simulation using line-wise (1), plane-wise (2), sphere-wise (3), and point-wise (4) spatial importance function</td><td>20</td></tr><tr><td>Figure 10</td><td>The collisions sites plot on XY axis of line-wise (1) and plane-wise (2) used in AMS simulation</td><td>21</td></tr><tr><td>Figure 11</td><td>Tested line-wise positions using AMS Spatial Dependence Only importance function</td><td>21</td></tr><tr><td>Figure 12</td><td>The evolution of detector 4001 relative standard deviation at various line positions</td><td>22</td></tr><tr><td>Figure 13</td><td>The neutron flux comparison of AMS using line-wise importance function to MCNP-ADVANTG with a confidence interval of 3 standard deviations</td><td>23</td></tr><tr><td>Figure 14</td><td>The FOM comparison of AMS using line-wise importance function to MCNP-ADVANTG</td><td>23</td></tr><tr><td>Figure 15</td><td>The equi-importance plot of INIPOND map when 'VOID' is not used (1) and used (2)</td><td>24</td></tr><tr><td>Figure 16</td><td>The collisions sites of AMS using INIPOND map when 7 detectors (1) and only detector 4001 (2) used in the INIPOND card</td><td>25</td></tr><tr><td>Figure 17</td><td>The neutron flux comparison of AMS using INIPOND map to MCNP-ADVANTG within a confidence interval of 3 standard deviations</td><td>26</td></tr><tr><td>Figure 18</td><td>The FOM comparison of AMS using INIPOND map to MCNP-ADVANTG</td><td>26</td></tr><tr><td>Figure 19</td><td>The neutron flux comparison of AMS using an ADVANTG map to MCNP-ADVANTG within a confidence interval of 3 standard deviations</td><td>28</td></tr><tr><td>Figure 20</td><td>The FOM comparison of AMS using an ADVANTG map to MCNP-ADVANTG</td><td>28</td></tr></table>
# Chapter 1 – Introduction

## I.1. Background

In the context of aging French PWRs, it is crucial to ensure the deteriorations of reactor structures' mechanical properties are still within acceptable criteria. Thus, it is important to be able to rely on calculation codes to estimate quantities of interest such as neutron fluence and activations of the reactor components.

In the particle transport problem, there are two ways of estimating quantities of interest: deterministic methods and stochastic methods (Monte Carlo simulation). In deterministic methods, the system will be discretized in space, energy, angular, and time to solve the transport problem. Typically, the solutions could be obtained faster. However, these solutions might contain errors due to the assumptions linked to the discretization [1]. Contrarily, a Monte Carlo is able to estimate the solution without any simplification to the system resulting in a more accurate system representation. The Monte Carlo simulation solves the transport problem by simulating a large number of particles from their birth to their death. The simulation of a large number of particles take time to finish. Furthermore, it might require a significant number of particles if the quantities of interest are associated with rare events. However, simulating a significant number of particles will increase the simulation time. Even then, only a few particles might contribute to the quantities of interest resulting in a high statistical error which makes the results inconclusive.

The use of a Variance Reduction Technique (VRT) could improve the efficiency of a Monte Carlo simulation. VRT works by altering the simulation process so that it would increase the occurrence probability of rare events that contribute to the quantities of interest. Thus, VRT plays a significant role in estimating the quantities of interest within a reasonable simulation time and standard deviation.

## I.2. Objective

In this study, Adaptive Multilevel Splitting (AMS), a newly developed Variance Reduction Technique in the Monte Carlo TRIPOLI-4 code, will be used to estimate the neutron flux at 7 detectors on the reactor structures. The objective of the internship is to compare the results and Figures of Merit from AMS to the previous results obtained from other VRTs in MCNP code. The internship is hosted by IRSN and collaborating with the CEA development team of TRIPOLI-4 code.

## I.3. Structure of the Report

The report is divided into 5 chapters. Chapter 1 consists of the background and the objective of the study. Chapter 2 consists of Monte Carlo simulation, variance reduction techniques, and Adaptive Multilevel Splitting theories. Chapter 3 presents the methodology of the study. Chapter 4 presents the results and discussion. Chapter 5 concludes on the key points of the study.

HAKIM Arief Rahman / Institut de Radioprotection et de Sûreté Nucléaire
Rapport non confidentiel et publiable sur internet
# Chapter 2 – Theories

## II.1. Monte Carlo Simulation

Monte Carlo simulation solves transport problems by simulating particles from their birth to their death. The simulation obtains the solution estimation by recording the average behavior of the particles. Furthermore, the solution could be estimated without any approximation and/or simplifications resulting in a more accurate system representation. In theory, the accuracy is only limited by the knowledge of the nuclear data used for the simulation.

There are two applicable laws to Monte Carlo simulation: Law of Large Numbers (LLN) and Central Limit Theorem (CLT). First, LLN states that the mean of the result obtained from a large number of independent trials will tend to become closer to the expected value as more trials are performed. Second, the CLT suggests that the distribution of a sample of independent and identically distributed means will tend to approximate a normal distribution as the sample size increases. As the sample size increases, the standard deviation will be reduced and the mean of sample data will be closer to the mean of the overall population.

The LLN and CLT play an important role to accurately predict the average behavior of particles in a Monte Carlo simulation. As a consequence of LLN and CLT, a Monte Carlo simulation requires simulating a sufficient number of particles and repetitions (batches) so that the average results are in an agreement with the expected results within the confidence interval. Furthermore, the mean of the results obtained from batches of Monte Carlo simulation will form a normal distribution. If insufficient particles and repetitions are used, the results obtained from a Monte Carlo simulation might be inaccurate to the expected results. Thus, a Monte Carlo simulation tends to be relatively computational costly when compared to a deterministic method.

## II.2. Variance Reduction Techniques

### II.2.1. Motivation

In a Monte Carlo simulation, it is important to obtain non-biased results (tallies) with acceptable standard deviations and simulation time. In a natural Monte Carlo simulation, the particles will be simulated without any alteration on the transport rules. However, a natural (analog) simulation might require a significant number of particles and long simulation time to obtain reliable tallies if associated with rare events. Furthermore, the tallies will be associated with high standard deviations since only very few simulated particles might contribute.

The correlation between the standard deviation to number of simulated particles is shown by Eq. 1.
$$S \propto \frac{1}{\sqrt{N}} \qquad (\text{Eq. 1})$$

where,

$S$ = standard deviation of the mean value of the tally $\bar{x}$

$N$ = number of simulated particles

From Eq. 1, the number of simulated particles should be increased to reduce the standard deviation of the tally. For instance, to reduce the standard deviation of the tally ($S$) by half, the number of simulated particles should be increased to 4 times. Thus, reducing the standard deviations by analog simulation could be computationally expensive and even prohibitive.

In such cases, a variance reduction technique might be used to improve the efficiency of the Monte Carlo simulation. The variance reduction technique works by biasing the transport rules to favor particles that could contribute to the tallies. The use of variance reduction technique(s) in a Monte Carlo simulation is called a non-analog Monte Carlo simulation.

## II.2.2. Figure of Merit

The Figure of Merit (FOM) describes the efficiency of a Monte Carlo simulation to obtain tallies in a given computing resource. The simulation becomes more efficient as the FOM increases. The FOM is defined in Eq. 2.

$$FOM = \frac{1}{\sigma^2 T} \qquad (\text{Eq. 2})$$

where,

$\sigma$ = relative standard deviation for a given quantity of interest

$T$ = simulation time

The FOM would be an important parameter when the user employs the VRTs. If the VRT is used appropriately, the user will be able to obtain a reliable tally while improving the FOM. The use of a VRT might require optimizations to achieve a satisfactory level of FOM. On the other hand, if used inappropriately, the user could end up not only decreasing the FOM but also obtaining a biased tally. Therefore, VRT(s) cannot be used as a 'black box' since users must ensure that it is still producing reliable results while improving the FOM.

The reliability of the tallies could be examined through statistical checks. There are several checks such as the behavior of the FOM and the relative standard deviation of the tally. During a simulation, the FOM should be approximately constant since the $\sigma^2 \propto 1/N$ and $T \propto N$. If there is a sharp FOM decrease, it indicates that a rarely sampled particle path has affected significantly the tally and standard deviation estimation [2]. Thus, the scores and confidence intervals obtained from the simulation might be not precise enough and too much fluctuations.

HAKIM Arief Rahman / Institut de Radioprotection et de Sûreté Nucléaire
Rapport non confidentiel et publiable sur internet
The FOM of a Monte Carlo simulation is affected by the computing power and its conditions. Therefore, it is important to ensure the simulations are conducted on the same computing power with the same conditions during the FOM comparison.

### II.2.3. Adaptive Multilevel Splitting

Recently, Adaptive Multilevel Splitting has been proposed in the literature of applied mathematics as a new variance reduction technique [3, cited in 4]. Initially, AMS was designed for the simulation of rare events associated with continuous Markov chains which later extended to discrete Markov chains. Later on, the AMS approach has been developed in TRIPOLI-4 as one of the VRTs of the code to solve particle transport problems [4]. AMS algorithm works on neutron, photon, and neutron-photon coupled simulations [4]. For the purpose of this study, the AMS algorithm explanation will focus only on a neutron transport problem.

The application of AMS as a VRT in the particle transport was made possible since the way particles interact share the Markov property. This refers to a memoryless property of a stochastic process. Thus, the future state of the particle does not depend on the past states but rather depends only on the present state.

For the purpose of further discussion, the particle's track is defined as the sequence of the phase-space coordinates of a particle (namely position, direction and energy) at each interaction with the medium [4]. In addition, if the CROSS_SPLIT option is activated, the track will also include the phase-space coordinates when the particle crosses frontiers between volumes [5]. Furthermore, the track's importance describes the likelihood of the particle to contribute to the tally. The higher the importance, the more likely the particle would contribute.

Figure 1 presents the flowchart of the AMS fundamental algorithm [4]. The AMS algorithm could be divided into 4 steps.

Step 1, the AMS algorithm performs the analog Monte Carlo simulation of a set of independent particles. The importance of each particle's track is then evaluated based on the importance function used. Currently, the important function could be determined from two options: from spatial dependence only or from an external pre-calculated importance map. The spatial importance function supports several options such as line-wise, plane-wise, sphere-wise, and point-wise. For the pre-calculated importance map, the user might generate the importance map from INIPOND module (see section III.2.) or from other external codes.

Step 2, the particles are sorted based on this importance function. After sorted, an importance splitting level is determined adaptively based on the user's splitting rate parameter. Step 3, particles that are below the importance splitting level are suppressed. In exchange, the suppressed particles are resampled by splitting the other tracks that are above the importance splitting level. The splitting point locations will be defined on the next step.
<figure><img src="image_1.png" /><figcaption>**Figure 1** : The flowchart of the AMS fundamental algorithm</figcaption></figure>

Step 4, the AMS iteration of steps 1 – 3 is performed again until the stopping criterion is met. Step 1 is then only concerning the new tracks after the splitting points. The stopping criterion is derived from the splitting rate and the chosen target(s). The iterations will stop when (n – k + 1) particles have reached the target (see definitions of k and n in Eq. 3.

When the particles contribute to the tally, the score is multiplied by the correction factor. The correction factor is defined in Eq. 3.

$$ \alpha = \left(1 - \frac{k}{n}\right)^N \qquad (\text{Eq. 3}) $$

where,

α = correction factor

k = number of suppressed particles at each iteration

n = number of particles per batch

k/n = splitting rate

N = number of AMS iterations until the stopping criterion is met

Figure 2 shows the illustration how AMS algorithm works based on a spatial importance function.

In step 1, neutrons are sampled through an analog Monte Carlo simulation for a given iteration. For each particle, a new track is generated and consists of particle's position, direction, and energy at each collision point. At Figure 2, track 1 consists of 1A, 1B, and 1C collision points. In each collision point, the importance will be
evaluated based on the importance function. The track's importance is the maximum importance of all collision points of the track. In Figure 2, the importance of track 1 is equal to the importance of particle at collision point 1B since it is the closest to the target.

<figure><img src="image_2.png" /><figcaption>**Figure 2**. The illustration of how AMS algorithm works in step 1 (1), step 2 (2), step 3 (3), and step 4 (4). Dashed-lines denote the suppressed particles.</figcaption></figure>

In step 2, particles are sorted based on their importance. The importance splitting level is defined adaptively, which distinguishes particles of interest among others. For a splitting rate ($k/n$) of 1/3, the splitting level will be adjusted so that 1 particle with the lowest importance will be suppressed. For the first iteration, the splitting level 1 is adjusted so that it will suppress particle 3.

In step 3, all particles that are below the splitting level are suppressed. In exchange, the suppressed particles are resampled at splitting points. The splitting point is the first collision site of a given particle after exceeding the importance splitting level. The code chooses randomly where particle 3 is resampled (either collision point 1A or 2A). In Figure 2, the code chose 1A as the splitting point.

In step 4, the iteration of steps 1 – 3 continues until the stopping criterion is met. If the stopping criterion is met, the AMS simulation of a given batch is completed. The next batch of AMS simulation is conducted again until all batches are simulated. For splitting rate ($k/n$) is 1/3, the stopping criterion requires ($n - k + 1$) particles to reach the target (blue region in Figure 2).
# Chapter 3 – Methodology

## III.1. Model Description

The reactor model is based on the reactor start-up data of the PWR Tihange-I 900 MWe. The model is precise up to the pressure vessel while the structures outside of the vessel were approximated based on the French PWR [6, cited in 7]. The core is modeled through pin by pin definition with three different $^{235}$U mass enrichments 1.95%, 2.55% and 3.10%, fresh fuel, and at hot zero power state (no thermal hydraulics feedback) [7].

There are 7 volumetric sphere detectors: 4001, 4002, 4003, 4004, 4005, 4006, 4007. Detectors 4002, 4003, and 4004 are located in the concrete while the detectors 4001, 4005, 4006, and 4007 are located in the air. The detectors are located far from the core to challenge the VRT to estimate the tally in an azimuthal and deep penetration problem. The main aim is to estimate the fast neutron flux ($E > 1$ MeV) of the most difficult detector, detector 4001, while using other detectors for analysis.

The model has been adapted to make it challenging for the VRT to obtain the solution. On the model, there are 2 paths for neutrons to reach the detector 4001 in the air: through the concrete and through the air channel. Figure 3 illustrates the paths for neutrons to reach detectors in the air in the Tihange-I reactor model.

<figure><img src="image_3.png" /><figcaption>**Figure 3** : The neutrons paths to reach the detectors in the air in the modified Tihange-I model. Circles represent the sphere detectors defined in the model.</figcaption></figure>
In Figure 3, there are two paths for neutrons to contribute to the detectors in the air. In the first path, neutrons will go through the first layer of concrete to reach the detectors in the air. The detectors in the concrete (4002, 4003, and 4004) help to identify the neutrons contribution to the detectors in the air from the concrete. In the second path, neutrons will go through the streaming path in the air channel. The detectors in the air (4001, 4005, 4006 and 4007) help to identify the neutrons contribution to the detectors in the air from the air channel.

### III.2. TRIPOLI-4 Model

TRIPOLI-4 is the fourth generation of the 3D continuous-energy Monte Carlo code developed by the Service d'Etudes des Réacteurs et de Mathématiques Appliquées (SERMA) at CEA Saclay [8]. Currently, TRIPOLI-4 can solve neutron, photon, electron, and positron transport problems. TRIPOLI-4 is designed for two major transport problems: radiation shielding and reactor physics. In version 11.0, there are two available VRTs: AMS and INIPOND. INIPOND uses the Exponential Transform to alter the transition kernel favoring particles' trajectories towards the detectors' locations [8].

Since the study will compare the results from TRIPOLI-4 to MCNP, it is important to ensure the existing TRIPOLI-4 input file from the previous study [7] is equivalent to the reference MCNP input file. The Verification and Validation (V&V) was conducted on geometry definition, material definition, and source definition (see Appendix A.3.). The initial input file was then revised so as to incorporate feedbacks found during the V&V. Furthermore, an energy cutoff was applied on the model since the main aim is to estimate only the fast neutron flux ($E > 1$ MeV).

In addition, there are other important parameters to consider: nuclear data, parallelization scheme, and the efficiency of the Monte Carlo simulation.

First of all, it is important to ensure both MCNP and TRIPOLI-4 codes are using the same nuclear data library, thermal library, the same nuclear data interpretation, and the same materials' temperatures. In this study, the same ENDF/B-VII.1 nuclear data library was used. Furthermore, the 'Force Unit Base Interpolation' option of TRIPOLI-4 simulation card was activated to force the same nuclear data interpretation [5]. The option will force TRIPOLI-4 to use the same nuclear data interpretation concerning the outgoing particle energy sampling.

Furthermore, during a TRIPOLI-4 parallel simulation with N number of CPUs, 1 CPU will be used as a collector, 1 other CPU will be used for monitoring, and N-2 CPUs will be used for particles simulation. Thus, it is important to correctly take into account the number of CPUs used for particles simulation (N-2 instead of N) for the calculation of total clock time and FOM.

Last but not least, it is important to investigate the efficiency of a Monte Carlo simulation in TRIPOLI-4 and MCNP in Tihange-I model. The efficiency of the Monte Carlo simulation in a given code will affect the FOM obtained from a non-analog
simulation. However, there is no identical VRT present in MCNP and TRIPOLI. Thus, a code to code FOM comparison in an analog simulation is prohibitive since it requires significant number of particles.

The investigation of the efficiency of the Monte Carlo simulation in TRIPOLI-4 and MCNP was conducted on a simplified model (see Appendix A.2.). The model is a sphere that consists of core, air, and concrete layers which tried to replicate the Tihange-I model.

### III.3. Workflow of the Study

During the internship, the work could be divided into 6 steps. Figure 4 presents the workflow of the study.

<figure><img src="image_4.png" /><figcaption>**Figure 4** : The workflow of the study</figcaption></figure>

In step 1, the Verification and Validation (V&V) was conducted to compare the available TRIPOLI-4 input file to the reference MCNP input file. At the end of step 1, the TRIPOLI-4 input file was revised accordingly.

In step 2, the investigation of AMS parameters was conducted. The AMS parameters that were investigated are the splitting rate, the resample collisions option, and the target(s) choice for stopping criterion. After the investigation of AMS parameters, the AMS simulation could begin.
There are 3 types of AMS simulations conducted: AMS using spatial dependence only importance function (step 3), AMS using an INIPOND map (step 4), and AMS using an ADVANTG map (step 5).

In step 3, the AMS simulation using spatial dependence only importance function was investigated. Some of AMS's built-in importance functions were explored: point-wise, line-wise, plane-wise, sphere-wise, and cylinder-wise. One option was chosen based on how efficient the particles could be pushed towards the detectors. Then, the position was varied to find the optimum.

In step 4, AMS simulation using an INIPOND map was investigated. Before running any AMS simulation, the importance map was generated from INIPOND. INIPOND parameters were optimized to find the best importance map candidate.

In step 5, AMS simulation using an importance map derived from ADVANTG was investigated. Automated VAriaNce reduction Generator (ADVANTG) is a software developed by Oak Ridge National Laboratory (ORNL). ADVANTG automates the generation of weight window input file for weight window technique in MCNP simulations of neutron, photon, coupled neutron-photon transport problems [9].

In the previous study, ADVANTG code was used to produce variance reduction parameters for Weight Window (one of the VRTs in MCNP) [9]. The weight window input based on the evaluation of a very finely defined in space-phase and energy-phase based on an adjoint flux calculation with a deterministic solver for neutron and gamma in the detectors.

Since TRIPOLI-4 version 11 is unable to read the weight window format, the existing weight window input file generated from the previous study [7] was converted into a TRIPOLI-4 importance map format using an external script. After the conversion script and the map have been checked, the AMS simulation using the importance map derived from ADVANTG could begin.

In step 6, the comparison of AMS results to the previous study could begin after the step 3, 4, and 5 are completed. Among other results on the previous study, MCNP-ADVANTG was chosen since it produced the highest FOM. The MCNP-ADVANTG calculation used Weight Window technique which the weight window input was provided by ADVANTG.
# Chapter 4 – Results and Discussion

## IV.1. AMS Parameters

### IV.1.1. Splitting rate

In the AMS algorithm, the splitting rate parameter determines the number of particles that will be resampled and the stopping criterion. There are consequences of the splitting rate parameter in AMS simulation.

If the splitting rate is relatively lower, AMS will resample fewer particles and have a more stringent stopping criterion. Since fewer particles are resampled at each iteration, there will be more AMS iterations until sufficient particles reach the targets (detectors). Therefore, AMS might spend more time on importance sorting. As a trade-off, the statistics of the results might be better since more particles would contribute to tally due to more stringent stopping criterion.

If the splitting rate is relatively higher, the AMS simulation will resample more particles with lower importance and have less stringent stopping criterion. Since AMS resamples more particles in each iteration, the particles will be pushed relatively faster to reach the detectors. Furthermore, since the stopping criterion is less stringent, a relatively smaller number of particles needs to reach the detectors and satisfy the stopping criterion. Thus, there will be fewer AMS iterations and less time spent on the importance sorting.

Figure 5 shows the total clock time vs splitting rate in the AMS simulation.

<figure><img src="image_5.png" /><figcaption>**Figure 5** : Total clock time vs splitting rate in AMS simulation using line-wise importance function</figcaption></figure>

Based on Figure 5, user might be tempted to use a high splitting rate since it will decrease the total clock time significantly. However, the statistics of the results might be relatively worse since fewer particles contribute to the score due to the less stringent stopping criterion.
Figure 6 shows the total evolution of detector 4004's relative standard deviation when low and high splitting rates are used.

<figure><img src="image_6.png" /><figcaption>**Figure 6** : The evolution of detector 4004's relative standard deviation</figcaption></figure>

In Figure 6, there are significant relative standard deviations jumps in the detector 4004 if a splitting rate of 40% was used. Relative standard deviation jumps indicate that a rarely sampled particle path has affected significantly the tally and relative standard deviation estimate due to the less stringent stopping criterion. These jumps should be avoided since the user could not be certain that the results are reliable. Even if there is no jump when using high splitting rate, the user cannot guarantee that there will be no relative standard deviation jumps if the simulation is extended. Thus, the use of high splitting rate in AMS simulation is tricky. Users should be careful when using high splitting rate in AMS simulation. Generally, it is recommended to not use splitting rate higher than 50% [5].

It could be inferred that the splitting rate parameter is a trade-off between pushing particles faster towards the targets but with relatively worse statistics against pushing particles slower towards the targets but with relatively better statistics. Thus, the determination of the splitting rate could be an iterative process requiring optimization. In this study, the splitting rate of 10% is chosen for all further AMS simulations because it still offers stringent stopping criterion while reducing the simulation time.

It would be interesting if the AMS module could set up a variable splitting rate once certain criteria have been met. For instance, the fixed-source simulation would have two steps: pushing the particles with higher splitting rate and recording the tallies with lower splitting rate.

In the first step, the variable splitting rate would set the value to be high so as to push particles faster towards the region of interest. During this mode, no tallies would be recorded since it would be less reliable due to the less stringent stopping criterion. In the second step, the variable splitting rate would set the value to be low
after the particles converged into specific region. During this mode, the tallies recording could begin since the stopping criterion would be more stringent.

#### IV.1.2. Resample Collisions

When splitting a particle's track, the regular AMS algorithm will duplicate the particle at the splitting point and sample a new flight length. The resample collisions feature will enable the AMS algorithm to evaluate the importance function before the collision occurs. If activated, the AMS algorithm does not duplicate the particle's outgoing energy and direction but rather resamples the collision which will determine the outgoing energy and direction.

Figure 7 shows the evolution of detector 4006's relative standard deviation if the resample collisions feature is on and off.

<figure><img src="image_7.png" /><figcaption>**Figure 7** : The evolution of detector 4006's relative standard deviation when the resample collisions feature is on (blue) and off (black).</figcaption></figure>

As previous studies suggested [4], the resample collisions feature in AMS could be advantageous or disadvantageous. The use of the resample collisions feature is less desired if the importance is correctly taken into account with the energies and directions of the kept particles. If the feature is activated, the collisions will be resampled during the resampling of the suppressed particles. It might change the energy and direction to a less desirable region in the system but it could also introduce more varieties which ends up with richer statistics.

Based on Figure 7, there are bigger relative standard deviations jumps if the resample collisions feature is activated. It seems that the importance function used in the study has correctly taken into account the energies and directions of the particles towards the detectors. The use of the resample collisions feature might then deteriorate the statistics of the tallies. Therefore, in this study, it seems to be beneficial not to use the resample collisions feature.
IV.1.3. Target(s) Choice for the Stopping Criterion

In the AMS card, the user needs to define the target(s) choice for the stopping criterion of the AMS simulation. In this section, variations of target(s) choices were studied.

Figure 8 shows the YZ axis plot of particles collisions sites in AMS line-wise simulation when 7 detectors and only detector 4001 were used as targets.

<figure><img src="image_8.png" /><figcaption>**Figure 8:** The YZ axis plot of particles collisions sites when 7 detectors (1) and only detector 4001 (2) were used as targets choice in an AMS line-wise simulation</figcaption></figure>

During the AMS simulation, the stopping criterion will determine how many AMS iterations are needed. Each time a new iteration undergoes, the tracks' importance will increase as the AMS pushes the particles towards the detectors. Based on Figure 8, if all detectors are defined as targets, the AMS simulation seems not to be able to push the particles sufficiently towards the detectors in the air. Later it was found out that defining 'easier' detectors (4002, 4003, and 4004) as targets choice will stop the iterations earlier since the stopping criterion has been met although the particles have not been pushed sufficiently towards the detectors in the air. Since fewer particles were coming to the difficult detectors (e.g. 4001), the standard deviation of the neutron flux at these detectors are relatively high.

Thus, it is the best user's interest to optimize the target(s) choice for AMS simulation in multiple detectors problem. In this study, users might consider only to define the detectors that are relatively more difficult to reach (e.g. detector 4001). In addition to defining the 'difficult' detectors, the user also should also consider defining the detectors on the path of the particles. In Figure 3, neutron could reach the detectors in the air through concrete and through the air channel. It might be interesting for the user to define an additional target in the channel (detector 4005). By defining these detectors, AMS iterations might populate the paths correctly. After some variations of targets choice explored, detector 4001 could be used as a single target in a multi detectors problem and seems to be the best candidate among other target(s).
IV.2. AMS Using Spatial Dependence Only Importance Function

In AMS's built-in spatial dependence only importance function, there are some options. In this study, the following importance functions were evaluated: point-wise, line-wise, plane-wise, sphere-wise, and cylinder-wise. The importance functions were used to push the particles toward the location of the importance function ('TOWARDS' option). The closer particles to the importance function's location, the higher the importance will be. For instance, in line-wise importance function, the particles will be pushed towards the YZ-axis line which goes through the center of detector 4001.

Figure 9 shows the YZ axis plot of the particle collisions sites in an AMS simulation using line-wise, plane-wise, sphere-wise, and point-wise spatial importance function.

<figure><img src="image_9.png" /><figcaption>**Figure 9** : The collisions sites plot on YZ axis of AMS simulation using line-wise (1), plane-wise (2), sphere-wise (3), and point-wise (4) spatial importance function</figcaption></figure>

Based on Figure 9, plane-wise and line-wise seem to be the best candidates of
spatial dependence importance functions that allow pushing particles efficiently
towards the detectors. Furthermore, plane-wise and line-wise seem to push neutrons
correctly towards the 2 paths: through concrete and through air channel. After the
particles' collisions sites on the YZ axis have been investigated, the particles' collisions sites on the XY axis were investigated.

Figure 10 shows the XY axis plot of the particle collisions sites in an AMS simulation using line-wise and plane-wise.

<figure><img src="image_10.png" /><figcaption>**Figure 10** : The collisions sites plot on XY axis of line-wise (1) and plane-wise (2) used in AMS simulation</figcaption></figure>

Based on Figure 10, it seems that the use of plane-wise causes AMS to push the particles towards not only the detectors but also towards other less desired regions. In contrast to plane-wise, the line-wise importance function pushes the particles in radial directions efficiently towards the detectors. Thus, the line-wise importance function seems to be the best candidate among AMS's built-in importance functions.

The optimization of the line positions was then conducted. Since the detectors in the air are more difficult to reach, it would be favorable to define the line's position near the detectors in the air.

Figure 11 illustrates the line-wise positions that were investigated using AMS spatial importance functions.

<figure><img src="image_11.png" /><figcaption>**Figure 11** : Tested line-wise positions using AMS Spatial Dependence Only importance function</figcaption></figure>

HAKIM Arief Rahman / Institut de Radioprotection et de Sûreté Nucléaire
Rapport non confidentiel et publiable sur internet
Figure 12 shows the evolution of detector 4001 relative standard deviation at various line positions.

<figure><img src="image_12.png" /><figcaption>**Figure 12** : The evolution of detector 4001 relative standard deviation at various line positions</figcaption></figure>

Based on Figure 12, the line position at -10 cm seems to be the best candidate because it has a relatively lower relative standard deviation and less relative standard deviation jumps. Furthermore, the performance of line-wise at -10 cm is consistent in other detectors which has lower relative standard deviation and less relative standard deviation jumps. After the best position was identified, the final AMS simulation was conducted. The parameters used for the final AMS simulation are presented in Appendix A.4.

Table 1 presents the results from AMS simulation using line-wise importance function compared to the MCNP-ADVANTG.

<table><caption>**Table 1** : The comparison of AMS using line-wise to MCNP-ADVANTG</caption><thead><tr><th rowspan="2">Detector</th><th colspan="2">Neutron flux (n/cm²s⁻¹)</th><th colspan="2">Relative standard deviation</th><th colspan="3">FOM (s⁻¹)</th></tr><tr><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>TRIPOLI4/ MCNP</th></tr></thead><tbody><tr><td>4001</td><td>5.57E+04</td><td>5.09E+04</td><td>2.98%</td><td>1.78%</td><td>1.24E-02</td><td>4.43E-03</td><td>0.356</td></tr><tr><td>4002</td><td>1.82E+09</td><td>1.74E+09</td><td>0.41%</td><td>1.24%</td><td>6.57E-01</td><td>9.17E-03</td><td>0.014</td></tr><tr><td>4003</td><td>1.21E+09</td><td>1.17E+09</td><td>0.43%</td><td>2.49%</td><td>5.98E-01</td><td>2.27E-03</td><td>0.004</td></tr><tr><td>4004</td><td>2.56E+08</td><td>2.70E+08</td><td>0.68%</td><td>3.00%</td><td>2.39E-01</td><td>1.56E-03</td><td>0.007</td></tr><tr><td>4005</td><td>2.86E+07</td><td>2.83E+07</td><td>0.87%</td><td>3.44%</td><td>1.46E-01</td><td>1.19E-03</td><td>0.008</td></tr><tr><td>4006</td><td>4.48E+06</td><td>4.71E+06</td><td>1.45%</td><td>8.19%</td><td>5.25E-02</td><td>2.09E-04</td><td>0.004</td></tr><tr><td>4007</td><td>9.12E+04</td><td>9.58E+04</td><td>2.67%</td><td>2.22%</td><td>1.55E-02</td><td>2.85E-03</td><td>0.184</td></tr></tbody></table>
Furthermore, Figure 13 shows the neutron flux comparison of AMS using line-wise importance function to MCNP-ADVANTG.

<figure><img src="image_13.png" /><figcaption>**Figure 13** : The neutron flux comparison of AMS using line-wise importance function to MCNP-ADVANTG with a confidence interval of 3 standard deviations</figcaption></figure>

Based on Figure 13, AMS using line-wise importance function shows a good agreement with the MCNP-ADVANTG results within a confidence interval of 3 standard deviations.

Figure 14 shows the FOM comparison between AMS using line-wise importance function and MCNP-ADVANTG.

<figure><img src="image_14.png" /><figcaption>**Figure 14** : The FOM comparison of AMS using line-wise importance function to MCNP-ADVANTG</figcaption></figure>

From the code to code FOM comparison between TRIPOLI to MCNP in an
analog calculation, a ratio of 0.4-0.8 is expected (see Appendix A.2). Based on Figure
14, the FOM ratios of the detector 4001 is close to the expected FOM ratio due to code
differences while other detectors are still outside of the expected FOM ratio. Thus,
AMS has achieved the goal to obtain a reliable neutron flux with satisfying FOM at
detector 4001 even when compared to the reference results. In addition, it shall be
noted that the FOM calculation for MCNP-ADVANTG has not taken into account the
time spent for the generation of ADVANTG map which is not negligible. Overall, AMS
using spatial dependence only importance function is proven able to estimate the tallies with satisfying statistics and total clock time.

AMS using spatial dependence only feature offers a powerful VRT option. The built-in spatial importance function requires less tuning effort and the syntax is simple and straightforward. The built-in spatial importance function might serve as a tempting option for beginner users who have a lack of prior VRT experience.

### IV.3. AMS Using Pre-calculated INIPOND Importance Map

Before AMS simulation begins, the importance map will be generated using the INIPOND module in TRIPOLI-4 code. Compared to the previous study [7], there were additional modification in INIPOND parameters: the use of 'VOID' in gas material's names, the targets list, and the use of energy biasing.

First, the use of 'VOID' in gas materials' names play an important role in INIPOND. By adding 'VOID' into the gas material definition, the bias parameter, 'k', will be set by taking half of the lowest value among the other values of 'k' [5]. Figure 15 shows the comparison of INIPOND equi-importance map before and after the use of 'VOID' in gas materials' name.

<figure><img src="image_15.png" /><figcaption>**Figure 15** : The equi-importance plot of INIPOND map when 'VOID' is not used (1) and used (2).</figcaption></figure>

Based on Figure 15, as suggested in TRIPOLI-4 manual, the use of 'VOID' in all gas material's name, even far from the detectors, helps the INIPOND module to correctly predict the neutron paths which will improve the quality of the importance map.

Second, the detectors target(s) in the INIPOND module was varied to check how it affects the collisions' sites. Figure 16 shows the collisions sites of AMS using INIPOND map when 7 detectors and only detector 4001 were used in the INIPOND card.
<figure><img src="image_16.png" /><figcaption>**Figure 16** : The collisions sites of AMS using INIPOND map when 7 detectors (1) and only detector 4001 (2) used in the INIPOND card</figcaption></figure>

After the optimization, the INIPOND map that uses ‘VOID’ and detector target 4001 is chosen as the best importance map candidate. The parameters used for the final AMS simulation are presented in Appendix A.4.

After the importance map was chosen, the AMS simulation using the INIPOND map could begin. Table 2 presents results from AMS simulation using INIPOND importance map compared to MCNP-ADVANTG.

<table><caption>**Table 2** : The comparison of AMS using INIPOND importance map to MCNP-ADVANTG</caption><thead><tr><th rowspan="2">Detector</th><th colspan="2">Neutron flux (n/cm²s⁻¹)</th><th colspan="2">Relative standard deviation</th><th colspan="3">FOM (s⁻¹)</th></tr><tr><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>TRIPOLI4/MCNP</th></tr></thead><tbody><tr><td>4001</td><td>5.57E+04</td><td>5.33E+04</td><td>2.98%</td><td>1.35%</td><td>1.24E-02</td><td>7.23E-04</td><td>0.359</td></tr><tr><td>4002</td><td>1.82E+09</td><td>1.78E+09</td><td>0.41%</td><td>0.38%</td><td>6.57E-01</td><td>3.12E-03</td><td>0.086</td></tr><tr><td>4003</td><td>1.21E+09</td><td>1.19E+09</td><td>0.43%</td><td>0.60%</td><td>5.98E-01</td><td>2.17E-02</td><td>0.038</td></tr><tr><td>4004</td><td>2.56E+08</td><td>2.72E+08</td><td>0.68%</td><td>1.63%</td><td>2.39E-01</td><td>9.77E-03</td><td>0.013</td></tr><tr><td>4005</td><td>2.86E+07</td><td>3.13E+07</td><td>0.87%</td><td>2.05%</td><td>1.46E-01</td><td>5.28E-03</td><td>0.013</td></tr><tr><td>4006</td><td>4.48E+06</td><td>5.02E+06</td><td>1.45%</td><td>2.36%</td><td>5.25E-02</td><td>3.76E-03</td><td>0.028</td></tr><tr><td>4007</td><td>9.12E+04</td><td>1.05E+05</td><td>2.67%</td><td>2.32%</td><td>1.55E-02</td><td>9.48E-04</td><td>0.098</td></tr></tbody></table>

Furthermore, Figure 17 shows the neutron flux comparison of AMS using an INIPOND map to MCNP-ADVANTG.
<figure><img src="image_17.png" /><figcaption>**Figure 17** : The neutron flux comparison of AMS using INIPOND map to MCNP-ADVANTG within a confidence interval of 3 standard deviations</figcaption></figure>

Based on Figure 17, AMS using INIPOND importance map shows a good agreement with the MCNP-ADVANTG results within a confidence interval of 3 standard deviations. It is observed that the neutron flux in detectors 4004, 4005, 4006, and 4007 are overestimated. It might be due to the specific optimization of the INIPOND map for the AMS simulation.

Furthermore, Figure 18 shows the FOM comparison.

<figure><img src="image_18.png" /><figcaption>**Figure 18** : The FOM comparison of AMS using INIPOND map to MCNP-ADVANTG</figcaption></figure>

As suggested on the code to code comparison (see Appendix A.2), a FOM ratio of 0.4-0.8 is expected between TRIPOLI-4 to MCNP. Based on Figure 18, AMS has achieved the goal to obtain a reliable tally with satisfying FOM at detector 4001. AMS is also still able to obtain reliable tallies on other detectors with acceptable FOM. Furthermore, the AMS FOMs seem to reproduce the same trend as the FOM of the MCNP-ADVANTG. It might be due to the fact that both of them are deterministic tools. The AMS using an INIPOND map has slightly higher FOMs compared to the AMS simulation using spatial importance function.

It shall also be noted that the generation of the INIPOND importance map requires expertise that even some experienced users might find challenging.
Nevertheless, the user might find it useful to choose AMS using the INIPOND map depending on the problem. Overall, AMS using the INIPOND map is proven able to estimate the tallies with satisfying statistics and total clock time.

#### IV.4. AMS Using the Importance Map Derived From ADVANTG

In the previous study, ADVANTG was used to generate Weight Window input file for MCNP. Due to the size of the ADVANTG Weight Window input file, in order to be used by TRIPOLI-4, the original file was then optimized only for detector 4001 by Weight Window Generator in MCNP. Since the TRIPOLI-4 version 11 is unable to read the Weight Window format, the existing Weight Window input file generated from the previous study [7] was converted into a TRIPOLI-4 importance map format using an external script. After the converted, the AMS simulation using an importance map derived from ADVANTG could begin. The parameters used for the final AMS simulation are presented on Appendix A.4.

Table 3 presents results from AMS simulation using ADVANTG importance map compared to MCNP-ADVANTG.

<table><caption>**Table 3** : The comparison of AMS using ADVANTG variance reduction parameters to MCNP-ADVANTG</caption><thead><tr><th rowspan="2">Detector</th><th colspan="2">Neutron flux (n/cm²s⁻¹)</th><th colspan="2">Relative standard deviation</th><th colspan="3">FOM (s⁻¹)</th></tr><tr><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>MCNP-ADVANTG</th><th>TRIPOLI-4-AMS</th><th>TRIPOLI4/ MCNP</th></tr></thead><tbody><tr><td>4001</td><td>5.57E+04</td><td>5.64E+04</td><td>2.98%</td><td>2.45%</td><td>1.24E-02</td><td>1.31E-03</td><td>0.105</td></tr><tr><td>4002</td><td>1.82E+09</td><td>1.78E+09</td><td>0.41%</td><td>0.41%</td><td>6.57E-01</td><td>4.60E-02</td><td>0.070</td></tr><tr><td>4003</td><td>1.21E+09</td><td>1.20E+09</td><td>0.43%</td><td>0.47%</td><td>5.98E-01</td><td>3.52E-02</td><td>0.059</td></tr><tr><td>4004</td><td>2.56E+08</td><td>2.72E+08</td><td>0.68%</td><td>0.74%</td><td>2.39E-01</td><td>1.46E-02</td><td>0.061</td></tr><tr><td>4005</td><td>2.86E+07</td><td>3.14E+07</td><td>0.87%</td><td>0.84%</td><td>1.46E-01</td><td>1.12E-02</td><td>0.077</td></tr><tr><td>4006</td><td>4.48E+06</td><td>4.94E+06</td><td>1.45%</td><td>1.02%</td><td>5.25E-02</td><td>7.54E-03</td><td>0.143</td></tr><tr><td>4007</td><td>9.12E+04</td><td>1.04E+05</td><td>2.67%</td><td>1.21%</td><td>1.55E-02</td><td>5.38E-03</td><td>0.347</td></tr></tbody></table>

Furthermore, Figure 19 shows the neutron flux comparison of AMS using ADVANTG map to MCNP-ADVANTG.

Based on Figure 19, AMS using ADVANTG importance map shows an agreement only in detectors 4001, 4002 and 4003 with the MCNP-ADVANTG. This is one of good examples how VRT should not be used as a 'black box'. Although the standard deviations obtained from the AMS simulation seem to be low, the result might be not reliable. The user should carefully investigate and compare the results to other VRTs or to the reference results.
<figure><img src="image_19.png" /><figcaption>**Figure 19** : The neutron flux comparison of AMS using an ADVANTG map to MCNP-ADVANTG within a confidence interval of 3 standard deviations</figcaption></figure>

In this study, the disagreement in detector 4004, 4005, 4006, and 4007 might be due to the fact that the ADVANTG map was optimized only for detector 4001 by Weight Window Generator in MCNP. Thus, the results from other detectors might be not reliable.

Furthermore, Figure 20 shows the FOM comparison.

<figure><img src="image_20.png" /><figcaption>**Figure 20** : The FOM comparison of AMS using an ADVANTG map to MCNP-ADVANTG</figcaption></figure>

As suggested on the code to code FOM comparison (see Appendix A.2), a FOM ratio of 0.4-0.8 is expected between TRIPOLI-4 to MCNP. In Figure 20, the FOMs of the AMS are still lower even after considering the expected FOM ratio. In addition, it is also observed that the AMS FOMs share the same trend due to the use of the same importance map.

In general, AMS using an ADVANTG map has obtained reliable tallies for the detector 4001, 4002 and 4003. However, there are disagreements on some detectors due to the optimization of ADVANTG file by Weight Window Generator in MCNP. Comparing the AMS results using an ADVANTG map to the AMS results using an INIPOND map, the AMS using an INIPOND map has the highest FOM.
# Chapter 5 – Conclusions

The comparison of Adaptive Multilevel Splitting to other VRTs in MCNP has been conducted. During the study, the AMS parameters have been explored to optimize the FOM. There are two types of importance function that could be used for AMS: a spatial dependence only or a pre-calculated importance map for space and energy.

AMS using the built-in spatial importance function produced results with an agreement to MCNP-ADVANTG. The study shows that the AMS using spatial importance function is able to obtain reliable neutron flux at all detectors. Furthermore, the FOM of the most difficult detector is satisfactory when compared to the previous study. It shall be noted that although AMS FOMs of other detectors are lower, ADVANTG requires an additional step of the deterministic solver which was not accounted during the MCNP-ADVANTG FOM calculation. Therefore, AMS offers an alternative powerful VRT option which is simple to use and uses a straightforward syntax. Thus, it might be tempting for beginner users who have a lack of prior VRT experience. Nonetheless, in spatial and energy dependence problems, it might be better to use a pre-calculated importance map rather than the spatial importance function.

The AMS simulation using an INIPOND importance map was conducted. The AMS using the INIPOND map seems to provide better importance map proven by the improvement of the FOM when compared to the spatial importance function. However, it shall be noted that the generation of the INIPOND importance map requires expertise that even some experienced users might find challenging. The AMS using an INIPOND map also provides the highest FOM compared to AMS simulations using spatial importance function and ADVANTG importance map.

The AMS simulation using ADVANTG importance map was also conducted. The results show an agreement only in 3 out of 7 detectors (including the most difficult one). This is a good example why users should not use the VRT as a 'black box'. Although the standard deviations obtained from the simulation seem to be low, the user should still investigate whether the results are reliable or not. In this study, the disagreements might be due to the optimization only for the most difficult detector.

Overall, Adaptive Multilevel Splitting offers alternative powerful VRT options. It is proven able to obtain reliable results with satisfying statistics and total clock time. While AMS is relatively simple to use, it is still the user's responsibility to ensure the use of AMS is still producing reliable results. Thus, AMS results need to be examined by further statistical checks to ensure its reliability.
Reference

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[4] H. Louvin, “Development of an adaptive variance reduction technique for Monte Carlo particle transport,” *Comput. Phys. Univ. Paris-Saclay*, 2017.

[5] TRIPOLI-4 Project Team, “Tripoli-4® Version 11 User Guide,” pp. 1–499, 2018.

[6] M. Brovchenko, B. Dechenaux, K. W. Burn, P. Console Camprini, I. Duhamel, and A. Peron, “Neutron-gamma flux and dose calculations in a Pressurized Water Reactor (PWR),” *EPJ Web Conf.*, vol. 153, 2017.

[7] M. Brovchenko, L. Tillard, K. W. Burn, B. Cochet, and A. Jinaphanh, “On the use of different variance reduction techniques within MCNP to calculate the flux on the concrete walls of a pressurized water reactor,” *Int. Conf. Math. Comput. Methods Appl. to Nucl. Sci. Eng. MC 2019*, pp. 104–119, 2019.

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# Appendix

## A.1. Simulation Results

In this section, the results from the previous study and AMS results are presented. It shall be noted that the TRIPOLI-4 INIPOND results are obtained from the revised TRIPOLI-4 input file using the same INIPOND importance map used in AMS (INIPOND).

Table A.1.1 presents the estimated neutron flux from the AMS and the previous study.

<table><caption>**Table A.1.1:** The estimated neutron flux from the AMS and the previous study</caption><thead><tr><th rowspan="3">Volume</th><th colspan="7">Neutron flux (n/cm²s⁻¹)</th></tr><tr><th colspan="3">MCNP</th><th colspan="4">TRIPOLI-4</th></tr><tr><th>ADVANTG</th><th>WWG scratch</th><th>DSA</th><th>INIPOND</th><th>AMS (line-wise)</th><th>AMS (INIPOND)</th><th>AMS (ADVANTG)</th></tr></thead><tbody><tr><td>4001</td><td>5.57E+04</td><td>6.74E+04</td><td>5.35E+04</td><td>5.37E+04</td><td>5.09E+04</td><td>5.33E+04</td><td>5.64E+04</td></tr><tr><td>4002</td><td>1.82E+09</td><td>1.81E+09</td><td>1.83E+09</td><td>1.78E+09</td><td>1.73E+09</td><td>1.78E+09</td><td>1.78E+09</td></tr><tr><td>4003</td><td>1.21E+09</td><td>1.19E+09</td><td>1.21E+09</td><td>1.20E+09</td><td>1.17E+09</td><td>1.19E+09</td><td>1.20E+09</td></tr><tr><td>4004</td><td>2.56E+08</td><td>2.35E+08</td><td>2.57E+08</td><td>2.70E+08</td><td>2.71E+08</td><td>2.72E+08</td><td>2.72E+08</td></tr><tr><td>4005</td><td>2.86E+07</td><td>2.57E+07</td><td>2.90E+07</td><td>3.05E+07</td><td>2.83E+07</td><td>3.13E+07</td><td>3.14E+07</td></tr><tr><td>4006</td><td>4.48E+06</td><td>3.96E+06</td><td>4.55E+06</td><td>4.70E+06</td><td>4.72E+06</td><td>5.02E+06</td><td>4.94E+06</td></tr><tr><td>4007</td><td>9.12E+04</td><td>5.87E+04</td><td>9.58E+04</td><td>1.01E+05</td><td>9.58E+04</td><td>1.05E+05</td><td>1.04E+05</td></tr></tbody></table>

Furthermore, the relative standard deviation is presented in Table A.1.2.

<table><caption>**Table A.1.2:** The neutron flux relative standard deviation from the AMS and the previous study</caption><thead><tr><th rowspan="3">Volume</th><th colspan="7">Neutron flux relative standard deviation</th></tr><tr><th colspan="3">MCNP</th><th colspan="4">TRIPOLI-4</th></tr><tr><th>ADVANTG</th><th>WWG scratch</th><th>DSA</th><th>INIPOND</th><th>AMS (line-wise)</th><th>AMS (INIPOND)</th><th>AMS (ADVANTG)</th></tr></thead><tbody><tr><td>4001</td><td>2.98%</td><td>35.97%</td><td>0.40%</td><td>4.35%</td><td>1.78%</td><td>1.35%</td><td>2.45%</td></tr><tr><td>4002</td><td>0.41%</td><td>1.63%</td><td>0.32%</td><td>0.15%</td><td>1.24%</td><td>0.38%</td><td>0.41%</td></tr><tr><td>4003</td><td>0.43%</td><td>1.37%</td><td>0.39%</td><td>0.28%</td><td>2.49%</td><td>0.60%</td><td>0.47%</td></tr><tr><td>4004</td><td>0.68%</td><td>2.40%</td><td>0.68%</td><td>0.64%</td><td>3.00%</td><td>1.63%</td><td>0.74%</td></tr><tr><td>4005</td><td>0.87%</td><td>4.39%</td><td>1.03%</td><td>1.17%</td><td>3.44%</td><td>2.05%</td><td>0.84%</td></tr><tr><td>4006</td><td>1.45%</td><td>6.31%</td><td>1.01%</td><td>2.39%</td><td>8.19%</td><td>2.36%</td><td>1.02%</td></tr><tr><td>4007</td><td>2.67%</td><td>9.27%</td><td>0.61%</td><td>4.57%</td><td>2.22%</td><td>2.32%</td><td>1.21%</td></tr></tbody></table>
In addition, the normalized FOM to the MCNP-ADVANTG is presented in Table A.1.3.

<table><caption>**Table A.1.3:** The normalized FOM to the MCNP-ADVANTG</caption><thead><tr><th rowspan="3">Volume</th><th colspan="7">FOM ... / FOM<sub>MCNP-ADVANTG</sub></th></tr><tr><th colspan="3">MCNP</th><th colspan="4">TRIPOLI-4</th></tr><tr><th>ADVANTG</th><th>WWG scratch</th><th>DSA</th><th>INIPOND</th><th>AMS (line-wise)</th><th>AMS (INIPOND)</th><th>AMS (ADVANTG)</th></tr></thead><tbody><tr><td>4001</td><td>1</td><td>0.00022</td><td>8.984</td><td>0.04550</td><td>0.356</td><td>0.359</td><td>0.105</td></tr><tr><td>4002</td><td>1</td><td>0.00205</td><td>0.266</td><td>0.68017</td><td>0.014</td><td>0.086</td><td>0.070</td></tr><tr><td>4003</td><td>1</td><td>0.00320</td><td>0.197</td><td>0.22458</td><td>0.004</td><td>0.038</td><td>0.059</td></tr><tr><td>4004</td><td>1</td><td>0.00260</td><td>0.162</td><td>0.11056</td><td>0.007</td><td>0.013</td><td>0.061</td></tr><tr><td>4005</td><td>1</td><td>0.00127</td><td>0.115</td><td>0.05351</td><td>0.008</td><td>0.013</td><td>0.077</td></tr><tr><td>4006</td><td>1</td><td>0.00171</td><td>0.334</td><td>0.03584</td><td>0.004</td><td>0.028</td><td>0.143</td></tr><tr><td>4007</td><td>1</td><td>0.00268</td><td>3.101</td><td>0.03316</td><td>0.184</td><td>0.098</td><td>0.347</td></tr></tbody></table>
## A.2. Comparison of the FOM in an Analog Monte Carlo Simulation

The FOM of the VRT in a different Monte Carlo codes will be affected by each code's efficiency in conducting a Monte Carlo simulation. Thus, it is important to compare the efficiency of an analog Monte Carlo simulation in TRIPOLI-4 and MCNP before comparing the efficiency of VRTs.

Unfortunately, the analog Monte Carlo simulation using the Tihange-I model is prohibitive since it requires significant number of particles to obtain reliable statistics. Thus, the comparison will be conducted on a simplified model. The model is a sphere that consists of core, air, and concrete layers which tried to replicate the Tihange-I model.

Figure A.2.1 presents the simplified Monte Carlo model.

<figure><img src="image_21.png" /><figcaption>**Figure A.2.1** : The simplified Monte Carlo model</figcaption></figure>

Based on the model, the analog Monte Carlo simulations are conducted with TRIPOLI-4 and MCNP to obtain the fast neutron flux (E > 1 MeV). Furthermore, the FOM is calculated and compared between the codes.

Figure A.2.2 shows the comparison of the FOM of the fast neutron flux (E > 1 MeV) between TRIPOLI-4 and MCNP.

<figure><img src="image_22.png" /><figcaption>**Figure A.2.2** : Comparison of the FOM of the fast neutron flux (E > 1 MeV) between TRIPOLI-4 and MCNP in the simplified model</figcaption></figure>

HAKIM Arief Rahman / Institut de Radioprotection et de Sûreté Nucléaire
Rapport non confidentiel et publiable sur internet
Since the study involves with the neutron flux with a distance of 2.69 m from
the center of the core, we shall focus on the variation of the FOM in the concrete part
of Figure A.2.2. It seems that the FOM of the neutron flux in concrete fluctuates about
40-80% lower than the MCNP's FOM. Thus, if an identical VRT is used with TRIPOLI-
4 and MCNP, it is expected that the TRIPOLI-4 will be lower about 40-80% in the
same model. So, if FOMs of the AMS are within 40-80% lower compared to non-analog
MCNP, AMS can be considered as efficient as VRTs in MCNP.
## A.3. Verification and Validation of the TRIPOLI-4 Input File

In this part, the Verification and Validation (V&V) document of the TRIPOLI-4 Tihange-I input file is summarized. The verification and validation of the input file was conducted by visually inspecting and comparing the geometry, material definition, and source definitions in the input file and several key parameters in the output files. Critical mistakes were found on the definition of FA 51 and the source energy spectrum definition. The revised input file was then incorporating the corrections of the mistakes found during the V&V.

### 1. Geometry

In this section, the V&V are performed on geometry section: fuel rod, guide tube, instrument tube, burnable poison tube, fuel assembly, core lattice, under core lattice, structures outside the core, and detectors.

During the V&V, a critical mistake was found on the fuel rod of Fuel Assembly (FA) 51. The height of the fuel pellet seems to be wrong. Figure A.3.1 presents the fuel rod definition in FA 51.

<figure><img src="image_23.png" /><figcaption>**Figure A.3.1** : The fuel rod definition in Fuel Assembly 51</figcaption></figure>

From Figure A.3.1, the fuel pellet has a height of (2 x 12.193333 cm) instead of (2 x 182.9 cm). Thus, the particles contribution to the tally from FA 51 might be under estimated. The TRIPOLI-4 input file was revised to change from 12.193333 cm to 182.9 cm.

### 2. Material

In this section, the V&V that performed on material cards are: compositions, temperature, density, thermal library and location. During the V&V on TRIPOLI-4 material card, there are mistakes on the temperature definition and nuclear data library and thermal library.

First, the temperature used in each material in TRIPOLI-4 should be identical to the MCNP reference input file. In MCNP, the temperature is defined directly on each cell (cell-basis). Later in the material card, the user should define which library
at what temperature the cross section will be calculated. Furthermore, the temperature is defined in kT (MeV). Thus, a conversion should be conducted by the user to convert Kelvin to MeV.

In MCNP and TRIPOLI-4 input files, all materials were set on the same temperature. Table A.3.1 presents materials' temperature in the initial model.

<table><caption>**Table A.3.1** : Materials' temperature in the initial model</caption><thead><tr><th rowspan="2"></th><th colspan="2">MCNP</th><th>TRIPOLI-4</th></tr><tr><th>T (MeV)</th><th>T (K)</th><th>T (K)</th></tr></thead><tbody><tr><th>Temperature</th><td>4.81838E-8</td><td>559.1714054</td><td>574</td></tr></tbody></table>

From Table A.3.1, we shall see that there is a roughly 14 Kelvin differences of temperature between MCNP input file and TRIPOLI-4 input file. This slight difference could affect the reaction rate for isotopes with resonance region such as <sup>238</sup>U. Since TRIPOLI-4 cannot use a float data type for the material's temperature, the TRIPOLI-4 input file was revised so that the code will interpolate the temperature at 559 K.

Second, TRIPOLI-4 should use the same nuclear data as the previous study used. In this study, both MCNP and TRIPOLI-4 used ENDF/B-VII.1 nuclear data library. Furthermore, it is found out that the existing TRIPOLI-4 input file does not use the thermal nuclear data library yet. Thus, the TRIPOLI-4 input file was revised to use the thermal library.

### 3. Source Energy Spectrum Definition

In this section, the comparison between the source energy spectrum definition in MCNP and TRIPOLI-4 input file is conducted. Based on the MCNP and TRIPOLI-4 input files, the source will be sampled from a Watt Energy Spectrum Equation. Table A.3.2 presents the Watt energy spectrum forms in MCNP and TRIPOLI-4.

<table><caption>**Table A.3.2** : Watt spectrum equations forms in MCNP and TRIPOLI-4</caption><thead><tr><th>MCNP</th><th>TRIPOLI-4</th></tr></thead><tbody><tr><td>f(E) = C exp(-E/a) sinh(√bE)</td><td>f(E) = C exp(-Ea) sinh(√bE)</td></tr></tbody></table>

While the equations above have the same form, there is a slight difference on the 'a' variable. In MCNP Watt Spectrum, the -E is multiplied by 1/a while in TRIPOLI-4 the -E is multiplied by a. Thus, it is important to properly convert the 'a' variable from MCNP format into TRIPOLI-4 format.

From the MCNP manual, the default value of a is 0.965. After the V&V, it is found out that in TRIPOLI-4 file, the value of a is still 0.965 following the Watt Spectrum Equation in MCNP format. It should have been 1.03627. Thus, the a values in TRIPOLI-4 are changed from 0.965 to 1.03627 to follow the TRIPOLI-4 Watt Spectrum Equation in format.
# A.4. Simulation Parameters

## 1. Parameter Descriptions

In this section, the simulation parameters used is presented. There are several key parameters presented here: simulation parameters, AMS parameters, and INIPOND parameters.

Table A.4.1 presents a summary of simulation parameter descriptions.

<table><caption>**Table A.4.1** : The simulation parameter descriptions</caption><thead><tr><td>Simulation parameters</td><td>Description</td></tr></thead><tbody><tr><td>BATCH</td><td>Number of batches for the simulation</td></tr><tr><td>SIZE</td><td>Number of particles per batch</td></tr><tr><td>EDITION</td><td>Print results per # batch</td></tr><tr><td>TEMPERATURE_INTERPOL</td><td>Perform temperature interpolation for the materials' cross section.</td></tr><tr><td>FORCED_UNIT_BASE_INTERPOL</td><td>Force TRIPOLI-4 to use the same nuclear data interpretation (see section III.2.)</td></tr><tr><td>PARTICULES</td><td>Types of transport problem (e.g. neutron transport problem)</td></tr><tr><td>ENERGY_INF</td><td>Energy cut-off (see section III.2.)</td></tr></tbody></table>

In AMS parameters, there are 'TARGET', 'SPLIT', 'CROSS_SPLIT', 'RESAMPLE_COLLISIONS', and 'IMPORTANCE' parameters.

Table A.4.2 presents a summary of AMS parameter descriptions.

<table><caption>**Table A.4.2** : AMS parameters descriptions</caption><thead><tr><td>AMS parameters</td><td>Description</td></tr></thead><tbody><tr><td>TARGET</td><td>The target(s) choice for the stopping criterion (see section IV.1.3.)</td></tr><tr><td>SPLIT</td><td>The AMS splitting rate (see Eq. 3)</td></tr><tr><td>CROSS_SPLIT</td><td>Enable AMS to evaluate the importance when a particle travels towards other volumes (see section II.2.3.)</td></tr><tr><td>RESAMPLE_COLLISIONS</td><td>Enable AMS to resample the collisions during the resampling of the suppressed particles (see section IV.1.2.)</td></tr><tr><td>IMPORTANCE</td><td>The AMS importance function :<br>1. Spatial dependence only (see section IV.2.) or<br>2. Pre-calculated importance map (see section IV.3. and section IV.4.)</td></tr></tbody></table>
In INIPOND card, there are several parameters. In this study, the key parameters are 'GRID', 'DETECTOR', 'MESH_VARIABLE', and 'STORE_IN_FILE', parameters.

Table A.4.3 presents a summary of INIPOND parameter descriptions.

<table><caption>**Table A.4.3** : The simulation parameter descriptions</caption><thead><tr><td>INIPOND parameters</td><td>Description</td></tr></thead><tbody><tr><td>GRID</td><td>Energy grid for the importance map</td></tr><tr><td>DETECTOR</td><td>Detector target(s) and its beta value(s) are defined.</td></tr><tr><td>MESH_VARIABLE</td><td>The mesh of the system is defined</td></tr><tr><td>STORE_IN_FILE</td><td>Export the importance map after the simulation</td></tr></tbody></table>

## 2. Parameters Used in the Study

In this section, the parameters used for the final AMS simulations are presented.

### A. AMS Spatial Dependence Only Importance Function

In this section, the parameters used for the final AMS simulation using spatial dependence only importance function is presented. Figure A.4.1 presents AMS parameters that were used.

<figure><img src="image_24.png" /><figcaption>**Figure A.4.1** : AMS parameters used for AMS simulation using spatial dependence only importance function</figcaption></figure>

In addition, Figure A.4.2 presents the simulation parameters that were used.

<figure><img src="image_25.png" /><figcaption>**Figure A.4.2** : Simulation parameters used for AMS simulation using spatial dependence only importance function</figcaption></figure>

HAKIM Arief Rahman / Institut de Radioprotection et de Sûreté Nucléaire
Rapport non confidentiel et publiable sur internet
## B. AMS using an INIPOND Importance Map

In this section, the parameters used for the final AMS simulation using an INIPOND importance map is presented. Figure A.4.3 presents the INIPOND parameters used to generate the importance map.

<figure><img src="image_26.png" /><figcaption>**Figure A.4.3** : INIPOND parameters for the importance map generation for AMS simulation using an INIPOND importance map</figcaption></figure>

Figure A.4.4 presents the AMS parameters used for the AMS simulation using an INIPOND map

<figure><img src="image_27.png" /><figcaption>**Figure A.4.4** : AMS parameters used for AMS simulation using an INIPOND importance map</figcaption></figure>

In addition, Figure A.4.5 presents the simulation parameters that were used:

<figure><img src="image_28.png" /><figcaption>**Figure A.4.5** : Simulation parameters used for AMS simulation using an INIPOND importance map</figcaption></figure>
### C. AMS using an ADVANTG Importance Map

In this section, the parameters used for the final AMS simulation using an ADVANTG importance map is presented. Figure A.4.6 presents the AMS parameters used for the AMS simulation using an ADVANTG map.

<figure><img src="image_29.png" /><figcaption>**Figure A.4.6** : AMS parameters used for AMS simulation using an ADVANTG importance map</figcaption></figure>

In addition, Figure A.4.7 presents the simulation parameters that were used:

<figure><img src="image_30.png" /><figcaption>**Figure A.4.7** : Simulation parameters used for AMS simulation using an ADVANTG importance map</figcaption></figure>