Enhancing fast neutron irradiation in thermal neutron spectrum reactors through python-based multi-objective optimization

Highlights

• Investigate candidate materials for establishing a fast neutron experiment irradiation in thermal neutron spectrum test reactors.
• Develop a multi-objective optimization workflow applicable to the ATR to enhance the fast neutron spectrum.
• Utilize open-source tools for high-fidelity Monte Carlo modeling with OpenMC, advanced machine learning, and multi-objective optimization in Python.

1 Introduction

Design optimization is a very challenging problem, especially when multiple design objectives need to be optimized. Multi-objective optimization is a field of mathematical optimization that addresses problems involving more than one objective function being optimized simultaneously. Unlike single-objective optimization, where only one criterion is optimized, multi-objective optimization aims to find a set of solutions that provide a trade-off among the various objectives, making it more robust by considering multiple design objectives (Deb, 2011).

In common practice, a simpler approach is adopted where multiple design objectives are normalized, weighted, and combined into a single objective to be optimized. In contrast to multi-objective optimization, single-objective optimization will always result in a single optimum. Thus, the result will change based on the weighting assigned to the different objectives, and no information will be gained concerning the trade-offs between them. For these reasons, multi-objective optimization is a more attractive choice for nuclear design optimization. Multi-objective optimization can find a wide range of globally optimal solutions while simultaneously considering all design objectives (Engineering Optimization, 2009).

In the context of nuclear reactor and shielding designs, multi-objective optimization plays a critical role in balancing conflicting objectives such as safety, cost, efficiency, and environmental impact. For example, in nuclear reactor design, objectives might include maximizing thermal efficiency, maximizing fuel burnup, and ensuring safety standards. In shielding design, objectives could include minimizing shield weight and volume while maximizing radiation protection and structural integrity. By employing advanced algorithms, one can explore a wide solution space and identify optimal designs that meet the multidimensional requirements of modern nuclear reactors and shielding systems (Wilding et al., 2020).

Reducing a multi-objective problem into a single-objective problem is used in the nuclear design field for various applications, such as fuel management (Khoshahval et al., 2014), fuel loading patterns (Zameer et al., 2014; Al-Dawood and Palmtag, 2025), and nuclear reactor design parameters (Kumar and Tsvetkov, 2015). Additionally, the Advanced Reactor Modeling Interface (ARMI) code has an integrated optimization module for advanced reactor optimization (Touran et al., 2017). More recently, multi-objective optimization was utilized for the nuclear power plant design process optimization (Wilding et al., 2020) and for in-core fuel management and cycle optimization of the SAFARI-1 research reactor (Schlünz et al., 2018). Moreover, multi-objective optimization has been used in shielding design to optimize shield weight and the radiation dose of microreactors, as provided in References (Liu et al., 2022; Wu et al., 2024; Stewart et al., 2024).

In this work, we investigate a simplified experiment design optimization to boost fast neutrons, filter out non-fast or low-energy neutrons, and allow for fast neutron irradiation in a thermal spectrum reactor such as the Advanced Test Reactor (ATR) by combining core and shielding design optimizations. The ATR, located at Idaho National Laboratory (INL), is one of the key nuclear engineering research and testing facilities of the US Department of Energy (DOE). The ATR is one of the few high-power research reactors in the world with diverse applications, including accelerated testing of nuclear fuel, materials irradiation in a very high neutron flux environment, and medical radioisotope production (Nigg and Steuhm, 2014). Additionally, the ATR offers opportunities for testing fast spectrum fission and fusion reactor materials (Parry, 2005; Medvedev et al., 2018). The key challenges in this area lie in further detailing and optimizing a fast spectrum environment within a thermal test reactor (Worrall et al., 2024). This challenge involves researching, developing, and testing novel concepts for multiplying neutron populations into ever higher energy spectra in high flux test reactors like the ATR (Jaradat and Brookman, 2024).

The main objective of this work is to investigate candidate materials for establishing a fast neutron experiment irradiation in thermal neutron spectrum test reactors. This can be accomplished by filtering thermal and epithermal neutrons and boosting fast neutrons at designated irradiation positions. However, adding these filters will alter the neutron spectrum and the criticality of the system. The selection of the thickness and material layers should be accomplished by developing an optimization design workflow that is applicable to the ATR to enhance fast neutron spectrum irradiation, utilizing high-fidelity Monte Carlo methods along with advanced machine learning capabilities. Performing high-fidelity calculations using Monte Carlo or neutron transport codes for multi-objective optimization requires running many simulations to achieve the desired solution, which is computationally expensive. To overcome this, we developed surrogate models of each of the objective functions based on high-fidelity Monte Carlo calculations to perform predictive analysis. Then, these surrogate models are utilized to perform multi-objective optimization in an easy and efficient way, identifying the optimum irradiation position, filter material combination, and thickness of each material.

This paper is structured as follows: Section 2 discusses the developed multi-objective optimization workflow and the tools utilized in the analysis. Section 3 provides a brief description of the ATR and its high-fidelity model, characteristics of the available irradiation holes in the ATR, the simplified experiment model, and candidate materials for neutron filtering. Section 4 presents the optimization results at various irradiation positions and considers different material combinations. Finally, Section 5 summarizes the findings of this work and proposes improvements to be considered in future work.

2 Analysis methodology

In a multi-objective optimization problem, the aim is to optimize multiple objectives simultaneously. These objectives often conflict with each other, meaning that improving one will result in the other being less optimal. To identify a state where all objectives are reasonably satisfied, a collection of optimal solutions, called the Pareto front (PF), is analyzed, where no single solution can improve one objective without worsening another. This analysis provides a range of equally optimal solutions to choose from. To explore the objective space, a set of design parameters is defined. The design space is the set of independent parameters used for the multi-objective optimization problem. For the experiment design of neutron filtering and boosting, irradiation hole, material type, and material thickness would be the main design parameters. In general, a multi-objective optimization problem can be defined by:where f m represents the m t h objective function, g j is the j t h inequality constraint, h k is the k t h equality constraint, x i is the i t h variable to be optimized, and x i L and x i U are the lower and upper bounds of x i. The objective functions f ( x ⃗ ) are supposed to be minimized by satisfying all equality and inequality constraints. If a specific objective function is maximized, it needs to be redefined to minimize its negative value instead.

m i n f x ⃗ = f 1 x ⃗ , f 2 x ⃗ , … , f m x ⃗ T g j x ⃗ , f x ⃗ ≤ 0 , j = 1,2 , … , J h k x ⃗ , f x ⃗ = 0 , k = 1,2 , … , K x i L ≤ x i ≤ x i U , i = 1,2 , … , I

Finally, to select the best solution option among the Pareto front solutions, we assign a weight (0.0–1.0) to each objective function and normalize each objective value to obtain an effective weight, as given in Equation 1:where w f is the total weight of the solution objectives, w k is the assigned arbitrary weight for objective k, and f k, f k min, and f k max are the k t h objective function with its own minimum and maximum values, respectively.

w f = ∑ k = 1 K w k × f k max − f k x ⃗ f k max − f k min (1)

The selection of the weights of the objective functions is arbitrary and up to the user to assign suitable values that reflect their optimization priorities. This approach serves as a decision-making aid, allowing the user to select a preferred candidate solution from the Pareto front. While different weight choices may lead to different preferred solutions, they do not alter the Pareto front itself.

To fulfill the main objective of this work, a Python-based calculation workflow was developed to perform multi-objective optimization analysis leveraging open-source tools, specifically the Monte Carlo code OpenMC (Romano et al., 2015), machine learning libraries in Python scikit-learn (Pedregosa et al., 2011), and the Python multi-objective optimization module Pymoo (Blank and Deb, 2020). In the following subsections, we will briefly introduce each component of the developed workflow followed by a detailed description of the solution algorithm.

2.1 OpenMC monte carlo code

OpenMC is an open-source Monte Carlo code designed for simulating neutron and photon transport. It is primarily used in the field of nuclear engineering for reactor physics calculations, including reactor design, safety analysis, and fuel cycle studies. OpenMC supports continuous-energy and multi-group transport calculations, leveraging high-fidelity nuclear data to provide accurate results. OpenMC’s user-friendly Python API allows for the streamlined setup, execution, and analysis of simulations, facilitating advanced research and development in nuclear technology (Romano et al., 2015).

In this work, we modified an ATR OpenMC model discussed in the following section to accommodate the proposed experiment design for various material combinations and thicknesses. Steady-state calculations were performed for several cases that cover the whole design space of the problem to generate training data sets for building surrogate models of each optimization objective function.

2.2 Python machine learning capabilities

Python is a powerful programming language widely used in machine learning due to its simplicity and rich ecosystem of libraries. Among these, CatBoost and scikit-learn stand out for their unique capabilities. CatBoost is a gradient-boosting library that excels in handling categorical features without extensive preprocessing, offering high performance, robustness against overfitting, and GPU support for faster training (Dorogush et al., 2018). On the other hand, scikit-learn provides a comprehensive suite of tools for various machine learning tasks, including regression, classification, clustering, and dimensionality reduction, enabling users to tackle various types of predictive tasks. It stands out as an essential tool for efficient and effective data analysis and machine learning (Pedregosa et al., 2011). Additionally, both libraries’ ensemble methods and pipeline utilities enhance the accuracy of machine learning workflows. In the current work, we utilized several regression models to build surrogate or reduced-order models of each objective function considered in the optimization workflow, utilizing training data sets obtained from OpenMC high-fidelity calculations. The main purpose of these reduced-order models is to reduce the computational cost of high-fidelity calculations while searching for the optimal experiment design.

2.3 Pymoo: multi-objective optimization module

Pymoo is a comprehensive open-source optimization module in Python designed specifically for multi-objective optimization problems. It provides a versatile framework that supports a variety of optimization algorithms, both classical and evolutionary. Pymoo is particularly useful for solving complex problems where multiple conflicting objectives need to be optimized simultaneously, as in the current design problem. The module offers tools for defining custom problems, handling constraints, and visualizing the Pareto front. It supports both single-objective and multi-objective optimization with several methodologies and features that focus on evolutionary algorithms. Pymoo provides implementations of popular evolutionary algorithms such as the Non-dominated Sorting Genetic Algorithm (NSGA-II, NSGA-III), Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), and Strength Pareto Evolutionary Algorithm (SPEA2). These algorithms are integral to Pymoo’s ability to handle complex multi-objective optimization problems, providing robust and efficient methods for finding optimal trade-offs between conflicting objectives (Blank and Deb, 2020).

In this work, we utilized the NSGA-II and SPEA2 algorithms. The NSGA-II algorithm provides a fast non-dominated sorting approach and maintains a diverse set of solutions with relatively low computational complexity. The SPEA2 algorithm offers a good balance between convergence and diversity and maintains a high-quality external archive of solutions. Both algorithms are widely used in various fields due to their robustness and efficiency in handling a wide range of problems.

2.4 Multi-objective optimization workflow

The developed workflow leverages Python API capabilities to establish a comprehensive workflow applicable to ATR experiment design optimization by connecting necessary components to perform an analysis in an efficient way. The Python-based multi-objective optimization workflow developed for ATR experiment design is shown in Figure 1. The workflow has four main components that can be summarized as follows:

• Design variable selection: generates a set of experiment design variables using either the stochastic method or the Latin hypercube method.
• Data generation model: performs OpenMC high-fidelity modeling of the ATR to generate training data sets for building surrogate models with different material thicknesses at a desired irradiation position.
• Reduced order model: builds surrogate models of each objective function using Python machine learning capabilities.
• Multi-objective optimization: performs optimization analysis to find a set of solutions that are non-dominated to each other but are superior to the rest of the solutions in the search space, or the Pareto front.

In the current work, we used Latin hypercube sampling to generate 130 combinations for the thicknesses of the thermal filter, epithermal filter, and/or fast booster for each material combination in each irradiation position. Then we performed OpenMC steady-state eigenvalue calculations of each case with coupled neutron/gamma transport using the ATR OpenMC model. In each calculation, we used 500 batches, including 50 inactive batches, and 1 0 5 particles were simulated per batch. Once the training data sets were generated, a surrogate model was built using different regression models for each of the objective functions, namely, fast neutron flux and fraction, thermal and epithermal neutron fluxes, total gamma flux, and heat deposition rate, which were then evaluated to select the best regression model that represented a certain objective. The tally-based data generated to train surrogate models from steady-state eigenvalue calculations, with a 10 pcm standard deviation, are within 1% statistical uncertainty. In our previous work (Jaradat et al., 2024; Jaradat et al., 2025), a similar approach was followed by utilizing surrogate models to perform a predictive transient analysis of the Transient Reactor Test Facility (TREAT) at INL. Finally, the optimized solution search was performed using the NSGA-II and SPEA2 algorithms from Pymoo with a fixed population size of 150 and 100 offspring, and the algorithm was terminated after 250 function evaluations. After that, a multicriteria decision-making method was used to select the optimized preferred solution from the Pareto front based on the assigned weights of each objective function.

3 Modeling approach

INL’s ATR is the world’s premier nuclear test reactor. It provides unmatched, national priority nuclear fuel and materials testing capabilities for military, federal, university, and industry partners and customers (Campbell, 2021). The ATR is the only U.S. research reactor capable of providing large-volume, high-flux thermal neutron irradiation in a prototype environment, allowing for the study of intense neutron and gamma radiation on reactor materials and fuels.

The ATR is a 250 MWth high-flux material testing reactor that operates at very low pressures and temperatures compared to commercial pressurized-water reactors. Instead of heat, the main product of a material test reactor is the neutrons it produces. The ATR is a highly heterogeneous plate-type reactor with fuel elements made of highly enriched uranium ( 93 % U 235 ) arranged in a serpentine pattern. The ATR is light-water cooled and moderated and uses a beryllium reflector to help concentrate neutrons in the core, where they are needed for fuels and materials testing (Advanced test reactor, 2025). The ATR is controlled by a combination of 16 rotating outer shim control cylinders (OSCCs) and 24 vertically moving neck shim rods. The beryllium OSCCs contain hafnium plates that can be rotated toward and away from the core and provide gross reactivity control. The hafnium neck shim rods can be individually inserted or withdrawn for minor power adjustments. The ATR’s unique design allows the reactor’s corner lobes to be operated at different power levels, which makes it possible to conduct multiple simultaneous experiments under different testing conditions.

The ATR offers 77 different experiment locations inside the core barrel. These positions consist of nine high-intensity flux traps, 32 drop-in positions within the area bounded by the serpentine fuel, and 36 positions in the outer reflector region. The available test positions in the ATR range in size from a 0.5 inch in diameter to 5.0 inches in diameter (Campbell, 2021).

3.1 ATR high-fidelity OpenMC model

A full core model of the ATR was developed with the OpenMC (Romano et al., 2015) Monte Carlo code and utilized in this work. The model details and description are discussed in Reference (Nguyen et al., 2025). The model was developed based on the detailed core description, geometry, and material benchmark specifications for the ATR outlined in the International Handbook of Evaluated Reactor Physics Benchmark Experiments (IRPhE) (Kim and Schnitzler, 2005), and it was verified against the benchmark reference solution. The developed benchmark adopts simplifications for the ATR, including homogenizing cooling channels (in the control drums and reflector regions) with the beryllium reflector, and homogenizing regions above and below the active fuel region with other materials as provided in the benchmark specifications. For this work, the OpenMC model of this simplified benchmark was utilized to generate data sets for irradiation hole parameters for the experiment design optimization calculations including absolute thermal, epithermal and fast neutron fluxes, fast neutron fraction, gamma flux, and total heat deposition.

The ATR offers several irradiation holes with different sizes and different neutron spectra. A cross-sectional view of the ATR OpenMC model core is shown in Figure 2, highlighting the six available irradiation holes that were investigated in this work. The irradiation holes are classified based on their location relative to the active core region and the available space for irradiation, as listed in Table 1. Three small irradiation positions, one in the active core region (A-11) and two outside the active core region (B-5 and B-11). The other three has are larger irradiation positions, the North-East (NE) flux trap in the active core region and the other two are in the reflector region (I-11 and I-14). The radius of each hole is provided in Table 1 with an assumed fixed thickness of the neutron filters. The thicknesses of the neutron filters and/or fast boosters were selected so a consistent volume ratio relative to the total volume of the irradiation hole would be maintained, thus securing about 47% of the irradiation hole volume for the experiment’s irradiation. These irradiation holes will be investigated for possible spectral modifications in the following sections.

TABLE 1. Characteristics of the available irradiation holes in the ATR.

The unmodified neutron flux spectrum at various irradiation holes in the ATR core is shown in Figure 3. Additionally, the absolute thermal, epithermal, and fast neutron fluxes in each hole are provided in Table 2. Thermal neutrons are defined as having an energy less than 0.5 eV, fast neutrons have an energy greater than 0.5 MeV, and neutrons in between 0.5 eV and 0.5 MeV are considered epithermal. The spectrum is very soft in the reflector region of large hole sizes (I-11 and I-14), with the fast neutron fraction being less than 6%. The spectrum shifts toward the epithermal range in out-of-core irradiation holes with small sizes (B-5 and B-11) and a slightly higher fast neutron fraction (20%) than in large irradiation holes in the reflector region. For in-core irradiation holes (A-11 and NE), the spectrum is much harder in positions closer to the core center; the fast neutron fraction is about 30% for both the small-sized A-11 irradiation hole and the large-sized NE flux trap. In this work, we investigated the possibility of hardening the spectrum and increasing the fast neutrons in these six irradiation holes, considering various combinations of neutron filtering materials and different material thicknesses. Figure 4 displays the flux maps integrated axially of the thermal and fast neutrons, along with gamma flux and deposited heat rate maps considering empty holes. Among the six irradiation positions, the absolute fast neutron flux in the NE flux trap is the highest at 1.87 × 1 0 14 n / c m 2 . s.

TABLE 2. Calculated neutron flux in the available irradiation holes of the ATR.

3.2 Fast neutron experiment modeling

The basic concept of designing a fast spectrum experimental location in a thermal spectrum reactor relies on the efficiency of rejecting thermal neutrons with a thermal absorber material, filtering epithermal neutrons with an epithermal absorber material, and converting thermal neutrons to fast neutrons via fission reactions in fissionable material.

In this work, we considered simplified models of the experiment for the purpose of testing and performing multi-objective optimization analysis to satisfy our main objectives. These simplified models have a cylindrical geometry with different layers for filtering or removing thermal and epithermal neutrons, or a layer to boost fast neutrons, as shown in Figure 5. Three experiment design cases were considered for this work.

For Case 1, which used the small irradiation holes A-11, B-5, and B-11, two layers for filtering thermal and epithermal neutrons were applied. Case 2 is similar to Case 1 but an additional layer of fuel was included to boost fast neutrons. The large irradiation holes NE, I-11, and I-14 were used for this case. Finally, for Case 3 the thermal and epithermal neutron filters were combined into a single layer made of material that can serve both purposes, along with an additional layer of fuel, as in Case 2. Various combinations of material filters and thicknesses were considered for each case in order to select the best irradiation position with the best combination of materials and thickness. For the small irradiation holes, two layers of filter/booster material were considered to maximize the available space for experiment irradiation, as in Cases 1 and 3. In contrast, the large irradiation holes allow for adding more than two layers without compromising the irradiation space. Also, the booster fuel layer was added after the filter layers in the outer regions to enhance the fission reaction rate and increase the fast neutron yield, while the innermost region experiences a much lower thermal neutron flux due to neutron filtering

3.3 Candidate neutron filter material

To select the candidate materials for filtering thermal and epithermal neutrons, we examined the total cross section of ten common elements across all energy ranges, as shown in Figure 6. These elements are boron (B), cadmium (Cd), indium (In), gadolinium (Gd), europium (Eu), iridium (Ir), hafnium (Hf), yttrium (Y), zirconium (Zr), and tungsten (W). Then we compared each element’s total cross section to that of boron. Table 3 lists the selected elements with their densities and applicability for thermal and epithermal neutron filtering and rejection. Boron, indium, and cadmium were selected for thermal neutron rejection, while europium, hafnium, and tungsten were selected for epithermal neutron filtering. Hafnium was eliminated from the list due to its higher cost. The estimated cost of each element ($/kg) was determined through a comprehensive web-based search. The selected elements were then used to test various combinations for neutron filtering in Cases 1 and 2. It should be noted that the estimated cost is based on a general search and is not meant to be very accurate, but rather to provide a value comparable to market prices. In this study’s analysis, the upper limit of each element’s cost range was used.

TABLE 3. Candidate elements for thermal and epithermal neutron filtering and rejection.

The best combination of elements for non-fast neutron filtering will be based on the results of Cases 1 and 2, and realistic materials from those combinations will be used for Case 3 to combine thermal and epithermal neutron rejection. The results of Case 3 will determine the best material choice for boosting the fast neutron spectrum in the desired irradiation location. In Cases 2 and 3, uranium nitride (UN) is considered for fast neutron boosting with a uranium enrichment of 19.995% and a mass density of 14.32 g / c m 3.

3.4 Reduced order modeling

In this study, several machine learning models are evaluated to predict various responses of interest, including neutron fluxes, gamma flux, heat deposition, and the effective multiplication factor ( k eff ). The models used include polynomial regression, Random Forest, MLPRegressor (modern feedforward neural network consisting of fully connected neurons with nonlinear activation functions), Gaussian Process Regressor (GPR) (Pedregosa et al., 2011), and the gradient-boosting algorithm optimized for categorical data (CatBoost) (Dorogush et al., 2018). Each model was trained using a pipeline that included preprocessing steps such as scaling and encoding of categorical and numerical features. In this analysis, no hyperparameter tuning was performed. All models were evaluated using default or manually specified hyperparameters, without the use of automated tuning frameworks. As a result, the performance metrics reported reflect the models’ behavior without optimization, and the relative rankings of model performance may change if hyperparameter tuning were applied. Future work could include systematic tuning to better understand each model’s potential. To assess surrogate model performance, the dataset was randomly split into 80% for training and 20% for validation, ensuring that reported metrics reflect predictions on validation data. While k-fold cross-validation model was not employed, the independent validation set provides a robust estimate of generalization error.

Among these, CatBoost and GPR emerged as the top performers, as shown in Figure 7 that compares the coefficient of determination ( R 2 ) and the normalized root mean square error ( R M S E ). CatBoost consistently delivered high accuracy with a low normalized R M S E, particularly for targets such as heat deposition and fast flux fraction. On the other hand, GPR consistently achieved near-perfect R 2 scores across all target variables. A closer look at the normalized R M S E reveals significant limitations in its practical predictive accuracy particularly for high-magnitude targets such as neutron flux. These large errors suggest that GPR may be overfitting or suffering from numerical instability despite its statistical fit. In contrast, CatBoost demonstrated a more balanced performance, delivering both a high R 2 and low normalized R M S E across several key parameters. Its robustness to feature types, native handling of categorical variables, and consistent generalization make it a more reliable choice. Figure 8 compares the reference calculated values with predicted values for absolute fast neutron flux with various models; it shows that the CatBoost and GPR models perform well with training data.

The observed near-perfect R 2 values for GPR alongside relatively large normalized R M S E for certain objectives can be attributed to differences in metric sensitivity: R 2 captures variance explained, whereas normalized R M S E is influenced by absolute prediction errors, which become more pronounced for objectives with narrow ranges near the Pareto front. GPR inherently provides predictive uncertainty estimates, which were leveraged during optimization to mitigate overconfidence in sparsely sampled regions. Nonetheless, surrogate accuracy and uncertainty near Pareto-front boundaries remain critical considerations, as these regions often exhibit extrapolation behavior. We acknowledge this limitation and recommend future work to incorporate adaptive sampling or uncertainty-aware acquisition strategies to further improve generalization in these regions.

4 Optimization results and discussion

In this section, we discuss the multi-objective optimization results of each case presented earlier. Calculations were performed for each case to find the best irradiation position with the optimum material combination and thicknesses, and a surrogate model of each objective function was generated considering the discrete (irradiation position and material combinations) and continuous (filter thicknesses) variables of each case. We will discuss the selected objective functions and constraints and explore the search domain and the Pareto front along with selected points that satisfied a certain outcome.

In this pilot study, several physical and engineering design constraints were intentionally omitted to focus on demonstrating the optimization framework. These include material irradiation effects on thermophysical properties, fuel burnup in UN booster layers, temperature rise due to neutron and gamma heating in filter materials, fabrication feasibility, and cooling requirements. Furthermore, the design must comply with ATR operational constraints, particularly regarding power peaking, neutron flux redistribution, and reactivity swings associated with introducing strong absorbers and/or booster materials. These considerations will be incorporated in future deployment-oriented design analyses.

Also, the estimated cost presented in this study is intended solely as a relative comparison metric to support design trade-offs and does not represent an accurate procurement estimate. Future work will incorporate more detailed and well-referenced cost data for deployment oriented analyses. For the small irradiation holes, two layers of filter booster material were considered to provide more space for experiment irradiation as in Case-1 and 3. While the large irradiation holes allows for adding more than two layers without compromising the available space for irradiation.

4.1 Case 1: two filter layers

Using this case we studied the possibility of increasing the fast neutron fraction in small irradiation positions, namely, A-11, B-5, and B-11, by filtering out thermal and epithermal neutrons with two layers of filter materials. We selected samples based on five variables: three irradiation positions, three thermal neutron filter materials (B, Cd, and Gd), two epithermal neutron filters (W and Eu), and the thickness of the thermal and epithermal filters. The main objective functions we were trying to optimize, along with the constraints, are listed in Table 4. We aimed to maximize the absolute fast neutron flux while minimizing the epithermal neutron flux, total cost, and total deposited heat. From the suggested Pareto front, we presented selected points that satisfy certain objective functions, such as the highest fast flux, highest fast fraction, lowest cost, and lowest heat generated. We also assigned arbitrary weights to each objective function, focusing more on the fast neutron flux to evaluate the suggested Pareto front.

TABLE 4. Objectives and constraints for Case 1 experiment optimization at the ATR.

Figure 9 shows the explored solution domain sampled in the small irradiation positions and the material combinations, with predicted values of the fast and epithermal neutron flux and the estimated cost. As can be seen the highest absolute fast neutron flux is achieved in irradiation position A-11. Figure 10 shows the search progress for the four objective functions at each generation. It indicates that the average value saturates at the optimum predicted level as the solution converges, and the absolute fast neutron flux saturates at around 1.6 × 1 0 14 [ n / c m 2 . s ].

To provide valuable insight into trade-offs and optimization priorities in experiment design, the correlation matrix of the Pareto front objective functions is depicted in Figure 11. It highlights several important relationships among design parameters and reveals a strong inverse relationship between the thermal and epithermal filter thicknesses when a fixed total thickness is maintained. Both epithermal and fast neutron fluxes are highly correlated, suggesting they respond similarly to changes in the system, mostly due to thermal neutrons being filtered out. These fluxes also show moderate to strong positive correlations with the estimated cost, implying that increasing the fast neutron population raises the estimated cost. Interestingly, thermal filter thickness is positively correlated with the estimated cost, while epithermal filter thickness is negatively correlated, reinforcing the trade-off between design parameters and cost impact. Heat generation correlates moderately with fast and epithermal neutron fluxes but shows a weak negative correlation with the estimated cost. Figure 12 shows the Pareto front solution for the first three objective functions—epithermal neutron flux, absolute fast neutron flux, and estimated cost—for irradiation positions and filter material combinations. The results suggest that irradiation position A-11 is the best candidate position.

For selecting the optimal filter material and thickness combinations, we present the best points based on the highest score, absolute fast neutron flux, fast neutron fraction, lowest estimated cost, and deposited heat in Tables 5, 6 using NSGA-II and SPEA2 optimization algorithms, which show consistent results. The highest score is achieved with the B / W filter combination with 0.205/0.215 cm thicknesses, a fast flux of 1.61 × 1 0 14 [ n / c m 2 . s ], and a fast fraction of 44.2 %. The C d / W filter combination shows the highest possible neutron flux with 1.66 × 1 0 14 [ n / c m 2 . s ] and a fast fraction of 36.6 %. The B / E u combination shows the highest fast fraction of 46.4 % with a very small layer of E u, the use of which also significantly reduces the estimated cost and heat deposition. Compared to the reference value of the absolute fast neutron flux at the A-11 irradiation position, the fast flux did not change significantly (0.5%–6.5%), while a large increase in the fast neutron fraction is observed (25%–60%) due to the filtering out of a large portion of the epithermal neutrons and nearly all the thermal neutrons. However, the suggested combinations show a small improvement in the magnitude of the absolute fast neutron flux but a greater improvement in the fast neutron fractions. This means that a booster layer would need to be added to enhance the fast neutron population.

TABLE 5. Predicted optimum filter thickness for Case 1 using NSGA-II optimization algorithm.

TABLE 6. Predicted optimum filter thickness for Case 1 using SPEA2 optimization algorithm.

4.2 Case 2: two filter layers and booster

As was shown with the previous case, it is very difficult to improve the magnitude of the absolute fast neutron flux without increasing neutron production to boost the number of fast neutrons. This can be achieved by adding a booster fuel material that contains a significant portion of a fissile isotope, such as U-235 in UN. For Case 2, we studied the possibility of increasing the absolute fast neutron flux and its fraction in large irradiation positions, namely, I-11, I-14, and NE, using two layers of filter materials with an additional layer of fast neutron boosting material. We selected samples based on six variables: three irradiation positions, three thermal neutron filter materials (B, Cd, and Gd), two epithermal neutron filters (W and Eu), the thickness of the thermal and epithermal filters, and the thickness od the fast neutron booster. The main objective functions we were trying to optimize, along with the constraints, are listed in Table 7. They are similar to those of Case 1. We aimed to maximize the fast neutron flux while minimizing the epithermal neutron flux, total cost, and total deposited heat, with additional constraints on the U-235 mass and booster material volume.

TABLE 7. Objectives and constraints for the Case 2 and 3 experiment optimization at the ATR.

Figure 13 shows the explored solution domain that was sampled in the large irradiation positions and the filter material combinations, as well as the predicted values of the fast and epithermal neutron flux and the estimated cost. The highest fast neutron flux was achieved in the NE flux trap. Figure 14 shows the search progress for all the objective functions at each generation. The average value saturates at the optimum predicted level as the solution converges, and the absolute fast neutron flux saturates at around 2.1 × 1 0 14 [ n / c m 2 . s ].

The correlation matrix of the Pareto front objective functions is shown in Figure 15. In this case, fast booster thickness plays a central role, showing strong positive correlations with fast neutron flux (0.71), estimated cost, and heat generation, indicating that increasing booster material significantly enhances neutron population at a high design cost, as expected. The fast neutron flux is also highly correlated with deposited heat, mainly due to fission in the UN booster layer. In contrast, thermal and epithermal filter thicknesses generally show negative correlations with most outputs, particularly with estimated cost and deposited heat, implying that increasing the neutron filter thicknesses may shift the neutron spectrum away from optimal conditions. The epithermal neutron flux maintains moderate correlations with heat generation and fast neutron flux but has little impact on estimated cost. Overall, this emphasizes the trade-off between boosting fast neutron flux and managing cost and heat, with booster design emerging as a key lever in system optimization. Figure 16 shows the Pareto front solution for the first three objective functions—epithermal neutron flux, fast neutron flux, and estimated cost—for irradiation positions and filter material combinations, suggesting that the NE flux trap is the best candidate position.

Tables 8, 9 show selected optimal filter material combinations using the NSGA-II and SPEA2 optimization algorithms based on a specified metric. The highest score is achieved with the B / E u / U N material combination with thicknesses of 0.438/1.523/1.639 cm, a fast flux of 2.02 × 1 0 14 [ n / c m 2 . s ], and a fast fraction of 52.6 %. The G d / W / U N material combination shows the highest possible neutron flux with 2.46 × 1 0 14 [ n / c m 2 . s ] and a fast fraction of 46.4 %. Additionally, the B / E u / U N combination shows the highest fast fraction of 53.4 % with thicknesses of 1.424/0.495/1.681 cm. Compared to the reference value of the absolute fast neutron flux at the NE flux trap, the fast flux increased by about 32 % for the G d / W / U N material combination and about 8 % for the B / E u / U N material combination, while the fast neutron fraction increased significantly (30 % to 68 %) due to the filtering out of a large portion of the epithermal neutrons and nearly all the thermal neutrons and the boosting of the fast neutrons. Both solution algorithms suggest the same material combinations with small variations in the thicknesses without significant bias in the predicted values.

TABLE 8. Predicted optimum filter thickness for Case 2 using NSGA-II optimization algorithm.

TABLE 9. Predicted optimum filter thickness for Case 2 using SPEA2 optimization algorithm.

4.3 Case 3: single filter layer with booster

The analysis performed in this case considered using filter materials that combine both thermal and epithermal neutron rejection. The optimization analyses conducted for Case 1 and Case 2 suggest that:

• It is necessary to use a booster material to increase the absolute fast neutron flux.
• The best irradiation locations with the highest absolute fast neutron flux are A-11 and the NE flux trap.
• The best filter materials that can be investigated further are B / W, G d / W, C d / W, and B / E u.

We investigated using a booster layer of UN fuel with a layer of thermal and epithermal filter material composed of B 4 C, W B 2, or W G d 2 O 3, as these filter materials combine most of the best elements identified in the previous two cases. We examined each of the six irradiation positions considering these three materials. The main objective functions we were trying to optimize, along with the constraints, are listed in Table 7 for Case 2 and Case 3.

Figure 17 shows the explored solution domain that was sampled in all available irradiation positions and for the filter/booster material combinations, as well as the predicted values of the fast and epithermal neutron flux and the estimated cost. The highest absolute fast neutron flux was achieved in irradiation positions A-11, B-11, and the NE flux trap. Figure 18 shows the Pareto front solution for the first three objective functions with irradiation positions and filter/booster material combinations, suggesting that irradiation position A-11 and the NE flux trap are the best candidates for this case.

Tables 10, 11 show selected optimal filter material combinations using the NSGA-II and SPEA2 optimization algorithms similar to Case 1 and Case 2. The highest score is achieved in the A-11 irradiation position with the B 4 C / U N filter/booster combination, with a thicknesses of 0.173/0.247 cm, a fast flux of 2.40 × 1 0 14 [ n / c m 2 . s ], and a fast fraction of 49.2 %. The highest neutron flux is predicted at A-11 with W B 2 / U N, with a fast flux of 2.43 × 1 0 14 [ n / c m 2 . s ] and a fast fraction of 48.3 %. The NE flux trap shows the highest fast fraction of 52.0 % with thicknesses of 1.781/1.819 cm for the B 4 C / U N filter/booster combination. Compared to the reference value of the absolute fast neutron flux at A-11, the fast flux increased by about 56 % for the W B 2 / U N material combination, while the fast neutron fraction increased significantly (57 % to 68 %).

TABLE 10. Predicted optimum filter thickness for Case 3 using NSGA-II optimization algorithm.

TABLE 11. Predicted optimum filter thickness for Case 3 using SPEA2 optimization algorithm.

4.4 Optimization results verification

We verified the predicted optimized experiment design of each case against the OpenMC results by comparing the value of the absolute fast neutron flux and fast neutron fraction. Table 12 presents the two best options of each case and shows a very good match with the OpenMC reference solution for the selected cases, with the absolute fast neutron flux maximum relative error within ± 2.8 % and the fast neutron fraction maximum relative error within ± 0.8 %.

TABLE 12. Predicted optimum absolute fast neutron flux for Case 2 in each irradiation position.

Figure 19 shows the neutron spectrum normalized per unit lethargy at position A-11 for Case 1 and Case 3 compared to the neutron spectrum for an empty hole. For Case 1, the added neutron filters resulted in a hardening of the neutron spectrum; a softer spectrum was obtained from the C d / W filter material combination compared to the B / W combination, with no improvement in the absolute fast neutron flux maximum value for both cases. In contrast, for Case 3, adding booster fuel material shows a harder spectrum and an increased absolute fast neutron flux peak, with comparable solutions obtained from both the B 4 C / U N and W B 2 / U N filter/booster combinations. Figure 20 shows the neutron spectrum and fast flux map for Case 2 in the NE flux trap for the B / E u / U N and G d / W / U N filter/booster combinations. Both options show a much harder spectrum with a significant increase in the absolute fast neutron flux magnitude with the G d / W / U N filter/booster combination than with the B / E u / U N, and both show an increased fast neutron peak compared to the reference case.

5 Summary

In this work, we presented an experiment design optimization workflow to boost fast neutron irradiation at different ATR irradiation positions. The developed workflow utilizes Python-based open-source tools to perform high-fidelity calculations to train surrogate models on the objective functions that will be optimized. We utilized the multi-objective optimization Python module Pymoo, along with the neutron transport Monte Carlo code OpenMC, to explore design options for making spectral changes in the ATR that maximize the absolute fast neutron flux and minimize the non-fast neutron flux, deposited heat, and estimated design cost.

First, we selected candidate materials with high absorption cross sections for thermal and epithermal neutrons to study three experiment design cases. In Case 1, we investigated filtering thermal and epithermal neutrons for irradiation positions with a harder spectrum (in-core)—that is, A-11, B-5, and B-11. The optimization analysis results showed that B / W and C d / W at the A-11 irradiation position were the best combinations to increase the fast neutron fraction with a slight increase in the absolute fast neutron flux. Then, for Case 2, a layer of U N fuel was added as a booster before the neutron filter layers to increase the absolute fast neutron flux. For this case, we added the booster fuel to the large irradiation holes of the I-11, I-14, and NE flux trap positions. The combinations of G d / W / U N and B / E u / U N at the NE flux trap were the best candidates, increasing the absolute fast neutron flux by about 32% with a fast neutron fraction of around 68%. However, the absolute fast neutron flux increased significantly in positions I-11 and I-14 but was still much lower than it was at the NE flux trap. To reduce the number of layers in the experiment design, for Case 3 we considered materials that combine the candidate elements obtained from the optimization analyses in Case 1 and Case 2. B 4 C, W B 2, and W-G d 2 O 3 were selected along with a layer of booster fuel. The B 4 C / U N and W B 2 / U N combinations at the A-11 irradiation position led to the best results in terms of absolute fast neutron flux and fast neutron fraction.

The developed multi-objective optimization workflow represents an initial step toward more sophisticated experimental designs for fast neutron irradiation. Future work will incorporate more comprehensive objective functions and constraints, including fuel booster enrichment, isotopic concentrations (e.g., B-10 enrichment), reactivity changes, and layer-specific temperatures, with the ultimate goal of validating design values against experimental data once it becomes available. Furthermore, the developed workflow could be utilized for multi-objective optimization studies in other fields of reactor physics and nuclear reactor design, as it provides the flexibility to replace the high-fidelity models used for data generation with available design tools.

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Footnotes

Edited by: Gert Van den Eynde, Belgian Nuclear Research Centre (SCK CEN), Belgium

Reviewed by: Taylor Kimball, Government of the United States of America, United States

Bhoomi Gajjar, Institute for Plasma Research (IPR), India