A Fast Variance Reduction Technique for Efficient Radiation Shielding Calculations in Nuclear Reactors

Abstract

The increasing demand for cleaner and more sustainable energy sources has sparked significant interest in small modular reactors (SMRs). Due to their compact and modular design, SMRs pose unique challenges in radiation shielding, requiring a more refined approach. This study focuses on developing a new variance reduction technique (VRT) for radiation shielding analysis, specifically tailored for SMRs, to address the limitations of traditional methods such as surface source write/surface source read (SSW/SSR). The proposed VRT supports multi-threading and enhances computational efficiency by redefining source particles using a two-step method. The analysis is conducted using the Monte Carlo radiation transport code, MCNP6, and the effectiveness of the new VRT is evaluated through sensitivity analyses across various energy and directional divisions.

1. Introduction

The growing demand for cleaner and more sustainable energy sources has sparked significant interest in the development of small modular reactors (SMRs). Unlike traditional large-scale nuclear reactors, SMRs are designed to be compact, modular, and capable of being deployed in a wide range of environments, from remote locations to densely populated areas. These characteristics, while offering considerable advantages in terms of flexibility and scalability, also present unique challenges, particularly in the area of radiation shielding. SMRs, with their compact size and modular construction, necessitate a more refined and comprehensive approach to radiation shielding. An analysis of efficient, lightweight, and compact radiation shielding is needed [1].

Radiation emitted from a reactor includes alpha particles, beta rays, gamma rays, and neutrons. Since alpha particles and beta rays have limited distance traveled, they can be excluded from shielding calculations, leaving only neutrons and gamma rays, which have high penetration power, to be considered [2]. SMRs emit high levels of radiation and heat, making radiation shielding essential. Especially for compact applications such as marine or space reactors, radiation shielding plays a critical role in operational safety [3]. To determine the shield thickness, the radiation dose at full power should be set to 10 µSv/hr behind the radiation shield [1]. Accordingly, this study proposes a methodology to reduce the time required for shielding design.

The reference reactor used in this study is a lead–bismuth eutectic (LBE)-cooled fast-spectrum SMR. The rated power of the reference reactor is 40 Mw_th and it has a minimum operational lifespan of at least 15 years [4]. LBE is an advantageous coolant for SMRs due to its high boiling point, which lowers the risk of coolant boiling, thus contributing to reactor safety. Its high thermal conductivity supports efficient heat transfer, ensuring stable operation. Unlike sodium coolant, however, LBE is chemically inert when in contact with water and air, ensuring stable operation [5].

For the reactor radiation shielding analysis, we utilize the Monte Carlo radiation transport code, MCNP6, to perform k-code criticality calculations for the LBE-cooled reactor. The k-code calculation provides not only the effective multiplication factor but also data on neutron flux, reaction rates, particle energy spectra, and more. The focus of this research is on calculating neutron flux and dose rates by simulating individual particle histories and recording key aspects to describe the average behavior of the reactor. To achieve this, we use a three-dimensional heterogeneous reactor model to accurately analyze the reactor’s behavior under different conditions.

Using k-code criticality calculations for primary radiation shielding design can be time-consuming due to the need for very low relative errors, making it impractical for detailed radiation mapping. To efficiently generate extensive radiation maps, VRTs are essential. Traditional methods such as SSW/SSR (surface source write/surface source read) record sources on a designated surface and use these surface sources instead of the original ones [6]. However, these methods have limitations, such as the inability to utilize multi-threading during recording and the potential for insufficient particle capture. Similarly, the WWG (weight window generation) technique may fail to record weights accurately [6]. To overcome these challenges and improve calculation efficiency, the aim of this study was to develop a new VRT that does not rely on surface sources and supports multi-threading.

This paper consists of four main sections. Following the introduction, Section 2 explores the methodology, starting with a description of the reference reactor, providing a detailed overview of the lead–bismuth eutectic (LBE)-cooled fast-spectrum SMR used as the reference reactor. Section 2.4 introduces the newly developed VRT, emphasizing its multi-threading capability, how it improves the efficiency and accuracy of radiation shielding analysis, and how it overcomes the limitations of the traditional SSW/SSR method. Section 3 presents the results and sensitivity analysis, where the performance of the new VRT is evaluated across various energy and direction divisions. The effectiveness of the two-step source in replicating the direct source is assessed, with a focus on the radiation dose and the computational efficiency. Finally, Section 4 provides the conclusion.

2. Methodology and Reactor Overview

2.1. MCNP

The MCNP Code is a globally recognized tool designed for Monte Carlo simulations that track neutron and gamma ray transport. This code is capable of handling neutron and gamma interactions, including the transport of secondary gamma rays produced by neutron reactions [6]. The MCNP program generates results for particle flux, current, dose, effective multiplication factor and energy deposition. Every MCNP result is normalized per starting particle history and is displayed in the output alongside a second value, representing the estimated relative error. The relative error (R) is defined as the ratio of an estimated standard deviation of the mean $S_{\tilde{X}}$ to the estimated mean $\tilde{X}$. The relative error indicates the statistical precision of the estimated mean [7].

$$R\mathit{=}{\displaystyle \frac{S_{\tilde{X}}}{\tilde{X}}}$$

(1)

To define a criticality calculation in MCNP, the user must use the k-code option. In a criticality calculation, MCNP simulates N neutrons generated by fission. These neutrons disappear through absorption or escape, and $N_1$ neutrons are simulated in the next cycle. This process is repeated until the fission neutron distribution converges. For this iterative process to converge, the number of inactive cycles must be defined. The calculation then begins for the active cycles. In MCNP, to perform this calculation, the source history N per cycle, an initial guess for the multiplication factor, the number of inactive cycles ($I_c$), and the total number of cycles ($I_t$), which is the sum of the inactive and active cycle numbers, must be defined [8].

2.2. Reference Reactor Description

The reference reactor, named NCLFR-Oil, is a lead–bismuth eutectic (LBE)-cooled fast-spectrum SMR, designed with a thermal power output of 40 MW_th and a minimum operational lifespan of 15 years [4].

The reactor core of the LBE-cooled SMR in this study is designed with specific structural and operational parameters, which are detailed in

Table 1. Description of main design parameters.

Parameters	Specification
Thermal power	40 Mw_th
Fuel	UO2
Enrichment (Innermost/Middle/Outermost)	13.5 wt%/16.5 wt%/18.5 wt%
Cladding	T91
Absorber	B4C
Reflector	YSZ
Primary coolant	LBE
Gap	Helium
Core lifetime	≥15 years
Assembly geometry	Hexagonal
Reactivity swing	5247 pcm
Secondary coolant	Rankine cycle with superheated steam
Core inlet temperature	405 °C
Core outlet temperature	545 °C
Primary cooling method	Natural circulation
Primary heat transfer system	Compact pool type

and

Table 2. Design parameters of fuel assembly.

Parameters	Specification
Number of fuel assemblies	37
Number of pins per assembly	198
Equivalent core diameter	180 (cm)
Active core height	90 (cm)
Pitch-to-diameter ratio	1.2
Fuel pin diameter	0.56 (cm)
Assembly geometry	Hexagonal

[4].

In Table 1, the main design parameters of the reactor are outlined, including, fuel type (UO₂), and enrichment levels, divided into three zones with enrichments of 13.5% for the innermost, 16.5% for the middle, and 18.5% for the outermost regions. This zoned enrichment is intended to optimize neutron economy by managing the radial power distribution, thereby reducing the peak power density and enhancing the thermal safety margins. The core is further supported by structural components, including T91 cladding and B₄C absorbers for safety, and is surrounded by a YSZ reflector. A unique feature of this design is its reliance on lead–bismuth eutectic (LBE) as the primary coolant, which provides strong neutron reflection and thermal stability [9]. The core lifetime is designed to extend at least 15 years, with a reactivity swing of 5247 pcm, and is further enhanced by a secondary cooling system using a Rankine cycle with superheated steam, which contributes to efficient heat transfer [4]. The core operates at an inlet temperature of 405 °C and an outlet temperature of 545 °C and relies on natural circulation as the primary cooling mode, minimizing mechanical complexity and improving passive safety [10].

Table 2 provides a detailed summary of the fuel assembly specifications. The reactor core consists of 37 hexagonal fuel assemblies, each containing 198 fuel pins arranged with a pitch-to-diameter ratio of 1.2 to balance neutron economy and thermal hydraulics. The active core height is 90 cm with an equivalent core diameter of 180 cm, providing a compact design that facilitates stable neutron flux distribution and effective heat removal. To maintain structural integrity and thermal efficiency, the core also incorporates a helium gap and a primary coolant (LBE), enhancing the cooling capacity while minimizing corrosion risks. The wide coolant channels between the fuel pins significantly improve natural convection, supporting stable long-term operation. The total uranium load in this design is substantial, providing ample fissile material for sustained operation over an extended period, contributing to a lower reactivity swing and minimizing control rod actuation needs.

These specifications collectively ensure a balance between safety, operational efficiency, and longevity, making the reactor suitable for applications requiring long refueling intervals and reliable power output.

As illustrated in Figure 1, the reactor core consists of 37 fuel assemblies surrounded by reflector assemblies, LBE coolant, a barrel, downcomer LBE, and the reactor pressure vessel.

Table 3. Specification of MCNP k-code simulation.

Number of source history N per cycle	500,000
Initial guess for the multiplication factor	1
Number of inactive cycles	100
Number of active cycles	150

shows details of MCNP K-code transport calculations. The standard deviation on the effective multiplication factor is 0.00005, which is less than 100 pcm and is considered acceptable [11].

2.3. Variance Reduction Technique (VRT)

Neutron penetration in the thick shield is one of the most difficult problems in dose calculation. In SMR shielding problems defined by the MCNP k-code option, particle transport through thick shields often leads to significant absorption and scattering, causing considerable changes in particle energy and direction. This can result in very low particle counts in narrow energy bins or localized regions, thus requiring a large number of histories and extended computation times to achieve values with a statistically reliable relative error of less than 5% [12]. So, the effective use of VRT is indispensable to obtain the particle flux or dose rate and also energy spectra with good statistical errors.

VRT in Monte Carlo calculations can significantly decrease the computational time required to achieve results with adequate precision. Precision, however, is only one aspect of an effective Monte Carlo simulation [13]. In complex deep penetration scenarios, applying variance reduction methods becomes essential to reduce the computation time and lower statistical errors. By focusing on sampling more particles in important areas instead of spreading samples evenly across all regions, these techniques effectively minimize variance within a given computational timeframe [14].

Another important statistic in MCNP is the figure of merit (FOM), which is defined as

$$FOM={\displaystyle \frac{1}{R^{2}T}}$$

(2)

T is the computational time. When the FOM is higher, this indicates better efficiency and results [15].

2.4. Two-Step Variance Reduction Technique

Traditional variance reduction techniques (VRTs) include the use of SSW (surface source write) and SSR (surface source read). SSW records the energy, position, and direction of particles at a specified surface to be used in subsequent simulation stages. SSR then retrieves this stored particle data, enabling it to be reused in further simulations [16].

For example, when simulating the surface of a reactor vessel and an outer shielding area separately, SSR can be used to load the surface source, allowing particle transport from the vessel to the shielding to be accurately reproduced. This approach helps to bypass eigenvalue or criticality calculations and skips computations within the vessel, conserving computational resources. SSW/SSR applications include nuclear waste, groups of tanks containing depleted or enriched uranium, and reactor fuel [17].

In reactor shielding analysis, the reactor is modeled as a cylindrical surface source directly close to the radiation shield using the SSW (Surface Source Writing) card [6]. However, SSW does not support multi-threading, which speeds up calculations, resulting in higher errors and longer computation times. For example, if we have access to 448 threads, the calculation speed difference between the SSW method and my method would be over 400 times. In deep penetration problems, where reducing error is crucial, additional source biasing techniques are needed to sample radiation sources in specific emission angles [18]. However, a cylindrical volume source cannot utilize this VRT effectively [19].

To reduce the relative error in criticality calculations and improve computational efficiency, it is essential to use a VRT. While the SSW/SSR VRT, which involves defining a surface around the reactor and recording source information on that surface, is available, it does not support multi-threading. To address this limitation, we developed a new VRT called the two-step method. The process for the new VRT is outlined in the flow chart presented in Figure 2.

In Step 1, for a conservative radiation dose calculation, the reactor is modeled under the assumption of all-rods-out and steady-state BOC conditions [20]. A k-code criticality source (direct source) is then generated for modeling radiation from the reactor by MCNP, and the radiation transport calculation is performed using this direct source. To ensure that particles reach the reactor perpendicularly, a large spherical tally surrounding the reactor is defined, as shown in Figure 3. The number of particles and the flux can be recorded on this sphere using an f1 or f4 tally. This sphere is divided into equal intervals based on the polar angle (0 < polar angle < π) in spherical coordinates as well as particle energy groups. By assuming azimuthal symmetry due to the reactor’s symmetrical structure [21], the flux distribution can be analyzed. By using the FS card and the Tally Energy card, the direction and energy distribution of particles reaching the spherical tally can be determined based on the polar angle [14]. To prevent overestimation in the tally from particles scattered outside the reactor vessel, the region outside the reactor vessel is set as void, meaning no material space [22].

In Step 2, a fixed source (two-step source) is defined using the direction and energy distribution recorded in Step 1. This source is made by the MCNP6′s SDEF card, which is used to specify the direction, energy, particle type, position, and shape of the radiation source in MCNP. Radiation shielding materials outside the reactor vessel, such as concrete, are then modeled while the reactor structures are removed. Radiation shielding analysis is performed using the two-step source.

In reactors, gamma rays are generated from prompt fission gamma rays, gamma rays from fission products, and gamma rays from radiative capture by various isotopes. Additionally, gamma rays exiting the vessel consist of direct gamma rays from the reactor and secondary gamma rays produced by interactions between neutrons that escape the vessel and the shielding material [23]. As seen in Figure 4, the radiation dose from direct gamma rays can be negligible in deep penetration problems as it decreases while passing through the shielding, allowing us to define the two-step source based on neutrons only [24].

Due to the significant attenuation of primary gamma rays by the thick LBE coolant and concrete, their contribution to the dose is negligible compared to neutrons. Therefore, the calculation is performed with the “mode n p” option [25], which means that MCNP transports neutrons and photons, but only neutrons are recorded in the tally.

However, it is important to note that the tally results must be scaled by factors such as C₁ and C₂ according to Equation (3). ϕ_Direct is the direct source’s dose or flux. ϕ_Sdef is the two-step source’s dose or flux. Step 1’s tally has to be scaled by C₁ because the direct source’s tally results are normalized per fission neutron in the MCNP K-code criticality calculation [26]. Similarly, Step 2’s tally has to be scaled by C₂ because the number of fixed source particles is normalized to 1.

$${\varnothing }_{Direct}={\varnothing }_{Sdef}{C}_1{C}_2$$

(3)

$$C_1={\displaystyle \frac{P\nu }{Q{k}_{eff}}}$$

(4)

$$C_2={\displaystyle \frac{{{\displaystyle \sum }}_i {\varnothing }_{Direct,i}}{{{\displaystyle \sum }}_j {\varnothing }_{Sdef,j}}}$$

(5)

C₁ is the number of fission neutrons, P is the reactor’s thermal power, ν is the average number of neutrons released per fission, Q is the average recoverable energy per fission, k_eff is the effective multiplication factor, and C₂ is the ratio of the number of neutrons from the direct source to the number of neutrons from the two-step source passing through the spherical tally. This can be calculated using the number of neutrons that passed through the Step 1 tally.

Let us consider an example where the direction and energy of particles are recorded in a spherical tally in the two-step method. The particle direction, represented by the polar angle (0 < polar angle < π), is divided into 16 intervals, and the energy is divided into 48 intervals. In Step 1, the recorded direction distribution of the radiation is redefined in Step 2 using the sdef command. Figure 5 shows a comparison between the direction distribution recorded in the tally during Step 1 and the redefined source’s direction distribution. The cosine-shaped graph represents the normalized particle count across the polar angle intervals. Figure 6 illustrates the energy distribution comparison, showing the direction distribution recorded in the tally from Step 1 and the redefined source’s energy distribution.

3. Results and Sensitivity Analysis

3.1. Comparison Between Direct Source Approach and Two-Step Source Approach

In this study, a sensitivity analysis of the source was performed using direction divisions of 64, 128, 256, and 500 and energy divisions of 48, 101, and 202. The direction and energy distributions stored in these divided tallies were used in Step 2 to define the energy and direction of the source. The methods used to define the energy intervals and directions of the source were the histogram and discrete modes, respectively. As shown in Figure 7, a histogram represents a continuous spectrum, while a discrete spectrum is represented by a line spectrum. For example, suppose five particles are recorded in the energy range 0 < E < 0.75. When defining the source using the histogram method, the source is redefined with random energies within this range. For example, particles with energies of 0.1, 0.3, 0.4, 0.6, and 0.7 MeV might be defined. However, when using the discrete method, all five particles would be defined with an energy of 0.75 MeV.

To verify that the two-step source in the two-step method accurately simulates the direct source, the dose ratios between the direct source and the two-step source are compared based on the concrete thickness. As shown in Figure 8, the dose ratios measured on the concrete surface are compared at 10 cm intervals. It is a structure where 100 cm thick concrete surrounds an area located 30 cm away from the reactor vessel. The two-step source models the neutron source by recording the neutron’s energy and direction in Step 1. For the analysis, F2 tallies were used with DE and DF cards for conversion, using the ICRP 116 (2010) dataset for neutrons and photons [27]. This converts the flux into radiation dose values in pSv/s. To further convert the units, the FM card was used to change the dose unit to µSv/hr.

To identify the optimized division, the energy was discretely divided into 101 segments as a baseline. Equations (6) and (7) were used to compare the different division methods. Equation (6) represents the dose ratio of the two-step source to the direct source, serving as an indicator of how accurately the two-step source replicates the original direct source. The value L in Equation (7) is the sum of the squared differences between the dose ratio and 1 across various concrete thicknesses. A smaller L indicates a more accurate division method.

$$Dose Ratio={\displaystyle \frac{Two-step source dose rate}{Direct source dose rate}}$$

(6)

$$L={{\displaystyle \sum }}_i({\left(Dose Rati{o}_i-1\right)}^2)$$

(7)

The goal of the sensitivity analysis is to identify a two-step source that is both conservative (yielding higher doses than the direct source) across all concrete thicknesses and closely replicates the direct source, indicated by a low L value. Figure 9 shows the baseline with the energy divided into 101 discrete segments. Among the direction divisions of 64, 128, 256, and 500, the 128 discrete direction division yields the lowest L value of 0.3305, indicating the best replication of the direct source. The average dose error is 18.4%, which is consistently overestimated compared to the direct source. As seen in Figure 9 and Figure 10, when the energy is sufficiently divided, there is little difference between dividing the direction using the discrete method or the histogram method. A comparison of Figure 9 and Figure 11 shows that when the energy is divided using the histogram method, underestimation occurs, resulting in doses lower than those from the direct source. Consequently, the two-step source becomes less reliable. When comparing Figure 9 and Figure 12, it is evident that, except for the 256 direction divisions, there is generally low accuracy. This indicates the need for more refined energy group divisions. Figure 13 represents the best approximation that most accurately replicates the direct source. In this division method, the energy is divided into 202 discrete groups, and the direction is also divided discretely. With 128 direction divisions, the L value is 0.1816, showing the closest similarity to the direct source. The average dose rate error compared to the direct source is 12.32%, and the required computation time to achieve the same relative error as the direct source is reduced by 89.8 times.

In the BOC criticality calculation, the average number of neutrons produced per fission was 2.494, the average recoverable energy per fission was 201 MeV, and the effective multiplication factor was 1.05247. However, since steady-state conditions were assumed, the effective multiplication factor was taken as 1. As a result, the calculated C_1 value from Equation (2), representing the number of fission neutrons, was 3.098 × 10¹⁸ neutrons. Additionally, the C_2 value was calculated to be 0.08216.

In this study, the sensitivity index L was used to quantify the difference between the dose ratios of the direct source and the two-step source. A lower L value indicates a closer approximation to the direct source, representing higher fidelity in shielding effectiveness.

Here is the sensitivity analysis result for the L values in configurations where no underestimation occurs. E48 d, E101 d, and E202 d represent cases where the energy is divided into 48, 101, and 202 segments, respectively, in the discrete mode. D64 d/h, D128 d/h, D256 d/h, and D500 d/h indicate cases where the direction is divided into 64, 128, 256, and 500 segments in both the discrete (d) and histogram (h) modes. It can be observed that the L value is minimized when the energy is divided into 202 discrete divisions and the direction into 256 discrete divisions. As shown in Figure 14, it can be observed that the L value is minimized when the energy is divided into 202 discrete divisions and the direction into 256 discrete divisions.

3.2. Efficiency of VRT

Table 4. Comparison of efficiency with different VRTs.

Method	FOM	Recording Time (min)	Computing Time (min)
Direct	1.336	-	188.58
SSW/SSR	23.995	1620	0.95
Two-step	12.097	16.5	2.1

compares the FOM, recording time, and computing time under three conditions based on a relative error of 0.05 for the dose measured at 100 cm of concrete: without applying VRT, with SSW/SSR VRT, and with the two-step method. The FOM in MCNP is one of the most important indicators for evaluating the efficiency and reliability of calculations [28]. A larger FOM value indicates greater computational efficiency [19].

The recording time refers to the time taken to record particle information in both the SSW/SSR VRT and two-step methods; therefore, the direct method, which does not use any VRT, has no recording time. The computing time represents the time required to perform the MCNP simulation. In this study, dose calculations were performed using an Intel 14th Generation i9 CPU, typically utilizing 20 tasks.

By recording particles on the outer wall of the reactor vessel using the VRT, a significant increase in the FOM was observed for both the SSW/SSR and two-step methods. Before performing the shielding calculation, the time required for particle recording in the two-step method is approximately 98 times faster than SSW/SSR. The two-step method supports multi-threading, so as the number of threads increases, the computation time further decreases as a function of the thread count [29]. By using the two-step method, the computing time increased by 89.8 times compared to the direct source, and this increase also scaled with the number of threads.

3.3. Potential Applications

In the context of marine nuclear reactor shielding challenges, the Small Modular Reactor (SMR) must be placed within the confined space of a nuclear merchant ship, where limitations in weight and volume are crucial [3]. Therefore, optimizing the shield to meet dose limits while minimizing both the weight and cost becomes essential [30]. Designing radiation shielding presents a complex optimization challenge with multiple objectives. Traditional approaches for optimization, which are often dependent on human expertise, tend to be inefficient and struggle to identify the most effective shielding configurations. As a result, conventional methods for radiation shielding design are generally time-intensive and lack efficiency [31].

When the new VRT is used, it allows for rapid calculations by ignoring neutron and photon tracking within the vessel and criticality calculations, which is particularly useful when the shielding material is modified. By reducing the computational load, the suggested method in this paper can streamline simulations for various shielding configurations, helping to identify cost-effective and efficient solutions for enhanced protection in critical environments like reactor control systems.

4. Conclusions

This study aimed to develop a new VRT that can replace the traditional SSW/SSR method for radiation shielding analysis in SMRs. Addressing limitations in traditional methods, such as the surface source write/read (SSW/SSR) approach, the proposed method records the particle energy and direction to redefine the source, allowing for efficient multi-threading and optimized computations. The core idea of the two-step-method is to record the energy and direction of particles in a large spherical tally surrounding the reactor from the k-code criticality source (direct source) and redefine them using a two-step source (fixed source, SDEF) for subsequent radiation shielding analysis. This approach is particularly beneficial for SMRs, where compact shielding and high computation performance are critical.

Our findings indicate that the best approximation, which closely replicates the direct source while conservatively estimating the dose, is achieved with a 202 discrete energy division and a 128 discrete direction division. This configuration not only provided the lowest L value of 0.1816, indicating a strong similarity to the direct source, but it also reduced the computation time by 89.8 times while maintaining an average dose rate error of 12.32%. However, it was observed that using histogram-based energy divisions can lead to underestimation, raising concerns about the reliability of the two-step source in certain configurations.

As shown in Figure 4, the primary gamma rays originating from the reactor are significantly lower than the secondary gamma rays generated from neutron interactions within the shielding material. Therefore, in Step 1 of the two-step method, it is not necessary to record primary gamma rays or redefine them through SDEF. Due to the presence of radiative capture, only neutrons are defined as the source in Step 2. However, the “mode n p” option in MCNP6 should be enabled to account for photon interactions.

Furthermore, the discrepancies in the dose rates between the direct and two-step sources were primarily attributed to scattering caused by the radiation shielding from concrete.

In this study, we did not include radiation shielding optimization. Future research could expand the applicability of the proposed two-step VRT to various reactor designs and conditions, optimizing radiation shield materials like concrete, water, steel, lead, and tungsten carbide in specific environments. This approach would allow for the optimization of the volume, weight, and shielding effectiveness based on the given conditions.

Overall, this study demonstrates the effectiveness of the newly proposed VRT in providing a reliable and efficient approach to radiation shielding analysis in SMRs, with significant improvements in the computation time and accuracy. However, the selection of appropriate energy and direction divisions remains crucial to ensuring the reliability of the results.