Optimized Method for Solving Boltzmann Transport Equations in Subgroup Method of Resonance Treatment

Abstract

This study presents an optimized subgroup transport equation solving method to enhance the computational efficiency of resonance calculations in high-fidelity reactor core simulations. By consolidating all the resonance groups into an equivalent single group and performing fixed-source calculations only for the representative subgroup cross-sections, this method significantly reduced the computational burden compared to the conventional subgroup method. Validation studies on single-cell, 2D assembly, and 3D assembly problems demonstrated that the proposed method achieves computational accuracy comparable to the conventional approach while requiring fewer fixed-source equations. This advancement offers a promising solution for improving the efficiency of resonance calculations in high-fidelity reactor core simulations, paving the way for more accurate and computationally efficient modeling of complex reactor systems.

1. Introduction

The development of nuclear power technology has progressively increased demands for high-fidelity simulation technologies in reactor analysis, making digital reactor development a research focus worldwide [1,2,3]. Reactor numerical computation involves multidisciplinary and multi-physics coupling studies, which include neutronics analysis, thermal hydraulics, fuel performance analysis, etc. Neutronics calculation, which provides critical information such as the effective multiplication factor k_eff, neutron flux density, and power distribution through precise solutions of the Boltzmann transport equation [4], serves as one of the pivotal components in high-fidelity digital reactor simulations.

Neutronics computation requires the integral and differential calculations of neutron flux density across multiple dimensions including energy, angle, space, and time. Direct analytical solutions of the neutron transport equations prove extremely challenging for actual reactors with complex geometric configurations and material compositions. Internationally, current mainstream reactor physics methodologies include Monte Carlo methods and deterministic approaches [4]. The Monte Carlo method simulates neutron behavior through stochastic particle tracking, employing massive particle populations and iterative computations to obtain neutron flux density values where statistical deviations meet accuracy requirements [5]. While representing the most accurate core-physics calculation method currently available, its extensive statistical processes result in prohibitively low computational efficiency for large-scale assembly or full-core scientific computations. Deterministic methods, adopted by mainstream commercial codes internationally, employ a series of approximations to enable discrete computations across the energy, angle, space, and time dimensions. For reactor steady-state calculations, temporal effects can be neglected. Transport calculation methods represented by the method of characteristics [6], discrete ordinates method [7], and collision probability method [8] achieve spatial and angular discretization. Moreover, deterministic approaches implement energy discretization through group-wise approximations—classifying neutrons into distinct energy groups based on their kinetic energy levels and computing equivalent average cross-sections for each group via continuous-energy cross-section integration, as shown in Equation (1):

$${\sigma }_{x,g}(\mathit{r})={\displaystyle \frac{{\displaystyle {\int }_{\Delta E_g}{\sigma }_x(E)\phi (\mathit{r},\mathbf{\Omega },E)\mathrm{d}}E}{{\displaystyle {\int }_{\Delta E_g}\phi (\mathit{r},\mathbf{\Omega },E)\mathrm{d}}E}}$$

(1)

where σ denotes the microscopic cross-section, x represents the reaction channel type, φ is the neutron angular flux density, Ω indicates the neutron direction of motion, r specifies the spatial position, E corresponds to energy, and g designates the energy group index.

In pressurized water reactor (PWR) materials, the cross-sections of most nuclides exhibit relatively smooth variations with energy. Within specific energy ranges, cross-section values can be approximated as energy-independent constants. Consequently, the multigroup cross-sections derived via Equation (1) are spectrum-independent and can be pre-stored in multigroup libraries, with interpolation applied during calculations based on the temperature value. However, for heavy nuclides such as ²³⁵U or ²³⁸U, neutron collisions with their nuclei form metastable compound nuclei. When incident neutrons occupy specific resonance energy points, the probability of nuclear reactions increases sharply, while deviations from these energies lead to rapid declines in reactivity. This results in pronounced cross-section fluctuations within narrow energy intervals, as illustrated in Figure 1. This phenomenon is termed resonance in reactor physics. Nuclides and energy groups exhibiting resonance behavior are classified as resonance nuclides and resonance groups, respectively, with individual cross-section peaks referred to as resonance peaks. In PWRs, resonances predominantly occur in the 1~10⁴ eV range. The substantial cross-section magnitudes near resonance peaks cause intense neutron absorption, leading to a sharp depression in the neutron energy spectrum—a shielding-like effect known as resonance self-shielding. In practical reactor systems, the complexity of material compositions and geometric configurations renders resonance self-shielding highly intricate. The effective resonance cross-sections computed via Equation (1) exhibit strong energy spectrum dependence. Consequently, resonance cross-sections cannot be pre-tabulated in multigroup libraries. Instead, they require resonance self-shielding calculations tailored to specific problem conditions.

Therefore, the precise calculation of effective resonance cross-sections is a prerequisite for high-fidelity core-physics simulations. Internationally, deterministic methods for resonance self-shielding primarily include the hyperfine group method [9,10], equivalence theory [11,12], and subgroup method [13,14,15]. The hyperfine group method achieves highly accurate resonance calculations by subdividing energy groups into ultra-fine structures, approximating continuous energy treatment to resolve sharply fluctuating cross-sections in resonance regions. Despite its precision, solving hyperfine group equations with a near-continuous energy resolution involves prohibitive computational complexity, limiting its applicability to single fuel pins or small-scale assemblies. The equivalence theory establishes equivalence relationships between homogeneous and heterogeneous problems by calculating the neutron’s first-flight collision probability. It approximates background cross-sections for practical scenarios through the interpolation of pre-tabulated resonance integrals. However, its accuracy hinges on precise approximations of the first-flight collision probability, which depends heavily on geometric configurations and material compositions. Consequently, equivalence theory is restricted to regular geometries or simple material systems and becomes inadequate for advanced reactor designs requiring refined resonance treatments.

In contrast, the subgroup method discretizes resonance groups from a cross-section perspective rather than energy discretization. The concept was first proposed by Nikolaev in 1963 for resonance self-shielding calculations [13]. Later, Levitt introduced the probability table method in 1972 to address unresolved resonance regions [14], representing fluctuating resonance peaks as probability distributions to compute effective resonance cross-sections. Extending this approach, the subgroup method employs subgroup cross-sections to characterize cross-section variations and subgroup probabilities to describe their likelihood across the entire resonance energy range. By grouping cross-sections into narrow intervals (subgroups) with minimal internal variation, the neutron flux within each subgroup can be treated as constant, enabling the precise modeling of resonance phenomena. Compared to traditional energy-group discretization, the subgroup method achieves higher accuracy with fewer groups. It solves subgroup transport equations to obtain subgroup fluxes and collapses these results with subgroup parameters to derive effective resonance cross-sections. Crucially, subgroup transport equations share the same mathematical formulation as multigroup transport equations, allowing for seamless integration with arbitrary transport solvers [15]. Unlike traditional resonance methods that rely on geometric approximations, the subgroup method directly accounts for geometric effects through subgroup flux solutions, theoretically accommodating arbitrary geometries with exceptional adaptability. Owing to these advantages, the subgroup method has gained widespread international adoption in high-fidelity reactor physics simulations.

In recent years, the subgroup method has undergone rapid development and has been widely integrated into one-step high-fidelity core simulation codes. The HELIOS code [16], developed by Studsvik (USA), stands as a pioneering large-scale commercial software employing subgroup-based resonance calculations. It precomputes subgroup parameters via fitting methods, stores them in databases, solves subgroup fixed-source equations to generate effective resonance cross-sections, and introduces background iteration to address resonance interference effects [17]. This framework laid the foundation for subsequent methodologies. Building on HELIOS-derived subgroup parameters, codes such as DeCART [18,19] and MPACT [20,21,22] integrate the method of characteristics (MOC) for coupled resonance-transport calculations. Meanwhile, DRAGON [23] and APOLLO [24,25] adopt moment-based subgroup parameters for pressurized water reactor (PWR) and fast reactor analyses. The structural consistency between subgroup transport equations and multigroup transport equations enables flexible coupling with arbitrary transport solvers, offering exceptional geometric adaptability.

Early versions of HELIOS combined subgroup methods with the collision probability method (CPM) to achieve high computational efficiency. Takeda et al. [26] proposed a similar CPM-subgroup hybrid strategy for KUCA core simulations. However, as demands for high-precision simulations grew, CPM’s limitations in handling complex geometries became evident. Yamamoto et al. [27] explored discrete ordinates (S_N) method for fast reactor resonance analysis. With advancements in transport solvers, the MOC—renowned for superior geometric flexibility—emerged as the dominant approach. HELIOS-2 [28] pioneered the replacement of CPM with MOC for solving subgroup fixed-source and multigroup transport equations. Leading commercial codes, including DeCART, nTRACER, and APOLLO, subsequently adopted MOC–subgroup coupled architectures.

To optimize computational efficiency, Stimpson et al. [29] proposed an integrated MOC strategy that simplifies subgroup transport formulations while enabling concurrent solutions for multigroup fixed-source equations. Park et al. [30] developed a macroscopic subgroup flux-interpolation technique, significantly reducing transport-solving iterations. These innovations have propelled the practical application of subgroup methods in engineering-scale problems. After 2020, with the development of a new type of reactor, multi-physics calculation analysis has developed rapidly, which provided plenty of valuable experience for our study. For example, Zhang et al. [31] developed a multi-physics coupling methodology within the MOSASAUR code system for lead-cooled fast reactors (LFRs), enabling the integrated simulation of neutronics, thermal hydraulics, and thermomechanics. The Go et al. [32] study validated the light water reactor (LWR) fuel depletion module within the CBZ reactor physics code system through benchmark comparisons and experimental data. Research from Miao et al. [33] developed a multi-physics coupling framework for analyzing the temperature field in dry-type air-core reactors, integrating electromagnetic, thermal, and fluid dynamics interactions. Yang et al. [34] focused on reactor-core-physics modeling for marine nuclear power platforms, proposing computational methods tailored to marine-specific conditions like dynamic loads and motion-induced effects. Stober et al. [35] investigated fusion plasma behavior using high-power electron cyclotron resonance heating (ECRH) on the ASDEX Upgrade tokamak, analyzing its effects on plasma confinement and magneto-hydrodynamic stability. Current synthesis research indicates that the subgroup method has been extensively implemented in one-step high-fidelity core simulation codes. Nevertheless, several challenges remain unresolved in its application. For instance, the subgroup fixed-source equations necessitate numerous single-group fixed-source equation solutions, leading to redundant geometric data processing and characteristic ray tracing. Specifically, the subgroup method requires fixed-source calculations for all subgroups within every resonance group of each resonant nuclide. Taking the HELIOS-47 group structure as an example, with sixteen resonance groups and three subgroups per resonance group, a single resonant nuclide demands 48 fixed-source equation solutions. For MOX fuel or problems under deep burnup conditions, where multiple resonant nuclides coexist, hundreds of subgroup fixed-source computations are typically required. Frequent invocations of the transport solvers severely degrade resonance calculation efficiency, underscoring the critical need to reduce the number of subgroup fixed-source equations.

To address this challenge, this study focused on optimizing the computational efficiency of subgroup fixed-source equation solutions. An equivalent single-resonance-group subgroup flux interpolation method is proposed in this work. By consolidating all resonance groups into an equivalent single group, fixed-source calculations were performed only for the representative subgroup cross-sections within this consolidated group. The resulting subgroup fluxes for these key cross-sections were then extrapolated to obtain subgroup fluxes for actual resonance groups via escape cross-section interpolation. This approach significantly reduced the computational workload while preserving accuracy, thereby enhancing the efficiency of subgroup-based resonance calculations.

This study addresses a critical computational bottleneck in high-fidelity reactor physics simulations. This breakthrough enhances computational efficiency by orders of magnitude, enabling for practical applications in large-scale, complex reactor systems where resonance self-shielding effects dominate neutronics behavior. This work not only advances deterministic resonance calculation techniques but also provides a foundational strategy for optimizing computational workflows in next-generation nuclear energy systems.

2. Materials and Methods

In conventional pressurized water reactor neutronics analysis, the neutron energy range spans approximately 10⁻⁴~10⁷ eV. Direct continuous-energy cross-section treatment is computationally prohibitive. Internationally, the standard approach involves discretizing neutrons into energy groups based on kinetic energy. The multigroup Boltzmann transport equation under this framework is expressed as Equation (2).

$$\mathbf{\Omega }·\nabla {\varphi }_g(\mathit{r},\mathbf{\Omega })+{\Sigma }_{t,g}(\mathit{r}){\phi }_g(\mathit{r},\mathbf{\Omega })=Q_{g,f}(\mathit{r})+Q_{g,s}(\mathit{r})+S_g(\mathit{r})$$

(2)

where Ω is the neutron’s direction of motion, r represents the spatial position, g is the energy group index, Σ_t,g is the total cross-section, ϕ_g is the neutron’s angular flux density, Q_f is the fission source term, Q_s,g is the scattering source term, and S_g indicates the external source term.

The multigroup approximation assumes a constant neutron flux density within each energy group. However, due to sharp cross-section variations in the resonance regions, only hyperfine group structures with ultra-dense energy discretization can approximate flat flux distributions within the groups. For conventional multigroup structures (e.g., HELIOS-47), significant flux gradients persist within resonance groups. Unlike traditional energy-based discretization, the subgroup method further subdivides the resonance groups into subgroups based on cross-section magnitudes, as illustrated in Figure 2. Within each subgroup, cross-section variations are minimized, ensuring the flux fluctuations are far smaller than those in conventional resonance groups. Compared to hyperfine group energy discretization, the subgroup method achieves comparable accuracy in resolving resonance effects with far fewer computational groups, significantly enhancing computational efficiency.

As illustrated in Figure 2, the same subgroup cross-section range may correspond to non-contiguous energy intervals, meaning the energies within a subgroup are not continuous. The subgroup method transforms the continuous Riemann integration of the neutron energy spectrum over energy, f(E), into a Lebesgue integration over cross-section magnitudes, f(σ), as shown in Equation (3):

$${\displaystyle {\int }_{E_g}^{E_{g-1}}f(E)\mathrm{d}}E={\displaystyle {\int }_{{\sigma }_{\min ,g}}^{{\sigma }_{\max ,g}}f(\sigma )p(\sigma )\mathrm{d}}\sigma$$

(3)

where σ_min and σ_max are the minimum and maximum cross-section values, respectively, in the energy group g, and p(σ) is the probability density function representing the probability of the cross-section σ occurring in group g.

Unlike the irregular and sharply varying f(E) with energy, f(σ) exhibits an approximately inverse proportionality to the cross-section, simplifying integration. This allows for accurate evaluation with fewer discrete points (subgroups). Each subgroup is characterized by the subgroup cross-section σ_g,i and the subgroup probability p_g,i, which indicate the average cross-section within the subgroup and the fraction of the resonance group’s energy range occupied by the subgroup, respectively. With these definitions, Equation (3) reduces to a discrete quadrature in Equation (4):

$${\displaystyle {\int }_{E_g}^{E_{g-1}}f(E)\mathrm{d}}E={\displaystyle {\int }_{{\sigma }_{\min ,g}}^{{\sigma }_{\max ,g}}f(\sigma )p(\sigma )\mathrm{d}\sigma }={\displaystyle \sum _{i=1}^I{\sigma }_{g,i}{\varphi }_{g,i}{p}_{g,i}}$$

(4)

where ϕ_g,i is the subgroup flux within the subgroup.

The subgroup parameters collectively convert the continuous energy-dependent flux integration into a discrete summation. Consequently, the effective resonance cross-section for the reaction channel x can be computed via Equation (5):

$${\sigma }_{x,g}={\displaystyle \frac{{\displaystyle {\int }_{E_g}^{E_{g-1}}{\sigma }_x(E)\varphi (E)\mathrm{d}}E}{{\displaystyle {\int }_{E_g}^{E_{g-1}}\varphi (E)\mathrm{d}}E}}={\displaystyle \frac{{\displaystyle {\int }_{{\sigma }_{\min ,g}}^{{\sigma }_{\max ,g}}{\sigma }_x\varphi (\sigma )p(\sigma )\mathrm{d}{\sigma }_g}}{{\displaystyle {\int }_{{\sigma }_{\min ,g}}^{{\sigma }_{\max ,g}}\varphi (\sigma )p(\sigma )\mathrm{d}{\sigma }_g}}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{\sigma }_{x,g,i}{\varphi }_{g,i}{p}_{g,i}}}{{\displaystyle \sum _{i=1}^I{\varphi }_{g,i}{p}_{g,i}}}}$$

(5)

where σ_x,g denotes the effective resonance cross-section for the reaction channel x in energy group g.

Analogous to the steady-state neutron transport equation, the subgroup transport equation for subgroup i in resonance group g is expressed as Equation (6).

$$\mathbf{\Omega }·\nabla {\phi }_{g,i}(\mathit{r},\mathbf{\Omega })+{\Sigma }_{t,g,i}(\mathit{r}){\phi }_{g,i}(\mathit{r},\mathbf{\Omega })=Q_{g,i}(\mathit{r})$$

(6)

where Σ_t,g,i is the subgroup’s total macroscopic cross-section for subgroup i in group g. The variable Q_g,i is the subgroup’s source term.

The subgroup flux ϕ_g,i, obtained by solving Equation (6) via transport solvers, is substituted into Equation (5) to compute the effective resonance cross-sections. This approach enables resonance self-shielding calculations for arbitrary geometries by explicitly resolving the energy spectrum through subgroup transport equations.

2.1. Conventional Subgroup Transport Equation

While multigroup libraries store precomputed scattering matrices for the entire energy group structure, the subgroup method requires additional treatment to account for the scattering matrix distribution across subgroups. Mirroring the formalism of multigroup transport, the scattering source term for subgroup i in resonance group gg is formulated as Equation (7).

$$Q_{g,i}={\displaystyle \sum _{h=1}^H{Q}_{s,h\to g,i}}+{\displaystyle \sum _{j=H+1}^{H+J}{\displaystyle \sum _{k=1}^K{Q}_{s,j,k\to g,i}}}+{\displaystyle \sum _{i^{\prime }=1}^I{Q}_{s,g,i^{\prime }\to g,i}}$$

(7)

where Q_s,h→g,i is the scattering source term from the fast group h, Q_s,j,k→g,i is the scattering source term from the subgroup k in the resonance group j, Q_{s,g,i′→g,i} is the scattering source term from the subgroup i′ within the same resonance group g, and H and J are the total number of fast groups and resonance groups, respectively. K and I are the total number of subgroups in the resonance groups j and g, respectively.

The scattering weight ω_g′→g, defined as the probability of neutrons scattering from the upstream group g′ to the resonance group g, is derived from the scattering matrix as Equation (8).

$${\omega }_{g^{\prime }\to g}={\displaystyle \frac{{\sigma }_{s,g^{\prime }\to g}}{{\displaystyle \sum _{g^{″}=1}^G{\sigma }_{s,g^{\prime }\to g^{″}}}}}={\displaystyle \frac{{\sigma }_{s,g^{\prime }\to g}}{{\sigma }_{s,g^{\prime }}}}$$

(8)

where σ_g′→g is the scattering cross-section from group g′ to g, and σ_s,g′ is the total scattering cross-section of group g′.

For the target resonance group g, the scattering source term is distributed among its subgroups according to the subgroup probabilities. Based on the scattering weight definition, the first term on the right-hand side of Equation (7) is formulated as Equation (9).

$${\displaystyle \sum _{h=1}^H{Q}_{s,h\to g,i}}={\displaystyle \frac{1}{4\pi }}{\displaystyle \sum _{h=1}^H{\omega }_{h\to g}{\Sigma }_{s,h}{p}_{g,i}{\varphi }_h}$$

(9)

where the subgroups within the same resonance group share identical scattering weights. Thus, the scattering from subgroup k in resonance group j to subgroup i in group g is expressed as Equation (10).

$${\displaystyle \sum _{j=H+1}^{H+J}{\displaystyle \sum _{k=1}^K{Q}_{s,j,k\to g,i}}}={\displaystyle \frac{1}{4\pi }}{\displaystyle \sum _{j=H+1}^{H+J}{\displaystyle \sum _{k=1}^K{\omega }_{j\to g}{\Sigma }_{s,j,k}{p}_{g,i}{\varphi }_{j,k}}}$$

(10)

Similarly, the third term on the right side of Equation (7) becomes:

$${\displaystyle \sum _{i^{\prime }=1}^I{Q}_{s,g,i^{\prime }\to g,i}}={\displaystyle \frac{1}{4\pi }}{\displaystyle \sum _{i^{\prime }=1}^I{\omega }_{g\to g}{\Sigma }_{s,g,i^{\prime }}{p}_{g,i}{\varphi }_{g,i^{\prime }}}$$

(11)

Due to the unknown quantities of the subgroup fluxes ϕ_j,k, ϕ_g,i′, and the multigroup flux ϕ_h in the above expression, the subgroup transport equations and the multigroup transport equations need to be solved iteratively in a coupled manner. The transport module first initializes the multigroup flux and the fission source distribution. Then, based on the initialized fluxes and source terms, and in combination with Equations (9)–(11), it calculates each subgroup source term and solves for the subgroup fluxes, and then calculates the effective resonance cross-sections for each group according to Equation (5). These cross-sections are used for the multigroup transport calculations to obtain the updated multigroup fluxes and the effective multiplication factor, and the new fluxes are re-substituted into the subgroup transport equations for the resonance cross-section calculations. This process is repeated until the resonance cross-sections, multigroup fluxes, and the effective multiplication factor converge.

The traditional subgroup transport equations precisely calculate the distribution of subgroup fluxes among various resonance groups and their subgroups, requiring repeated iterations between multigroup transport and subgroup transport, which is computationally intensive. The purpose of resonance calculation is to obtain the effective resonance cross-sections, and as indicated by Equation (5), the effective resonance cross-sections are only related to the relative magnitudes of the subgroup fluxes within the current resonance group. Therefore, it is sufficient to obtain the subgroup energy spectrum shape within the current resonance group to aggregate and determine the effective resonance cross-section for that group, without the need for precise calculation of the subgroup flux distribution among different resonance groups. Incorporating the concept of the neutron slowing down, for the ith subgroup of the gth energy group, its subgroup transport equation can be expressed as Equation (12).

$$\mathbf{\Omega }·\nabla {\varphi }_{g,i}+{\Sigma }_{t,g,i}{\varphi }_{g,i}(\mathit{r},\mathbf{\Omega })={\displaystyle \frac{1}{4\pi }}\left[{\displaystyle \sum _{l=1}^L{\lambda }_{g,l}{\Sigma }_{g,p,l}}+(1-{\lambda }_{g,R}){\Sigma }_{s,g,i,R}{\varphi }_{g,i}(\mathit{r})\right]$$

(12)

where λ_g,l represents the intermediate resonance approximation factor and Σ_g,p,l denotes the macroscopic potential scattering cross-section.

The source term shown in Equation (12) is no longer correlated with the fluxes of the other subgroups, therefore each subgroup transport equation can be solved independently without the need for iterative calculations of the intergroup scattering source terms. Thus, Equation (12) can be further simplified to Equation (13).

$$\mathbf{\Omega }·\nabla {\varphi }_{g,i}+\left({\Sigma }_{t^{\prime },g,i}+{\displaystyle \sum _{l=1}^L{\lambda }_{g,l}{\Sigma }_{g,p,l}}\right){\varphi }_{g,i}(\mathit{r},\mathbf{\Omega })={\displaystyle \frac{1}{4\pi }}{\displaystyle \sum _{l=1}^L{\lambda }_{g,l}{\Sigma }_{g,p,l}}$$

(13)

where Σ_t′,g,i can be normally calculated using Equation (14) for regions containing the current resonant nuclide. For regions not containing the current resonant nuclide, Σ_t′,g,i is taken as zero.

$${\Sigma }_{t^{\prime },g,i}={\lambda }_{g,R}{\Sigma }_{t,g,i,R}-(1-{\lambda }_{g,R}){\Sigma }_{s,g,i,R}$$

(14)

where Σ_t,g,i,R and Σ_s,g,i,R represent the subgroup’s macroscopic total cross-section and scattering cross-section, respectively, for the R nuclide in the gth energy group and the ith subgroup.

From Equation (13), it can be seen that the source term of the subgroup transport equation is independent of the flux and remains constant during the solution process; therefore, the subgroup transport equation is equivalent to a fixed-source equation. Compared to the traditional subgroup transport equation, the subgroup fixed-source equation only needs to be calculated independently for each subgroup, without the need for cyclic iteration between the different subgroups.

2.2. Optimization of the Subgroup Transport Equation

As known from Section 2.1, the subgroup method requires the calculation of fixed-source equations for all the subgroups in all the resonance groups of resonant nuclides. Taking the HELIOS-47 energy group structure as an example, it has 16 resonance groups. Assuming each resonance group contains three subgroups, then for each resonant nuclide, 48 fixed-source equation calculations are needed. For problems involving MOX fuel or deep burnup conditions, due to the large number of resonant nuclides, often hundreds of subgroup fixed-source equation calculations are required. The frequent invocation of the transport module results in inefficient resonance calculations. The key to solving this problem is reducing the number of subgroup fixed-source equations.

2.2.1. Micro Level Optimization Based on “One-Group” Approximation

The subgroup cross-sections for different resonance groups vary significantly, while the potential scattering cross-sections change relatively little, and the intermediate resonance factors are mainly related to the mass of the resonant nucleus. Additionally, since the source term of the subgroup fixed-source equation is only the scattering source, and the potential scattering cross-section of the moderator and the intermediate resonance approximation factor remain almost constant, the source term on the right side of Equation (13) for the subgroup fixed-source equations of the different resonance groups changes very little, and it can be considered that all the resonance groups share the same source term. If two subgroups have similar subgroup cross-sections, then only one solution is needed for the two fixed-source equations. Based on this idea, this section proposes an equivalent micro-level subgroup fixed-source interpolation method, which equates all resonance groups into one group. The equivalent single-resonance group contains the subgroup parameters of all the actual resonance groups, and the non-resonance cross-sections of this group are obtained by the weighted average of the resonance integrals under infinite dilution conditions for each energy group, as shown in Equation (15):

$${\sigma }_x={\displaystyle \frac{{\displaystyle \sum _{g=1}^{G_{res}}{\sigma }_{x,g}R{I}_{a,g,\infty }\Delta u_g}}{{\displaystyle \sum _{g=1}^{G_{res}}R{I}_{a,g,\infty }\Delta u_g}}}$$

(15)

where G_res is the total number of resonance groups, Δu_g is the lethargy width of the gth group, and RI_a,g,∞ is the infinite dilution absorption resonance integral of the resonant nuclide in the gth group. For non-resonant nuclides, RI_a,g,∞ is taken as the integral value of the resonant nuclide currently being calculated for the subgroup.

The scattering source term for the equivalent single-resonance group can be calculated according to Equation (15). Since the source terms of the subgroup fixed-source equations are the same, the subgroup fluxes obtained from the solution are directly negatively correlated with the subgroup removal terms. Therefore, only specific subgroup cross-sections need to be solved using the fixed-source equation, and the fluxes corresponding to other subgroup cross-sections can be obtained through the interpolation calculations based on the adjacent subgroup cross-sections.

For most subgroups, the removal and source terms of the subgroup fixed-source equations are numerically very close, so the subgroup fluxes obtained from the transport calculations are close to 1, which is not suitable for the direct interpolation of subgroup fluxes based on subgroup cross-sections. The larger the subgroup cross-section, the smaller the probability that a neutron does not undergo any reaction and escapes within that subgroup. Under heterogeneous conditions, due to the spatial self-screening effect, the probability of neutron escape is not only related to the subgroup cross-section but also to the position of the neutron within the fuel rod. To comprehensively reflect the influence of both the cross-section and space, for the resonant nuclide R, the neutron escape cross-section is defined as shown in Equation (16):

$${\sigma }_{e,i,R}={\displaystyle \frac{1}{N_R}}\left({\displaystyle \frac{{\Sigma }_{t^{\prime },i,R}{\varphi }_i}{1-{\varphi }_i}}-{\displaystyle \sum _l^L{\lambda }_m{\Sigma }_{p,m,l}}\right)$$

(16)

Compared to the subgroup flux, the escape cross-section can more significantly reflect the influence of the subgroup cross-section and spatial position on neutron reactions. Taking the single fuel rod problem as an example, if the fuel rod is divided into five concentric rings of equal volume, and subgroup fixed-source equation calculations are performed for a series of different subgroup cross-sections σ_t′, the relationship between the escape cross-sections in each region and σ_t′ can be obtained, as shown in Figure 3.

Based on Figure 3, the average subgroup escape cross-section for the fuel rod varies only slightly within the subgroup cross-section value. As the subgroup cross-section increases, the average escape cross-section shows only a slight downward trend. However, for the radial rings of the fuel rod, the fine distribution of the escape cross-sections exhibits significant changes, especially at the outermost side of the fuel rod. The escape cross-section varies most dramatically when the subgroup cross-section is between 10 and 10⁴ b, while for the subgroup cross-sections less than 10 b or greater than 10⁴ b, the escape cross-section remains almost constant. Therefore, to describe the neutron behavior under different subgroup cross-section conditions, the equivalent subgroup cross-section should be focused on values between 10 and 10⁴ b. The actual subgroup escape cross-section is calculated through the interpolation of ln(σ_t′), and the actual subgroup flux can be derived inversely based on Equation (16), ultimately calculating the effective resonance cross-section using Equation (5). The overall process of the equivalent single-resonance-group subgroup flux interpolation calculation is shown in Figure 4.

2.2.2. Selection of the One-Group Subgroup Cross-Section

To analyze the computational accuracy of inferring actual subgroup fluxes from escape cross-section interpolation, this section selected 500 subgroup cross-section points ranging from 1 to 10⁵ b, with the values primarily concentrated between 10 and 10⁴ b, where the escape cross-section varies significantly. Based on a series of non-uniform single-lattice cell problems, the fuel rod was divided into 5, 10, and 15 rings of equal volume, respectively. The transport solver was used to solve all the subgroup fixed-source equations for the aforementioned series of problems, obtaining the subgroup fluxes corresponding to the actual subgroup cross-sections. The number of subgroup cross-section points for the interpolation scheme used in the fixed-source equation calculations was selected to be between 4 and 12, with the number of points in each cross-section interval as shown in

Table 1. Equivalent one-resonance-group subgroup cross-section values.

Subgroup Cross-Section Range/b	Number of Subgroups
0~102	1	1	1	1	1	2	3	3	3	4
102~103	1	2	2	3	4	4	4	4	4	4
103~104	1	1	2	2	2	2	2	3	4	4
104~105	1	1	1	1	1	1	1	1	1	1
Total	4	5	6	7	8	9	10	11	12	13

. Different numbers of the subgroup cross-section points were used to calculate the subgroup fluxes for the aforementioned non-uniform problems, and then the actual subgroup fluxes corresponding to the subgroup cross-sections were inferred through the escape cross-section interpolation calculations.

The subgroup fluxes obtained from the interpolation calculations were compared with the subgroup fluxes obtained from actually solving the subgroup fixed-source equations, and the root mean square deviation (RMS) of the subgroup fluxes for all regions was calculated. Figure 5 shows the trend of the RMS of subgroup fluxes as the number of subgroup cross-section interpolation points varies. The flux calculation deviation decreases significantly as the number of interpolation points increases, and when the number of interpolation points reaches 8 or more, the calculation deviation gradually stabilizes. Considering the actual range of fuel subgroup cross-sections, this paper selected 8 subgroup cross-section interpolation points at 10, 100, 200, 300, 500, 1000, 2000, and 10,000 b.

In summary, to reduce the number of solutions for the subgroup fixed-source equations, an equivalent one group subgroup flux interpolation method was proposed. All the resonance groups were combined into an equivalent single group, and on the basis of this single resonance group, fixed-source equations were solved only for specific subgroup cross-sections. After obtaining the subgroup fluxes for specific cross-sections, the subgroup fluxes for the actual resonance groups were then obtained through escape cross-section interpolation, improving the efficiency of subgroup resonance calculations while ensuring accuracy.

3. Results

Based on the theoretical framework presented above, the optimized subgroup transport equation solving method was applied to practical problem solving (hereafter referred to as the optimized subgroup method, OSM), and comparative analyses were conducted with the conventional subgroup method (CSM) described in Section 2.1. To validate that OSM could effectively improve computational efficiency while maintaining equivalent accuracy to CSM, three progressively complex cases were sequentially selected for verification: a single fuel cell problem, a 2D assembly problem, and a 3D assembly problem. All validations were performed using the high-fidelity core-physics code developed by the method in this work [36], which employed the method of characteristics (MOC) for neutron transport equation solving and achieved 3D core calculations through the MOC-EX method. To facilitate comparative analysis, identical computer hardware configurations were maintained for all computational cases in this section.

3.1. Single-Cell Problem

This section constructed a pressurized water reactor (PWR) single-cell problem as illustrated in Figure 5, based on the JAEA benchmark problems [37,38]. The fuel rod radius r_f in the cell was 0.4095 cm, with a cell half-side length l_c of 0.63 cm. The moderator was H₂O, while the cladding and gas gap structures were omitted.

This case served as a reference cell, where all regions were maintained at 300 K under fully reflective boundary conditions. Four single-resonant-nuclide single-cell problems were established by loading the fuel with ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu respectively. The material compositions are listed in

Table 2. Material composition of the single resonant nuclide problem.

Material	Nuclide Number Density/(1024 cm−3)
	235U	238U	239Pu	241Pu	1H	16O
Single 235U	3.8879 × 10−5	\	\	\	\	4.5945 × 10−2
Single 238U	\	1.9159 × 10−2	\	\	\
Single 239Pu	\	\	2.1706 × 10−3	\	\
Single 241Pu	\	\	\	3.6732 × 10−4	\
Moderator	\	\	\	\	6.6630 × 10−2	3.3315 × 10−2

.

3.1.1. Analysis of Calculation Accuracy

The aforementioned problem was calculated using both OSM and CSM. The multigroup structure selected for this example consisted of 408 groups with 289 resonance energy groups. First, this section verified the application of the subgroup flux interpolation in fine-group calculations. Under this energy group structure, each resonant nuclide had approximately 2–3 subgroups per resonance fine group, and strictly solving the fixed-source subgroup equations according to this subgroup configuration required significant computational effort. The total numbers of subgroups in the resonance groups for ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu were 611, 475, 540, and 553 respectively. Traditional methods required solving fixed-source equations for each subgroup individually. Through equivalent single-resonance-group subgroup flux interpolation calculations, only eight fixed-source subgroup equations needed to be solved for each resonant nuclide, greatly reducing the computational burden.

Taking the absorption cross-sections as an example, Figure 6 shows the computational deviations of the fuel rod absorption cross-sections in the resonance fine groups calculated by CSM and OSM compared to the Monte Carlo benchmark solution. For ²³⁸U, the maximum relative deviation, average deviation, and root mean square (RMS) deviation of the absorption cross-sections calculated by CSM across all the resonance groups were −2.14%, −0.01%, and 0.43% respectively, while those calculated by OSM were −2.17%, −0.004%, and 0.44% respectively, indicating nearly identical computational accuracy between the two methods. Significant deviations only occurred in very few energy groups and had minimal impact on the overall results, with computational deviations for the vast majority of energy groups remaining within ±0.5%, demonstrating high computational precision. Compared with ²³⁸U, ²³⁵U, and ²³⁹Pu, ²⁴¹Pu exhibited even higher accuracy, with RMS deviations of 0.27%, 0.28%, and 0.26%, respectively, in OSM calculations. Through this analysis, the OSM results demonstrated comparable accuracy to CSM calculations, validating the effectiveness of subgroup flux interpolation in ensuring accurate resonance fine-group cross-section calculations within the fine-group/subgroup methodology.

3.1.2. Analysis of the Calculating Efficiency

Figure 7 illustrates the subgroup quantities per resonance fine group for the resonance nuclides including ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu. Most resonance groups employed 2–3 subgroups, while the energy groups with sufficiently discrete spacing or indistinct resonance peaks utilized a single subgroup configuration. In such cases, temperature-interpolated fine-group cross-sections were directly adopted as effective resonance cross-sections. The substantial total subgroup counts—611, 475, 540, and 553 subgroups for ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu, respectively—imposed significant computational burdens when rigorously solving subgroup fixed-source equations across all the fine-group structures. Traditional methods required fixed-source equation solutions for every subgroup. Through equivalent single-resonance-group subgroup flux interpolation, this approach reduced the required fixed-source equation solutions to merely eight subgroups per resonance nuclide, achieving dramatic computational load reduction. For these four nuclides, the total computational efficiency improvements reached were 98.6%, 98.3%, 98.5%, and 98.6%, respectively.

3.2. 2D Assembly Problem

3.2.1. Complex Single Assembly Configuration Problem

In typical pressurized water reactors (PWRs), fuel rods generally consist of identical fuel types. However, with the emergence of novel reactor designs, configurations containing multiple fuel types have become increasingly common. For instance, high conversion ratio reactors may employ assemblies with alternating UO₂ and MOX fuel arrangements. Figure 8 illustrates two such interleaved fuel assembly configurations: a 3 × 3 lattice featuring alternating UO₂ and MOX fuel cells, and a 14 × 14 assembly composed of four interleaved 7 × 7 UO₂ and MOX sub-assemblies. The resonance cross-section calculations for these mixed fuel configurations become significantly more complex due to the combined effects of the fuel-specific resonance self-shielding and mutual interference between adjacent fuel rods.

This section employed the JAEA benchmark problems involving the cold-state UO₂ and MOX single-cell configurations to construct the assembly cases shown in Figure 8, with all systems maintained at 300 K. Detailed material compositions and geometric specifications are provided in Reference [39]. For computational efficiency in assembly calculations, the HELIOS-47 group structure was adopted, consisting of nine fast groups, sixteen resonance groups, and twenty-two thermal groups. The method of characteristics (MOC) parameters were configured as follows: sixteen azimuthal angles and three polar angles per octant, with ray spacings of 0.01 cm for the 3 × 3 assembly and 0.03 cm for the 14 × 14 assembly. Benchmark values were obtained from Monte Carlo simulations.

For the 3 × 3 assembly, Figure 9 presents the OSM-calculated absorption cross-sections of ²³⁵U and ²³⁸U in Fuel Rods 1 and 2. The maximum computational deviations for the ²³⁸U absorption cross-sections in the UO₂ and MOX rods were 1.26% and −1.43% respectively, with most energy group deviations remaining within ±0.8%. ²³⁵U demonstrated relatively higher computational accuracy, showing maximum deviations of 0.83% and −0.87% in the UO₂ and MOX rods respectively. While relative deviations in resonance cross-sections increased compared to single-cell benchmarks due to combined intra-rod resonance interference and inter-rod spatial effects, the overall computational precision remained high. Figure 10 displays the relative deviations for the ²³⁹Pu and ²⁴¹Pu absorption cross-sections in Fuel Rod 2, with maximum deviations of −1.03% and −1.67%, respectively.

Table 3. Calculated results of the resonance cross-section of pin 1 and 2 in the 3 × 3 lattice.

Absorption XS Error	UO2 Pin 1		MOX Pin 2
	238U	235U	238U	235U	239Pu	240Pu	241Pu	242Pu
MAX	1.26%	0.83%	−1.43%	−0.87%	−1.03%	1.12%	−1.67%	−1.27%
AVG	0.18%	−0.09%	−0.12%	−0.14%	−0.03%	0.02%	−0.17%	−0.23%
RMS	0.52%	0.37%	0.55%	0.52%	0.60%	0.64%	0.53%	0.34%

summarizes the overall computational deviations for the resonance cross-sections in both fuel rods, demonstrating high accuracy across all resonant nuclides. The reference k_eff for the 3 × 3 assembly was 1.35551, with OSM yielding 1.35591 (absolute deviation: 40 pcm).

Regarding the 14 × 14 assembly, Figure 11 shows the normalized power distribution and absolute relative deviation magnitudes. MOX sub-assemblies exhibited peak normalized power at the fuel interfaces while showing lower power at the corners, contrasting with UO₂ sub-assemblies. The higher ²³⁵U enrichment in UO₂ fuel resulted in generally elevated power levels throughout the UO₂ regions. At the fuel interfaces, MOX rods created spatial mutual shielding effects—their strong resonance peaks significantly increased neutron absorption in these areas, reducing UO₂ fuel reactivity at the boundaries. OSM accurately captured this normalized power distribution, with the maximum deviation (1.69%) occurring in the interfaced MOX rods. The assembly exhibited RMS and average power deviations of 0.98% and 0.94%, respectively. While the power calculation deviations increased compared to conventional PWR assemblies, the overall precision remained satisfactory. The reference value k_eff at 1.38457 compared favorably with OSM’s 1.38467 (absolute deviation: 10 pcm). Comprehensive analysis confirmed OSM’s capability for the precise modeling of interleaved fuel configurations.

3.2.2. SCWR Problem

The SCWR assembly represents a simplified supercritical water reactor structure [40]. Due to its unique characteristics, the SCWR exhibited a mixed neutron energy spectrum combining thermal and fast neutrons, thereby demonstrating the capability of the proposed method in calculating fast neutron spectra. The geometric configuration is illustrated in Figure 12. The material compositions and geometric parameters of the fuel rods, guide tubes, and moderators in the two assemblies were obtained from reference [40], with the system temperature set to 300 K. Computational parameters were configured as follows: sixteen azimuthal angles and three polar angles per octant, with a characteristic line spacing of 0.03 cm. Benchmark values were provided by Monte Carlo methods.

Figure 13 presents the normalized power distribution and absolute relative deviation distribution of the SCWR assembly. Higher power densities were observed near the water cavity boundaries within the assembly, while lower power levels characterized the outer boundaries, demonstrating an overall radially decreasing power profile from the interior to the periphery. The OSM accurately resolved the power distribution of the SCWR assembly, with the absolute relative deviations in fuel rod power remaining below 0.5% across most regions. Slightly increased power deviations occurred at the water–cavity interface points and assembly corner regions. The maximum relative deviation of 0.87% was localized to the extreme lower-right corner of the assembly. The power calculation exhibited a mean deviation of 0.24% and a root mean square (RMS) deviation of 0.32%, indicating high computational accuracy. The benchmark k_eff value for this assembly was 1.50329, while the OSM-calculated k_eff was 1.50273, yielding an absolute deviation of −56 pcm. These results collectively demonstrated the capability of the OSM code’s resonance treatment module to achieve precise solutions for SCWR problems.

3.2.3. Multi-Assembly Configuration Problem

This section employed the VERA benchmark problem 4-2D (multi-assembly configuration) for analysis and validation. The assembly configuration was derived from the Unit 1 core of the Watts Bar Nuclear Plant in the United States, comprising nine 17 × 17 fuel assemblies arranged in a 3 × 3 checkerboard pattern. As shown in Figure 14, the configuration contained two fuel assembly types with enrichments of 2.11 wt% and 2.619 wt%, respectively, with each single assembly containing 25 guide tubes and 20 Pyrex rods. The core represented a fresh-fuel hot zero-power state with a uniform temperature distribution maintained at 600 K.

In this section, we performed computational analysis on the components under two operational conditions. The first condition was with the control rods withdrawn, where all the guide tubes in the components were filled with moderator. The second condition was with the control rods inserted, featuring the B₄C control rods placed in 24 guide tubes of the central assembly in the core. The computational parameters were set as follows: eight azimuthal angles and three polar angles selected per octant, with a characteristic ray spacing of 0.03 cm. Benchmark values were provided by reference [41].

Figure 15 and Figure 16 present the power calculation results for the VERA 4-2D 3 × 3 assembly under the control-rod-out and B₄C control-rod-in conditions, respectively. The maximum power deviations for the two scenarios were 1.32% and 1.37%, both occurring in the corner lattice cells. For the majority of the fuel rods, deviations remained below 1%, with average deviations of 0.18% and 0.23% for the respective conditions. The effective multiplication factor (k_eff) calculation deviations were 77 pcm and 113 pcm for the two configurations, which are shown in

Table 4. Calculated results for VERA 4-2D 3 × 3 lattice.

Type	Normalized Power Error			keff		keff Error/pcm
	MAX	AVG	RMS	Reference	OSM
No control rod	1.32%	0.18%	0.25%	1.01024	1.01101	77
With control rod	1.37%	0.23%	0.32%	0.98029	0.98142	113

. The rod-in condition exhibited steeper power distribution gradients, leading to relatively increased computational deviations while maintaining high accuracy. Overall, OSM demonstrated precise modeling capabilities for both control-rod-out and rod-in configurations in realistic 2D multi-assembly core problems.

3.3. 3D Assembly Problem

This section selected the VERA 3(a) and 3(b) benchmark problems for three-dimensional computational analysis. The benchmark geometry, shown in Figure 17, consisted of a typical Westinghouse single assembly that was axially extended, comprising fuel rods, spacer grids, upper/lower core plates, nozzle structures, and plenum regions. The fuel was UO₂ with a Zr-4 alloy cladding, spacer grids combining Zr-4 and Inconel alloys, and the moderator was borated water. Two configurations (Case 3A and 3B) were analyzed: Case 3A represented a standard assembly without absorber rods, while Case 3B incorporated 16 Pyrex rods within its guide tubes. Key parameters included fuel enrichments of 3.1 wt.% and 2.619 wt.%, system temperatures of 600 K and 565 K, and moderator boron concentrations of 1300 ppm and 1066 ppm for the respective cases. Computational parameters were set to eight azimuthal angles and three polar angles per octant, with a ray spacing of 0.05 cm. Benchmark values were sourced from reference [41].

Figure 18 and Figure 19 display the radially integrated power distributions for the 3A and 3B configurations. In Case 3A (Figure 14), the radial power profile decreased radially outward, showing a maximum deviation of 0.81% at the corner positions. Case 3B (Figure 15) exhibited a sharp power depression near the Pyrex rods and elevated power at the assembly peripheries, with a maximum deviation of 1.06% while maintaining sub−0.5% deviations across most regions. Both configurations demonstrated high fidelity in power prediction accuracy.

Figure 19 presents the axial normalized power distribution calculation results for Cases 3A and 3B. Both cases exhibited significant axial power variations, with power progressively increasing from both ends toward the central region. Localized power depressions occurred at the spacer grid positions due to the absorption effects of grid materials. The OSM code demonstrated accurate modeling of the three-dimensional axial power distributions, with computational deviations below 1% for most fuel layers. However, larger deviations emerged at the axial boundary regions. However, significant power calculation deviations were observed at the upper and lower axial boundaries of the assembly. On the one hand, the normalized power values at the axial boundaries were extremely low, which amplified relative deviations, though their impact on the global power distribution remained negligible. On the other hand, complex geometric features (e.g., gas plenums, springs, and nozzles) at the axial extremities introduced challenges in resonance self-shielding and neutron transport modeling accuracy.

Table 5. Calculated results for VERA 3-D benchmark.

VERA 3D Assembly	Radial Normalized Power Error			keff		keff Error/pcm
	MAX	AVG	RMS	Reference	OSM
3A	0.81%	0.15%	0.22%	1.17572	1.17488	−84
3B	1.06%	0.16%	0.25%	1.00015	1.00052	36

summarizes the deviations in radially integrated power and the effective multiplication factor. Both test cases demonstrated high overall computational fidelity, with a k_eff absolute deviation value of −84 and 36 pcm, respectively. It should be noted that −84 pcm and +36 pcm deviations in k_eff calculations are considered highly precise in reactor physics, as 1 pcm equals a 0.00001 reactivity change—these deviations fall well within industry-accepted margins (±200–300 pcm) and have negligible operational impact, demonstrating state-of-the-art computational accuracy for engineering applications.

4. Discussion

The optimized subgroup method (OSM) proposed in this work significantly enhanced computational efficiency for resonance self-shielding calculations by consolidating the resonance groups into an equivalent single group and interpolating subgroup fluxes through escape cross-sections. This approach reduced the number of fixed-source equations by up to 80% compared to conventional methods, as validated across single-cell, 2D assembly, and 3D assembly benchmarks. Results demonstrated high accuracy, with resonance absorption cross-section deviations consistently below 1% and k_eff errors within 120 pcm, meeting rigorous reactor physics standards. This method’s geometric adaptability was evident in its ability to resolve spatial self-shielding effects, such as power depression near the MOX fuel interfaces in the mixed lattices. A key limitation lies in the interpolation accuracy for regions with steep escape cross-section gradients, where the sampling density directly impacts precision. Future efforts could focus on adaptive subgroup sampling strategies or by extending the method to unstructured meshes for broader applicability. By drastically lowering the computational costs while maintaining accuracy, OSM enables high-fidelity resonance calculations for large-scale reactor systems, advancing the feasibility of detailed digital reactor simulations.

5. Conclusions

The present study successfully developed an optimized subgroup transport equation solving method, addressing the computational bottleneck associated with resonance calculations in high-fidelity reactor core simulations. By adopting an equivalent single-resonance-group subgroup flux interpolation approach, the method significantly reduced the number of fixed-source equations required, leading to substantial improvements in computational efficiency. Validation studies on various problem scales, including single-cell, 2D assembly, and 3D assembly cases, demonstrated that the optimized method maintained computational accuracy comparable to the conventional subgroup method. This breakthrough paves the way for more efficient and accurate modeling of complex reactor systems, particularly those with intricate material compositions and geometric configurations. The proposed optimization holds great promise for advancing the field of reactor physics and contributing to the development of next-generation nuclear energy technologies.