Analysis of the Ill-Posedness in Subgroup Parameter Calculation Based on Pade Approximation and Research on Improved Methods

Abstract

This paper addresses the ill-posed problem in calculating subgroup parameters for resonance self-shielding within nuclear reactor physics. The conventional Pade approximation method often yields negative subgroup cross-sections lacking physical meaning due to its treatment of overdetermined nonlinear systems, making the subgroup transport equations unsolvable. To overcome this, an optimized Pade approximation method is proposed: a resonance factor criterion is used to select energy groups requiring calculation; a systematic procedure dynamically traverses background cross-section combinations starting from a minimal subgroup number, incrementally increasing it until solutions meeting accuracy constraints with positive parameters are found; and, given the insufficiency of background points, a high-resolution resonance integral table is constructed, particularly for ranges exhibiting significant cross-section variations. Numerical validation confirms the method eliminates negative parameters, ensures physical validity, and significantly improves accuracy across benchmark cases including typical fuel pins, burnt pellets, and Gd-bearing lattices. This approach effectively resolves the ill-posedness of the traditional method, offering a more robust and precise subgroup resonance treatment for high-fidelity core neutronics.

1. Introduction

Nuclear reactor physics calculations are of great significance for the safe operation of reactors and the efficient and economical utilization of nuclear fuel [1]. The core objective of nuclear reactor physics calculations is to rapidly and accurately determine the distribution of neutron flux density inside the reactor, providing precise parameters for reactor core design, operation under various conditions, and control. Nuclear reactor physics calculation forms the foundation for core nuclear design, thermal–hydraulic design, and core safety analysis, and its central task is solving the neutron transport equation. The neutron transport equation is a balance equation describing neutron migration within a medium, governing the variation in neutrons with space, energy, direction of motion, and time. It is an integro-differential equation involving multi-dimensional variables. Due to the complexity and heterogeneity of geometry, materials, and structures in practical problems, obtaining exact solutions to the neutron transport equation for real-world scenarios is extremely difficult. Therefore, a series of approximations are typically introduced during actual calculations, employing corresponding numerical methods for solution. Among these, the most effective approximation for neutron energy is to divide neutrons into different energy groups based on energy magnitude. The neutron transport equation is then solved using group-wise cross-section parameters corresponding to these energy groups.

To ensure the accuracy of deterministic neutronics calculations, accurate multi-group effective cross-sections must be assigned to each energy group when employing the multi-group approximation. In many cases, the neutron reaction cross-sections of nuclides vary smoothly within a single energy group; hence, the effective multi-group cross-sections can be approximately considered independent of the actual neutron spectrum. In practical deterministic neutronics calculations, multi-group constant libraries are typically generated based on specific energy group structures. These libraries store the multi-group reaction cross-sections for different nuclides to be used in subsequent calculations. However, for neutrons in the energy range of 1 eV to 10,000 eV, their reaction cross-sections with several important reactor nuclides (such as U, Pu, Th, etc.) exhibit very strong variations with energy, as shown in Figure 1. This phenomenon of a sharp rise in neutron reaction cross-section at specific incident neutron energies is called a resonance. The strong resonance peaks in neutron reaction cross-sections cause a rapid decrease in neutron flux density, manifesting a “self-shielding” effect. Therefore, the resonance phenomenon is also referred to as resonance self-shielding, and the energy region where resonance self-shielding is particularly strong is called the resonance energy range. Due to the presence of resonance self-shielding, problem-dependent resonance self-shielded cross-sections must be precisely calculated for the target problem. This is the origin of resonance self-shielding calculation.

The resonance self-shielded cross-section is linked to various factors, such as the geometric shape of the actual problem, fuel composition and arrangement, temperature, and burn-up. This significantly increases the complexity of resonance self-shielding calculation. Simultaneously, as a front-end module for transport calculations, the computational time occupied by resonance self-shielding calculation should be minimized, imposing high demands on its efficiency. For over half a century, numerous studies on resonance self-shielding calculation methods have been conducted, which can be broadly classified into the following three categories:

(1) Ultra-fine-group method [2].

The ultra-fine-group method is a relatively direct approach. Its basic principle involves dividing the resonance energy range into extremely fine energy groups to avoid handling resonance peaks within a group. A simplified neutron transport equation, also called the neutron slowing-down equation, is established within each fine energy group. Subsequently, the problem-dependent ultra-fine-group neutron spectrum is obtained by calculating the neutron flux distribution for each energy group and each flat source region from top to bottom, enabling precise treatment of resonance self-shielding. The ultra-fine-group method began with Kier [3], who proposed calculating resonance integrals for two-region cells on fine energy groups. Building on this, the first practically applicable ultra-fine-group code, RABBLE [4], emerged. Since then, the ultra-fine-group method has found wide application in neutronics codes [5,6,7,8]. However, because it requires solving the slowing-down equation on ultra-fine (or continuous energy) groups, its calculation is relatively time-consuming. Ultra-fine-group methods represented by the PEACO [9] code significantly improve computational efficiency by pre-tabulating collision probabilities, but their geometric handling capabilities are limited. The geometric adaptability of the ultra-fine-group method can be enhanced by solving the slowing-down equation using MOC fixed-source calculations [10], although this approach is less computationally efficient compared to the method used in the PEACO code. The ultra-fine-group method cannot be directly applied to assembly or core-scale neutronics calculations. Nevertheless, it remains the foundation for contemporary multi-group resonance self-shielding calculation methods, as is evident in two aspects: First, the resonance integral tables used in modern multi-group libraries are mostly generated through ultra-fine-group slowing-down calculations. Second, the ultra-fine-group method can accurately handle multi-nuclide resonance interference effects, which are intractable in multi-group resonance self-shielding calculations. Based on this, ultra-fine-group calculations still play a key role in high-fidelity resonance self-shielding calculations.

(2) Equivalence theory [11]

The equivalence theory method is a resonance self-shielding calculation method that is widely used in industrial applications. In equivalence theory, a virtual cross-section called the background cross-section (also known as the dilution cross-section) is first defined based on the slowing-down effect experienced by the resonant nuclide in a homogeneous system. A series of homogeneous problems with different background cross-sections are constructed by setting different nucleon density ratios between non-resonant nuclei (usually H) and the resonant nucleus. Resonance self-shielded cross-sections for the resonant nuclide are obtained by solving these homogeneous problems using the ultra-fine-group method, establishing a one-to-one resonance integral table relating background cross-section to resonance self-shielded cross-section. When treating a heterogeneous problem, a background cross-section considering the slowing-down effect from both inside and outside the resonance region is calculated using the rational approximation for the first-flight collision probability [12] and the narrow resonance approximation [13]. This background cross-section is then used to rapidly interpolate the resonance integral table to obtain the resonance self-shielded cross-section within the heterogeneous system. Due to its numerical stability and high computational efficiency, equivalence theory has a very broad range of applications in reactor physics calculations and has undergone the most extensive verification. However, because equivalence theory introduces multiple approximations or assumptions during its derivation, it suffers from problems such as limited accuracy and poor geometric handling capability. These limitations restrict the direct application of equivalence theory in high-fidelity resonance self-shielding calculations. Nevertheless, as mentioned earlier, equivalence theory also has advantages, like high computational efficiency and rich application/verification experience. Numerous international studies have focused on improving equivalence theory [14,15,16,17], so its role in full-core high-fidelity resonance self-shielding calculations remains significant.

(3) Subgroup method [18]

The subgroup method divides a broad resonance energy group into several subgroups based on the magnitude of the cross-section values, reducing the fluctuation in cross-sections within each subgroup. Within each subgroup, subgroup parameters must first be calculated. The calculation of subgroup parameters is primarily carried out through fitting based on resonance integral tables [19] or by processing continuous energy point cross-sections to preserve cross-section moments [20]. After obtaining reliable subgroup parameters, the subgroup transport equation can be established and solved to yield the subgroup flux. The final resonance self-shielded cross-section is then obtained based on the combined results of the subgroups within the broad group. In terms of accuracy, the subgroup method is superior to equivalence theory, and its integration with transport calculations enhances its geometric adaptability. In terms of computational efficiency, although lower than equivalence theory, the subgroup method is significantly more efficient than the ultra-fine-group method because only a few subgroups need to be calculated within a broad resonance energy group. However, a key step in the subgroup method, namely solving for the subgroup parameters, inevitably suffers from numerical instability. Furthermore, similar to equivalence theory, the subgroup method must rely on additional techniques, such as Bondarenko iteration [21], to handle multi-nuclide resonance interference effects.

Due to its relatively high computational accuracy and efficiency, along with its strong geometric adaptability, the subgroup method has become the most commonly used resonance calculation method internationally. The subgroup method requires first calculating subgroup parameters, which include subgroup cross-sections and subgroup probabilities, described in the form of probability tables to represent the sharply varying resonance cross-sections. To calculate subgroup parameters, Cullen [22] proposed a fitting method. This method’s subgroup parameters are no longer constrained by the actual shape of resonance peaks but are solved by establishing a correspondence between background cross-section and effective resonance cross-section. Yamamoto [23] utilized this correspondence, selecting different background cross-section points to establish a nonlinear well-posed system of equations, and then solved for subgroup parameters using the Pade approximation. Ribon [24] proposed the “moment method,” calculating subgroup parameters by defining cross-section moments of different orders. Hébert [25,26] and others improved the moment method, optimizing subgroup parameter calculation for the resolved resonance region using the root mean square method and enhancing its accuracy. Due to the complexity and time consumption of calculating cross-section moments, most commercial codes employ the fitting method to compute subgroup parameters. However, the mathematical process of subgroup parameter calculation introduces numerous approximations, and the subgroup parameters obtained by solving the well-posed system of equations can deviate from their actual physical meaning. For example, negative subgroup cross-section values can be solved, rendering the subgroup transport equation unsolvable [27]. To address this issue, the HELIOS code [28] fixes the subgroup absorption cross-section values based on trial and error and computational experience, and then calculates other subgroup parameters based on these fixed values, circumventing the ill-posedness. Similar to HELIOS, Joo et al. [29] proposed a method to fit subgroup probabilities by fixing subgroup cross-section values. This method predetermines subgroup cross-section values and then calculates subgroup probabilities using constrained least-squares fitting. By predefining subgroup cross-sections, this method avoids generating negative values, but the process of determining these cross-section values is complex and lacks a theoretical basis. Peng et al. [30] proposed a constrained optimization method to calculate subgroup parameters. This method treats subgroup parameters as variables to be solved and calls the numerical computation program MATHEMATICA to optimize the fit by minimizing the deviation in reproducing resonance integrals.

However, synthesizing the above research, existing subgroup parameter calculation methods have deficiencies. The most significant is the lack of clarity on how to avoid ill-posedness in the subgroup-parameter-solving process. Current computational processes still rely on empirical optimization or trial and error to determine parameter values, resulting in poor applicability [31]. Therefore, addressing the limitations of the traditional subgroup method, this paper analyzes the ill-posed problem in the conventional Pade approximation for calculating subgroup parameters and optimizes its computational workflow to ensure the physical meaning of the subgroup parameters.

2. Methods

2.1. Subgroup Method and Subgroup Parameters

According to the multi-group approximation hypothesis, the neutron flux density within a single energy group is approximated as the total value over the energy grid. In reality, due to the sharp variations in cross-sections within the resonance energy range, only the dense group structure of the ultra-fine-group method can strictly guarantee the accuracy of the multi-group approximation. For commonly used multi-group structures, significant flux gradient variations still exist within the energy segments of a resonance group. Unlike traditional discretization in the energy dimension, the subgroup method further divides the resonance group into several subgroups based on the magnitude of the cross-section values, as shown in Figure 2. The variation in the cross-section within each subgroup is relatively small; therefore, the flux variation within the same subgroup is much smaller than that within a traditional resonance group. Compared to the energy group structure of the ultra-fine-group method, the subgroup method achieves precise discretization of the resonance group using only a few subgroups, offering higher computational efficiency [32,33].

As can be seen from Figure 2, the same range of subgroup cross-section values may correspond to different energy intervals; therefore, the energies within the same subgroup are not continuous. The subgroup method transforms the continuous Riemann integral of the neutron spectrum with respect to energy, f(E), into a Lebesgue integral with respect to the vertical cross-section, f(σ), as shown in Equation (1):

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

(1)

where σ_min and σ_max are the minimum and maximum cross-section values within energy group g, respectively, and p(σ) is the probability that the neutron reaction cross-section value in group g is exactly σ.

The transformation from a Riemann integral over energy to a Lebesgue-like integral over cross-sectional values in Equation (1) can be formally justified through both mathematical and physical arguments. Mathematically, this relies on the variable substitution theorem in integration theory: since the neutron reaction cross-section σ is a measurable function of energy E (i.e., σ = σ(E) for E within the resonance group g), the integral over energy can be reparameterized by σ when σ(E) is invertible over its range [σ_min, σ_max]. Physically, this transformation is justified by the nature of resonance phenomena: while the neutron flux f(E) varies sharply with E due to resonance peaks, the flux as a function of cross-section f(σ) exhibits smoother behavior (approximately inversely proportional to σ). This regularity makes the Lebesgue-like integral over σ more tractable, as it avoids resolving the fine-scale energy oscillations that complicate the Riemann integral. Additionally, the normalization of p(σ) (i.e., ${\int }_{{\sigma }_{min}}^{{\sigma }_{max}}p\left(\sigma \right)\mathrm{d}\sigma =1$) ensures the total probability over all cross-section values within the group equals the total energy fraction, maintaining consistency with the original Riemann integral’s physical meaning.

Compared to the irregular and sharp variation in f(E) with energy, f(σ) exhibits an approximately inverse proportional relationship with the cross-section, making the integral calculation simpler. It can be computed using fewer discrete points, each point representing a subgroup. The subgroup cross-section is the average cross-section value for that subgroup, and the subgroup probability reflects the proportion of the total energy range of the resonance group occupied by the energy intervals belonging to that subgroup. The subgroup cross-section, subgroup probability, and subgroup flux for the same subgroup are all constant values. Therefore, Equation (1) can be simplified to Equation (2):

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

(2)

where σ_g,i is the subgroup cross-section; p_g,i is the subgroup probability; ϕ_g,i is the subgroup flux; and I is the number of subgroups.

From the definition of subgroup probability, the sum of all subgroup probabilities should always equal 1. The subgroup cross-section and subgroup probability are collectively referred to as subgroup parameters. Equation (2) transforms the continuous integral of flux with respect to energy into a discrete quadrature set concerning the subgroup parameters. Mathematically, the range of values for subgroup cross-sections is not limited to σ_min~σ_max, and subgroup probabilities no longer correspond to physically meaningful energy ranges. Therefore, the effective resonance cross-section for reaction channel *x* can be calculated using Equation (3):

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

(3)

Similar to the steady-state neutron transport equation, the neutron transport equation for subgroup i of resonance group g can be expressed by Equation (4):

$$\mathsf{\Omega }\cdot \nabla {\phi }_{g,i}(\mathit{r},\mathsf{\Omega })+{\sum }_{t,g,i}(\mathit{r}){\phi }_{g,i}(\mathit{r},\mathsf{\Omega })=Q_{g,i}(\mathit{r})$$

(4)

The subgroup flux is obtained by solving Equation (4) using a transport module, and then substituting it into Equation (3) yields the effective resonance cross-section. By calculating the actual problem’s neutron spectrum through the subgroup transport equation, resonance self-shielding calculations can be performed for arbitrary geometric problems.

In a homogeneous system, for the resonant nuclide R, the background cross-section σ_b is defined as in Equation (5):

$${\sigma }_b={\displaystyle \frac{{\displaystyle \sum _l^L{\lambda }_l{\sum }_{p,l}}}{N_R}}$$

(5)

where N_R is the nucleon density of the resonant nuclide, and L is the number of nuclide types in the system.

The background cross-section reflects the degree to which the resonant nuclide is influenced by the scattering effects of the entire system. When the moderator increases, the background cross-section increases, neutron moderation becomes more thorough, interactions with the resonant nuclide become more intense, and consequently, the resonance cross-section also increases. The intermediate resonance approximation cross-section σ_int is defined as in Equation (6):

$${\sigma }_{int}={\sigma }_t-\left(1-\lambda \right){\sigma }_s-\lambda {\sigma }_p={\sigma }_a+\lambda \left({\sigma }_s-{\sigma }_p\right)$$

(6)

Under homogeneous problem conditions, combining Equations (5) and (6), the formula for the scalar neutron flux can be expressed as Equation (7):

$$\varphi ={\displaystyle \frac{{\sigma }_b}{{\sigma }_{int}+{\sigma }_b}}$$

(7)

Substituting Equation (7) into Equation (3), the correspondence between the effective intermediate resonance approximation cross-section for energy group g, and the subgroup cross-sections, subgroup probabilities, and background cross-section can be obtained, as shown in Equation (8):

$${\sigma }_{int,g}={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{\sigma }_{int,g,i}{p}_{g,i}\frac{{\sigma }_{b,g}}{{\sigma }_{int,g,i}+{\sigma }_{b,g}}}}{{\displaystyle \sum _{i=1}^I{p}_{g,i}\frac{{\sigma }_{b,g}}{{\sigma }_{int,g,i}+{\sigma }_{b,g}}}}}$$

(8)

Equation (8) establishes the relationship between the effective intermediate resonance cross-section ${\sigma }_{int,g}$, subgroup parameters (subgroup cross-sections ${\sigma }_{int,g,i}$ and subgroup probabilities $p_{g,i}$), and the background cross-section ${\sigma }_{b,g}$ in a homogeneous system. Based on this relationship, the subgroup parameters can be derived. However, this equation is inherently a nonlinear rational function of σ_b, where both the numerator and denominator are weighted sums of subgroup terms. Direct linear solving for the subgroup parameters {${\sigma }_{int,g,i}$, $p_{g,i}$} is not feasible, necessitating polynomial transformation to apply Pade approximation.

2.2. Subgroup Parameter Calculation Based on Pade Approximation and Analysis of Its Ill-Posedness

In a homogeneous system composed of resonant nuclide R and hydrogen nuclide H, by continuously varying the nucleon density and solving the ultra-fine-group slowing-down equation, a series of background cross-sections and their corresponding effective resonance cross-sections can be obtained. Combining Equation (8) under different conditions yields a system of nonlinear equations with subgroup cross-sections and subgroup probabilities as unknowns. Solving this nonlinear system directly is very complex. Using the Pade approximation method, the nonlinear system can be transformed into a linear system of equations that is easier to solve [34]. To simplify the presentation, the energy group subscript g will be omitted in the following text of this section. For given positive integers L and M, if a function f(x) satisfies Equation (9), then P_L(x)/Q_M(x) is called the [L, M]-order Pade approximant of f(x).

$$f(x)-o\left(x^{L+M+1}\right)={\displaystyle \frac{P_L\left(x\right)}{Q_M\left(x\right)}}={\displaystyle \frac{a_0+a_{1}x+a_2{x}^2+\dots +a_L{x}^L}{b_0+b_{1}x+b_2{x}^2+\dots +b_M{x}^M}}$$

(9)

Analogous to Equation (9), to eliminate the fractional structure, both the numerator and denominator of the fraction on the right side of Equation (8) can be multiplied by ${\prod }_i^I\left({\sigma }_{int,i}+{\sigma }_b\right)$ and the yields simplified:

$$f\left({\sigma }_b\right)={\displaystyle \frac{{\displaystyle \sum _i^I\left[{\sigma }_{int,i}{p}_i{\displaystyle \prod _{j=1,j\ne i}^I\left({\sigma }_{int,j}+{\sigma }_b\right)}\right]}}{{\displaystyle \sum _i^I\left[p_i{\displaystyle \prod _{j=1,j\ne i}^I\left({\sigma }_{int,j}+{\sigma }_b\right)}\right]}}}$$

(10)

Both sides of Equation (10) are polynomials in σ_b, which can be expanded and combined into standard polynomial forms. The value of the function f(σ_b) is σ_int. Expanding the polynomial in the fraction on the right side of Equation (10) and combining like terms containing σ_b simplifies it to Equation (11):

$${\sigma }_{int}=f\left({\sigma }_b\right)={\displaystyle \frac{P_{I-1}\left({\sigma }_b\right)}{Q_{I-1}\left({\sigma }_b\right)}}={\displaystyle \frac{a_0+a_1{\sigma }_b+a_2{{\sigma }_b}^2+\dots +a_{I-1}{{\sigma }_b}^{I-1}}{b_0+b_1{\sigma }_b+b_2{{\sigma }_b}^2+\dots +b_{I-1}{{\sigma }_b}^{I-1}}}$$

(11)

where a_n and b_n are both polynomials concerning σ_int,i and p_i; $b_{I-1}={\sum }_i^I{p}_i=1$ (due to the normalization of subgroup probabilities), and lower-degree coefficients, $b_i$, depend on both σ_int,j and p_i.

Equation (11) is the [I−1, I−1]-order Pade approximant of the function f(σ_b). Both sides of Equation (8) are multiped by Q_I−1(σ_b), and the constant term is moved to the right side of the equation. Q_I−1(σ_b) approximates the denominator of Equation (8) via the polynomial expansion of weighted subgroup terms, with its leading coefficient fixed at 1 to ensure physical normalization. P_I−1(σ_b) corresponds to the polynomial approximation of the numerator, enabling the transformation of the nonlinear rational equation into a linear system solvable for subgroup parameters. Combined with Equation (11), Equation (8) simplifies to the linear equation shown in Equation (12):

$${\sigma }_b^{I-1}{a}_{I-1}+{\sigma }_b^{I-2}{a}_{I-2}+\cdots +{\sigma }_b^0{a}_0-{\sigma }_{int}{\sigma }_b^{I-2}{b}_{I-2}-\cdots -{\sigma }_{int}{\sigma }_b^0{b}_0={\sigma }_{int}{\sigma }_b^{I-1}$$

(12)

Since the value of b_I−1 is known to be 1, the total number of unknowns in Equation (12) is 2I − 1. Selecting 2I − 1 background cross-sections and their corresponding intermediate resonance approximation cross-sections, combining Equation (12) yields a well-posed system of linear equations, as shown in Equation (13):

$$\left[\begin{array}{ccccccc}{\sigma }_{b_1}^{I-1}& {\sigma }_{b_1}^{I-2}& \cdots & {\sigma }_{b_1}^0& {\sigma }_{in{t}_1}{\sigma }_{b_1}^{I-2}& \cdots & {\sigma }_{in{t}_1}{\sigma }_{b_1}^0\\ {\sigma }_{b_2}^{I-1}& {\sigma }_{b_2}^{I-2}& \cdots & {\sigma }_{b_2}^0& {\sigma }_{in{t}_2}{\sigma }_{b_2}^{I-2}& \cdots & {\sigma }_{in{t}_2}{\sigma }_{b_2}^0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {\sigma }_{b_{I-1}}^{I-1}& {\sigma }_{b_{I-1}}^{I-2}& \cdots & {\sigma }_{b_{I-1}}^0& {\sigma }_{in{t}_{I-1}}{\sigma }_{b_{I-1}}^{I-2}& \cdots & {\sigma }_{in{t}_{I-1}}{\sigma }_{b_{I-1}}^0\end{array}\right]\left[\begin{array}{c}{a}_{I-1}\\ a_{I-2}\\ \cdots \\ a_0\\ b_{I-2}\\ \cdots \\ b_0\end{array}\right]=\left[\begin{array}{c}{\sigma }_{in{t}_1}{\sigma }_{b_1}^{I-1}\\ {\sigma }_{in{t}_2}{\sigma }_{b_2}^{I-1}\\ \cdots \\ {\sigma }_{in{t}_{I-1}}{\sigma }_{b_{I-1}}^{I-1}\end{array}\right]$$

(13)

Solving this linear system yields the coefficients a_n and b_n. Similar to Equation (12), multiplying both sides of Equation (8) by the denominator on the right side and combining like terms gives:

$${\displaystyle \sum _{i=1}^I\frac{p_i\left[f\left({\sigma }_b\right)-{\sigma }_{int,i}\right]}{{\sigma }_{int,i}+{\sigma }_b}}=0$$

(14)

Physically, the background cross-section σ_b ranges over (0, +∞). Mathematically, the domain of σ_b in Equation (14) is (−∞, −σ_int,i) ∪ (−σ_int,i, +∞), and the function is continuously differentiable within both intervals. Therefore, when σ_b approaches −σ_int,i infinitely closely, the numerator of Equation (14) must be a higher-order infinitesimal relative to the denominator. This leads to

$$\underset{{\sigma }_b\to -{\sigma }_{int,i}}{\lim }f\left({\sigma }_b\right)={\sigma }_{int,i}$$

(15)

Substituting Equation (15) into Equation (11) and simplifying yields the I-th degree polynomial shown in Equation (16):

$${\left(-{\sigma }_{int,i}\right)}^I+\left(b_{I-2}-a_{I-1}\right){\left(-{\sigma }_{int,i}\right)}^{I-1}+\cdots +\left(b_0-a_1\right)\left(-{\sigma }_{int,i}\right)-a_0=0$$

(16)

All real roots of Equation (16) are the negatives of the intermediate resonance approximation subgroup cross-sections for each subgroup. Since the right sides of Equations (10) and (11) are identical, Q_I−1(σ_b) can be expressed as shown in Equation (17):

$$Q_{I-1}\left({\sigma }_b\right)=b_0+b_1{\sigma }_b+b_2{{\sigma }_b}^2\cdots +b_{I-1}{{\sigma }_b}^{I-1}={\displaystyle \sum _i^I\left[p_i{\displaystyle \prod _{j=1,j\ne i}^I\left({\sigma }_{int,j}+{\sigma }_b\right)}\right]}$$

(17)

Setting σ_b = −σ_int,i and substituting into Equation (17) simplifies to yield the subgroup probability for subgroup i, as shown in Equation (18):

$$p_i={\displaystyle \frac{{\left(-{\sigma }_{int,i}\right)}^{I-1}+{\displaystyle \sum _{j=0}^{I-2}{b}_j{\left(-{\sigma }_{int,i}\right)}^j}}{{\displaystyle \prod _{j=1,j\ne i}^I\left({\sigma }_{int,j}-{\sigma }_{int,i}\right)}}}$$

(18)

The calculation process for subgroup cross-sections of other reaction channels, such as the absorption cross-section or fission cross-section, is similar to the method described above. Taking the absorption cross-section as an example, combining Equations (3) and (7), the effective resonance absorption cross-section is given by Equation (19):

$${\sigma }_a={\displaystyle \frac{{\displaystyle \sum _{i=1}^I{\sigma }_{a,i}{p}_i\frac{{\sigma }_b}{{\sigma }_{int,i}+{\sigma }_b}}}{{\displaystyle \sum _{i=1}^I{p}_i\frac{{\sigma }_b}{{\sigma }_{int,i}+{\sigma }_b}}}}$$

(19)

Similar to the treatment of Equations (10) and (11), the Pade approximant expression for the absorption cross-section is

$$f\left({\sigma }_b\right)={\displaystyle \frac{R_{I-1}\left({\sigma }_b\right)}{Q_{I-1}\left({\sigma }_b\right)}}={\displaystyle \frac{c_0+c_1{\sigma }_b+c_2{{\sigma }_b}^2+\cdots +c_{I-1}{{\sigma }_b}^{I-1}}{b_0+b_1{\sigma }_b+b_2{{\sigma }_b}^2+\cdots +b_{I-1}{{\sigma }_b}^{I-1}}}$$

(20)

Since the terms in the denominator on the right side of Equation (20) have already been obtained from solving Equation (13), the numerator can be transformed into a polynomial coefficient fitting problem. Using a series of background cross-sections and their corresponding effective absorption cross-sections, the polynomial coefficients, c_n, can be obtained via least-squares fitting. Simultaneously, when σ_b approaches −σ_int,i infinitely closely, Equation (21) must hold:

$$\underset{{\sigma }_b\to -{\sigma }_{int,i}}{\lim }{f}_a\left({\sigma }_b\right)={\sigma }_{a,i}$$

(21)

After obtaining c_n, substituting Equation (21) into Equation (20) and simplifying yields the formula for calculating the subgroup absorption cross-section:

$${\sigma }_{a,i}={\displaystyle \frac{{\displaystyle \sum _{j=1}^{I-1}{c}_j{\left(-{\sigma }_{int,i}\right)}^j}}{{\left(-{\sigma }_{int,i}\right)}^{I-1}+{\displaystyle \prod _{j=1}^{I-2}{b}_j{\left(-{\sigma }_{int,i}\right)}^j}}}$$

(22)

The calculation methods for other subgroup partial cross-sections are the same as those used for the absorption cross-section, and so on, until all subgroup parameters are solved. At this point, for subgroup i in energy group g, its subgroup transport equation can be expressed as

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

(23)

For regions containing the current resonant nuclide, $\sum$_t’,g,i can be obtained from Equation (24). For regions not containing the current resonant nuclide, $\sum$_t’,g,i is set to zero.

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

(24)

As seen from Equation (23), the source term in 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. The subgroup fixed-source equation only needs to be solved independently for each subgroup, without requiring iterative cycling between different subgroups.

The Pade approximation method solves the nonlinear system through variable transformation. The range of its numerical solution only satisfies the mathematical domain definition and is no longer constrained by actual physical meaning. Consequently, under specific conditions, the Pade approximation method may compute subgroup parameters with negative values, rendering the subgroup transport equation unsolvable. Ensuring the physical meaning of subgroup parameters is a crucial prerequisite for subgroup resonance calculations.

Taking the energy range 6.48–7.34 eV as an example, ²³⁸U exhibits significant resonance peaks within this group. The absorption cross-section of ²³⁸U rises sharply around 6.7 eV, with large variations in cross-section values within the resonance group. Figure 3 shows the variation trend of the effective absorption cross-section with background cross-section for this group. The effective absorption cross-section changes most dramatically within the background cross-section range of 10~10⁵ b. As the system approaches infinite dilution conditions (background cross-section > 10⁵ b), the effective absorption cross-section stabilizes.

Using the Pade approximation method to calculate the subgroup parameters for ²³⁸U in this energy group. To adequately describe the resonance peak variation, the number of subgroups is chosen as 5, requiring 9 background cross-section points to form a well-posed equation system. The WLUP format database typically provides resonance integrals for 10 background cross-sections: 10, 28, 52, 64, 140, 260, 1000, 3600, 20,000, and 10¹⁰ b. Selecting the last 9 background cross-section points, the calculated subgroup parameters for ²³⁸U are shown in

Table 1. Subgroup parameters for ²³⁸U in energy range of 6.48~7.34 eV, as determined by conventional Pade approximation.

Subgroup No.	Subgroup Weight	Subgroup XS Type
		Total XS/b	Scattering XS/b	Absorption XS/b	Fission Production/b
1	1.46798 × 10−1	1.25201 × 103	7.86523 × 101	1.17336 × 103	1.28242 × 10−3
2	1.35515 × 10−1	6.16003 × 103	3.81676 × 102	5.77835 × 103	6.25391 × 10−3
3	3.01309 × 10−5	−5.97155 × 102	−5.75200 × 102	−2.19581 × 101	9.52770 × 10−4
4	2.93851 × 10−1	1.56231 × 102	2.32727 × 101	1.32958 × 102	1.37348 × 10−4
5	4.23806 × 10−1	2.94230 × 101	1.61830 × 101	1.32400 × 101	6.66726 × 10−6

.

As seen in Table 1, the subgroup cross-section for subgroup 3 appears as a negative value, which contradicts the physical meaning of a cross-section. A negative cross-section will cause the subgroup fixed-source equation to diverge, preventing the solution for the subgroup flux. The reason for the negative cross-section lies in the process of solving the nonlinear system. The Pade approximation method requires selecting a specific number of background cross-section points to establish a well-posed equation system. However, since every background cross-section point yields a nonlinear equation, the actual nonlinear system has more equations than unknowns. Therefore, calculating subgroup parameters requires solving an overdetermined nonlinear system, which is inherently an ill-posed problem and cannot be solved directly through analytical methods. The subgroup parameters obtained by the Pade approximation method by establishing a well-posed system only satisfy the variation pattern for a subset of the background cross-sections and deviate from the actual overdetermined system. Consequently, it is prone to yielding subgroup parameters that violate physical meaning.

2.3. Improvement Method for Ill-Posedness of Pade Approximation Method

To avoid negative subgroup parameters caused by ill-posedness, traditional methods are used to select different combinations of subgroup numbers or background cross-section points to obtain various subgroup parameters, and then the solution that conforms to physical meaning is chosen. However, the selection rules for subgroup numbers or background cross-section points are not clear. For I subgroups, the Pade approximation method requires selecting 2I−1 background cross-section points to establish a system of nonlinear equations. Multi-group databases usually store resonance integrals under different background cross-sections, and the selection of background cross-section points has a significant impact on subgroup parameters. As mentioned above, more subgroups can more accurately describe the severe fluctuations in resonance cross-sections, but this will increase the number of solutions for the subgroup fixed-source equation and reduce computational efficiency. Meanwhile, a larger number of subgroups will increase the number of unknowns in the nonlinear equations composed of subgroup parameters, leading to more severe ill-posedness. Therefore, the number of subgroups should be minimized under the premise of meeting accuracy requirements. Additionally, in the resonance energy range, there may be energy segments where the cross-section of resonant nuclides changes relatively gently. For example, the resonance cross-section of ²³⁸U in the range of 9.88~16.0 eV is almost constant. Thus, a criterion for subgroup calculation is defined as shown in Equation (25):

$$R_f=\left|{\displaystyle \frac{{\sigma }_{t,g,10}}{{\sigma }_{t,g,\infty }}}-1\right|$$

(25)

where σ_t,g,10 and σ_t,g,∞ represent the total cross-sections under the background cross-section of 10 b and infinite dilution condition, respectively.

The subgroup method only performs calculations when R_f is greater than or equal to 0.01. If R_f is less than 0.01, it is considered that the resonance phenomenon of the nuclide in this energy group is not significant, and the resonance cross-section under the infinite dilution condition can be directly used as the effective cross-section.

After calculating the subgroup parameters, according to the conversion relationship between resonance integrals and resonance cross-sections, the resonance integral or effective resonance cross-section of reaction channel x is back-calculated using the subgroup parameters. The calculation deviations are shown in Equations (26) and (27), respectively:

$$R_{error,RI}=\left|{\displaystyle \frac{{\displaystyle \sum _{i=1}^I{\sigma }_{x,i}{p}_i\frac{{\sigma }_b}{{\sigma }_{int,i}+{\sigma }_b}}-R{I}_x\left({\sigma }_b\right)}{R{I}_x\left({\sigma }_b\right)}}\right|\times 100\%$$

(26)

$$R_{error,\sigma }=\left|{\displaystyle \frac{\frac{{\displaystyle \sum _{i=1}^I{\sigma }_{x,i}{p}_i\frac{{\sigma }_b}{{{\sigma }_{int,}}_i+{\sigma }_b}}}{{\displaystyle \sum _{i=1}^I{p}_i\frac{{\sigma }_b}{{{\sigma }_{int,}}_i+{\sigma }_b}}}-{\sigma }_x({\sigma }_b)}{{\sigma }_x({\sigma }_b)}}\right|\times 100\%$$

(27)

In the process of calculating subgroup parameters, the number of subgroups is first set to 2, and at this time, 3 background cross-section points need to be selected to establish equations for solving subgroup parameters. In the database, resonance integrals under different background cross-sections are stored. All combinations of 3 background cross-sections are traversed, subgroup parameters are calculated, and then the resonance integral table provided by the database is back-calculated using these subgroup parameters. When the calculation deviation shown in Equation (26) or (27) is less than 1% for all background cross-sections, and the subgroup parameters are all positive, the traversal calculation can be terminated. If no combination of background cross-sections under the current number of subgroups can meet the calculation requirements, the number of subgroups is increased by 1, and the above process is repeated. To ensure computational efficiency, the maximum number of subgroups is limited to 5. If no qualified subgroup parameters can be obtained when the maximum number of subgroups is reached, the deviation criterion shown in Equation (26) or (27) is increased by 0.1%, and the calculation is restarted with the number of subgroups set to 2. The above process is repeated continuously until subgroup parameters with positive values that meet the deviation criterion are obtained.

Taking the WLUP format database as an example, the number of background cross-section points for resonance integrals provided by default is only 10. According to the Padé approximation method, if the number of subgroups is 5, 9 background cross-section points are needed to establish a system of nonlinear equations. At this time, even if all combinations of background cross-section selections are traversed, only 10 different selection schemes can be provided, which may easily lead to a situation where all schemes cannot meet the calculation requirements. Therefore, it is necessary to supplement the background cross-section points of the resonance integrals in the database.

The resonance integral table can be obtained by solving the ultra-fine-group equations of a homogeneous system or using a Monte Carlo program. For reaction channel x, the corresponding relationship between the resonance integral and the resonance cross-section is shown in Equation (28):

$$R{I}_x\left({\sigma }_p\right)={\displaystyle \frac{{\sigma }_p{\sigma }_x\left({\sigma }_p\right)}{{\sigma }_p+{\sigma }_a\left({\sigma }_p\right)}}$$

(28)

In a homogeneous system composed of resonant nuclide R and hydrogen nucleus H, the number density of the resonant nuclide is fixed, and the number density of hydrogen nuclei is continuously adjusted to obtain effective resonance cross-section values under different background cross-sections. After calculating the effective resonance cross-sections under various background cross-section conditions using the ultra-fine group or Monte Carlo program, the intermediate resonance approximation cross-section is calculated using Equation (24). Combined with Equation (28), the resonance integral under the intermediate resonance approximation condition is shown in Equation (29):

$$R{I}_x\left({\sigma }_b\right)={\displaystyle \frac{{\sigma }_b{\sigma }_x\left({\sigma }_b\right)}{{\sigma }_b+{\sigma }_{int}\left({\sigma }_b\right)}}$$

(29)

where RI_x is the resonance integral of reaction channel x; σ_b is the background cross-section; and σ_int is the intermediate resonance approximation cross-section.

It can be seen from Figure 3 that the effective resonance cross-section changes most drastically when the background cross-section is between 10 and 10⁵ b. After exceeding 10⁵ b, the background cross-section is close to the infinite dilution condition, and the effective resonance cross-section almost no longer changes. Therefore, during the production of the resonance integral table, the number of background cross-section points between 10 and 10⁵ b should be increased. In this paper, the number of included background cross-section points is 40, and their values are shown in

Table 2. Background cross-sections of resonance integral.

Background XS Range/b	Number	Background XS/b
10~102	9	10, 20, 30, 40, 50, 60, 70, 80, 90
102~103	11	100, 120, 150, 160, 200, 250, 300, 500, 750, 800, 900
103~104	9	1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000
104~105	7	10,000, 20,000, 30,000, 40,000, 50,000, 75,000, 90,000
105~1010	4	105, 106, 107, 1010
Total	40	\

.

3. Numerical Validation

3.1. Subgroup Parameters of ²³⁸U in the 6.48~7.34 eV Energy Range

To further validate the effectiveness of the optimized Padé approximation method in solving the ill-posedness of subgroup parameter calculation, this section applies the improved method to calculate the subgroup parameters of ²³⁸U in the 6.48~7.34 eV energy group, which is characterized by significant resonance peaks. Taking five subgroups as an example, the calculated subgroup parameters are listed in

Table 3. Subgroup parameters for ²³⁸U in energy range of 6.48~7.34 eV by optimized Pade approximation.

Subgroup No.	Subgroup Weight	Subgroup XS Type
		Total XS/b	Scattering XS/b	Absorption XS/b	Total XS/b
1	1.48010 × 10−1	2.02922 × 103	1.31704 × 102	1.89752 × 103	2.05550 × 10−3
2	1.00289 × 10−1	7.02553 × 103	4.31070 × 102	6.59446 × 103	7.14170 × 10−3
3	2.43853 × 10−1	1.06825 × 102	2.44676 × 101	8.23577 × 101	7.51630 × 10−5
4	1.41928 × 10−1	2.69198 × 102	1.87610 × 101	2.50437 × 102	2.84496 × 10−4
5	3.65920 × 10−1	2.60996 × 101	1.55207 × 101	1.05789 × 101	4.60302 × 10−6

.

Table 3 provides detailed information on the subgroup parameters of ²³⁸U in the 6.48~7.34 eV energy group obtained by the optimized Padé approximation method. Specifically, the table includes five subgroups, each with a subgroup weight and four types of cross-section data. A detailed analysis of the data in Table 3 reveals two key improvements compared to the subgroup parameters calculated by the traditional Padé approximation method (as shown in Table 1). First, the optimized method imposes constraints on the value range of subgroup parameters. All subgroup parameters in Table 3 are positive, completely eliminating the negative cross-section values that appeared in the third subgroup of Table 1 (e.g., the total cross-section of the third subgroup in Table 1 was −5.97155 × 10² b, which is physically meaningless). This ensures that the subgroup fixed-source equation can be solved normally, laying a solid foundation for subsequent resonance self-shielding calculations. Second, the optimized subgroup parameters exhibit high computational accuracy. For instance, the subgroup probabilities in Table 3 sum to 1, which is consistent with the physical definition that the sum of subgroup probabilities must equal 1. In terms of cross-section values, the cross-sections of each subgroup all show reasonable magnitude distributions. For example, the second subgroup has the largest total cross-section (7.02553 × 10³ b) and absorption cross-section (6.59446 × 10³ b), which aligns with the characteristic that the resonance peak of ²³⁸U in this energy group is concentrated around 6.7 eV, where the absorption capacity is significantly enhanced. The fifth subgroup, with the smallest total cross-section and absorption cross-section, corresponds to the energy segments in the resonance range where the cross-section varies gently, reflecting the rationality of the subgroup division.

To quantitatively evaluate the accuracy of the optimized subgroup parameters, Figure 4 presents the relative deviation of the effective resonance cross-sections back-calculated using the subgroup parameters in Table 3 under different background cross-sections. The horizontal axis of Figure 4 represents the background cross-section σ_b, covering a wide range from 10 b to 10⁵ b. The vertical axis represents the relative deviation between the back-calculated effective resonance cross-sections and the reference values (obtained from ultra-fine-group calculations).

As shown in Figure 4, the relative deviations of all types of cross-sections are almost within ±0.8% for almost all background cross-sections. The maximum deviation occurs in the absorption cross-section when the background cross-section is 10 b, but it still does not exceed 1%. For most background cross-sections (especially in the range of 100~10⁴ b), the relative deviations are within ±0.5%, demonstrating excellent agreement with the reference values. This indicates that the optimized subgroup parameters not only avoid unphysical negative values but also maintain high accuracy in reproducing the variation trend of effective resonance cross-sections with background cross-sections.

In summary, compared with the traditional Padé approximation method, the optimized method effectively solves the ill-posed problem in subgroup parameter calculation. The obtained subgroup parameters are physically meaningful, with high accuracy in describing resonance characteristics, providing a reliable basis for subsequent resonance self-shielding calculations.

3.2. Typical Single-Cell Problem

To further validate the applicability of the optimized Padé approximation subgroup method (SIPM) in practical reactor physics calculations, this section describes the numerical verification of typical pressurized water reactor (PWR) single-pin problems. The 47-energy group structure from the HELIOS-1.11 code is adopted, where the resonance energy range covers groups 10 to 25, with an energy span from 1.855 eV to 9188 eV. In the numerical simulations, the transport calculation for the subgroup method employs the method of characteristics, and the results of the optimized Padé approximation subgroup method (SIPM) are compared with those of the conventional Padé approximation subgroup method (SCPM), using the results from the Monte Carlo code OpenMC [35] as the reference benchmark.

This section includes a UO₂ fuel pin from the Japan Atomic Energy Agency (JAEA) benchmark [36]. The geometric configuration and dimensions of this case are shown in Figure 5, which depicts a cylindrical fuel pin consisting of a fuel pellet and cladding. The radial dimensions and material distributions of the fuel pellet are detailed in the figure. The resonant isotopes of UO₂ fuel only include ²³⁸U and ²³⁵U, which have distinct cross-section characteristics in the resonance energy range: ²³⁸U has sharp and densely distributed resonance peaks (e.g., the strong resonance peak around 6.7 eV), while ²³⁵U has relatively flat resonance peaks. Figure 6, Figure 7 and Figure 8 show the calculation errors of the absorption cross-section of ²³⁸U, the absorption cross-section of ²³⁵U, and the fission production cross-section of ²³⁵U, respectively.

The horizontal axis of Figure 6 represents neutron energy (in eV), covering the resonance energy range from 1 eV to 10,000 eV; the vertical axis represents the relative error (in %) between the results of the SIPM/SCPM and the reference values. It should be noted that the points in Figure 6, Figure 7 and Figure 8 correspond to discrete energy groups (group energy range), and lines are used only to visualize trends. It can be clearly observed from the figure that the calculation error of ²³⁸U absorption cross-section by the SIPM is generally controlled within ±0.5%. Even in the energy intervals with dense resonance peaks (5~7 eV and 10~100 eV, especially around the strong resonance peak at 6.7 eV), the error only increases moderately, with the maximum relative error not exceeding 0.5%. This indicates that the SIPM can accurately capture the sharp cross-section variations in ²³⁸U caused by strong resonance peaks. In contrast, the error of the SCPM increases significantly in these high-resonance regions, with errors at some energy points exceeding 1%, reflecting the limitations of the conventional method in handling strong resonant isotopes.

Figure 7 focuses on the error of ²³⁵U absorption cross-section, with the same energy range as Figure 6. Since the resonance peaks of ²³⁵U are relatively flat compared to those of ²³⁸U, the calculation error of the SIPM is better: the error in most energy ranges is within ±0.1%, and the maximum relative error does not exceed 0.2%. However, the error of the SCPM increases obviously, reaching over 1.5% at some energy points (e.g., 20~50 eV), further verifying the stability of the optimized method in handling isotopes with different resonance characteristics.

Figure 8 shows the error distribution of the ²³⁵U fission production cross-section, with a trend highly consistent with Figure 7. The calculation error of the SIPM is generally controlled within ±0.15%, with the maximum error not exceeding 0.2%, while the error of the SCPM increases significantly to 1.8% in the high-resonance energy region (e.g., 100~500 eV), once again demonstrating that SIPM has higher accuracy in describing the energy dependence of fission reaction cross-sections.

In summary, in typical UO₂ fuel pin problems, the optimized Padé approximation subgroup method (SIPM) shows significant advantages over the conventional method (SCPM) in resonance cross-section calculations: it can not only accurately capture the sharp cross-section variations and resonance interference effects of strong resonant ²³⁸U but also maintain high accuracy in the calculation ²³⁵U, fully verifying its effectiveness and reliability in resonance self-shielding calculations at the single-pin scale.

3.3. Burn-Up Condition Problem

To further validate the applicability of the optimized Padé approximation subgroup method (SIPM) under complex nuclide compositions and non-uniform burn-up distributions, this section describes the verification of resonance self-shielding calculations for UO₂ fuel pellets under burn-up conditions. The burn-up process significantly increases the number of nuclides in the fuel and leads to radial non-uniformity in nuclide density (known as the “rim effect”), which imposes higher requirements on the accuracy and stability of resonance self-shielding calculations. The nuclear density during the depletion process is provided by the inhouse code [37].

The calculation results for the UO₂ fuel pellet under burn-up conditions are presented in Figure 9, Figure 10 and Figure 11. Due to the large number of resonant isotopes (16 out of 133 total nuclides), individual analysis is impractical. Figure 9 shows the relative errors of the macroscopic absorption cross-sections for all 133 nuclides. The horizontal axis represents the group numbers in the resonance energy range (corresponding to 1.855 eV to 9188 eV), while the vertical axis denotes the relative error compared to the reference values. It can be observed that the relative errors of the SIPM are within 1% across the entire resonance energy range, with a uniform distribution and no significant fluctuations. In contrast, the errors of the conventional method, the SCPM, vary considerably with energy groups, reaching up to 2% in some groups. Additionally, the overall error of macroscopic cross-sections is lower than that of individual nuclides, as the flat cross-section variations of most non-resonant nuclides partially “average out” the total error.

To analyze the impact of radial burn-up non-uniformity, Figure 10 focuses on the radial distribution errors of resonance cross-sections within the 5.72 eV to 7.34 eV energy range, where there exists a critical interval containing strong resonance peaks of ²³⁸U. The fuel rod is radially divided into eight equal-volume rings (numbered one to eight, with one being the outermost and eight the center) to simulate the rim effect. The outer rings, which are subjected to higher neutron flux, undergo more significant burn-up, resulting in lower concentrations of heavy nuclides, weaker resonance self-shielding, and relatively higher cross-sections; the opposite trend occurs in inner rings. The horizontal axis of Figure 10 is the radial ring number, and the vertical axis is the relative error of resonance absorption cross-sections. The results indicate that the SIPM maintains errors within ±0.8% for all rings and accurately captures the decreasing trend of cross-sections from the outer to inner rings. In contrast, the SCPM exhibits significantly increased errors in rings 1–3 (near the outer layer), with a maximum deviation of 1.5%, failing to correctly reflect the influence of the rim effect. This demonstrates that the SIPM’s subgroup parameters not only achieve high accuracy in the energy dimension but also adapt to complex self-shielding effects caused by spatial non-uniformity.

As a core parameter characterizing reactor criticality, the calculation accuracy of the effective multiplication factor k_eff directly affects the safety and economy of core design. Figure 11 shows the deviation of k_eff calculated by the SIPM and the SCPM at different burn-up depths, where the deviation is measured in pcm (1 pcm = 10⁻⁵). The horizontal axis represents burn-up depth, and the vertical axis represents the k_eff calculation deviation. The results show that the k_eff deviation calculated by the SIPM is consistently within ±120 pcm and remains stable with increasing burn-up depth, while it is within ±50 pcm for most of the depletion process. This further verifies the reliability of the SIPM in handling complex burn-up problems. By accurately calculating resonance cross-sections, the SIPM can precisely reproduce changes in neutron multiplication capacity during burn-up, providing high-precision numerical support for core lifetime analysis.

3.4. Lattice Problem

There are two kinds of lattice cell problems. One is a four-by-four Gd-bearing lattice, and the other is the JAEA UO₂ lattice at a two-dimensional scale. The aim of these lattice benchmarks is to examine the ability of the subgroup method based on the improved Pade approximation used in this work to capture the resonance effect in strong absorbers and lattice scale problems. In the calculating procedure, the boundary condition is set to be reflective on all four sides.

3.4.1. 4 × 4 Gd-Bearing Lattice

To verify the applicability of the optimized Padé approximation subgroup method (SIPM) in lattice structures containing strong absorbers (e.g., Gadolinium, Gd), this section describes the numerical validation of a 4 × 4 boiling water reactor (BWR) Gd-bearing lattice problem. This lattice structure is a typical complex geometric case in reactor fuel assembly design. Due to the presence of Gd₂O₃ as a burnable poison, its resonance self-shielding effect and multi-nuclide interference effect are more significant, imposing strict requirements on the accuracy of the calculation method.

Figure 12 shows the geometric configuration of the 4×4 Gd-bearing lattice. The lattice consists of 16 fuel pins, among which the 5th and 10th fuel pins in the central area are Gd-bearing fuel pins, containing 3 wt% UO₂ and 3 wt% Gd₂O₃. Gadolinium is a burnable poison, has a strong neutron absorption capacity, and can be used to suppress initial core reactivity. The remaining surrounding fuel pins are conventional fuel pins with 3 wt% UO₂. The boundary condition of the lattice is set to be reflective on all four sides to simulate the neutronic characteristics of an infinite lattice.

To accurately capture the spatial self-shielding effect and cross-section distribution differences caused by asymmetric geometry, the fuel pins are finely partitioned in the calculation: Gd-bearing fuel pins (e.g., pin 5) are radially divided into 9 equal-volume rings, and each ring is circumferentially divided into 8 equal-volume sectors, forming 72 flat source regions; conventional fuel pins (e.g., pin 11) are divided into 3 radial rings to balance calculation accuracy and efficiency. The flat-source-region numbers increase from the outside to the inside, aiming to clearly reflect the trend of neutron flux and cross-sections from the fuel edge to the center. Due to the space self-shielding effect, the neutron flux in the edge region is high, the resonance cross-section is weakly “shielded”, and the cross-section value is relatively high. In the central region, due to neutron absorption by the outer layer, the flux is low, and the cross-section value is low. In addition, angular discretization uses 16 azimuthal angles and 3 polar angles per octant, with a ray spacing of 0.02 cm to ensure spatial resolution in transport calculations.

The 13.7 eV to 29.0 eV energy range (which contains strong resonance peaks of ²³⁸U, ²³⁵U, and gadolinium isotopes) is selected to analyze the absorption cross-section calculation errors of ²³⁸U, ²³⁵U, ¹⁵⁵Gd, and ¹⁵⁷Gd in the fifth Gd-bearing fuel pin. The results are shown in Figure 13, Figure 14, Figure 15 and Figure 16, respectively. The horizontal axis of each figure is the flat-source-region number from outside to inside, and the vertical axis is the relative error (%) between the calculated values of the SIPM and reference value.

Figure 13 displays the error of the ²³⁸U absorption cross-section. ²³⁸U has multiple obvious resonance peaks in this energy range, and its absorption cross-section decreases gradually from the outer layer to the center, reflecting a significant space self-shielding effect. The relative error calculated by the SIPM is controlled within ±0.4% and accurately captures the radial decreasing trend of the cross-section; the error difference between different sectors in the same ring is less than 0.2%, indicating its adaptability to asymmetric geometry.

Figure 14 shows the error of the ²³⁵U absorption cross-section. The resonance peak intensity of ²³⁵U is weaker than that of ²³⁸U, but its cross-section distribution is more complex due to the strong absorption interference of gadolinium isotopes. The error of the SIPM is generally within ±0.6%, and the error in the outer layer is slightly higher than that in the inner layer, which is related to the neutron flux depression effect caused by the high concentration of gadolinium in the outer layer.

Figure 15 displays the calculation results of the ¹⁵⁵Gd absorption cross-section. ¹⁵⁵Gd is a strong absorber, and its absorption cross-section reaches 10⁴ barns in this energy range, with an extremely strong space self-shielding effect, and the outer cross-section is more than twice as large as that of the center. The error of the SIPM is within ±0.8%, accurately reflecting the steep decline in the cross-section from the outside to the inside.

Figure 16 displays the calculation results of the ¹⁵⁷Gd absorption cross-section: ¹⁵⁷Gd has a stronger absorption capacity than ¹⁵⁵Gd, with sharper resonance peaks. The error of the SIPM is controlled within ±1% and remains stable even in the regions with dense resonance peaks, verifying the accuracy advantage of the SIPM in handling strongly resonant nuclides.

The verification results of the 4 × 4 Gd-bearing lattice show that the SIPM performs excellently in handling complex geometric problems containing strong absorbers, whether for conventional uranium isotopes like ²³⁸U and ²³⁵U, or strong absorbing poisons like ¹⁵⁵Gd and ¹⁵⁷Gd. The calculation error of their absorption cross-sections is controlled within the engineering acceptable range (±1%), and it can accurately capture the cross-section distribution differences caused by the space self-shielding effect and asymmetric geometry. The calculation results provide reliable numerical support for the design of cores containing burnable poisons.

3.4.2. 17 × 17 UO₂ Lattice

To further verify the applicability of the optimized Padé approximation subgroup SIPM in large-scale complex lattice structures, this section describes the numerical validation of the two-dimensional JAEA 17 × 17 lattice problem. This lattice model is a typical simplified model of a pressurized water reactor fuel assembly, consisting of 264 3.3 wt% UO₂ fuel pins and 25 guide tubes, with its geometric configuration shown in Figure 17. The guide tubes are uniformly distributed in the lattice to accommodate control rods or measuring instruments, which cause significant local perturbations to the neutron flux distribution. In such large-scale lattice structures, neutron migration paths are longer, and space self-shielding effects and inter-lattice interference effects are more complex, imposing higher requirements on the stability and efficiency of resonance self-shielding calculation methods.

To accurately capture the neutron flux gradient in the complex lattice, a refined spatial discretization strategy is adopted in the calculation: each fuel cell is divided into 48 flat source regions, and a single fuel pin is circumferentially divided into 24 equal-volume sectors to reflect the asymmetric neutron flux distribution; angular discretization uses 16 azimuthal angles and 3 polar angles per octant, with a ray spacing of 0.02 cm to ensure an accurate description of neutron motion directions. The red rectangular box marked in Figure 17 designates the target fuel pin for error analysis in this section. This fuel pin is located in the central area of the lattice, and its resonance cross-section distribution is representative due to the combined influence of the surrounding fuel pins and guide tubes.

Figure 18 and Figure 19 show the calculation errors of the resonance absorption cross-sections of ²³⁸U and ²³⁵U in the target fuel pin, respectively. The horizontal axis is the flat-source-region number of the fuel pin, which increases from the edge to the center, and the vertical axis is the relative error (%) of the SIPM relative to the reference value.

Figure 18 displays the error of the ²³⁸U absorption cross-section. ²³⁸U has strong, dense resonance peaks in the resonance energy region, and its absorption cross-section is significantly affected by the space self-shielding effect from the edge to the center of the fuel pin. The cross-section value shows a downward trend due to the gradual decrease in neutron flux (a decrease of about 40%). The relative error calculated by the SIPM is controlled within ±0.6% and remains stable in all flat source regions, accurately reproducing the radial variation law of the cross-section; the error fluctuation in the same sector is less than 0.3%, indicating its good adaptability to local flux perturbations caused by guide tubes.

Figure 19 displays the error of the ²³⁵U absorption cross-section. The resonance peak intensity of ²³⁵U is weaker than that of ²³⁸U, but it is more significantly affected by neutron shielding from surrounding fuel pins in the 17 × 17 lattice, resulting in a complex radial non-uniformity in cross-section distribution. The calculation error of the SIPM is generally controlled within ±0.6%. The error in the edge region is slightly higher than that in the central region due to the higher neutron flux, but it still remains within the engineering acceptable range.

The verification results of the 17 × 17 UO₂ lattice show that the SIPM can still maintain high accuracy in handling large-scale complex lattice structures. Whether for strong resonant nuclides ²³⁸U or ²³⁵U, the calculation error of their absorption cross-sections is significantly lower than that of the traditional method, and it can accurately capture the cross-section distribution characteristics caused by the space self-shielding effect and lattice interference. This benefits from the optimized subgroup parameters, which accurately fit the cross-section changes in the resonance energy region, and the efficient coupling with the method of characteristics, providing a reliable resonance self-shielding solution for full-core scale neutronics calculations.

4. Conclusions

This study addresses the ill-posedness in subgroup parameter calculation using the traditional Padé approximation method within the framework of resonance self-shielding treatment in nuclear reactor physics. To resolve the issue of non-physical negative subgroup cross-sections and improve calculation reliability, an optimized Padé approximation method is proposed, with three key technical improvements: (1) A resonance factor criterion is introduced to screen energy groups requiring detailed subgroup calculations, avoiding unnecessary computations for groups with weak resonance effects. (2) A dynamic solution strategy is adopted, starting from the minimum number of subgroups, systematically traversing background cross-section combinations and incrementally increasing the subgroup number until positive parameters meeting accuracy constraints are obtained. (3) A high-resolution resonance integral table with 40 background cross-section points is constructed, particularly densifying points in the 10~10⁵ b range, where cross-sections vary drastically, to compensate for insufficient background data.

Numerical validations across various benchmark cases demonstrate the effectiveness of the proposed method: For the strong resonance energy range of ²³⁸U (6.48~7.34 eV), the optimized method eliminates negative subgroup parameters and controls the relative deviation of back-calculated effective cross-sections within ±0.5%; in typical single-fuel-pin calculations, errors in the ²³⁸U and ²³⁵U cross-sections are reduced to <0.5% and <0.2%, respectively; for burn-up pellet problems, macroscopic absorption cross-section errors remain within 1%, and k_eff deviations are stable within ±30 pcm; in complex lattice structures, the method accurately captures spatial self-shielding effects, with cross-section errors for strong absorbers controlled within ±1%. These results confirm that the optimized method effectively resolves the ill-posedness of the traditional Padé approximation, ensuring the physical validity of subgroup parameters while significantly improving calculation accuracy and stability. The proposed approach provides a robust and efficient solution for high-fidelity resonance self-shielding treatment, supporting more reliable reactor core neutronics calculations. In future work, the method proposed in this work will be applied to fast-neutron reactors, which often adopt highly enriched uranium or fuels with Pu-isotopes, like ²³⁹Pu in MOX fuel. The capability of the SCPM to capture the resonance effect in the fast neutron spectrum will be tested in the following work.