Generalized Heterogeneous Resonance Integral Method Based on Particle Swarm Optimization

Accurate resonance self-shielding treatment is essential for reliable prediction of the effective multiplication factor (k_eff), power distribution, and safety characteristics of advanced reactor cores. Conventional homogeneous resonance integral (HRI) tables rely on equivalence theory and therefore cannot distinguish the volumetric effect of non-resonant nuclides inside fuel from the leakage effect associated with neutron escape into the moderator. Heterogeneous resonance integral tables generated by the three-phase method (TRI) improve the macroscopic treatment of lattice heterogeneity, but the spatial self-shielding distribution inside a fuel pin cannot be represented completely. This study aims to develop a generalized heterogeneous resonance integral (GRI) method with high accuracy, broad applicability, and unchanged online computational cost. The construction of resonance integrals was reformulated as a high-dimensional optimization problem within a posteriori framework. Instead of expanding the independent variables of the resonance table, the tabulated resonance-integral values were optimized directly under explicit spatial self-shielding constraints, thereby retaining a conventional data structure and interpolation procedure. Forty fixed background cross-section points from 10 to 10¹⁰ b were used to represent each resonance group, and the HELIOS-47 group structure, containing 16 resonance groups from 1.855 to 9 118 eV, was adopted. HRIs were used as the initial particle positions, and the corresponding resonance-integral values at the 40 background points were treated as the coordinates of each particle. A benchmark database was generated by a three-phase procedure, with systematic variations in moderator density, lattice size, resonant-nuclide density, and fuel radius. The resulting cases covered loose and tight lattice arrangements and a broad range of background cross sections. Twenty heterogeneous pin-cell problems and one homogeneous problem were included, each heterogeneous fuel pin was divided into 10 equal-volume radial rings. For every candidate particle, subgroup parameters were fitted from the perturbed resonance integrals, and both pin-averaged and ring-dependent effective resonance cross sections were calculated. The fitness was defined as the maximum absolute relative deviation from continuous-energy Monte Carlo reference results over all problems, radial regions, and reaction cross sections. Particle swarm optimization was then performed independently for each resonance group, using 1 000 particles, learning factors of 2.0, a linearly decreasing inertia weight from 0.9 to 0.4, and at most 500 iterations. Convergence was accepted when the fitness reached 1.5% or less. Segev interpolation was applied between tabulated background points, and OpenMC reference solutions were produced with 100 000 particles per cycle and 400 active cycles, keeping the statistical uncertainty of k_eff below 50 pcm. Numerical verification on VERA single-pin problems 1A-1D and two-dimensional assembly problems 2A-2D demonstrates the effectiveness of the proposed GRI. For the single-pin cases, GRI keeps the k_eff deviation within ±30 pcm across variations in fuel temperature, moderator temperature, and moderator density, whereas the TRI gives deviations of approximately 89-120 pcm and HRI gives errors exceeding 200 pcm. GRI also predicts the radial ²³⁸U absorption cross section from the pin center to the periphery within ±1.5%. In contrast, HRI introduces systematic deviations of about 5%-10%, while TRI overestimates the inner-region cross section by approximately 2%-4% and underestimates the outer-region value by about 2%-3%. The maximum root-mean-square deviation among the 16 resonance groups remains below 1%. For the 17×17 PWR assembly cases, GRI limits the k_eff deviation to −32 to +34 pcm, compared with −157 to +232 pcm for the TRI and −368 to −297 pcm for HRI. This stability indicates good transferability from isolated pins to assembly environments containing center, edge, corner, and poison-adjacent fuel rods. The optimized resonance integrals are obtained offline and therefore introduce no additional online computational complexity. The proposed GRI provides accurate and general resonance data for high-fidelity core neutronics calculations, particularly for advanced fuel lattices with strong intra-pin spatial self-shielding effects.