Fourier Analysis of CMFD Method in Cylindrical Geometry

The coarse mesh finite difference (CMFD) method has been widely adopted as an acceleration algorithm for neutron transport calculations, effectively reducing computational costs and improving convergence. While its convergence properties have been thoroughly established in Cartesian geometries, where the optical thickness of coarse meshes critically impacts acceleration performance that the extension of CMFD. While coarse meshes have been established in cylindrical geometry, it will face problem that the circumferential difference length gradually increase along radial direction, which also raise concerns about convergence stability. This study aims to theoretically and numerically investigate the convergence effect and stability of the CMFD method in cylindrical geometries under diverse optical thickness conditions. A Fourier analysis framework was developed for CMFD in two-dimensional (r-θ) cylindrical geometry to systematically evaluate its spectral properties based on optical thickness of each coarse mesh. Theoretical derivations reveal that the radially increasing mesh size in cylindrical geometries introduces spatially varying optical thicknesses, which destabilize the conventional CMFD iteration process. To address this instability, an optical diffusion CMFD method (odCMFD) strategy was implemented to optimize the original CMFD formulation based on cylindrical geometry. Numerical experiments were conducted using both homogeneous and heterogeneous cylindrical models to validate theoretical findings. Results demonstrate that the odCMFD modification ensures unconditional convergence across all tested optical thicknesses, resolving divergence issues observed in conventional CMFD for large optical thickness regimes. However, the acceleration efficiency of CMFD diminishes progressively as the model scale increases, particularly in systems with high radius. This degradation is attributed to the amplified inconsistency between fine-mesh transport solutions and coarse-mesh diffusion approximations in larger geometries. In conclusion, while odCMFD stabilizes CMFD convergence in cylindrical geometries, its practical efficacy is constrained by geometric scaling effects. The findings emphasize the necessity for further algorithmic enhancements, such as adaptive mesh refinement or hybrid acceleration schemes, to maintain CMFD’s performance advantages in large-scale cylindrical reactor simulations. This study provides critical insights into the geometric adaptability of acceleration methods for high-fidelity neutron transport computations. Building on the aforementioned Fourier analysis, this study develops a cylindrical pebble-bed reactor model for steady-state 3D neutron simulation. The results demonstrate that the acceleration efficiency of the CMFD method aligns with the phenomena predicted by the Fourier analysis in cylindrical geometry. These findings provide theoretical guidelines for optimizing CMFD parameter selection in cylindrical geometries, ensuring convergence and computational efficiency.