Improved Variational Nodal Method Based on Symmetry Group Theory

To reduce the calculation effort and memory requirement for high-order P_N expansion calculation in the Variational Nodal Method (VNM), the surficial irreducible basis functions based on the symmetry group theory have been employed to block-diagonalize one of the four nodal response matrices. Its effectiveness encourages our further investigation on the application of the symmetry group theory to volumetric expansion to block-diagonalize the remaining three of the nodal response matrices in this paper. By using the symmetry group theory, the neutron transport problem for each node can be decoupled into several independent subproblems as long as both the geometry and the material distribution of the node are symmetric. Each of these subproblems can be solved by using variational principles as in the traditional VNM, providing their nodal response matrices as the diagonal blocks of the corresponding entire ones. For hexagonal-z node, each nodal response matrix can be reduced into 16 diagonal blocks, among which only 12 have to be calculated due to the properly selected irreducible basis functions. In addition, it is also proved that the response matrices with anisotropic scattering can also be block-diagonalized as the same. Calculation results based on typical problems demonstrate that the new method reduces the time cost for the response matrice calculation by one order of magnitude compared with our previous work. For the total computing time, the speedup ratio is about 2 for P₃ calculation and 4 for P₅ calculation. Furthermore, almost 40% of the memory requirement can be saved.