Block-diagonalization of the variational nodal response matrix using the symmetry group theory

To further improve the efficiency of the Variational Nodal Method (VNM) for solving the neutron transport equation in hexagonal-z geometry, the nodal response matrix is further block-diagonalized by utilizing the symmetry group theory to decompose the surface basis functions into irreducible components. The block-diagonal property of the nodal response matrix is determined by the symmetry properties of the hexagonal node in geometry, material and basis functions, including both reflection and rotation symmetries. To fully utilize those properties, the symmetry group theory is employed to analyze the symmetry property of the nodal response matrices. It is mathematically proved that the nodal response matrix can be further block-diagonalized into 16 diagonal blocks instead of the current 4 ones by using the symmetry group theory. Numerical comparisons demonstrate that the new approach can reduce the memory storage and computing time by a factor of 2∼3 for P7 angular approximation, compared with the currently employed variables transformation algorithm.