Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

The discrete ordinates（SN） method requires numerous angular unknowns to achieve the desired accuracy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in the SN method to automatically optimize the angular distribution and minimize angular discretization errors with lower expenses. The proposed method enables linear discontinuous finite element quadrature sets over an icosahedron to vary their quadrature orders in a one-twentieth sphere so that fine resolutions can be applied to the angular domains that are important. An error estimation that operates in conjunction with the spherical harmonics method is developed to determine the locations where more refinement is required. The adaptive quadrature sets are applied to three duct problems, including the Kobayashi benchmarks and the IRI-TUB research reactor, which emphasize the ability of this method to resolve neutron streaming through ducts with voids. The results indicate that the performance of the adaptive method is more efficient than that of uniform quadrature sets for duct transport problems. Our adaptive method offers an appropriate placement of angular unknowns to accurately integrate angular fluxes while reducing the computational costs in terms of unknowns and run times.