ANS is committed to advancing, fostering, and promoting the development and application of nuclear sciences and technologies to benefit society.
Explore the many uses for nuclear science and its impact on energy, the environment, healthcare, food, and more.
Explore membership for yourself or for your organization.
Conference Spotlight
2026 Nuclear Energy Conference & Expo (NECX)
August 24–27, 2026
Dallas, TX|Hilton Anatole
Latest Magazine Issues
Jul 2026
Jan 2026
2026
Latest Journal Issues
Nuclear Science and Engineering
September 2026
Nuclear Technology
August 2026
Fusion Science and Technology
Latest News
In transition: Commercializing fusion power
Commercial fusion power is closer than ever. There are now around 30 U.S. fusion companies, several of which claim to be on track to connect to the grid as early as the 2030s.
Tokamak and laser inertial confinement approaches benefit from decades of research at facilities such as the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory and ITER, with alternative concepts including stellarator, magnetic mirror, and Z-pinch confinement also making notable progress as private and government funding for fusion increases.
Richard Sanchez, Jean Ragusa
Nuclear Science and Engineering | Volume 169 | Number 2 | October 2011 | Pages 133-154
Technical Paper | doi.org/10.13182/NSE10-31
Articles are hosted by Taylor and Francis Online.
An angular approximation of the transport equation based on a collocation technique results as an intermediary step in the search for a set of modified discrete ordinates (DO) equations, which eliminates ray effects. The collocation equations are similar to the DO ones with the only difference being that the scattering term is evaluated with a full Galerkin matrix instead of with the DO quadrature formula. The Galerkin quadrature offers the advantage of a better treatment of scattering anisotropy and a correct evaluation of the singular scattering associated to multigroup transport correction. However, the construction of the Galerkin matrix requires the existence of two equivalent bases in a final-dimensional representation space: an interpolatory basis to retain the collocative nature of the DO approximation and a spherical harmonic basis to represent scattering terms accurately. Up to now, the relationship between these two bases was heuristic, stemming from trial and errors. In this work we analyze the symmetries of the angular direction set and also use the factorized form of the spherical harmonics to derive a set of necessary conditions for the construction of the spherical harmonic basis. These conditions give an analytical explanation to previous heuristic techniques and fully extend them to three-dimensional geometries. We have adopted an assembling method for which extensive numerical tests show that the necessary conditions permit the construction of the Galerkin quadrature from level-symmetric, triangular, and product direction sets up to a high number of polar cosines. Our results can also be generalized to calculate Galerkin matrices for nonregular quadrature formulas. However, these necessary conditions are not sufficient, and we give numerical proof of this fact using different assembling techniques. Our assembling technique allows for the construction of Galerkin matrices from triangular direction sets (for which the DO quadrature is notoriously poor), which have positive weights for up to 44 polar cosines. In three dimensions this quadrature has 2024 angular directions and is able to exactly integrate scattering of anisotropy order 24.
