Application of Quadruple Range Quadratures to Three-Dimensional Model Shielding Problems

A widely used numerical method for discretizing the direction variable in the transport equation is the discrete ordinates technique. Here, we test various discrete ordinates quadrature sets on two three-dimensional (3-D) (X-Y-Z) shielding problems: the doglegged void neutron model and the pool critical assembly model. Commonly used quadrature sets, including the standard level symmetric sets and double Gauss-Chebyshev sets, produce significant ray effects associated with material discontinuities in both models. Abu-Shumays designed the quadruple range (QR) sets specifically for these types of problems and showed that they perform well in two-dimensional X-Y geometry. Here, we show that compared to more commonly used quadrature sets, the 3-D QR sets substantially reduce ray effects associated with material discontinuities in 3-D X-Y-Z discrete ordinates calculations.