Discontinuous Finite Elements Angular Quadrature Sets For Discrete Ordinates Transport Calculation

Particle transport calculation is widely used in many fields such as engineering.Discrete ordinates method(SN)is one of the main methods for solving particle transport equation in the world,which has obvious advantages for deep penetration problems.The discrete ordinate method directly discretizes the angle variables in the transport equation.It chooses a set of discrete directions and gives them corresponding weights which called quadrature sets.However,traditional quadrature sets methods have larger angular discretization errors when solving strong anisotropic deep penetration shielding problems,which may reduce the reliability of shielding calculation.Constructing the quadrature sets based on discontinuous finite element theory can reduce angular discretization errors and have great meanings for practical engineering problems.Therefore,it is necessary to conduct a comprehensive and in-depth study and analysis on discontinuous finite element quadrature sets.This paper firstly summarizes the angular discretization methods,introduces two different constructions of discontinuous finite element quadrature sets in detail,and analyzes the advantages of the methods above mentioned.Multi-variate secant method is used to optimize the discrete directions and corresponding weights of quadrature sets.It adjusts the distribution of discrete directions in the discrete direction domain to make them approximately uniform,and eliminate negative weights.The spherical harmonic function expansion on the angular flux can verify the integration accuracy and visually reflect the applicable conditions of each quadrature sets.Taking Kobayashi benchmark as an example,this paper compares the effects of different quadrature sets an tests the accuracy of discontinuous finite element quadrature sets in solving the duct problems.The calculation accuracy and efficiency of cubic discontinuous finite element quadrature sets are verified by solving the neutron flux in the cross section of DLVN model.The numerical results show that the integration accuracy of spherical harmonic function of discontinuous finite element quadrature sets is less than 3%.It declares that the discontinuous finite element quadrature sets is more adaptable to the problems with duct,and have higher calculation accuracy at the interface of different materials than traditional quadrature sets.This paper extended the types of quadrature sets according to study the construction and application of discontinuous finite element quadrature sets.The application of discontinuous finite element quadrature sets can effectively reduce the angular discretization error,which is of great significance to improve the accuracy and efficiency of shielding calculation.