TU-AB-BRC-01: Spherical Harmonic Based Finite Element Method (SHFEM): A New Angular Discretization of Linear Boltzmann Transport Equation for Accurate Dose Calculation




As a deterministic method for dose calculation, the linear Boltzmann transport equation (LBTE) can reach the same accuracy as the Monte Carlo method (MC). In terms of speed, LBTE can be significantly accelerated through various state-of-art numerical techniques. The angular discretization is crucial since the accuracy of LBTE highly depends on it and the computational time of LBTE is quadratic with respect to number of discretized angles. This work proposes a new angular discretization scheme that synergizes spherical harmonics method and finite element method.


The LBTE is solved by the discrete ordinate method with a new quadrature scheme, i.e., Spherical Harmonics based Finite Element Method (SHFEM) for discretizing the angular variable. The proposed SHFEM is free from unphysical negative weights that often appear in a Level-Symmetric (LQn) quadrature set when the order is beyond 20. Moreover, the proposed SHFEM can accurately deal with directional sources. The spatial variables are discretized on the structured grid using the diamond-difference scheme. The Source Iteration method (SI) is utilized to solve the discretized LBTE, with the acceleration by the Diffusion Synthetic Acceleration method (DSA).


The SHFEM is compared with LQn and Legendre-Chebyshev (PN-TN) quadrature set respectively. With a three-dimensional (3D) MC method as the benchmark, the proposed SHFEM had the best performance, especially for the directional-source problems.


SHFEM, a novel angular discretization method for LBTE that integrates spherical harmonics method and finite element method, is proposed for dose calculation with improved accuracy from LQn and PN-TN, particularly in the presence of directional sources.

The authors were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000), and the Shanghai Pujiang Talent Program (#14PJ1404500)