Spherical Harmonics Finite Element Solution of the Least-squares Neutron Transport Equation

To solve the neutron Boltzmann equation in anisotropic or void regions, one may use deterministic methods such as the discrete ordinates method or the even-parity approach. Both methods have shown numerous applications and developments in the last 30 years. Nevertheless both exhibit drawbacks in three-dimensions void regions that require special treatments, such as using characteristics method or first order spherical harmonic method. The basic idea of the work presented here is to define a unique approach that can be use in 3D void regions without any special care. The least-squares technique is applied to the neutron transport equation. Combined with a spherical harmonic expansion of the angular flux, this approach leads to a variational formulation suitable for continuous finite elements. Extensions are proposed to allow for the 3D transport in void regions and complete derivation of discrete equations is given for anisotropic media. Numerical results show that ARTEMIS, a transport solver based on this method, gives solutions free of ray effects. These results include 3D tests on the Kobayashi first benchmark problem.