A Finite Element Projection Method for the Solution of Particle Transport Problems

A method for solving particle transport problems has been developed. In this method the particle flux is expressed as a linear and separable sum of odd and even components in the direction variables. Then a Bubnov-Galerkin projection technique and an equivalent variational Raleigh-Ritz solution are applied to the second-order transport equation. A dual finite element basis of polynomial splines in space and spherical harmonics in angle is used. The general theoretical and numerical problem formalism is carried out for a seven-dimensional problem with anisotropic scattering, time dependence, three spatial and two angular variables, and with a multigroup treatment of the energy dependence. The boundary conditions for most physical problems of interest are dealt with explicitly and rigorously by a classical minimization (variational) principle. Finally, the computational validation of the method is obtained by a computer solution to the monoenergetic steady-state air-over-ground problem in a cylindrical (r, z) geometry and with an exponentially varying atmosphere.