The Application of the Finite Element Method to the Neutron Transport Equation

Abstract

This paper examines the theoretical and practical application of the finite element method to the neutron transport equation. The theoretical examination which is applicable to the general transport equation in arbitrary geometry includes a derivation of the equivalent integral law (or weak form) of the first order neutron transport equation, to which the finite element method (Galerkin approach) is applied, resulting in a system of algebraic equations. We show that in principle the system of equations can be solved with certain physical restrictions concerning the criticality of the medium. The convergence of this approximate solution to the exact solution with mesh refinement is examined, and a non-optimal estimate of the convergence rate is obtained analytically. It is noted that the numerical results indicate a faster convergence rate and several approaches to obtain this result analytically are outlined. The practical application of the finite element method involved the development of a computer code capable of solving the neutron transport equation in l-D plane geometry. Vacuum, reflecting, or specified incoming boundary conditions may be analyzed, and all are treated as natural boundary conditions. The incorporation of the reflecting boundary conditions is seen to result in an ambiguity, which must be resolved by consideration of the direction in which neutrons travel. Discontinuous phase space finite elements are introduced, and it is seen that discontinuous angular elements effectively match the analytical discontinuities in the angular flux at mu = 0 for plane geometry. In addition, the use of discontinuous spatial elements is shown to result in treating continuity of the angular flux at an interface as-a natural interface condition in the direction of neutron travel. The time-dependent transport. equation is also examined and it is shown that the application of the finite element method in conjunction . with the Crank-Nicholson time discretization method results in a system of algebraic equations which is readily solved. Numerical results are given for several critical slab eigenvalue problems, including anisotropic scattering, and the results compare extremely well with benchmark results. It is seen that the finite element code is more efficient than a standard discrete ordinates code for certain problems. Precise numerical tests are made on the convergence rate of the approximate solution (L2 norm) with mesh refinement and also with the eigenvalue error. These results indicate O[h^(k+l )] solution error in the L2 norm and O(h 2k+l ) error in the eigenvalue, where h is the mesh spacing and k the degree of the finite element. A problem with severe heterogeneities is considered and it is shown that the use of discontinuous spatial and angular elements results in a marked improvement in the results. Finally, time-dependent problems are examined and it is seen that the phenomenon of angular mode separation makes the numerical treatment of the transport equation "in slab geometry a considerable challenge, with the result that the angular mesh has a dominant effect on obtaining acceptable solutions to the time-dependent transport equation.