Finite element methods for space-time reactor analysis

Abstract

Finite element methods are developed for the solution of the neutron diffusion equation in space, energy and time domains. Constructions of piecewise polynomial spaces in multiple variables are considered for the approximation of a general class of piecewise continuous functions such as neutron fluxes and concentrations of nuclear elements. The approximate solution in the piecewise polynomial space is determined by applying the Galerkin scheme to a weak form of the neutron diffusion equation. A piecewise polynomial method is also developed for the solution of first-order ordinary differential equations. The numerical methods are applied to neutron slowing-down problems, static neutron diffusion problems, point kinetics problems and time-dependent neutron diffusion problems. The uniqueness, stability and approximation error of the numerical methods are considered. The finite element methods yield high-order accuracy, depending on the degree of the polynomials used, and thereby permit coarse-mesh calculations. The conventional multigroup method, the Crank-Nicolson and the Pad6 schemes are shown to be special cases of the finite element methods. Numerical examples are presented which confirm the truncation error and demonstrate the utility of the finite element methods in reactor problems.

Linear triangle finite element formulation for multigroup neutron transport analysis with anisotropic scattering

“…As noted earlier, the variational formulation generates symmetric system matrices. This is easily seen by replacing i by j, j by i, d by d' and d' by d in Equations (3,5) and (3.7). Consequently in practice, the assemblying procedure is performed only when q' >^ k' in Equation (3.11).…”

Finite element method for neutron diffusion problems in hexagonal geometry

“…We now proceed to expound upon the weak form. 1,6 Let W 1 (0) be the class of functions which are continuous and have square integrable first derivatives, that is,…”

“…In passing let us note that 'I' is identical to the Ig u+ (y)* of Kang's 1 -D cubic Hermite set. 1 This draws attention to the possibility of using the three natural axes of hexagonal/triangular geometry to derive element sets. More will be said about this possibility in the next section.…”

“…We shall only mention one of them, namely the finite element method. 1 ' 7 As with the finite difference…”

Finite element method applied to neutron diffusion problems.

“…6 and ref. 7 present a very good discussion of the method when applied to static and time-dependent neutron diffusion problems, and to slowing down problems in infinite homogeneous media, , using Hermite piecewise expansion. Numerical examples are presented for the one and two-dimensional static diffusion equations, as well 1 as for the zero, one and two-dimensional kinetics equations.…”