Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method

Abstract

Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.

Development of a finite volume inter-cell polynomial expansion method for the neutron diffusion equation

“…If one applies the FVM to Equation 2, Equation 3 is obtained [3]. In this equation, the face-averaged values of the neutron flux gradient ( ⃗⃗ , , ) have to be determined.…”

“…In this equation, the face-averaged values of the neutron flux gradient ( ⃗⃗ , , ) have to be determined. In the reference mentioned [3], ⃗⃗ , , is calculated by means of Arb [5] algorithm, which calculates it as a weighted sum of the cell averaged values of the neutron flux of the neighbouring cells ( , ) as in Equation 4 [3,5]. Nonetheless, it works well for fine meshes, but it requires high computational time [3].…”

“…In the reference mentioned [3], ⃗⃗ , , is calculated by means of Arb [5] algorithm, which calculates it as a weighted sum of the cell averaged values of the neutron flux of the neighbouring cells ( , ) as in Equation 4 [3,5]. Nonetheless, it works well for fine meshes, but it requires high computational time [3]. Besides, the neutron flux continuity is not applied in this method and the neutron current continuity is imposed by several approximations [3], since they imply an excess of equations in comparison with the unknowns, which are the cell values.…”

“…Nonetheless, it works well for fine meshes, but it requires high computational time [3]. Besides, the neutron flux continuity is not applied in this method and the neutron current continuity is imposed by several approximations [3], since they imply an excess of equations in comparison with the unknowns, which are the cell values.…”

Development of a finite element method for neutron transport equation approximations

“…Also, a nodal collation method based on the expansion of the neutron in terms of orthogonal polynomials has been extensively used to deal with rectangular geometries (Verdú et al, 1994;Verdú et al, 1999;Ginestar, Marín, and Verdú, 2001;Verdú et al, 2005). The Finite Volume Method, FVM, has also been considered to calculate the neutronic steady state of a nuclear power (Bernal et al, 2014).…”