Alain Hébert

Development of the nodal collocation method for solving the neutron diffusion equation

Abstract The nodal collocation method is a new technique for the discretization of the multidimensional neutron diffusion equation where the solution sought is expressed in the form of tensorial expansions of Legendre polynomials defined over homogeneous parallelepipeds. In this study, we have truncated the tensorial expansions using the serendipity approximation in an attempt to reduce the total number of unknowns and improve the effectiveness of the discretization. The remaining Legendre coefficients are then determined in order to preserve selected moments of the neutron conservation equation in each parallelepiped. This approach allows a variable order of convergence without sacrificing the consistency peculiar to full tensorial expansions and generates matrix systems which may be resolved by an Alternating Direction Implicit algorithm. Furthermore, we have proved that the linear nodal collocation method and the mesh centered finite difference method are equivalent. Validation results are given for the IAEA 2-D and 3-D benchmarks and for a 2-D representation of a pressurized water reactor (PWR).

Development of a third-generation superhomogénéisation method for the homogenization of a pressurized water reactor assembly

Proposals are made for improving current second-generation superhomogeneisation (SPH) methods in three different ways and to use them in heterogeneous and homogeneous diffusion procedures for react...

Application of the Hermite Method for Finite Element Reactor Calculations

AbstractA number of improvements have been made to the Hermite method in order to obtain a high order finite element method capable of solving the neutron diffusion equation. First, a variational f...

The ribon extended self-shielding model

Abstract Improvement of the lattice code component related to resonance self-shielding calculations is described. The proposed self-shielding model is based on a subgroup flux equation with probability tables, as implemented in the CALENDF approach of P. Ribon. A new type of correlated two-dimensional probability table is introduced for the representation of the slowing-down effect in the resolved energy domain. The resulting formalism makes possible a better representation of distributed self-shielding effects. A new numerical scheme is also proposed to represent the mutual shielding effect of overlapping resonances between different isotopes in the context of the Ribon subgroup equations. The interference effects between two resonant isotopes are represented by a correlated weight matrix also computed using a CALENDF approach. The model was designed with the primary goal of allowing the straightforward replacement of legacy self-shielding components in typical lattice codes to gain improved accuracy without any noticeable increase in CPU resources. Finally, a validation is presented where the absorption rates are compared with exact values obtained using a fine-group elastic slowing-down calculation in the resolved energy domain. Other results, relative to Rowland’s pin-cell benchmarks, are also presented. The need to represent mutual shielding effects, at least for mixed-oxide fuel is demonstrated.

A Transport Method for Treating Three-Dimensional Lattices of Heterogeneous Cells

AbstractA new ray-tracing method for the calculation of collision probabilities within arbitrary three-dimensional geometries has been developed. This method is used to discretize the neutron transport equation for heterogeneous rectangular cells containing zones of mixed cylindrical and rectangular geometry. For multicell applications, the interface current (IC) method provides the coupling between cells. The solution to the IC equations over multicell domains consisting of rectangular three-dimensional cells is improved by using an alternate direction implicit iteration scheme with variational acceleration. Results include comparisons of this technique with SHETAN for simple geometries and the analysis of a three-dimensional extension of a two-dimensional 15 × 15 pressurized water reactor benchmark problem.

Development of the Subgroup Projection Method for Resonance Self-Shielding Calculations

Abstract We investigate a new approach for resonance self-shielding calculations, based on a simplified and straightforward subgroup model, used in association with an improved Santamarina-Hfaiedh energy mesh. This subgroup model relaxes the need to represent the correlated slowing-down effects by optimizing the energy mesh. The resulting equations become sufficiently simple to reintroduce an accurate representation of other physical effects that are generally neglected, namely, the mutual shielding effect between different isotopes and the temperature correlation effect caused by an explicit temperature gradient in a resonant isotope. The resulting self-shielding model is shown to reach levels of accuracies that are similar to those of a Monte Carlo method.

Application of a dual variational formulation to finite element reactor calculations

Abstract A number of improvements have been made to the nodal collocation method in order to obtain a high-order nodal technique capable of solving the neutron diffusion equation over full-core three-dimensional pressurized water reactors. First, the nodal collocation method is derived formally from a dual variational principle, using Gauss-Lobatto quadratures. Analytical integration and Gauss-Legendre quadratures are next applied to the same dual functional in order to obtain more accurate discretizations. An efficient ADI numerical technique with a supervectorization procedure was set up to solve the resulting matrix system. Validation results are given for the IAEA 2-D, Biblis and IAEA 3-D benchmarks and for a typical full-core 3-D representation of a pressurized water reactor at the beginning of the second cycle.

Computing moment-based probability tables for self-shielding calculations in lattice codes

Abstract As part of the self-shielding model used in the APOLLO2 lattice code, probability tables are required to compute self-shielded cross sections for coarse energy groups (typically with 99 or 172 groups). This paper describes the replacement of the multiband tables (typically with 51 subgroups) with moment-based tables in release 2.5 of APOLLO2. An improved Ribon method is proposed to compute moment-based probability tables, allowing important savings in CPU resources while maintaining the accuracy of the self-shielding algorithm. Finally, a validation is presented where the absorption rates obtained with each of these techniques are compared with exact values obtained using a fine-group elastic slowing-down calculation in the resolved energy domain. Other results, relative to the Rowland’s benchmark and to three assembly production cases, are also presented.

The Search for Superconvergence in Spherical Harmonics Approximations

Abstract The occurrence of superconvergence in various first-order spherical harmonics approximations of the neutral particle transport equation is being investigated. Superconvergence refers to the added accuracy gained in evaluating the solution of the transport equation at optimally chosen base points of the finite element trial functions. It has been observed that this phenomenon is happening when primal and dual discretizations in space and angle lead to the same numerical result, a property also referred as primal-dual agreement. A systematic search is presented for primal-dual agreement on one-dimensional slab, tube, and spherical geometries and on Cartesian two-dimensional geometries based on complete and simplified Pn approximations. Primal-dual agreement was successfully obtained in all Cartesian geometries but not in tube and spherical geometries, due to the angular redistribution term.

An Improved Algebraic Collapsing Acceleration with General Boundary Conditions for the Characteristics Method

Abstract A detailed derivation of the algebraic collapsing acceleration (ACA), a synthetic acceleration of the characteristics method, is presented. An improvement of the synthetic hypothesis is proposed, and the corrective system is derived for general boundary conditions. Both Fourier and direct spectral analyses of the accelerated iterations for a one-dimensional slab geometry are given. The solving strategy for the corrective system along with implementation details about the method of characteristics is discussed. Numerical results for a one-group, two-dimensional benchmark are provided to illustrate the basic synthetic hypothesis and the enhancement of its robustness with the proposed two-step collapsing hypothesis. The practical performance of ACA is illustrated on a pressurized water reactor–type assembly in the context of multigroup eigenvalue calculations.