R. Le Tellier

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.

Reactivity Perturbation Formulation for a Discontinuous Galerkin-Based Transport Solver and Its Use with Adaptive Mesh Refinement

Abstract Within the framework of a Discontinuous Galerkin spatial approximation of the multigroup discrete ordinates transport equation, we present a generalization of the exact standard perturbation formula that takes into account spatial discretization-induced reactivity changes. It encompasses in two separate contributions the nuclear data-induced reactivity change and the reactivity modification induced by two different spatial discretizations. The two potential uses of such a formulation when considering adaptive mesh refinement are discussed, and numerical results on a simple two-group Cartesian two-dimensional benchmark are provided. In particular, such a formulation is shown to be useful to filter out a more accurate estimate of nuclear data-related reactivity effects from initial and perturbed calculations based on independent adaptation processes.

A Wavelet-Based Finite Element Method for the Self-Shielding Issue in Neutron Transport

Abstract This paper describes a new approach for treating the energy variable of the neutron transport equation in the resolved resonance energy range. The aim is to avoid recourse to a case-specific spatially dependent self-shielding calculation when considering a broad group structure. This method consists of a discontinuous Galerkin discretization of the energy using wavelet-based elements. A Σt-orthogonalization of the element basis is presented in order to make the approach tractable for spatially dependent problems. First numerical tests of this method are carried out in a limited framework under the Livolant-Jeanpierre hypotheses in an infinite homogeneous medium. They are mainly focused on the way to construct the wavelet-based element basis. Indeed, the prior selection of these wavelet functions by a thresholding strategy applied to the discrete wavelet transform of a given quantity is a key issue for the convergence rate of the method. The Canuto thresholding approach applied to an approximate flux is found to yield a nearly optimal convergence in many cases. In these tests, the capability of such a finite element discretization to represent the flux depression in a resonant region is demonstrated; a relative accuracy of 10–3 on the flux (in L2-norm) is reached with less than 100 wavelet coefficients per group.

Anisotropy and Particle Conservation for Trajectory-Based Deterministic Methods

Abstract Necessary and sufficient constraints are derived for the solid angle quadrature of a ray-tracing procedure to unconditionally ensure the particle conservation when anisotropic scattering is considered. As an application of this result, a discussion on the choice of a polar quadrature in two-dimensional (2-D) calculations with the method of characteristics is provided. A new quadrature based on the derived constraints is introduced and compared with other optimized quadratures on a simple 2-D benchmark.

A Modeling of BWR-MOX Assemblies Based on the Characteristics Method Combined with Advanced Self-Shielding Models

Abstract Calculations based on the characteristics method and different self-shielding models are presented for 9 × 9 boiling water reactor (BWR) assemblies fully loaded with mixed-oxide (MOX) fuel. The geometry of these assemblies was recovered from the BASALA experimental program. We have focused our study on three configurations simulating the different voiding conditions that an assembly can undergo in a BWR pressure vessel. A parametric study was carried out with respect to the spatial discretization, the tracking parameters, and the anisotropy order. Comparisons with Monte Carlo calculations in terms of keff, radiative capture, and fission rates were performed to validate the computational tools. The results are in good agreement between the stochastic and deterministic approaches. The mutual self-shielding model recently introduced within the framework of the Ribon extending self-shielding method appears to be useful for this type of assemblies. Indeed, in the calculation of these MOX benchmarks, the overlapping of resonances, especially between 238U and 240Pu, plays an important role due to the spectral strengthening of the flux as the voiding percentage is increased. The method of characteristics is shown to be adequate to perform accurate calculations handling a fine spatial discretization.

An Adaptive Energy Discretization of the Neutron Transport Equation Based on a Wavelet Galerkin Method

of interactions. The unknown is the so-called neutron flux φ( r, Ω, E) = v(E)n( r, Ω, E) with n( r, Ω, E) the neutron density and v(E) the neutron velocity. The problem is defined in terms of the neutron interaction properties of the different materials i.e. the cross sections. The solution of this equation in a deterministic way proceeds by the successive discretization of the three variables: energy, angle and space. The treatment of the energy variable invariably consists in a multigroup discretization which considers the cross sections and the flux to be constant within a group (i.e. a cell of the 1D energy mesh). A pre-homogenization of the cross sections is performed at the library processing level using a spatially independent weighting flux (e.g. 1/E spectrum in the epithermal range). With a broad group structure (≈ 100 to 2000 energy groups), this prior homogenization is unsufficient to take into account the case-specific, spatially-dependent, self-shielding effect i.e. the flux local depression in the vicinity of resonances that largely affects the neutron balance. As a consequence, a neutron transport calculation has to incorporate a so-called self-shielding model to correct the group cross sections of resonant isotopes. This homogenization stage of a neutron transport calculation is known to be a main source of errors for deterministic methods; as a consequence, an important work has been carried out to improve it. An optimized energymesh structure (Mosca et al., 2011) in addition to an advanced self-shielding model (Hebert, 2007) is incorporated in state-of-the-art transport codes. A different treatment for the energy variable based on a finite element approach is the basis of the present work. Such an avenue was proposed in the past by (Allen, 1986) but seldom used in practice. Indeed, finite element methods are commonly based on polynomial function bases which are not appropriate for non-smooth behavior. Recently, two independent works by (Le Tellier et al., 2009) and (Yang et al., 2010) have proposed wavelet-Galerkin methods to overcome this issue. In this chapter, after a review An Adaptive Energy Discretization of the Neutron Transport Equation Based on a Wavelet Galerkin Method 16