A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes

International audienceThis document presents a new numerical scheme dealing with the Boltzmann transport equation. This scheme is based on the expansion of the angular flux in a truncated spherical harmonics function and the discontinuous finite element method for the spatial variable. The advantage of this scheme lies in the fact that we can deal with unstructured, non-conformal and curved meshes. Indeed, it is possible to deal with distorted regions whose boundary is constituted by edges that can be either line segments or circular arcs or circles. In this document, we detail the derivation of the method for 2D geometries. However, the generalization to 2D extruded geometries is trivial

Analyse d'une méthode numérique combinée d'harmoniques sphériques et de discrétisation de Galerkin discontinue pour l'équation de transport de Boltzmann

In [11], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In this paper, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa's type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on $P_k$ discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order $O( N^{−t} + h^k )$ for a smooth solution which admits partial derivatives of order k + 1 and t with respect to the spatial and angular variables, respectively. For k = 0 (piecewise constant finite elements) we also obtain a convergence result of order $O( N^{−t} + h^{1/2} )$. Numerical experiments in 1, 2 and 3 dimensions are provided, showing a better convergence behavior for the L 2-norm, typically of one more order, $O( N^{−t} + h^{k+1} )$

Analyse d'une méthode numérique combinée d'harmoniques sphériques et de discrétisation de Galerkin discontinue pour l'équation de transport de Boltzmann

In [11], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In this paper, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa's type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on $P_k$ discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order $O( N^{−t} + h^k )$ for a smooth solution which admits partial derivatives of order k + 1 and t with respect to the spatial and angular variables, respectively. For k = 0 (piecewise constant finite elements) we also obtain a convergence result of order $O( N^{−t} + h^{1/2} )$. Numerical experiments in 1, 2 and 3 dimensions are provided, showing a better convergence behavior for the L 2-norm, typically of one more order, $O( N^{−t} + h^{k+1} )$

Precise 3D reactor core calculation using spherical harmonics and Discontinuous Galerkin finite element methods

International audienceWe study the use of PN method in angle and Discontinuous Galerkin in space to solve 3D neutron transport problem. PN method consists in developing the angular flux on truncated spherical harmonics basis. In this paper, we couple this method with the discontinuous finite elements in space to obtain a complete discretisation of the multigroup neutron transport equation. To investigate its precision, the method was applied to Takeda and C5G7 benchmark problems. These calculations point out that the proposed PN-DG method is capable of producing accurate solutions in small computational time, and that it is able to handle complex 3D geometries

Spherical Harmonics and Discontinuous Galerkin Finite Element Methods for the Three-Dimensional Neutron Transport Equation: Application to Core and Lattice Calculation

The spherical harmonics or P N method is intended to approximate the neutron angular flux by a linear combination of spherical harmonics of degree at most N. In this work, the P N method is combined with discontinuous Galerkin finite elements method and yield to a full discretization of the multigroup neutron transport equation. The employed method is able to handle all geometries describing the fuel elements without any simplification nor homogenisation. Moreover the use of a matrix assembly-free method avoids building large sparse matrices, which enables to produce high-order solutions in small computational time and less storage usage. The resulting transport solver called NYMO has a wide range of applications: it can be used for a core calculation as well as for a precise 281-groups lattice calculation accounting anisotropic scattering. To assess the accuracy of this numerical scheme, it was applied to 3D reactor core and fuel assembly calculations. These calculations point out that the proposed P N-DG method is capable of producing precise solutions while the developed solver is able to handle complex 3D core and assemblies geometries

APOLLO3®: Overview of the new code capabilities for reactor physics analysis

International audienceAPOLLO3® is the French deterministic code for lattice and core calculations, it is developed at the CEA with the financial and technical support of EDF and Framatome since 2007.The main goal of APOLLO3® is the development of a unique deterministic tool which includes the capabilities of lattice and core French codes of previous generation, e.g. APOLLO2, CRONOS2, ECCO and ERANOS, while providing large improvements on reactor physical modeling in a modern and flexible software architecture platform for R&D and industrial activities.This paper presents an overview of the new main capabilities of the deterministic APOLLO3® code in the lattice and core components