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Adaptive mesh refinement on Cartesian meshes applied to the mixed finite element discretization of the multigroup neutron diffusion equations

Patrick Ciarlet,, Minh-Hieu Do, François Madiot

TL;DR

This work develops an AMR framework for the mixed finite element discretization of multigroup neutron diffusion on Cartesian meshes, guided by a posteriori estimators. It presents a robust reconstruction-based error estimation scheme, including average, average+, and a post-processing reconstruction, to drive refinement while preserving Cartesian structure. The MINOS solver combines outer energy-group iterations with inner directional sweeps to solve the discretized mixed problem, and AMR uses a directional marking strategy to maintain mesh regularity. Numerical tests on Takeda benchmarks demonstrate that post-processing reconstruction yields substantial mesh reductions with accurate k_eff predictions, underscoring the practical impact for reactor-scale simulations.

Abstract

The multigroup neutron diffusion equations are often used to model the neutron density at the nuclear reactor core scale. Classically, these equations can be recast in a mixed variational form. This chapter presents an adaptive mesh refinement approach based on a posteriori estimators. We focus on refinement strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications.

Adaptive mesh refinement on Cartesian meshes applied to the mixed finite element discretization of the multigroup neutron diffusion equations

TL;DR

This work develops an AMR framework for the mixed finite element discretization of multigroup neutron diffusion on Cartesian meshes, guided by a posteriori estimators. It presents a robust reconstruction-based error estimation scheme, including average, average+, and a post-processing reconstruction, to drive refinement while preserving Cartesian structure. The MINOS solver combines outer energy-group iterations with inner directional sweeps to solve the discretized mixed problem, and AMR uses a directional marking strategy to maintain mesh regularity. Numerical tests on Takeda benchmarks demonstrate that post-processing reconstruction yields substantial mesh reductions with accurate k_eff predictions, underscoring the practical impact for reactor-scale simulations.

Abstract

The multigroup neutron diffusion equations are often used to model the neutron density at the nuclear reactor core scale. Classically, these equations can be recast in a mixed variational form. This chapter presents an adaptive mesh refinement approach based on a posteriori estimators. We focus on refinement strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications.

Paper Structure

This paper contains 24 sections, 1 theorem, 61 equations, 6 figures, 19 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. The neutron diffusion model
  4. Mixed variational formulation
  5. Discretization
  6. Algorithmic motivations
  7. Outer iteration
  8. Inner iteration
  9. Global description
  10. A posteriori error estimators for a mixed finite element discretization
  11. Reconstruction of the discrete solution
  12. Average and average,+ reconstructions
  13. Post-processing reconstruction
  14. A posteriori error estimate
  15. Adaptive mesh refinement
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let ${\mathbb D}$ and ${\mathbb T}_e$ satisfy (MG_Pos). Then, the bilinear symmetric form $c$ fulfills an inf-sup condition:

Figures (6)

  • Figure 1: Description of the AMR process.
  • Figure 2: Small FBR geometry
  • Figure 3: Total estimator on the initial mesh for Model 2 - Post-processing reconstruction
  • Figure 4: Configuration of Model 3 case 1 and case 2
  • Figure 5: Configuration of Model 3 case 3
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Remark 3.3
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1
  • Remark 5.2