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Finite element analysis of an eigenvalue problem arising from neutron transport

Nicolás A. Barnafi, Felipe Lepe, Francisca Muñoz Riquelme

TL;DR

This work analyzes a non-selfadjoint eigenvalue problem arising from a two-group neutron transport model using finite elements. By formulating the problem in a compact-operator framework with appropriate diffusion-capacity and boundary terms, the authors establish convergence and a priori error estimates for FE discretizations with degree $k\ge1$ and prove spectral convergence without spurious eigenvalues. They derive a robust variational formulation, prove compactness of the solution and adjoint operators, and show that all eigenvalues are real under the model's self-adjoint structure when appropriate. The methodology is validated through extensive numerical tests in 2D and 3D geometries, including a realistic IAEA benchmark, demonstrating optimal convergence on smooth domains, with expected deterioration in non-convex or curved domains due to geometric approximations. Overall, the paper provides a rigorous mathematical basis for FE approximations of neutron-transport eigenproblems and confirms the practical reliability of the method for reactor-type applications.

Abstract

In two and three dimensions, we analyze a finite element method to approximate the solutions of an eigenvalue problem arising from neutron transport. We derive the eigenvalue problem of interest, which results to be non-symmetric. Under a standard finite element approximation based on piecewise polynomials of degree $k \geq 1$, and under the framework of the compact operators theory, we prove convergence and error estimates of the proposed method. We report a series of numerical tests in order confirm the theoretical results.

Finite element analysis of an eigenvalue problem arising from neutron transport

TL;DR

This work analyzes a non-selfadjoint eigenvalue problem arising from a two-group neutron transport model using finite elements. By formulating the problem in a compact-operator framework with appropriate diffusion-capacity and boundary terms, the authors establish convergence and a priori error estimates for FE discretizations with degree and prove spectral convergence without spurious eigenvalues. They derive a robust variational formulation, prove compactness of the solution and adjoint operators, and show that all eigenvalues are real under the model's self-adjoint structure when appropriate. The methodology is validated through extensive numerical tests in 2D and 3D geometries, including a realistic IAEA benchmark, demonstrating optimal convergence on smooth domains, with expected deterioration in non-convex or curved domains due to geometric approximations. Overall, the paper provides a rigorous mathematical basis for FE approximations of neutron-transport eigenproblems and confirms the practical reliability of the method for reactor-type applications.

Abstract

In two and three dimensions, we analyze a finite element method to approximate the solutions of an eigenvalue problem arising from neutron transport. We derive the eigenvalue problem of interest, which results to be non-symmetric. Under a standard finite element approximation based on piecewise polynomials of degree , and under the framework of the compact operators theory, we prove convergence and error estimates of the proposed method. We report a series of numerical tests in order confirm the theoretical results.

Paper Structure

This paper contains 38 sections, 12 theorems, 113 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The model problem
  3. Generation rate
  4. Fission
  5. In-Scattering
  6. Loss rate
  7. Leakage
  8. Absorption
  9. Out-scattering
  10. Multi-group equation at steady state
  11. Fast Group
  12. Thermal Group
  13. Variational Formulation
  14. The source problem
  15. The eigenvalue problem
  16. ...and 23 more sections

Key Result

Lemma 3.1

\newlabellemma:source-invertibility0 Let $D_1,D_2,\Sigma_{a1},\Sigma_{a2},\Sigma_{1\to 2}$ be strictly positive constants. Then, there exists a unique solution $(\widetilde{\phi}_1,\widetilde{\phi}_2)$ in $V$ to eq:source-problem such that for $\widetilde{\alpha}_1\coloneqq \min\{D_1,\Sigma_{a1} + \Sigma_{1\to 2}\}$, and with $\widetilde{\alpha}_2 \coloneqq \min\{D_2, \Sigma_{a2}\}$. In particu

Figures (5)

  • Figure 1: Eigenfunctions associated to the first four eigenvalues on the unit square domain with $k=1$.
  • Figure 2: Eigenfunctions associated to the first four eigenvalues on the unit circle d with $k=1$.
  • Figure 3: Results with IAEA 2D benchmark
  • Figure 4: Results with IAEA 2D benchmark for horizontal cross section
  • Figure 5: Results with IAEA 2D benchmark for vertical cross section

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Lemma 4.1
  • Corollary 4.2
  • Proof 3
  • Lemma 4.3
  • ...and 11 more