[Paper Review] Numerical modeling of neutron transport in SP3 approximation by finite element method

A. V. Avvakumov, V. F. Strizhov+2 more|arXiv (Cornell University)|Mar 27, 2019

Nuclear reactor physics and engineering19 references4 citations

TL;DR

This paper presents a finite element method (FEM) implementation of the SP₃ neutron transport approximation for reactor core analysis, demonstrating improved accuracy over diffusion theory with minimal computational overhead. It validates the method on IAEA-2D and HWR benchmarks, showing that both diffusion and SP₃ solutions can yield complex eigenvalues in certain configurations, particularly in the HWR case, while maintaining close agreement with reference transport solutions for k- and α-eigenvalues.

ABSTRACT

The SP3 approximation of the neutron transport equation allows improving the accuracy for both static and transient simulations for reactor core analysis compared with the neutron diffusion theory. Besides, the SP3 calculation costs are much less than higher order transport methods (SN or PN). Another advantage of the SP3 approximation is a similar structure of equations that is used in the diffusion method. Therefore, there is no difficulty to implement the SP3 solution option to the multi-group neutron diffusion codes. In this work, the application of the SP3 methodology based on solution of the λ- and α-spectral problems has been tested for the IAEA-2D and HWR reactor benchmark tests. The FEM is chosen to achieve the 3D geometrical generality, using GMSH as a generic mesh generator. The results calculated with the diffusion and SP3 methods are compared with the reference transport calculation results. It was found for the HWR reactor test that some eigenvalues are complex when calculating using both diffusion and SP3 options.

Motivation & Objective

• To develop and validate a finite element method (FEM) for solving the SP₃ approximation of the neutron transport equation in 3D geometries.
• To assess the accuracy and computational efficiency of SP₃ relative to neutron diffusion theory in whole-core reactor simulations.
• To investigate the occurrence and implications of complex eigenvalues in both diffusion and SP₃ solutions for reactor spectral problems.
• To compare SP₃ results with reference transport solutions for k-effective and α-eigenvalues in benchmark problems.
• To enable integration of SP₃ into existing diffusion code frameworks by preserving equation structure similarity.

Proposed method

• Application of the SP₃ approximation to reduce the neutron transport equation to a system of two coupled diffusion-type equations using spherical harmonics expansion.
• Spatial discretization via high-order Lagrange finite elements with mesh generation using GMSH for 3D geometric flexibility.
• Solution of λ- and α-spectral problems using the SLEPc library for large-scale eigenvalue computations.
• Use of condensed grids for accuracy verification and convergence control in eigenvalue calculations.
• Implementation of both prompt and delayed neutron models to compute α-eigenvalues with realistic reactor dynamics parameters.
• Comparison of results from SP₃ and diffusion methods against high-fidelity reference solutions from fine-mesh transport calculations.

Experimental results

Research questions

• RQ1 Does the SP₃ approximation significantly improve accuracy over diffusion theory in whole-core reactor simulations while maintaining low computational cost?
• RQ2 Can the SP₃ method reproduce reference transport results for k-effective and α-eigenvalues in standard benchmark problems like IAEA-2D and HWR?
• RQ3 Under what conditions do complex eigenvalues emerge in diffusion and SP₃ solutions, and what is their physical significance?
• RQ4 How do the spectral properties (eigenvalues and eigenfunctions) of SP₃ compare to those of diffusion theory in symmetric and asymmetric configurations?
• RQ5 To what extent can SP₃ be seamlessly integrated into existing diffusion code architectures due to structural similarity?

Key findings

• The SP₃ method reduces computation time to approximately 1.5 times that of diffusion while improving accuracy over diffusion, especially in pin-by-pin and control rod reactivity calculations.
• For the HWR benchmark, both diffusion and SP₃ solutions produced complex eigenvalues with small imaginary parts, indicating oscillatory behavior in neutron flux dynamics.
• In contrast, the IAEA-2D benchmark showed only real eigenvalues, even in the non-symmetric case, suggesting that complex eigenvalues are configuration-dependent.
• The fundamental α-eigenvalue for the SP₃ method was 0.04395 s⁻¹, closely matching the reference value of 0.04395 s⁻¹ at high mesh resolution (n=96, p=3), with relative error below 0.1%.
• The prompt neutron generation time Λ_pr was consistently estimated as 1.917×10⁻⁴ s for both diffusion and SP₃, confirming consistency in dynamic behavior modeling.
• Minor eigenvalues (e.g., α₂ to α₅, α₉ to α₁₀) exhibited complex conjugate pairs in both methods, with small imaginary components, indicating damped or growing oscillatory modes in the neutron flux.