Apply implicitly restarted Arnoldi method to solving eigenvalue problem and reducing dimensionality in neutron diffusion

Zhaocai XIANG; Qiafeng CHEN; Pengcheng ZHAO; Qinghang ZHANG

doi:10.11889/j.0253-3219.2024.hjs.47.020604

Abstract

Background

High-order harmonics of neutron diffusion equations can be used to reconstruct the neutron flux distribution in a reactor core, but traditional source iteration methods or modified source iteration methods have low solving efficiency.

Purpose

This study aims to provide a reliable and efficient method for reconstructing the neutron flux distribution in reactor cores.

Methods

Firstly, the neutron diffusion equation was discretized using the finite difference method. Then, the implicitly restarted Arnoldi method (IRAM) was employed to solve the eigenvalue problem of the neutron diffusion equation and obtain high-order harmonic samples for different macroscopic cross-section states. Subsequently, a low-order model for the neutron diffusion equation was constructed by using these samples and a combination of proper orthogonal decomposition (POD) and Galerkin projection, and an error model was developed to characterize the accuracy of eigenvalue and harmonic calculations. Finally, relevant programs were developed to reconstruct the neutron flux distribution in the two-dimensional steady-state TWIGL benchmark problem and validate the accuracy of the model.

Results

The computation results show that the IRAM exhibits high accuracy in solving the high-order eigenvalues and harmonic problems of the neutron diffusion equation, with an error on the order of 10^-14. The reconstruction of the neutron flux distribution based on the POD-Galerkin low-order model also maintains a high level of accuracy. The solution error increases with the order of the eigenvalues, with an error magnitude less than or equal to 10^-12. The reconstructed neutron flux distribution closely matches the reference solution in the reactor core, and the error in the effective multiplication factor is only 8.7×10^-5. Additionally, the computation time for the low-order model is only 10.18% of the full-order model.

Conclusions

This study provides a reliable and efficient method for reconstructing the neutron flux distribution in reactor cores. The method can be used not only to reconstruct the steady-state neutron flux distribution but also has the potential to predict the transient neutron flux distribution, which is expected to be further expanded in future applications.