A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

Abstract

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation. Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method, several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in the small modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.