Unstructured Discrete Ordinates Method Based on Radial Basis Function Approximation

Abstract

Generally, solving a neutron transport equation on a domain with curved boundaries by using mesh-based methods requires to approximate the domain geometry. This increases the computation time in some specified cases that a very fine regular mesh must be used to approximate geometry. Several unstructured mesh methods have been developed, but all of these methods suffer from limitations related to the shape of the used unstructured mesh. In this study, a new 2D unstructured discrete ordinates method (Sn) is developed to solve discrete ordinates equation on a generally unstructured mesh. By using radial basis function (RBF), it provides a flexible technique for discretizing the spatial variable of the Sn equation without any requirement related to the shape of unstructured mesh as well as the number of spatial dimensions. The work in this study includes a two-step approach. First, the two types of meshfree approximation methods, moving least squares (MLS) and radial basis function (RBF), are investigated for implementation capability and accuracy for solving a diffusion equation in a weak-strong form. Attention is paid to the approximation quality on the boundaries and the stability of the discrete equation system which are affected by support domain of shape function of the MLS and RBF. This step lays the groundwork for developing an unstructured discrete ordinates method based on the RBF approximation. In the developed method, the discretization of the spatial variable in Sn equation is carried out on the polygons which overlap each other. The flux within each polygon is approximated only by nodes which belong to the polygon. It is intended to prevent oscillation in the flux due to Gibb phenomenon which appears when a support radius of the RBF is large. By doing so, it allows the balance the accuracy of discretization, but still retaining the flexibility of unstructured mesh in the shape. Several benchmarks are implemented and compared to the standard Sn method to verify the accuracy and stability of the developed method. The yielded results are in good agreement with the standard Sn results. The discrete equation system based on this approach also achieves the stability in the condition of the average density of solution nodes is not quasi-uniform. Therefore, it is expected that the unstructured discrete ordinates method developed in this study is possible to be implemented in the neutron transport calculation.