Title: 101st NIA CFD Seminar: Meshless Methods for the Neutron Transport Equation
Speaker: Brody Bassett, PhD Candidate, University of Michigan
Date: Wednesday, May 2, 2018
Location: NIA, Room 137
Mesh-based methods for the numerical solution of partial differential equations (PDEs) require the division of the problem domain into non-overlapping, contiguous subdomains that conform to the problem geometry. The mesh constrains the placement and connectivity of the solution nodes over which the PDE is solved. In meshless methods, the solution nodes are independent of the problem geometry and do not require a mesh to determine connectivity. This allows the solution of PDEs on geometries that would be difficult to represent using even unstructured meshes. The ability to represent difficult geometries and place solution nodes independent of a mesh motivates the use of meshless methods for the neutron transport equation, which often includes spatially-dependent PDE coefficients and strong localized gradients. The meshless local Petrov-Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. The MLPG method uses weighted residuals of the transport equation to solve for basis function expansion coefficients of the neutron angular flux. Connectivity of the solution nodes is determined by the shared support domain of overlapping meshless functions, such as radial basis functions (RBFs) and moving least squares (MLS) functions.
To prevent oscillations in the neutron flux, the MLPG transport equations are stabilized by the streamline-upwind Petrov-Galerkin (SUPG) method, which adds numerical diffusion to the streaming term. Global neutron conservation is enforced by using MLS basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. Two integration schemes for the basis and weight functions in the MLPG method are presented, including a background mesh integration and a fully meshless integration approach.
The method of manufactured solutions (MMS) is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations. Finally, meshless heat transfer equations are derived using a similar MLPG approach and verified using the MMS. These heat equations are coupled to the MLPG neutron transport equations and results for the power coefficient of reactivity are compared to values from a commercial pressurized water reactor.
Brody Bassett received his BS in Physics from Brigham Young University and his MS in Nuclear Engineering from Oregon State University. He is currently working on his PhD in Nuclear Engineering and Radiological Sciences at the University of Michigan, where his dissertation is focused on the application of meshless methods to the neutron transport equation.