Adaptive Mesh Refinement Solution Techniques for the Multigroup SN Transport Equation Using a Higher-Order Discontinuous Finite Element Method

摘要

In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniquesfor the steady-state multigroup SN neutron transport equation using a higher-orderDiscontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations,a projection-based estimator and a jump-based indicator, both of whichare shown to reliably drive the spatial discretization error down using h-type AMR.Algorithms to treat the mesh irregularity resulting from the local refinement areimplemented in a matrix-free fashion. The DGFEM spatial discretization schemeemployed in this research allows the easy use of adapted meshes and can, therefore,follow the physics tightly by generating group-dependent adapted meshes. Indeed,the spatial discretization error is controlled with AMR for the entire multigroup SNtransportsimulation, resulting in group-dependent AMR meshes. The computingefforts, both in memory and CPU-time, are significantly reduced. While the convergencerates obtained using uniform mesh refinement are limited by the singularityindex of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous),the convergence rates achieved with mesh adaptivity are superior. Theaccuracy in the AMR solution reaches a level where the solution angular error (or rayeffects) are highlighted by the mesh adaptivity process. The superiority of higherordercalculations based on a matrix-free scheme is verified on modern computing architectures.A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA)scheme is devised for the DGFEM-discretized transport equation using a variationalargument. The Modified Interior Penalty (MIP) diffusion form used to accelerate theSN transport solves has been obtained directly from the DGFEM variational form ofthe SN equations. This MIP form is stable and compatible with AMR meshes. Becausethis MIP form is based on a DGFEM formulation as well, it avoids the costlycontinuity requirements of continuous finite elements. It has been used as a preconditionerfor both the standard source iteration and the GMRes solution techniqueemployed when solving the transport equation. The variational argument used indevising transport acceleration schemes is a powerful tool for obtaining transportconformingdiffusion schemes.xuthus, a 2-D AMR transport code implementing these findings, has been developedfor unstructured triangular meshes.