Higher-Order DGFEM Transport Calculations on Polytope Meshes for Massively-Parallel Architectures

In this dissertation, we develop improvements to the discrete ordinates (S_N) neutron transport equation using a Discontinuous Galerkin Finite Element Method (DGFEM) spatial discretization on arbitrary polytope (polygonal and polyhedral) grids compatible for massively-parallel computer architectures. Polytope meshes are attractive for multiple reasons, including their use in other physics communities and their ease in handling local mesh refinement strategies. In this work, we focus on two topical areas of research. First, we discuss higher-order basis functions compatible to solve the DGFEM S_N transport equation on arbitrary polygonal meshes. Second, we assess Diffusion Synthetic Acceleration (DSA) schemes compatible with polytope grids for massively-parallel transport problems. We first utilize basis functions compatible with arbitrary polygonal grids for the DGFEM transport equation. We analyze four different basis functions that have linear completeness on polygons: the Wachspress rational functions, the PWL functions, the mean value coordinates, and the maximum entropy coordinates. We then describe the procedure to extend these polygonal linear basis functions into the quadratic serendipity space of functions. These quadratic basis functions can exactly interpolate monomial functions up to order 2. Both the linear and quadratic sets of basis functions preserve transport solutions in the thick diffusion limit. Maximum convergence rates of 2 and 3 are observed for regular transport solutions for the linear and quadratic basis functions, respectively. For problems that are limited by the regularity of the transport solution, convergence rates of 3/2 (when the solution is continuous) and 1/2 (when the solution is discontinuous) are observed. Spatial Adaptive Mesh Refinement (AMR) achieved superior convergence rates than uniform refinement, even for problems bounded by the solution regularity. We demonstrated accuracy in the AMR solutions by allowing them to reach a level where the ray effects of the angular discretization are realized. Next, we analyzed DSA schemes to accelerate both the within-group iterations as well as the thermal upscattering iterations for multigroup transport problems. Accelerating the thermal upscattering iterations is important for materials (e.g., graphite) with significant thermal energy scattering and minimal absorption. All of the acceleration schemes analyzed use a DGFEM discretization of the diffusion equation that is compatible with arbitrary polytope meshes: the Modified Interior Penalty Method (MIP). MIP uses the same DGFEM discretization as the transport equation. The MIP form is Symmetric Positive De_nite (SPD) and e_ciently solved with Preconditioned Conjugate Gradient (PCG) with Algebraic MultiGrid (AMG) preconditioning. The analysis from previous work was extended to show MIP's stability and robustness for accelerating 3D transport problems. MIP DSA preconditioning was implemented in the Parallel Deterministic Transport (PDT) code at Texas A&M University and linked with the HYPRE suite of linear solvers. Good scalability was numerically verified out to around 131K processors. The fraction of time spent performing DSA operations was small for problems with sufficient work performed in the transport sweep (O(10^3) angular directions). Finally, we have developed a novel methodology to accelerate transport problems dominated by thermal neutron upscattering. Compared to historical upscatter acceleration methods, our method is parallelizable and amenable to massively parallel transport calculations. Speedup factors of about 3-4 were observed with our new method

Characterization of a Stochastic Procedure for the Generation and Transport of Fission Fragments within Nuclear Fuels

With the ever-increasing demands of the nuclear power community to extend fuel cycles and overall core-lifetimes in a safe and economic manner, it is becoming more necessary to extend the working knowledge of nuclear fuel performance. From the atomistic to the macroscopic level, great morphological changes occur within the fuel over its lifetime. The main initial damaging events produced by fuel recoils from fast neutrons and fission fragment spiking leads to the onset of grain growths and fuel restructuring. Therefore, it is desirable to have a more detailed understanding of the initial events leading to fuel morphology changes at the atomistic level. However, this is difficult to achieve with the fission fragments due to the wide variability of their species (charge, mass, and energy) and the large averaging of their relative yields in the nuclear data files. This work is our first iteration at developing a general methodology to characterize a procedure, based on Monte Carlo principles, for generating individual fission event result channels and analyzing their specific response in the fuel. We utilized the nuclear reaction simulation tool, TALYS, to generate energy-dependent fission fragment yield distributions for different fissile/fissionable isotopes. These distributions can then be used in conjunction with fuel isotopics and a neutron energy spectrum to generate a fission-reaction-rate-averaged distribution of the fission fragment yields. We then used Monte Carlo sampling to generate the result channels from individual fission events, using the Q-value of the prompt fission system to either accept or reject. The simulation tool: Transport of Ions in Matter (TRIM) was used to characterize the general response of the fission fragment species within Uranium Dioxide (UO2), including the range, energy loss, displacements, recoils, etc. These responses were then correlated which allowed for the quick calculation of the response of the individual fission fragment species generated from the Monte Carlo sampling. As an example of this strategy, we calculated the response on a PWR fuel pin where MCNP was used to generate a high-fidelity neutron energy spectrum