| dc.description.abstract | This research explores the application of a reduced order modeling technique known as proper generalized decomposition (PGD) to models commonly employed in nuclear science and engineering. PGD is an a priori reduced order modeling technique that seeks a separated representation of a multi-dimensional variable. A separated representation involves decomposing a multi-dimensional variable into a sum of products of 1-D dimensional functions. It is conjecture that this representation can significantly reduce the burden of evaluating multi-dimensional linear systems. To investigate PGD’s capability for this computational expediency and reduction in dimensionality, this research applies a PGD approach to four different types of problems: nuclear reactor criticality, multigroup neutron diffusion, neutron transport, and parameterized neutron diffusion. This dissertation first discusses the impetus of reduced order modeling and the methodology behind PGD. It then details the mathematics of the PGD algorithm and its application to several simple examples, including tailoring the algorithm for heterogeneous domains. The rest of the dissertation discusses the various new applications of this PGD approach. In the criticality application, PGD is utilized to reduce the computational burden of evaluating multigroup neutron diffusion eigenvalue problems. In this application, each multigroup flux is sought as a finite sum of separable one-dimensional functions. With this representation, PGD is used to evaluate the linear systems within the power iteration process of the eigenvalue problem. The dissertation discusses the implementation of PGD to these eigenvalue systems including a derivation of PGD operators for multigroup neutron diffusion problems with standard power iteration and power iteration accelerated with adaptive Wielandt shift. To illustrate PGD’s effectiveness, the implementation is applied to eigenvalue problems ranging from homogeneous to highly heterogeneous geometries with one-, two-, and four-group material properties. With comparison to full-order model evaluation with MOOSE, the effectiveness of PGD is found to be problem-dependent. PGD always out performs the full-order model with close to homogeneous problems, but its performance degrades with more realistic reactor problems.
In the space-energy approach, two different approaches are analyzed that utilize PGD to evaluate multigroup neutron diffusion problems, or more generally, coupled diffusion-reaction problems. This dissertation gives an overview of the PGD methodology and neutron diffusion with multigroup energy discretization. The first PGD approach performs a space-only decomposition, where a spatial separated representation is sought for each multigroup flux. The second approach is a full space-energy decomposition, where the energy dimension is included in the PGD separated representation. The dissertation also explores the prospect of performing a decomposition for different energy regions, effectively creating macro groups that retain fine-group structure. An algorithm for decomposing the linear operators to create an efficient PGD iteration process is explained for each of the approaches. The results include two 2-D, two-group examples and a 3-D seven-group example. When comparing with the full-order model, evaluated using MOOSE, both PGD approaches prove effective for mildly heterogeneous geometries, but show difficulty when dealing with more complex geometries. Furthermore, the space-energy representation is much slower than the space-only approach for the two-group problem, but proves more effective for the seven-group problem. The results also include a 145-group graphite block example, where PGD with space-energy separation significantly reduces the computational time compared to a specialized deal.II implementation. In the neutron transport application, two different PGD approaches are utilized to evaluate the linear systems involved with SN neutron transport. In the first approach, each SN angular flux is sought as a finite sum of separable one-dimensional functions. In the second appoach, a space-angle decomposition is investigated, whereby including the angular decomposition in the separated representation. PGD has been applied extensively to advection-diffusion problems, but none that include pure advection and scattering-type variable coupling. This discussion discusses these implementations of PGD to the source iteration strategy for solving the neutron transport equation. To illustrate the effectiveness of PGD to evaluate these problems, it is applied a two-dimensional homogeneous example with a volumetric source with various scattering ratios. It is found that PGD is ineffective for pure absorption problems due to the extensive number of terms required in the separated representation, which is verified by singular value decomposition of the full-order model. However, potential is found in utilizing PGD for problems requiring source iteration where the difference in two iterations’ solution is much more separable. In the parameterization application, a PGD approach is employed for uncertainty quantification purposes. The neutron diffusion equation with external sources, a diffusion-reaction problem, is used as the parametric model. The uncertainty parameters include the zone-wise constant material diffusion and reaction coefficients as well as the source strengths, yielding a large uncertain space in highly heterogeneous geometries. The PGD solution, parameterized in all uncertain variables, can then be used to compute mean, variance, and more generally probability distributions of various quantities of interest. In addition to parameterized properties, parameterized geometrical variations of 3D models are also considered. To achieve and analyze a parametric PGD solution, algorithms are developed to decompose the model’s parametric space and semi-analytically integrate solutions for evaluating statistical moments. Varying dimensional problems are evaluated in order to showcase PGD’s ability to solve high-dimensional problems and analyze its convergence. | en |
