A Non Intrusive Reduced Order Model via Sparse Grids for Parametric Neutron Transport

Abstract

The simulation of radiation neutron transport involves a complex system of partial differential equations. Methods in solving these systems of equations can be categorized into two areas of concern, stochastic and deterministic. In stochastic approaches, like that of a Monte Carlo method, the probabilistic nature of neutrons to interact with surrounding media is leveraged to simulate in-dividual particles along its trajectory through the model, essentially tracking and collecting particle histories. In a deterministic approach the system of linear equations are discretized about several characteristics including combinations of space, angle and energy. Deterministic methods are an area of a extensive research with its implementation taking many forms. Modern deterministic approaches may involve the use of higher-order polynomials and or Gauss quadratures with SN, PN and characteristic methods among the most common deterministic approaches for solving the transport equations. In general, dealing with parametric or geometric variations to the neutron transport equation would require a complete construction and solve of the system of equations. This process, in many cases, would depend on the assistance of multiple computationally large machines, a resource that is not only expensive but not readily available. In an attempt to remedy this constraint the application of model-order reduction was applied to the governing equations. The work presented in this thesis delves into the implementation of a nonintrusive Reduced-Order Model (ROM) that utilizes the orthogonality of a reduced basis. In particular, a sparse grid implementation represented the basis of the ROM. By proving effective for problems of higher dimensionality and requiring no information regarding the underlying physics of the problem, sparse grids are shown to be a reliable approach for parametric simulations. In this thesis, Oak Ridge National Laboratory’s (ORNL) sparse grid toolkit Tasmanian was employed to generate and interpolate sampling points. A large-scale scientific simulation engine, Chi-Tech, actively being developed at Texas A&M University handled solving the neutron transport equation generating values for the neutron scalar flux. A python wrapper was developed to handle the transition of information between production codes while producing the visualization and organization of results. While a sparse grid can be used to interpolate snapshots, solutions to the Full-Order Model (FOM) for parameter variations, it may prove beneficial to instead interpolate based on a reduced operator that retains the underlying physics of the model. The class of numerical methods be-longing to this form of reduction are referred to as Model Order Reduction (MOR) techniques. In particular, Proper Orthogonal Decomposition (POD) is a method that has shown success in applications of computational fluid dynamics and structural analysis. With success in similarly complex systems of linear equations, the successful application of POD methods towards deterministic radiation neutron transport holds merit. By utilizing a Galerkin approach a general function can be rewritten as a combination of linear basis functions, ui, weighted by expansion coefficients, ci. By multiplying the set of coefficients by their respective basis functions and summed together, an ap-proximation to the system of linear equations is produced. The values of these coefficients are not constant and are to be resolved for every solution to the system of equations. In order to compute these expansions coefficients the Singular Value Decomposition (SVD) of the snapshot matrix is taken. The resulting singular vectors are projected onto the snapshots resulting in the expansion coefficients. The expansion coefficients are of size N × N where as the snapshots themselves would be of size M × N, M being the degrees of freedom and N being the number of snapshots. Given that M is often much larger than the number of snapshots utilized in a reduced order model, interpolating base upon expansions coefficients rather than the snapshots themselves holds merit. It can readily be seen that after some simple matrix multiplication the interpolated expansions coefficients can be used to retrieve solutions to the FOM, be it that the parameter variations are within the sampled range that the ROM was built with. The effectiveness of reconstructing solutions to the FOM with interpolated expansion coefficients from the ROM were demonstrated through several benchmarks. The L2 relative error norm in the reproduction of the scalar flux and specified quantities of interest were also explored.