Pebble bed reactors (PBRs) are expected to display excellent heat removal characteristics due to graphite's capability for storing and transferring heat, the high failure temperatures of particle fuel, and the low power densities involved. However, a major challenge associated with the modeling of PBRs is the complex fuel-coolant structure in the core. Thermal-hydraulic (T/H) modeling of PBRs requires consideration of thermal and flow effects over five orders in spatial magnitude - from 5e9 fuel particles, each about 1 mm in size, to 5e5 pebbles, each about 5 cm in size, within a 10 m-size reactor core in the larger context of a power generating system. This research develops and applies multiscale methods to the thermal analysis of PBRs. By decomposing the complex PBR geometry into coupled models for three characteristic length scales - the particle, pebble, and core - efficient predictions of core T/H relevant to reactor design are achieved.

These multiscale models are implemented in a new finite element software application built on the open source Multiphysics Object-Oriented Software Environment (MOOSE). By leveraging state-of-the-art numerical methods, solvers, and meshing tools, this dissertation enables rapid design and analysis for scoping studies, fast-turnaround design, and multiphysics coupling to a comprehensive reactor analysis framework.

Application of multiscale analysis to a wide variety of flows demonstrates the software tool's capabilities as a general flow solver and the applicability of these models to both porous and open flows. Verification for open flows shows that the multiscale model reduces to the Navier-Stokes equations in regions such as reactor plena, where prediction of mixing and the ensuing thermal stresses are essential to the design of reactor internals. Application to the SANA experiments, a gas-cooled scaled PBR facility, demonstrates that the multiscale model predicts the passive conduction cool-down heat removal process with an average solid temperature error of 22.6 C. Statistical analysis as a function of position within the bed and other experimental characteristics highlights limitations of model closures and simplifications that are useful in guiding further macroscale analysis of gas-cooled PBRs.

Supported by the verification and validation for open flows and gas systems, full-core steady-state T/H analysis is performed for the Mark-1 Pebble Bed Fluoride-Salt-Cooled High-Temperature Reactor (PB-FHR). The unconventional reflector block design, uniquely thin fuel-matrix annulus, and non-uniform flow boundary conditions (BCs) highlight the new capabilities enabled by this research for PBR industrial analysis. Two multiscale fuel models are compared against full-resolved PB-FHR fuel pebbles for a wide range in thermal conditions. While a homogeneous layer model is characterized by errors in excess of 200 C, a linear superposition method is shown to predict average and maximum temperatures to within 10 C.

Early models of PBRs have struggled to accurately characterize the core bypass fraction, with significant implications on fuel temperature predictions. A porous media model is constructed of the reflectors corresponding to the maximum-bypass end-of-life condition with friction factor correlations generated using COMSOL Multiphysics. A tensor representation of the friction factor shows that the momentum loss is significantly higher in the radial than the axial direction. These drag models are combined with the multiscale fuel verification for full-core analysis of the PB-FHR. A parametric study varying the reflector block gap distribution and the inflow port design demonstrates that the inflow BC has a significant effect on the core bypass fraction and that the bypass fraction is a strong function of the reflector block gap distribution. The maximum bypass fraction is predicted to be within the range of 11.9%-14.0% depending on the inflow BC and gap distribution. For a bottom-heavy center reflector inlet, fuel and reflector temperature predictions are provided. The primary effect of the core bypass is to uniformly raise core temperatures; for all reflector gap distributions, the maximum kernel temperature is approximately 93 C higher than the maximum fluid temperature, which remains far below the fuel failure limit.

This work demonstrates the utility of multiscale methods to thermal analysis of PBRs. In conjunction with the larger scientific community, this research enables fast-turnaround design and analysis of all single-phase PBRs to facilitate the contribution of advanced nuclear reactors to a clean energy future.

The development of methods for deep-penetration radiation transport is of continued importance for radiation shielding, nonproliferation, nuclear threat reduction, and medical applications. As these applications become more ubiquitous, the need for transport methods that can accurately and reliably model the systems’ behavior will persist. For these types of systems, hybrid methods are often the best choice to obtain a reliable answer in a short amount of time. Hybrid methods leverage the speed and uniform uncertainty distribution of a deterministic solution to bias Monte Carlo transport to reduce the variance in the solution. At present, the Consistent Adjoint-Driven Importance Sampling (CADIS) and Forward-Weighted CADIS (FW-CADIS) hybrid methods are the gold standard by which to model systems that have deeply-penetrating radiation. They use an adjoint scalar flux to generate variance reduction parameters for Monte Carlo. However, in problems where there exists strong anisotropy in the flux, CADIS and FW-CADIS are not as effective at reducing the problem variance as isotropic problems.

This dissertation covers the theoretical background, implementation of, and characteri- zation of a set of angle-informed hybrid methods that can be applied to strongly anisotropic deep-penetration radiation transport problems. These methods use a forward-weighted adjoint angular flux to generate variance reduction parameters for Monte Carlo. As a result, they leverage both adjoint and contributon theory for variance reduction. They have been named CADIS-Ω and FW-CADIS-Ω.

To characterize CADIS-Ω, several characterization problems with flux anisotropies were devised. These problems contain different physical mechanisms by which flux anisotropy is induced. Additionally, a series of novel anisotropy metrics by which to quantify flux anisotropy are used to characterize the methods beyond standard Figure of Merit (FOM) and relative error metrics. As a result, a more thorough investigation into the effects of anisotropy and the degree of anisotropy on Monte Carlo convergence is possible.

The results from the characterization of CADIS-Ω show that it performs best in strongly anisotropic problems that have preferential particle flowpaths, but only if the flowpaths are not comprised of air. Further, the characterization of the method’s sensitivity to deterministic angular discretization showed that CADIS-Ω has less sensitivity to discretization than CADIS for both quadrature order and PN order. However, more variation in the results were observed in response to changing quadrature order than PN order. Further, as a result of the forward-normalization in the Ω-methods, ray effect mitigation was observed in many of the characterization problems.

The characterization of the CADIS-Ω-method in this dissertation serves to outline a path forward for further hybrid methods development. In particular, the response that the Ω- method has with changes in quadrature order, PN order, and on ray effect mitigation are strong indicators that the method is more resilient than its predecessors to strong anisotropies in the flux. With further method characterization, the full potential of the Ω-methods can be realized. The method can then be applied to geometrically complex, materially diverse problems and help to advance system modelling in deep-penetration radiation transport problems with strong anisotropies in the flux.

Neutral particle radiation transport simulations are critical for radiation shielding and deep penetration applications. Arriving at a solution for a given response of interest can be computationally difficult because of the magnitude of particle attenuation often seen in these shielding problems. Hybrid methods, which aim to synergize the individual favorable aspects of deterministic and stochastic solution methods for solving the steady-state neutron transport equation, are commonly used in radiation shielding applications to achieve statistically meaningful results in a reduced amount of computational time and effort. The current state of the art in hybrid calculations is the Consistent Adjoint-Driven Importance Sampling (CADIS) and Forward-Weighted CADIS (FW-CADIS) methods, which generate Monte Carlo variance reduction parameters based on deterministically-calculated scalar flux solutions. For certain types of radiation shielding problems, however, results produced using these methods suffer from unphysical oscillations in scalar flux solutions that are a product of angular discretization. These aberrations are termed “ray effects”.

The Lagrange Discrete Ordinates (LDO) equations retain the formal structure of the traditional discrete ordinates formulation of the neutron transport equation and mitigate ray effects at high angular resolution. In this work, the LDO equations have been implemented in the Exnihilo parallel neutral particle radiation transport framework, with the deterministic scalar flux solutions passed to the Automated Variance Reduction Generator (ADVANTG) software and the resultant Monte Carlo variance reduction parameters’ efficacy assessed based on results from MCNP5. Studies were conducted in both the CADIS and FW-CADIS contexts, with the LDO equations’ variance reduction parameters seeing their best performance in the FW-CADIS method, especially for photon transport.

The alpha- and k-effective eigenproblems describe the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. The k-effective eigenvalue problem has generally been solved using power iteration. For problems with dominance ratios close to one, however, power iteration can be intractably slow.

We present the Rayleigh Quotient Fixed Point (RQFP) methods, nonlinear fixed-point methods that are applied to the primitive discretizations of the neutron transport eigenvalue equations. We prove that the discretized eigenvalue equations form a primitive linear system for one-dimensional slab geometry where the unique, positive angular flux eigenvectors are guaranteed to exist from the Perron-Frobenius Theorem for primitive matrices. These methods are capable of solving subcritical and supercritical alpha- and k-effective eigenvalue problems. The derived eigenvalue updates are proven to be optimal in the least squares sense and positive eigenvector updates are guaranteed from the properties of primitive matrices. We consider infinite-medium, one-dimensional slabs and spheres, two-dimensional cylinders, and three-dimensional quarter core benchmark problems and show the ability of the Rayleigh Quotient Fixed Point method to obtain the fundamental eigenvalue and eigenvector of these systems, even when the discretized eigenvalue equations no longer form a primitive system. We also consider the use of Anderson acceleration to accelerate the convergence of the Rayleigh Quotient Fixed Point methods.

We demonstrate that for alpha-eigenvalue problems, the Rayleigh Quotient Fixed Point method substantially reduces the number of iterations required for convergence when compared to traditional alpha-eigenvalue methods such as the critical search method. For a wide variety of problems, the RQFP method for alpha-eigenvalue problems reduces the number of iterations required by up to factors of 50 and converges problems that other methods are unable to converge. For $k$-effective problems, the RQFP method provides moderate reductions of iterations required for convergence when compared to power iteration. In particular, the RQFP method does well for infinite-medium problems or problems where the eigenvector is highly localized. We also demonstrate acceleration of the RQFP method by Anderson acceleration. For slowly converging alpha-eigenvalue problems solved using the RQFP method, Anderson acceleration can provide acceleration of the linear fixed-point method convergence by a factor of up to ten.

By looking at the linear algebraic structure of the discretized neutron transport eigenvalue problems, the RQFP method guarantees the existence of the positive angular flux eigenvector and its corresponding eigenvalue. By examining a wide variety of problems of interest to nuclear engineers, we show that the RQFP method is robust, easily implementable in neutron transport codes, and an efficient solution method for eigenvalue problems in neutron transport.

Nuclear safeguards are a key component in international efforts to prevent the proliferation of nuclear weapons. The International Atomic Energy Agency (IAEA) and its nuclear inspectors are tasked with administering these safeguards measures, ensuring to the best of their ability that weaponizable nuclear material is not created or diverted without the knowledge of the international community. However, no safeguards system is invincible, and so it is necessary for the IAEA to be constantly advancing and expanding its capabilities. At the same time, despite the vast majority of the international community agreeing that the proliferation of nuclear weapons should be avoided via their commitments permitting the IAEA to administer safeguards within their jurisdictions, it is expected that some States will be hesitant to allow the agency to significantly expand the scope of implemented safeguards. This leaves large quantities of potentially insightful data unavailable to safeguards administrators evaluating compliance with international regulations.

This dissertation proposes a solution to reconcile these competing interests in the form of privacy- preserving computation. Using privacy-preserving algorithms, IAEA inspectors and IAEA Mem- ber States may realize a new paradigm where nuclear facility data can be analyzed for safeguards purposes without that data ever being revealed to anyone other than the facility operators. These algorithms open the possibility that entirely new data streams may become accessible to IAEA inspectors, especially those that lend themselves to modern data analytics techniques.

This work represents the culmination of recent efforts to showcase for the first time how nuclear safeguards could be implemented in a privacy-preserving context. It includes a proof-of-concept demonstration of how the privacy-preserving technique of garbled circuits could be used for a safe- guards analysis of real nuclear data, highlighting the present feasibility of such algorithms. It also introduces CypherCircuit, a new software framework for building and evaluating garbled circuits that is intended to facilitate the adoption of privacy-preserving technologies by skeptical or non- expert audiences. The CypherCircuit Python package is designed to cultivate an understanding of garbled circuits through an intuitive, transparent, and accessible design, encouraging the application of privacy-preserving techniques to novel challenges—in this case nuclear safeguards.