There is a significant need for 3-D steady-state and transient neutron transport formulations and codes that yield accurate, high-fidelity solutions with reasonable computing resources and time. These tools are essential for modeling innovative nuclear systems, such as next-generation reactor designs. The existing methods generally compromise heavily between accuracy and affordability in terms of computation times. In this dissertation, novel algorithms for simulation of reactor transient conditions have been developed and implemented into the RAPID code system. In addition, extensive computational verification and experimental validation of RAPID's steady-state and transient algorithms was performed, and a novel virtual reality system (VRS) web-application was developed for the RAPID code system. The new algorithms, collectively referred to as tRAPID, are based on the Transient Fission Matrix (TFM) methodology. By decoupling the kinetic neutron transport problem into two different stages (an accurate pre-calculation to generate a database and an on-line solution of linear partial differential equations) the method ensures the preservation of the highest level of accuracy while also allowing for high-fidelity modeling and simulation of nuclear reactor kinetics in a short time with minimal computing resources. The tRAPID algorithms have been computationally verified using several computational benchmarks and experimentally validated using the JSI TRIGA Mark-II reactor. In order to develop these algorithms, first the steady-state capabilities of RAPID have been successfully benchmarked against the GBC-32 spent fuel cask system, also highlighting issues with the standard eigenvalue Monte Carlo calculations that our code is capable of overcoming. A novel methodology for accounting for the movement of control rods in the JSI TRIGA reactor has been developed. This methodology, referred to as FM-CRd, is capable of determining the neutron flux distribution changes due to the presence of control rod in real-time. The FM-CRd method has been validated with successfully using the JSI TRIGA reactor. The time-dependent, kinetic capabilities of the new tRAPID algorithm have been implemented based on the Transient Fission Matrix (TFM) method. tRAPID has been verified and validated using the Flattop-Pu benchmark and reference calculations and measurements using the JSI TRIGA reactor. In addition to the main tRAPID algorithms development and benchmarking efforts, a new web-application for the RAPID Code System for input preparation and interactive output visualization was developed. VRS-RAPID greatly enhances the usability, intuitiveness, and outreach possibilities of the RAPID Code System. / Doctor of Philosophy / The simulation of the behavior of nuclear systems (such as power reactors) relies on the development of innovative software that allows for calculating nuclear-relevant quantities in support of the design, operation, and safety of said systems. Traditional codes are often very complex and need to rely on approximations and/or require a very large amount of time to perform even a single calculation. The RAPID Code System is based on a methodology that allows for pre-calculation of a database that can later be used to simulate nuclear systems in real-time while achieving high levels of accuracy. For this dissertation, several new algorithms for simulation of equilibrium and transient conditions of nuclear systems have been developed for the RAPID Code System. In particular, the main features and additions are the ability of simulating the insertion of control rods (devices that are used to control the fission chain reaction) in nuclear reactors and the ability of analyzing the kinetics of nuclear systems. This latter feature, in particular, is extremely important and difficult to simulate, as it involves the fast variation in time of the nuclear quantities under analysis. Finally, a Virtual Reality System (VRS) is embedded with RAPID for easy utilization of the code through a web-application. All these new algorithms and tools have been benchmarked and validated, against reference high-fidelity computational predictions and experimental data. This dissertation demonstrates RAPID's ability of achieving accurate high quality solutions in a rather short time.

The subgroup decomposition method (SDM) has recently been developed as an improvement over the consistent generalized energy condensation theory for treatment of the energy variable in deterministic particle transport problems. By explicitly preserving reaction rates of the fine-group energy structure, the SDM directly couples a consistent coarse-group transport calculation with a set of fixed-source "decomposition sweeps" to provide a fine-group flux spectrum. This paper will outline the implementation of the SDM into the three-dimensional, discrete ordinates (SN) deterministic transport code TITAN. The new version of TITAN, TITAN-SDM, is tested using 1-D and 2-D benchmark problems based on the Japanese designed High Temperature Engineering Test Reactor (HTTR). In addition to accuracy, this study examines the efficiency of the SDM algorithm in a 3-D SN transport code. / Master of Science

Fuel burnup calculations provide material concentrations and intrinsic neutron and gamma source strengths as a function of irradiation and cooling time. Detailed, full-core 3D burnup calculations are critical for nuclear fuel management studies, including core design and spent fuel storage safety and safeguards analysis. For core design, specifically during refueling, full- core pin-wise, axially-dependent burnup distributions are necessary to determine assembly positioning to efficiently utilize fuel resources. In spent fuel storage criticality safety analysis, detailed burnup distributions enable best-estimate analysis which allows for more effective utilization of storage space. Additionally, detailed knowledge of neutron and gamma source distributions provide the ability to ensure nuclear material safeguards. The need for accurate and efficient burnup calculations has become more urgent for the simulation of advanced reactors and monitoring and safeguards of spent fuel pools. To this end, the Virginia Tech Transport Theory Group (VT3G) has been working on advanced computational tools for accurate modeling and simulation of nuclear systems in real-time. These tools are based on the Multi-stage Response-function Transport (MRT) methodology. For monitoring and safety evaluation of spent fuel pools and casks, the RAPID (Real-time Analysis for Particle transport and In-situ Detection) code system has been developed. This dissertation presents a novel methodology and algorithm for performing 3D fuel bur- nup calculations, referred to as bRAPID- Burnup with RAPID . bRAPID utilizes the existing RAPID code system for accurate calculation of 3D fission source distributions as the trans- port calculation tool to drive the 3D burnup calculation. bRAPID is capable of accurately and efficiently calculating assembly-wise axially-dependent fission source and burnup dis- tributions, and irradiated-fuel properties including material compositions, neutron source, gamma source, spontaneous fission source, and activities. bRAPID performs 3D burnup calculations in a fraction of the time required by state-of-the-art methodologies because it utilizes a pre-calculated database of response functions. The bRAPID database pre-calculation procedure, and its automation, is presented. The ex- isting RAPID code is then benchmarked against the MCNP and Serpent Monte Carlo codes for a spent fuel pool and the U.S. Naval Academy Subcritical Reactor facility. RAPID is shown to accurately calculate eigenvalue, subcritical multiplication, and 3D fission source dis- tributions. Finally, bRAPID is compared to traditional, state-of-the art Serpent Monte Carlo burnup calculations and its performance will be evaluated. It is important to note that the automated pre-calculation proceedure is required for evaluating the performance of bRAPID. Additionally, benchmarking of the RAPID code is necessary to understand RAPID's ability to solve problems with variable burnups distributions and to asses its accuracy. / Ph. D. / In a nuclear reactor, the energy released from a fission reaction, the splitting of an atomic nucleus into smaller parts, is harnessed to generate electricity. Nuclear reactors rely on fuel, typically comprised of uranium oxide (UO₂). While the reactor is operating and the fuel is being used, or “burned”, for power production it is undergoing numerous nuclear reactions, including fission, and radioactive decays which alter the material composition. Knowing the time evolution of fuel as it is burned in the reactor, i.e., concentration of isotopes and sources of radiation, is critical. Nuclear reactor designers and operators use this information to optimize power production and perform safety analysis of used nuclear fuel. By performing fuel burnup calculations, material concentrations and radiation source strengths are obtained as a function of time in an operating nuclear reactor. Using traditional computational techniques, these calculations are extremely time consuming and, for certain problems, can be difficult to obtain an accurate solution. Ideally, a reactor designer would like to know the three-dimensional (3D) distribution of material compositions and sources; however this level of detail would require an excessive amount of calculation time, therefore simplified models and assumptions are used. For the design of the new generation of nuclear reactors, and monitoring and safeguards analysis, this level of detail will be required in lieu of the availability of experimental facilities which do not currently exist. This dissertation presents a novel methodology and algorithm for performing accurate 3D fuel burnup calculations in real-time, referred to as bRAPID (Burnup with RAPID). bRAPID utilizes an existing nuclear software, RAPID (Real-time Analysis for Particle transport and In-situ Detection), developed in the Virginia Tech Transport Theory Group (VT3G), which has been shown to accurately solve time-independent nuclear calculations in significantly less time than traditional approaches. bRAPID is capable of accurately calculating 3D material and source distributions as a function of time in an operating nuclear reactor, and requires significantly less time and computational resources than traditional approaches. To ensure that bRAPID is relatively easy to use, a number of automated routines have been developed and are presented. RAPID is benchmarked against the traditional code systems MCNP (Monte Carlo N-Particle) and Serpent, both of which are widely used in the nuclear community, for a spent fuel storage pool and the U.S. Naval Academy subcritical nuclear reactor facility. RAPID is shown to accurately calculate system parameters (eigenvalue and subcritical multiplication factor) and 3D fission source distributions. Finally, bRAPID is compared to the traditional burnup approach, using the Serpent code system. bRAPID is shown to accurately calculate system parameters and 3D material and source distributions in significantly less time than the traditional approach.

This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.

Un certain nombre de domaines tels que les études de criticité requièrent le calcul de certaines grandeurs neutroniques d'intérêt. Il existe deux types de code : les codes déterministes et les codes stochastiques. Ces derniers sont réputés simuler la physique de la configuration traitée de manière exacte. Toutefois, le temps de calcul nécessaire peut s'avérer très élevé.Le travail réalisé dans cette thèse a pour but de bâtir une stratégie d'accélération de la convergence de la criticité dans le code de calcul TRIPOLI-4®. Nous souhaitons mettre en œuvre le jeu à variance nulle. Pour ce faire, il est nécessaire de calculer le flux adjoint. L'originalité de cette thèse est de calculer directement le flux adjoint par une simulation directe Monte-Carlo sans passer par un code externe, grâce à la méthode de la matrice de fission. Ce flux adjoint est ensuite utilisé comme carte d'importance afin d'accélérer la convergence de la simulation. / Fields such as criticality studies need to compute some values of interest in neutron physics. Two kind of codes may be used : deterministic ones and stochastic ones. The stochastic codes do not require approximation and are thus more exact. However, they may require a lot of time to converge with a sufficient precision.The work carried out during this thesis aims to build an efficient acceleration strategy in the TRIPOLI-4®. We wish to implement the zero variance game. To do so, the method requires to compute the adjoint flux. The originality of this work is to directly compute the adjoint flux directly from a Monte-Carlo simulation without using external codes thanks to the fission matrix method. This adjoint flux is then used as an importance map to bias the simulation.

Benchmark calculations are performed for neutron transport in a two material (binary) stochastic multiplying medium. Spatial, angular, and energy dependence are included. The problem considered is based on a fuel assembly of a common pressurized water nuclear reactor. The mean chord length through the assembly is determined and used as the planar geometry system length. According to assumed or calculated material distributions, this system length is populated with alternating fuel and moderator segments of random size. Neutron flux distributions are numerically computed using a discretized form of the Boltzmann transport equation employing diffusion synthetic acceleration. Average quantities (group fluxes and k-eigenvalue) and variances are calculated from an ensemble of realizations of the mixing statistics. The effects of varying two parameters in the fuel, two different boundary conditions, and three different sets of mixing statistics are assessed. A probability distribution function (PDF) of the k-eigenvalue is generated and compared with previous research. Atomic mix solutions are compared with these benchmark ensemble average flux and k-eigenvalue solutions. Mixing statistics with large standard deviations give the most widely varying ensemble solutions of the flux and k-eigenvalue. The shape of the k-eigenvalue PDF qualitatively agrees with previous work. Its overall shape is independent of variations in fuel cross-sections for the problems considered, but its width is impacted by these variations. Statistical distributions with smaller standard deviations alter the shape of this PDF toward a normal distribution. The atomic mix approximation yields large over-predictions of the ensemble average k-eigenvalue and under-predictions of the flux. Qualitatively correct flux shapes are obtained, however. These benchmark calculations indicate that a model which includes higher statistical moments of the mixing statistics is needed for accurate predictions of binary stochastic media k-eigenvalue problems. This is consistent with previous findings. / Graduation date: 2005

The subject of this thesis is the development of a nodal discretization of the low-order quasi-diffusion (QDLO) equations for global reactor core calculations. The advantage of quasi-diffusion (QD) is that it is able to capture transport effects at the surface between unlike fuel assemblies better than the diffusion approximation. We discretize QDLO equations with the advanced nodal methodology described by Palmtag (Pal 1997) for diffusion. The fast and thermal neutron fluxes are presented as 2-D, non-separable expansions of polynomial and hyperbolic functions. The fast flux expansion consists of polynomial functions, while the thermal flux is expanded in a combination of polynomial and hyperbolic functions. The advantage of using hyperbolic functions in the thermal flux expansion lies in the accuracy with which hyperbolic functions can represent the large gradients at the interface between unlike fuel assemblies. The hyperbolic expansion functions proposed in (Pal 1997) are the analytic solutions of the zero-source diffusion equation for the thermal flux. The specific form of the QDLO equations requires the derivation of new hyperbolic basis functions which are different from those proposed for the diffusion equation. We have developed a discretization of the QDLO equations with node-averaged cross-sections and Eddington tensor components, solving the 2-D equations using the weighted residual method (Ame 1992). These node-averaged data are assumed known from single assembly transport calculations. We wrote a code in "Mathematica" that solves k-eigenvalue problems and calculates neutron fluxes in 2-D Cartesian coordinates. Numerical test problems show that the model proposed here can reproduce the results of both the simple diffusion problems presented in (Pal 1997) and those with analytic solutions. While the QDLO calculations performed on one-node, zero-current, boundary condition diffusion problems and two-node, zero-current boundary condition problems with UO₂-UO₂ assemblies are in excellent agreement with the benchmark and analytic solutions, UO₂-MOX configurations show more important discrepancies that are due to the single-assembly homogenized cross-sections used in the calculations. The results of the multiple-node problems show similar discrepancies in power distribution with the results reported in (Pal 1997). Multiple-node k-eigenvalue problems exhibit larger discrepancies, but these can be diminished by using adjusted diffusion coefficients (Pal 1997). The results of several "transport" problems demonstrate the influence of Eddington functionals on homogenized flux, power distribution, and multiplication factor k. / Graduation date: 2003

Several computer codes based on one and two-group diffusion theory models were developed for SHUFFLE. The programs were developed to calculate power distributions in a two-dimensional quarter core geometry of a pressurized power reactor. The various coarse-mesh numerical computations for the power calculations yield the following: the Borresen's scheme applied to the modified one-group power calculation came up with an improved power distribution, the modified Borresen's method yielded a more accurate power calculations than the Borresen's scheme, the face dependent discontinuity factor method have a better prediction of the power distribution than the node averaged discontinuity factor method, Both the face dependent discontinuity factor method and the modified Borresen's methods for the two-group model have quite attractive features. / Graduation date: 1992

A condensed multigroup formulation is developed which maintains direct consistency with the continuous energy or fine-group structure, exhibiting the accuracy of the detailed energy spectrum within the coarse-group calculation. Two methods are then developed which seek to invert the condensation process turning the standard one-way condensation (from fine-group to coarse-group) into the first step of a two-way iterative process. The first method is based on the previously published Generalized Energy Condensation, which established a framework for obtaining the fine-group flux by preserving the flux energy spectrum in orthogonal energy expansion functions, but did not maintain a consistent coarse-group formulation. It is demonstrated that with a consistent extension of the GEC, a cross section recondensation scheme can be used to correct for the spectral core environment error. A more practical and efficient new method is also developed, termed the "Subgroup Decomposition (SGD) Method," which eliminates the need for expansion functions altogether, and allows the fine-group flux to be decomposed from a consistent coarse-group flux with minimal additional computation or memory requirements. In addition, a new whole-core BWR benchmark problem is generated based on operating reactor parameters in 2D and 3D, and a set of 1D benchmark problems is developed for a BWR, PWR, and VHTR core.

This work presents a whole-core benchmark problem based on a 2-loop pressurized water reactor with both UO₂and MOX fuel assemblies. The specification includes heterogeneity at both the assembly and core level. The geometry and material compositions are fully described and multi-group cross section libraries are provided in 2, 4, and 8 group formats. Simplifications made to the benchmark specification include a Cartesian boundary, to facilitate the use of transport codes that may have trouble with cylindrical boundaries, and control rod homogenization, to reduce the geometric complexity of the problem. These modifications were carefully chosen to preserve the physics of the problem and a justification of these modifications is given. Detailed Monte Carlo reference solutions including core eigenvalue, assembly averaged fission densities and selected fuel pin fission densities are presented for benchmarking diffusion and transport methods. Three different core configurations are presented in the paper namely all-rods-out, all-rods-in, and some-rods-in.