A Robust Second-Order Multiple Balance Method and Alpha-Weighted Multigroup Constants for Time-Dependent Nuclear Reactor Simulations

Time-dependent nuclear reactor simulations are essential in improving the safety, effectiveness, and efficiency of nuclear reactor designs, experiments, and operations. This thesis proposes, implements, and tests two new methods designed to improve two different aspects of time-dependent reactor simulation: (1) Multiple Balance Time-Discretization (MBTD), a robust second-order accurate time-stepping method, an alternative to the highly reliable Backward Euler (BE); and (2) α-Weighted Multigroup Constants (α-MGXS), an alternative formulation of multigroup constants that offers advantages over the traditionally-used k-Weighted Multigroup Constants (k-MGXS) for time-dependent neutron transport simulations. Despite being only first-order accurate, BE has been the primary time-discretization method in reactor simulations due to its simplicity and robustness (unconditionally stable and free of spurious oscillations). The Multiple Balance method [Morel & Larsen 1990] was originally introduced as a spatial discretization for neutron transport methods. We show that its application to time-discretization (MBTD) yields a method that is not only robust like BE but also second-order accurate. MBTD consists of solving two coupled balance equations at each time step. In this thesis, three general strategies for solving these coupled equations are explored. MBTD adaptations are made for (1) the finite difference method (FDM) applied to the neutron diffusion equation and for (2) several techniques for the neutron transport equation, including Source Iteration (SI), applied to the Diamond-Difference (SN-DD) and Method of Characteristics (MOC). By exploiting the results of Fourier convergence analysis, an effective Diffusion Synthetic Acceleration (DSA) method for MBTD-SI is developed. Four representative kinetic problems are devised to test and assess the relative efficiency of MBTD versus BE. It is found that MBTD is about 2, 2.5, and 3 times computationally more expensive than BE for neutron diffusion with FDM, neutron transport DSA with SN-DD, and MOC, respectively, given the same uniform time-step size. However, due to its higher-order accuracy, MBTD is generally more efficient than BE: a larger time step can be used to achieve a certain accuracy. Finally, a similar trend is observed in a neutronics/thermal-hydraulics tight-coupling multi-physics application, where MBTD is more efficient than BE for reasonably accurate simulations (relative error less than ~10%). Multigroup neutron transport methods remain as essential tools for reactor simulations, but their accuracy can only be as good as their multigroup constants (MGXS). Estimation of MGXS is traditionally based on the solution of the k-eigenvalue neutron transport calculation. However, the k-eigenfunction is not physically representative for systems that are far from critical, which is the case in many reactor transient simulations. Representing the asymptotic behavior of time-dependent transport problems, the α-eigenfunction may be a better alternative for the calculation of MGXS. In this thesis, physics-preserving MGXS for time-stepping methods are derived. A review of α-eigenvalue iteration methods is presented. A relaxed α-k Iteration developed to simulate the fundamental α-mode is implemented in the open-source Monte Carlo code OpenMC and verified with several benchmark problems. Results from four kinetics problems simulating absorber injection and removal to initially-critical infinite-medium fast and thermal systems emphasize that the fundamental α-eigenfunction—as a multigroup constant weighting spectrum—offers physical characteristics that make it advantageous (in producing accurate solutions) over the typically used fundamental k-eigenfunction.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/167951/1/ilhamv_1.pd

Exploring One-Cell Inversion Method for Transient Transport on GPU

To find deterministic solutions to the transient $S_N$ neutron transport equation, iterative schemes are typically used to treat the scattering (and fission) source terms. We explore the one-cell inversion iteration scheme to do this on the GPU and make comparisons to a source iteration scheme. We examine convergence behavior, through the analysis of spectral radii, of both one-cell inversion and source iterations. To further boost the GPU parallel efficiency, we derive a higher-order discretization method, simple corner balance (in space) and multiple balance (in time), to add more work to the threads and gain accuracy. Fourier analysis on this higher-order numerical method shows that it is unconditionally stable, but it can produce negative flux alterations that are critically damped through time. We explore a whole-problem (in all angle and all cell) sparse linear algebra framework, for both iterative schemes, to quickly produce performant code for GPUs. Despite one-cell inversion requiring additional iterations to convergence, those iterations can be done faster to provide a significant speedup over source iteration in quadrature sets at or below $S_{128}$. Going forward we will produce a two-dimensional implementation of this code to experiment with memory and performance impacts of a whole-problem framework including methods of synthetic acceleration and pre-conditioners for this scheme, then we will begin making direct comparisons to traditionally implemented source iteration in production code.Comment: 11 pages, 4 figures, M\&C 2023 ANS conferenc

Analysis of Population Control Techniques for Time-Dependent and Eigenvalue Monte Carlo Neutron Transport Calculations

An extensive study of population control techniques (PCTs) for time-dependent and eigenvalue Monte Carlo (MC) neutron transport calculations is presented. We define PCT as a technique that takes a censused population and returns a controlled, unbiased population. A new perspective based on an abstraction of particle census and population control is explored, paving the way to improved understanding and application of the concepts. Five distinct PCTs identified from the literature are reviewed: Simple Sampling (SS), Splitting-Roulette (SR), Combing (CO), modified Combing (COX), and Duplicate-Discard (DD). A theoretical analysis of how much uncertainty is introduced to a population by each PCT is presented. Parallel algorithms for the PCTs applicable for both time-dependent and eigenvalue MC simulations are proposed. The relative performances of the PCTs based on runtime and tally mean error or standard deviation are assessed by solving time-dependent and eigenvalue test problems. It is found that SR and CO are equally the most performant techniques, closely followed by DD.Comment: 51 pages (double-spaced, 5-page appendix), 20 figures, submitted to Nuclear Science and Engineerin

Development of Machine Learning Model for Neutron Clustering Classification in Monte Carlo Criticality Simulation

This study presents the ongoing development of a machine learning model as a diagnostic tool to identify particle cluster(s) formation during Monte Carlo criticality simulations. The particle distribution is essentially spatial-temporal features???evolves in space and simulation cycle???and can be directly extracted from Monte Carlo simulation for machine-learning model training without the use of intermediary/silhouette metrics???potentially losing information. The machine learning model uses a combination of convolutional neural network and long short-term memory (Many-to-One variant) to map the spatial-temporal features into higher time-independent features for the classification task. The model was trained with synthetic dataset generated by 3D Gauss distribution to mimic particle cluster(s) formation, and later evaluated with the test set generated from OpenMC simulations. The model achieves an accuracy of 99% for the synthetic test set but only 11% for the OpenMC dataset, revealing the limitations of current features extraction method. Furthermore, the direct use of particle distribution as spatial-temporal features is not viable for a large-scale machine learning model due to the prohibitively large dataset space requirement, so potential spatial-temporal features are discussed for further study

SIMULASI DINAMIKA REAKTOR TITIK UNTUK REAKTOR PRODUKSI ISOTOP BERBAHAN BAKAR CAIR (LFIPR) BERBAHAN BAKAR URANIL NITRAT

Liquid-Fueled Isotope Production Reactor (LFIPR) is one of Aqueous Homogeneous Reactor (AHR)-typed reactors being developed. Modelling and simulation of reactor dynamics play important roles in achieving insight regarding responses of the design for implementations during operation. Utilizing point reactor approximation, research on reactor dynamics of LFIPR has been performed. The main purpose o f the research was developing simulator and simulating point reactor dynamics of uranyl nitrate fuel-based LFIPR on burnup level of 0 MWd/t (Begin of Life, BOL), 20677.5 MWd/t (Middle of Life, MOL), and 41355 MWd/t (End of Life, EOL). The reactor dynamics model comprises two groups point reactor kinetics (thermal and fast neutron fluxes, 6 delayed neutron precursor group and 4 neutron poison with the parent concentrations, and fuel and coolant temperatures), feedback and control of kinetic properties (reactivity, average neutron lifetime, macroscopic cross sections, and diffusion coefficient), and kinetic properties. Simulation with its initial condition and kinetic parameters are provided for burnup level of BOL, MOL, and EOL. The track ed neutron poisons are xenon-135 and samarium-149. Numerical model of reactor kinetics is solved using Runge-Kutta-Fehlberg 45 (RKF45). Reactor core kinetic properties are obtained by utilizing PIJ and CITATION code in SRAC2006 code package. Feedback parameters (temperature, void, neutron poison coefficient) and control rod parameters (central and peripheral control rod coefficient) are obtained from gradient of polinomial regression model between kinetic property and related parameters. Burnup levels are obtained by utilizing BURN code in SRAC2006 code package. Overall heat transfer coefficient is assumed to be 1000 W/m 2 s. Simulator is built in Python programming language. The simulation results show that the reactor dynamic respons toward implementations during simu lation conform with the theory. The simulated implementations comprise insertion of positive and negative step reactivit ies on critical zero power and power level, engineered reactivity accident, shutdown, and loss of coo lant accident

A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS

A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed

A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS

A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed