A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation

The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation

Verification of the 3D Method of characteristics solver in OpenMOC

The Method of Characteristics (MOC) has seen wide interest in full-core reactor physics analysis due to its computational efficiency and ability to easily treat complex geometries. Recently, the OpenMOC reactor physics code was extended to include 3D MOC capability. In this study, we present verification for the 3D MOC solver in OpenMOC and sensitivity of 3D MOC to the axial geometry discretization and axial track laydown. Results for the Takeda Model 1 benchmark show excellent agreement with the reference eigenvalues. A sensitivity study was conducted on a UO [subscript 2] quarter-assembly extracted from the C5G7 3D unrodded benchmark geometry in order to show the effect of the axial MOC parameters on the solution eigenvalue for a heterogeneous problem. The sensitivity results demonstrated that the solution accuracy was highly dependent on the axial source region discretization, but insensitive to axial spacing between tracks below ~0.2 cm. Using the equal angle quadrature set, at least 10 and 18 polar angles were required to converge the problem to with 100 and 10 pcm, respectively. These results both verify the 3D MOC solver in OpenMOC and provide inUnited States. Office of the Assistant Secretary for Nuclear Energy (Nuclear Energy Uni- versity Programs Fellowship)Center for Exascale Simulation of Advanced Reactors (Contract No. DE-AC02-06CH11357

An Investigation of 2D/1D Approximations to the 3D Boltzmann Transport Equation.

A new class of "2D/1D" approximations is proposed for the 3D linear Boltzmann equation. These approximate equations preserve the exact transport physics in the radial directions x and y and employ approximate diffusion physics in the axial direction z. Thus, the 2D/1D equations are more accurate approximations of the 3D Boltzmann equation than the conventional 3D diffusion equation. The 2D/1D equations can be systematically discretized, to yield accurate simulation methods for 3D reactor core problems. The resulting solutions are more accurate than 3D diffusion solutions, and less expensive to generate than standard 3D transport solutions. In this work, we (i) introduce several new "2D/1D equations" as accurate approximations to the 3D Boltzmann transport equation, (ii) show that the 2D/1D equations have certain desirable properties, (iii) systematically discretize the equations, and (iv) derive a stable iteration scheme for solving the discrete system of equations. Additionally, we give numerical results that confirm the theoretical predictions of accuracy and iterative stability.PhDNuclear Engineering & Radiological Sciences and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113576/1/kelleybl_1.pd

A 2D/1D Neutron Transport Method with Improved Angular Coupling

Developing efficient and accurate three-dimensional (3D) neutron transport methods for nuclear reactor applications has long been a major objective for nuclear scientists in the field of reactor physics and radiation transport. Even with the large computers available today, exact 3D neutron transport methods are often too costly to be used for practical core design or safety analysis. Several methods have been developed that use various approximations to the neutron transport equation so that the calculations can be performed on commonly available computing platforms. One such method is the 2D/1D method, which decomposes 3D geometries into several 2D domains wherein 2D transport equations are solved. These 2D transport equations are coupled to one another through transverse, 1D, approximate transport solutions in the axial direction. The 2D/1D method is best suited for problems where the axial gradient of the solution is relatively weak, such as Light Water Reactor (LWR) problems. The 2D/1D method uses an accurate 2D transport solution to resolve the highly heterogeneous radial geometry, and treats the axial dimension with a lower-fidelity, more coarsely discretized solution, which is usually appropriate. Some of the typical assumptions made in many 2D/1D methods can negatively affect the accuracy of the solution in a non-negligible way. Two of the most significant are the isotropic approximations made to the transverse leakage (TL) and homogenized total cross section (XS) used to couple the 2D and 1D equations. In cases where the axial gradients are relatively strong, these assumptions are detrimental to the accuracy. The isotropic TL approximation was corrected in previous work. In this work, the XS is also allowed to be anisotropic. The results show that with both anisotropic TL and XS, the accuracy of 2D/1D is improved significantly. The 2D/1D methods with anisotropic TL and XS are significantly more expensive than the isotropic TL and XS method, which is the standard in the Michigan Parallel Characteristics Transport (MPACT) code. In this work, a 2D/1D method with polar angle parity is developed to significantly reduce the run time of the anisotropic TL and XS method while still significantly improving the accuracy compared to the isotropic TL and XS method. The theoretical accuracy limit of the 2D/1D methods are analyzed and compared to the 3D Simplified P3 (SP3) method. We find that the 2D/1D method with anisotropic TL preserves the 3D SP3 limit with only a few anisotropic TL moments, while the 2D/1D method with isotropic TL does not. As a result, the isotropic TL method is less accurate in problems where there are strong spatial gradients in the radial and axial dimensions.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147498/1/jarremic_1.pd

An Azimuthal, Fourier Moment-Based Axial SN Solver for the 2D/1D Scheme.

Despite the incredible advancements in computing power in recent decades, using explicitly three-dimensional neutron transport methods is still very computationally expensive. Several alternative methods have been developed to make computation more tractable, such as the “2D/1D” scheme, which decomposes three-dimensional geometries into an axial stack of radial planes. In this scheme, one common approximation is to assume that the radial and axial transverse leakages that couple the axial and radial solvers are isotropic, which means that all angular dependence of the leakage is neglected. For more complicated problems, such as those with control rods or mixed oxide (MOX) fuels, higher fidelity treatment of the axial and radial leakages is needed to better capture the relationship between the solvers. The first objective of the work presented here investigates incorporating full angular dependence of both the azimuthal and polar angles into the transverse leakages. Fully explicit angular dependence is shown to be particularly burdensome, both in terms of memory and run time requirements. The second, more novel objective uses a Fourier series expansion to account for the azimuthal dependence, requiring the formulation of a new axial SN solver to generate angular fluxes for the axial transverse leakage construction. In several test cases analyzed, which include cases with both control rods and MOX fuels, noteworthy accuracy gains are observed by including the angular dependence of the leakages. The Fourier moment-based approach performs very well, accurately capturing the azimuthal dependence with only a few moments. Overall, the Fourier moment-based approach reduces the run time by roughly a factor of 1.5, the aggregate memory footprint by a factor of 3 to 4, and angle-dependent variables by an order of magnitude. Other test problems highlight one of the remaining sources of error relating to the spatial distribution of the axial transverse leakage, which is introduced because the axial solver operates on a coarse radial grid. The results suggest that by including a more accurate angular representation, some cancellation of error between the spatial and angular treatments is removed, indicating that future work focusing on improving the spatial distribution should be pursued.PhDNuclear Engineering & Radiological Sciences & Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111446/1/sgstim_1.pd

Robust and Efficient Methods in Transient Whole-Core Neutron Transport Calculations

Modeling the time-dependent transient behavior of nuclear reactors with high-fidelity pin-resolved detail has increased importance when the operating power of the reactor is increased to improve the economic performance. In previous research, the efficiency of the solution of the steady-state neutron transport equation, which provides the initial condition for the transient, was improved with the development of advanced methods such as the Multilevel-in-Space-and-Energy Diffusion (MSED). However, the application of the MSED method was ultimately limited by numerical instabilities in the presence of cross section feedback. The first objective of this research is to improve the efficiency of the steady-state solution by investigating and eliminating the numerical instability of accelerated neutron transport iterations when there is cross section feedback. The second objective of the research here is to address the computational costs of performing transient simulations by improving the performance of the Transient Multilevel (TML) method in the MPACT code. Specifically, the run time of the CMFD solver in TML dominates the run time, so a one-group acceleration method is developed and added. Automated time-stepping methods were also not previously available for TML. The research here significantly improves the efficiency of the transient calculation by accelerating the CMFD solver and using adaptive time-stepping methods. Improving the stability and efficiency of the transient whole-core neutron transport calculations is the main significant and original contribution of this work. The specific contributions of this thesis for the steady-state calculation are the theory, development, and implementation of the nearly-optimally partially converged CMFD (NOPC-CMFD) method and the X-CMFD method in MPACT. As its name suggests, the NOPC-CMFD method stabilizes the iteration scheme by determining and utilizing the nearly-optimal partial convergence of the diffusion solutions. The X-CMFD method is an original method that stabilizes the iteration by applying the feedback at the power iteration level of the low-order diffusion eigenvalue problem. Compared to the default iteration scheme in MPACT, the methods developed here demonstrate the same stability compared to CMFD-accelerated transport iterations in problems without feedback, and reduce the overall run time of the full-core multi-state depletion problem by ∼43%. The principal original work of this thesis for the transient simulations is the introduction of a one-group CMFD (1GCMFD) acceleration method and the development of adaptive time-stepping methods to further accelerate the TML scheme. The 1GCMFD method is shown to reduce the overall computational time of CMFD by as much as 50% for practical large-scale applications. The adaptive time-stepping method introduced adjusts the time step so that the maximum magnitude of the relative error is smaller than 1% for the applications considered in this research. Other innovative methods include the usage of the Spectral Deferred Correction (SDC) method to solve the point-kinetics equation and the use of Strang Splitting (SS) to replace Lie Splitting for coupling the neutronics and the TH solvers. The implemented SDC method is A-stable for orders up to 8, and the SS addresses the inconsistency between the error and time step size when the time step size is varied. When the 1GCMFD acceleration and the adaptive methods are applied together, the performance of the TML scheme for the SPERT test 86 problem is reduced by ∼22%, and the maximum magnitude of the relative error is reduced from ~1.8% to ~0.4%, compared to the use of TML with the default parameters.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168050/1/qicangsh_1.pd

Parallel 3-D Method of Characteristics with Linear Source and Advanced Transverse Integration

In the design and analysis of nuclear fission reactor systems, simulations are an essential tool for improving efficiency and safety. Neutronics simulations have always been limited by the available computational resources. This is because of the large discretizations necessary for the neutron transport equation, which has a 6-dimensional phase space for steady-state eigenvalue problems. The “gold standard” for 3-D neutron transport simulations is Monte Carlo with explicit geometry representation because it treats all dependent variables continuously. However, there are significant remaining challenges for Monte Carlo methods that prohibit widespread use and put them at a disadvantage compared to deterministic methods. The “gold standard” for deterministic 3-D neutron transport is the MoC. Numerous deterministic methods exist for solving the transport equation. Each of them has their own drawback. MoC is considered the “best” due to its ability to accurately model the exact geometry and approximate anisotropic scattering (other methods do just one of these well or become undesirably complex). The downside of the 3-D MoC method is the substantial computational resources required to discretize the problem. There has been renewed interest in assessing the state of the art for MoC and the tractability of this problem on the newest computer architectures. Previous work made significant strides in parallelizing the 3-D MoC algorithm for 100,000’s of processors, but ultimately did not prove viable due to the extreme compute resources required. Since then there has been progress in making 3-D MoC less computationally burdensome by adopting more advanced discretization methods that lead to fewer spatial mesh regions and rays; namely the linear-source approximation (LSA), and chord-classification or on-the-fly ray-tracing. The goal of this thesis is to continue progress in reducing the computational burden of MoC calculations, with a focus on three-dimensional calculation. This thesis tries to reach this goal through three related contributions: the utilization of graph-theory for spatial decomposition, improvements to the LSA for Multiphysics calculations, and a novel 3-D ray-tracing method with advanced transverse integration. Spatial decomposition is typically very beneficial, if not necessary, for whole-core direct transport methods. Previous works on 3-D MoC calculations have used simple spatial decomposition schemes, that often resulted in poor load-balancing, particularly when using the LSA. This work addresses this issue by utilizing graph partitioning methods to give better load-balance, even in cases where the number of computational cells is very different in different regions of the reactor. The LSA has previously been shown to allow for the use of a coarser mesh while maintaining accuracy in pure neutronics calculations. However, typically the problems of interest involve multiple physics such as isotopic depletion and thermal-hydraulic (T/H) feedback. This work improves the LSA method for such problems by re-formulating the equations to eliminate an inefficiency in cases with non-constant cross sections. This is shown to significantly improve run-times and reduce memory usage, even in such cases. Finally, a novel 3-D ray-tracing method, based on the macroband, is developed to reduce the number of characteristic tracks necessary for accurate results. The method is compared against a traditional ray-tracing method for several benchmark problems. In several of these cases, the method is shown to significantly reduce the number of segments necessary for similar accuracy. The ray-tracing method is also shown to have very desirable properties such as near-monotonic convergence, and can act as more of a “black-box” solver.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155260/1/apfitzge_1.pd

A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation

The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation

Multiphysics Simulation of Fission Gas Production and Release in Light Water Reactor Fuel

Along with a recent trend in nuclear engineering of coupling codes to perform high-fidelity, multiphysics simulations, the MOOSE application Redwing was developed to couple the neutron transport and core simulator MPACT and the fuel performance program BISON in order to simulate light water reactor (LWR) fuel pins. Redwing enables two-way data transfer of intrapin fields such as power density and temperature in order to improve the prediction of fission gas release and the overall accuracy of the simulation. An original algorithm was developed to enable transfer of fission gas data between MPACT and BISON, referred to as fission gas coupling. A fuel pin model based on the Watts Bar Nuclear 1 reactor was created, and several aspects of the model were studied: radial mesh and time step sensitivity, the effect of fission gas coupling on a single pin at constant power, and the effect of fission gas coupling on a fuel pin array that undergoes a shutdown. The results show that fission gas coupling has a significant effect on the solution for fuel pins at high power and high burnup, causing an approximate 9% increase in fission gas released to the plenum. It was hypothesized that changes in output quantities of interest (QOIs) were mainly due to a change in the fission gas source. So, a fissionable nuclide-dependent fission gas source was implemented in the Sifgrs module of BISON to test this; results showed some improvement in QOIs compared to standard BISON simulations. Although the fissionable nuclide-dependent source did not have the same effects as enabling fission gas coupling in Redwing, results showed that improving the fission gas source prescription for BISON can capture some effects of fission gas coupling. The fissionable nuclide-dependent source requires further study to validate it. Apart from fission gas coupling, this research illustrated a few little-discussed ways that coupling neutron transport, nuclide depletion, and fuel performance simulations can effect QOIs usually associated with fuel performance. For one, capturing the time dependence of the recoverable energy released per fission has a significant effect on several QOIs in high-burnup fuel. Another important physical quantity derived from the neutron transport solution is the fast neutron flux in the cladding, which has a large effect on cladding creep rate.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140807/1/mprose_1.pd

Linear Diffusion Acceleration for Neutron Transport Problems

Nuclear engineers are interested in solutions of the Neutron Transport Equation (NTE), with the goal of improving the safety and efficiency of reactors and critical nuclear systems. Complex simulations are used to obtain detailed solutions of the NTE, and can require immense computational resources to execute. A variety of methods have been developed to ease the computational burden of simulating full-scale, whole-core reactor problems. Among these is transport acceleration, which improves the convergence rate of iterative transport calculations. In addition to the use of acceleration methods, certain approximations are often made when solving the NTE. The 2D/1D approximation is used to generate a 3D solution of the NTE by iteratively solving coupled 2D radial and 1D axial equations. This method is one of the foundational techniques used in the neutronics code MPACT. Also, the Transport-Corrected P0 (TCP0) approximation for neutron scattering is often used in reactor analysis codes to simplify higher-order scattering physics. Unfortunately, both of these approximations allow for non-positive flux solutions of the NTE. More importantly, some spatial discretizations of the NTE also permit negative solutions. Under certain conditions, this can cause instability for nonlinear acceleration methods such as Coarse Mesh Finite Difference (CMFD). In this thesis, we propose a novel acceleration scheme called Linear Diffusion Acceleration (LDA) that does not possess the nonlinearities present in CMFD. This thesis work presents LDA as an alternative acceleration scheme to CMFD. As the name suggests, the LDA method is linear with respect to the scalar flux. Therefore, LDA is not susceptible to the same nonlinear modes of numerical failure as CMFD. In addition, LDA is shown to possess similar convergence properties as CMFD for practical problems that have no negative scalar fluxes. Transport acceleration with LDA allows for the use of some of the aforementioned approximations, in which the positivity of the scalar flux is not guaranteed. Fourier analysis of CMFD and LDA is performed to compare the theoretical convergence rates of the two methods for simple, spatially-heterogeneous problems. In addition, simple and practical case studies are presented in which CMFD fails due to nonlinearity. For these cases, LDA is shown to retain stability. Certain other advantages of LDA, which are a consequence of its mathematical structure, are also discussed.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169898/1/zdodson_1.pd