Development of a polynomial nodal method with flux and current discontinuity factors

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Engineering, 1992.Includes bibliographical references (leaves 129-131).by Michael L. Zerkle.Ph.D

Low-Order Multiphysics Coupling Techniques for Nuclear Reactor Applications

The accurate modeling and simulation of nuclear reactor designs depends greatly on the ability to couple differing sets of physics together. Current coupling techniques most often use a fixed-point, or Picard, iteration scheme in which each set of physics is solved separately, and the resulting solutions are passed between each solver. In the work presented here, two different coupling techniques are investigated: a Jacobian-Free Newton-Krylov (JFNK) approach and a new methodology called Coarse Mesh Finite Difference Coupling (CMFD-Coupling). What both of these techniques have in common is that they are applied to the low-order CMFD system of equations. This allows for the multiphysics feedback effects to be captured on the low-order system without having to perform a neutron transport solve.The JFNK and CMFD-Coupling approaches were implemented in the MPACT (Michigan Parallel Analysis based on Characteristic Tracing) neutron transport code, which is being developed for the Consortium for Advanced Simulation of Light Water Reactors (CASL). These methods were tested on a wide range of practical reactor physics problems, from a 2D pin cell to a massively parallel 3D full core problem. Initially, JFNK was implemented only as an eigenvalue solver without any feedback enabled. However this led to greatly increased runtimes without any obvious benefit. When multiphysics problems were investigated with both JFNK and CMFD-Coupling, it was concluded that CMFD-Coupling outperformed JFNK in terms of both accuracy and runtime for every problem. When applied to large full core problems with multiple sources of strong feedback enabled, CMFD-Coupling reduced the overall number of transport sweeps required for convergence

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

Discontinuous space-time dependent nodal synthesis method for reactor analysis

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.Includes bibliographical references (leaves 110-112).by Chi H. Kang.M.Eng

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

A Multilevel in Space and Energy Solver for Multigroup Diffusion and Coarse Mesh Finite Difference Eigenvalue Problems

In reactor physics, the efficient solution of the multigroup neutron diffusion eigenvalue problem is desired for various applications. The diffusion problem is a lower-order but reasonably accurate approximation to the higher-fidelity multigroup neutron transport eigenvalue problem. In cases where the full-fidelity of the transport solution is needed, the solution of the diffusion problem can be used to accelerate the convergence of transport solvers via methods such as Coarse Mesh Finite Difference (CMFD). The diffusion problem can have O(108) unknowns, and, despite being orders of magnitude smaller than a typical transport problem, obtaining its solution is still not a trivial task. In the Michigan Parallel Characteristics Transport (MPACT) code, the lack of an efficient CMFD solver has resulted in a computational bottleneck at the CMFD step. Solving the CMFD system can comprise 50% or more of the overall runtime in MPACT when the de facto default CMFD solver is used; addressing this bottleneck is the motivation for our work. The primary focus of this thesis is the theory, development, implementation, and testing of a new Multilevel-in-Space-and-Energy Diffusion (MSED) method for efficiently solving multigroup diffusion and CMFD eigenvalue problems. As its name suggests, MSED efficiently converges multigroup diffusion and CMFD problems by leveraging lower-order systems with coarsened energy and/or spatial grids. The efficiency of MSED is verified via various Fourier analyses of its components and via testing in a 1-D diffusion code. In the later chapters of this thesis, the MSED method is tested on a variety of reactor problems in MPACT. Compared to the default CMFD solver, our implementation of MSED in MPACT has resulted in an ~8-12x reduction in the CMFD runtime required by MPACT for single statepoint calculations on 3-D, full-core, 51-group reactor models. The number of transport sweeps is also typically reduced by the use of MSED, which is able to better converge the CMFD system than the default CMFD solver. This leads to a further savings in overall runtime that is not captured by the differences in CMFD runtime.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146075/1/bcyee_1.pd

Orthogonal-Mesh, 3D Sn with Embedded 2-D Method of Characteristics for Whole-Core, Pin-Resolved Reactor Analysis

Solutions to the Boltzmann transport equation are an important starting point for many aspects of nuclear reactor design and analysis. Many interesting physical phenomena depend on minute variations in neutron flux at scales well below the fuel pin level, requiring solutions to the transport equation that resolve this intra-pin behavior. Being large systems, direct, naive solutions to the transport equation would be impractical, even with the fastest computers available today. Solution techniques must therefore make appropriate approximations and simplifications to produce tractable yet sufficiently accurate solutions. The Method of Characteristics (MoC) is a well-known approach to solving fine-mesh 2-D reactor problems, but has proven computationally impractical when extended directly to 3-D. Instead, a series of 2-D MoC solvers, which treat the radial dimensions, are coupled with a 1-D solver, which treats the axial dimension. This is called the 2-D/1-D method, and is the mainstay of several reactor analysis tools in use today. These methods sometimes suffer poor performance when strong axial heterogeneities are present, or when very fine axial refinement is desired. Another technique, orthogonal-mesh SN with homogenized cross sections, while efficient in 3-D, suffers poor accuracy at reasonable mesh refinements because it cannot resolve pin geometries. This work develops a new method, called 2-D/3-D, in which 2-D MoC solvers are used to inform an orthogonal-mesh, 3-D SN solver such that accuracy is maintained on a very coarse mesh. The 2-D/3-D method uses a Corrected Diamond Difference scheme for solving the SN equations using information extracted from the 2-D MoC solvers. This preserves important streaming and collision behavior from the fine-mesh solution. A conventional differencing scheme is used to treat the axial dimension. Transverse leakage sources are used to provide axial streaming information back to the MoC solvers. This method was applied to the three C5G7 rodded benchmark problems, and the results were compared to those of commonly-used 2-D/1-D methods. It was found that 2-D/3-D was capable of producing smaller error in most cases, at an increased memory and computational cost. These costs are likely to be minor for realistic problems and parallel decompositions.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135759/1/youngmit_1.pd

Implementation of the analytic nodal method in the ndf code and applications to candu reactors

Review of the analytic nodal method for 3-D static neutron diffusion equation -- Description of the coarse mesh finite difference method in the code NDF and static applications -- Numerical considerations of the analytic nodal method -- Static application of the analytic nodal method -- Analytic nodal method for 3-D space-time kinetics neutron diffusion equation -- Numerical results for 3D space-time kinetics neutron diffusion calculations

Investigation of a discrete ordinates method for neutron noise simulations in the frequency domain

During normal operations of a nuclear reactor, neutron flux measurements show small fluctuations around mean values, the so-called neutron noise. These fluctuations may be driven by a variety of perturbations, e.g., mechanical vibrations of core components. From the analysis of the neutron noise, anomalous patterns can be identified at an early stage and corrected before they escalate. For this purpose, the modelling of the reactor transfer function, which describes the core response to a possible perturbation and is based on the neutron transport equation, is often required. In this thesis a discrete ordinate method is investigated to solve the neutron noise transport equation in the frequency domain. When applying the method, two main issues need to be considered carefully, i.e., the performance of the numerical algorithm and possible numerical artifacts arising from the discretization of the equation. For an efficient numerical scheme, acceleration techniques are tested, namely, the synthetic diffusion acceleration and various forms of the coarse mesh finite difference method. To reduce the possible numerical artifacts, the impact of the order of discrete ordinates and the use of a fictitious source method are studied. These analyses serve to develop the higher-order neutron noise solver NOISE-SN. The solver is compared with different solvers and used to simulate neutron noise experiments carried out in the research reactor CROCUS (at EPFL). The solver NOISE-SN is shown to provide results that are consistent with the results obtained from other higher-order codes and can reproduce features observed in neutron noise experiments

A transient, quadratic nodal method for triangular-Z geometry

Many systematically-derived nodal methods have been developed for Cartesian geometry due to the extensive interest in Light Water Reactors. These methods typically model the transverse-integrated flux as either an analytic or low order polynomial function of position within the node. Recently, quadratic nodal methods have been developed for R-Z and hexagonal geometry. A static and transient quadratic nodal method is developed for triangular-Z geometry. This development is particularly challenging because the quadratic expansion in each node must be performed between the node faces and the triangular points. As a consequence, in the 2-D plane, the flux and current at the points of the triangles must be treated. Quadratic nodal equations are solved using a non-linear iteration scheme, which utilizes the corrected, mesh-centered finite difference equations, and forces these equations to match the quadratic equations by computing discontinuity factors during the solution. Transient nodal equations are solved using the improved quasi-static method, which has been shown to be a very efficient solution method for transient problems. Several static problems are used to compare the quadratic nodal method to the Coarse Mesh Finite Difference (CMFD) method. The quadratic method is shown to give more accurate node-averaged fluxes. However, it appears that the method has difficulty predicting node leakages near reactor boundaries and severe material interfaces. The consequence is that the eigenvalue may be poorly predicted for certain reactor configurations. The transient methods are tested using a simple analytic test problem, a heterogeneous heavy water reactor benchmark problem, and three thermal hydraulic test problems. Results indicate that the transient methods have been implemented correctly