Implementation and Verification of Three-Dimensional MOC Transient Solver in Whole-core Transport code STREAM

Department of Nuclear EngineeringIn the state of the art of computer technology, the transient analysis in the nuclear reactor has become on demand of the nuclear engineering field. This thesis presents the development and the preliminary validation of the transient transport capability in STREAM code. The Theta method with the well-known Crank-Nicholsen scheme providing a second-order accurate is then applied to tackle the time integration in the right-hand side of the time-dependent neutron transport. Eventually, the Methods Of Characteristic (MOC) solver in the steady state with excellent performance and accuracy in STREAM is modified with the delayed neutron term to solve the transient cases. Additionally, a multi-group Coarse Mesh Finite Difference (CMFD) accelerator is introduced to alleviate the computational burden of the simulation. The transient problems, namely TD0, TD1, TD2, TD3, TD4 and TD5 in the C5G7-TD benchmark suite are used for verification. The STREAM time-dependent MOC calculation results show good agreement with results of a deterministic transport analysis code, nTRACER with the maximum disparity of the total power level change is 1.22 % for the TD3-4 case. In the 3D cases, with a proposed decusping method, the maximum amplitude of the relative error is about 3.5% between the two deterministic codes. The disparities induced by decusping methods and whole-core transport method are obviously observed and thus are the primary source causing the discrepancies. With this high fidelity to replicate the solution in the time-dependent transport equation, the transient analysis capability of STREAM code has been proved. This work plays as a foundation for the later 3D whole core transient solver accompanied by TH1D feedbacks for practical transient problems.ope

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 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

Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) method must be employed. In this work an h-p Finite Element Method (FEM) is used to obtain the dominant Lambda mode associated with a configuration of a reactor core using the SPN approximation. The performance of the SPN (N= 1, 3, 5) approximations has been tested for different reactor benchmarks.[ES] La ecuación de la difusión neutrónica es una aproximación de la ecuación del transporte de neutrones que describe la población de neutrones en el núcleo de un reactor nuclear. En particular, consideraremos reactores de tipo VVER y para simular su comportamiento se utilizará la ecuación de la difusión neutrónica para cuya discretización se hace uso de mallas hexagonales. La mayoría de los códigos de simulación de reactores nucleares utilizan aproximación multigrupo de energía de la ecuación de la difusión neutrónica para describir la distribución de neutrones en el interior del núcleo del reactor. Para estudiar el estado estacionario del reactor, es posible forzar la criticidad del reactor de forma artificial modificando las secciones eficaces de forma que se obtiene un problema de valores propios diferencial, conocido como el problema de los Modos Lambda, que se resuelve para obtener los valores propios dominantes del reactor y sus correspondientes funciones propias. Para discretizar este modelo se ha hecho uso de un método de elementos finitos con adaptabilidad h-p. Este método permite el uso de mallas heterogéneas, y de diferentes refinamientos como el uso mallas h-adaptativas, reduciendo el tamaño de los distintos nodos, y el p-refinado, aumentando el grado del polinomio de las funciones básicas utilizado en los desarrollos de la solución en los diferentes nodos. Se ha desarrollado un código basado en un método de elementos finitos de alto orden para resolver el problema de los Modos Lambda en un reactor con geometría hexagonal y se han obtenido los Modos dominantes para distintos problemas de referencia. Una vez que se ha obtenido la solución para la distribución de neutrones en estado estacionario, ésta se utiliza como condición inicial para la integración de la ecuación de difusión neutrónica dependiente del tiempo. Para simular el comportamiento de un reactor nuclear para un determinado transitorio, es necesario ser capaz de integrar la ecuación de la difusión neutrónica dependiente del tiempo en el interior del núcleo del reactor. La discretización espacial de esta ecuación se hace usando un método de elementos finitos de alto orden que permite refinados de tipo h-p para distintas geometrías. Los transitorios que implican el movimiento de los bancos de las barras de control tienen el problema conocido como el efecto 'rod-cusping'. Estudios anteriores, por lo general, han abordado este problema utilizando una malla fija y definiendo propiedades promedio para los materiales correspondientes a las celdas donde se tiene la barra de control parcialmente insertada. En el presente trabajo se propone el uso de un esquema de malla móvil, de forma que en mallado espacial va cambiando con el movimiento de la barra de control, evitando la necesidad de utilizar secciones eficaces equivalentes para las celdas parcialmente insertadas. El funcionamiento de este esquema de malla móvil propuesto se estudia resolviendo distintos problemas tipo. La precisión obtenida mediante de la teoría de la difusión en los cálculos de reactores es limitada cuando se tienen elementos de combustible complejos o se pretenden realizar cálculos en malla fina. Para mejorar estos resultados, es necesario disponer de un método que incorpore aproximaciones de orden superior de la ecuación del transporte de neutrones. Una posibilidad es hacer uso de las ecuaciones PN simplificadas (SPN ). En este trabajo se utiliza un método de elementos finitos h-p para obtener los modos dominantes asociados con una configuración dada del núcleo de un reactor nuclear con geometría hexagonal usando la aproximación SPN . El funcionamiento de las aproximaciones SPN (N = 1, 3, 5) se ha estudiado para distintos problemas de referencia.[CA] L'equació de la difusió neutrònica és una aproximació de l'equació del transport de neutrons que descriu la població de neutrons en el nucli de un reactor nuclear. En particular, considerarem reactors de tipus VVER i per a simular el seu comportament s'utilitzarà l'equació de la difusió neutrónica que es discretitza fent ús de malles hexagonals. La majoria dels codis de simulació de reactors nuclears utilitzen l'aproximació multigrup d'energia de l'equació de la difusió neutrónica per a descriure la distribució de neutrons a l'interior del nucli del reactor. Per a estudiar l'estat estacionari del reactor, és possible forçar la seua criticitat de forma artificial modificant les seccions eficaces de manera que s'obté un problema de valors propis diferencial, conegut com el problema dels Modes Lambda, que es resol per a obtenir els valors propis dominants del reactor i les seues corresponents funcions pròpies. Per a discretitzar aquest model s'ha fet ús d'un mètode d'elements finits amb adaptabilitat h-p. Aquest mètode permet l'ús de malles heterogènies, i de diferents refinaments com l'ús malles h-adaptatives, reduint la grandària dels diferents nodes, i el p-refinat, augmentant el grau del polinomi de les funcions bàsiques utilitzat en els desenvolupaments de la solució en els diferents nodes. S'ha desenvolupat un codi basat en un mètode d'elements finits d'alt ordre per a resoldre el problema dels Modes Lambda en un reactor amb geometria hexagonal i s'han obtingut els Modes dominants per a diferents problemes de referència. Una vegada que s'ha obtingut la solució per a la distribució de neutrons en estat estacionari, aquesta s'utilitza com a condició inicial per a la integració de l'equació de difusió neutrònica depenent del temps. Per a simular el comportament d'un reactor nuclear per a un determinat transitori, és necessari ser capaç d'integrar l'equació de la difusió neutrónica depenent del temps a l'interior del nucli del reactor. La discretitzación espacial d'aquesta equació es fa usant un mètode d'elements finits d'alt ordre que permet refinats de tipus h-p per a diferents geometries. Els transitoris que impliquen el moviment dels bancs de les barres de control tenen el problema conegut com l'efecte 'rod-cusping'. Estudis anteriors, en general, han abordat aquest problema utilitzant una malla fixa i definint propietats equivalents per als materials corresponents a les cel·les on es té la barra de control parcialment inserida. En el present treball es proposa l'ús d'un esquema de malla mòbil, de manera que en mallat espacial va canviant amb el moviment de la barra de control, evitant la necessitat d'utilitzar seccions eficaces equivalents per a les cel·les parcialment inserides. El funcionament de aquest esquema de malla mòbil s'estudia resolent diferents problemes tipus. La precisió obtinguda mitjançant de la teoria de la difusió en els càlculs de reactors és limitada quan es tenen elements de combustible complexos o es pretenen realitzar càlculs en malla fina. Per a millorar aquests resultats, és necessari disposar d'un mètode que incorpore aproximacions d'ordre superior de l'equació del transport de neutrons. Una possibilitat és fer ús de les equacions PN simplificades (SPN ). En aquest treball s'utilitza un mètode d'elements finits h- p per a obtenir els modes dominants associats amb una configuració donada del nucli de un reactor amb geometria hexagonal usant l'aproximació SPN . El funcionament de les aproximacions SPN (N = 1, 3, 5) s'ha estudiat per a diferents problemes de referència.Fayez Moustafa Moawad, R. (2016). Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/65353TESI

Homogenization Methods for Full Core Solution of the Pn Transport Equations with 3-D Cross Sections.

The design of advanced light water reactors having reduced moderation and axially varying fuel has exposed the limitations of diffusion theory methods based on two-dimensional homogenized group constants. Instead, an alternative approach is required to account for the axial neutron streaming within the core. The development of three-dimensional Monte Carlo cross sections and axial discontinuity factors has improved upon the lower-order diffusion solution, but numerical instabilities can still arise from large discontinuity factors. Therefore the use of higher-order transport corrections is necessary for the solution of full core problems. In this thesis the spherical harmonics (Pn) and Quasidiffusion equations are derived for one-dimensional applications to improve upon the angular approximation implicit in diffusion. Discontinuity factors for each method were determined based on finite difference, as well as the generation of Eddington factors from Monte Carlo results. A subplane method based on refining the Pn and Quasidiffusion solution was introduced to reduce the spatial discretization error. An alternative definition of the discontinuity factor based on an additive relation was investigated to improve the numerical stability. Numerical results based on an axially heterogeneous assembly demonstrate that P3 and Quasidiffusion improved the accuracy of the spatial flux distribution the most. Discontinuity factors allowed each of the lower-order methods to reproduce the reference Monte Carlo eigenvalue and flux. Minor improvement was seen when bounding the discontinuity factors compared to diffusion. The addition of the subplane method reduced the spatial discretization error and improved the range of discontinuity factors seen for all four methods at the cost of increased computational run time. Additive discontinuity factors for each method eliminated the possibility of large discontinuity factors and were able to reproduce the reference solution. The combination of the Quasidiffusion and subplane methods provided the most accurate axial solution.PhDNuclear Engineering and Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116697/1/halland_1.pd

Quasi-Normal Modes of Stars and Black Holes

Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstr\"om, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.Comment: 74 pages, 7 figures, Review article for "Living Reviews in Relativity

Nuclear rocket shielding methods, modification, updating, and input data preparation. Volume 1 - Synopsis of methods and results of analysis Final progress report

Analysis of data systems and computer programs for nuclear rocket shielding methods, modification, updating, and data input preparation - Vol.

Gravitating discs around black holes

Fluid discs and tori around black holes are discussed within different approaches and with the emphasis on the role of disc gravity. First reviewed are the prospects of investigating the gravitational field of a black hole--disc system by analytical solutions of stationary, axially symmetric Einstein's equations. Then, more detailed considerations are focused to middle and outer parts of extended disc-like configurations where relativistic effects are small and the Newtonian description is adequate. Within general relativity, only a static case has been analysed in detail. Results are often very inspiring, however, simplifying assumptions must be imposed: ad hoc profiles of the disc density are commonly assumed and the effects of frame-dragging and completely lacking. Astrophysical discs (e.g. accretion discs in active galactic nuclei) typically extend far beyond the relativistic domain and are fairly diluted. However, self-gravity is still essential for their structure and evolution, as well as for their radiation emission and the impact on the environment around. For example, a nuclear star cluster in a galactic centre may bear various imprints of mutual star--disc interactions, which can be recognised in observational properties, such as the relation between the central mass and stellar velocity dispersion.Comment: Accepted for publication in CQG; high-resolution figures will be available from http://www.iop.org/EJ/journal/CQ