Inexact and Nonlinear Extensions of the FEAST Eigenvalue Algorithm

Eigenvalue problems are a basic element of linear algebra that have a wide variety of applications. Common examples include determining the stability of dynamical systems, performing dimensionality reduction on large data sets, and predicting the physical properties of nanoscopic objects. Many applications require solving large dimensional eigenvalue problems, which can be very challenging when the required number of eigenvalues and eigenvectors is also large. The FEAST algorithm is a method of solving eigenvalue problems that allows one to calculate large numbers of eigenvalue/eigenvector pairs by using contour integration in the complex plane to divide the large number of desired pairs into many small groups; these small groups of eigenvalue/eigenvector pairs may then be simultaneously calculated independently of each other. This makes it possible to quickly solve eigenvalue problems that might otherwise be very difficult to solve efficiently. The standard FEAST algorithm can only be used to solve eigenvalue problems that are linear, and whose matrices are small enough to be factorized efficiently (thus allowing linear systems of equations to be solved exactly). This limits the size and the scope of the problems to which the FEAST algorithm may be applied. This dissertation describes extensions of the standard FEAST algorithm that allow it to efficiently solve nonlinear eigenvalue problems, and eigenvalue problems whose matrices are large enough that linear systems of equations can only be solved inexactly

Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation

[ES] Uno de los objetivos más importantes en el análisis de la seguridad en el campo de la ingeniería nuclear es el cálculo, rápido y preciso, de la evolución de la potencia dentro del núcleo del reactor. La distribución de los neutrones se puede describir a través de la ecuación de transporte de Boltzmann. La solución de esta ecuación no puede obtenerse de manera sencilla para reactores realistas, y es por ello que se tienen que considerar aproximaciones numéricas. En primer lugar, esta tesis se centra en obtener la solución para varios problemas estáticos asociados con la ecuación de difusión neutrónica: los modos lambda, los modos gamma y los modos alpha. Para la discretización espacial se ha utilizado un método de elementos finitos de alto orden. Diversas características de cada problema espectral se analizan y se comparan en diferentes reactores. Después, se investigan varios métodos de cálculo para problemas de autovalores y estrategias para calcular los problemas algebraicos obtenidos a partir de la discretización espacial. La mayoría de los trabajos destinados a la resolución de la ecuación de difusión neutrónica están diseñados para la aproximación de dos grupos de energía, sin considerar dispersión de neutrones del grupo térmico al grupo rápido. La principal ventaja de la metodología que se propone es que no depende de la geometría del reactor, del tipo de problema de autovalores ni del número de grupos de energía del problema. Tras esto, se obtiene la solución de las ecuaciones estacionarias de armónicos esféricos. La implementación de estas ecuaciones tiene dos principales diferencias respecto a la ecuación de difusión neutrónica. Primero, la discretización espacial se realiza a nivel de pin. Por tanto, se estudian diferentes tipos de mallas. Segundo, el número de grupos de energía es, generalmente, mayor que dos. De este modo, se desarrollan estrategias a bloques para optimizar el cálculo de los problemas algebraicos asociados. Finalmente, se implementa un método modal actualizado para integrar la ecuación de difusión neutrónica dependiente del tiempo. Se presentan y comparan los métodos modales basados en desarrollos en función de los diferentes modos espaciales para varios tipos de transitorios. Además, también se desarrolla un control de paso de tiempo adaptativo, que evita la actualización de los modos de una manera fija y adapta el paso de tiempo en función de varias estimaciones del error.[CA] Un dels objectius més importants per a l'anàlisi de la seguretat en el camp de l'enginyeria nuclear és el càlcul, ràpid i precís, de l'evolució de la potència dins del nucli d'un reactor. La distribució dels neutrons pot modelar-se mitjançant l'equació del transport de Boltzmann. La solució d'aquesta equació per a un reactor realístic no pot obtenir's de manera senzilla. És per això que han de considerar-se aproximacions numèriques. En primer lloc, la tesi se centra en l'obtenció de la solució per a diversos problemes estàtics associats amb l'equació de difusió neutrònica: els modes lambda, els modes gamma i els modes alpha. Per a la discretització espacial s'ha utilitzat un mètode d'elements finits d'alt ordre. Algunes de les característiques dels problemes espectrals s'analitzaran i es compararan per a diferents reactors. Tanmateix, diversos solucionadors de problemes d'autovalors i estratègies es desenvolupen per a calcular els problemes obtinguts de la discretització espacial. La majoria dels treballs per a resoldre l'equació de difusió neutrònica estan dissenyats per a l'aproximació de dos grups d'energia i sense considerar dispersió de neutrons del grup tèrmic al grup ràpid. El principal avantatge de la metodologia exposada és que no depèn de la geometria del reactor, del tipus de problema d'autovalors ni del nombre de grups d'energia del problema. Seguidament, s'obté la solució de les equacions estacionàries d'harmònics esfèrics. La implementació d'aquestes equacions té dues principals diferències respecte a l'equació de difusió. Primer, la discretització espacial es realitza a nivell de pin a partir de l'estudi de diferents malles. Segon, el nombre de grups d'energia és, generalment, major que dos. D'aquesta forma, es desenvolupen estratègies a blocs per a optimitzar el càlcul dels problemes algebraics associats. Finalment, s'implementa un mètode modal amb actualitzacions dels modes per a integrar l'equació de difusió neutrònica dependent del temps. Es presenten i es comparen els mètodes modals basats en l'expansió dels diferents modes espacials per a diversos tipus de transitoris. A més a més, un control de pas de temps adaptatiu es desenvolupa, evitant l'actualització dels modes d'una manera fixa i adaptant el pas de temps en funció de vàries estimacions de l'error.[EN] One of the most important targets in nuclear safety analyses is the fast and accurate computation of the power evolution inside of the reactor core. The distribution of neutrons can be described by the neutron transport Boltzmann equation. The solution of this equation for realistic nuclear reactors is not straightforward, and therefore, numerical approximations must be considered. First, the thesis is focused on the attainment of the solution for several steady-state problems associated with neutron diffusion problem: the $\lambda$-modes, the $\gamma$-modes and the $\alpha$-modes problems. A high order finite element method is used for the spatial discretization. Several characteristics of each type of spectral problem are compared and analyzed on different reactors. Thereafter, several eigenvalue solvers and strategies are investigated to compute efficiently the algebraic eigenvalue problems obtained from the discretization. Most works devoted to solve the neutron diffusion equation are made for the approximation of two energy groups and without considering up-scattering. The main property of the proposed methodologies is that they depend on neither the reactor geometry, the type of eigenvalue problem nor the number of energy groups. After that, the solution of the steady-state simplified spherical harmonics equations is obtained. The implementation of these equations has two main differences with respect to the neutron diffusion. First, the spatial discretization is made at level of pin. Thus, different meshes are studied. Second, the number of energy groups is commonly bigger than two. Therefore, block strategies are developed to optimize the computation of the algebraic eigenvalue problems associated. Finally, an updated modal method is implemented to integrate the time-dependent neutron diffusion equation. Modal methods based on the expansion of the different spatial modes are presented and compared in several types of transients. Moreover, an adaptive time-step control is developed that avoids setting the time-step with a fixed value and it is adapted according to several error estimations.Carreño Sánchez, AM. (2020). Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/144771TESI

High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis

The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis

Matrix-based techniques for (flow-)transition studies

In this thesis, numerical techniques for the computation of flow transitions was introduced and studied. The numerical experiments on a variety of two- and three- dimensional multi-physics problems show that continuation approach is a practical and efficient way to solve series of steady states as a function of parameters and to do bifurcation analysis. Starting with a proper initial guess, Newton’s method converges in a few steps. Since solving the linear systems arising from the discretization takes most of the computational work, efficiency is determined by how fast the linear systems can be solved. Our home-made preconditioner Hybrid Multilevel Linear Solver(HYMLS) can compute three-dimensional solutions at higher Reynolds numbers and shows its robustness both in the computation of solutions as well as eigenpairs, due to the iteration in the divergence-free space. To test the efficiency of linear solvers for non-flow problems, we studied a well-known reaction-diffusion system, i.e., the BVAM model of the Turing problem. The application to the Turing system not only proved our program’s ability in doing nonlinear bifurcation analysis efficiently but also provided insightful information on two- and three- dimensional pattern formation