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

Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations

[EN] The evolution of the neutronic power inside of a nuclear reactor core can be approximated by means of the diffusive time-dependent simplified spherical harmonics equations (SPN). For the spatial discretization of these equations, a continuous Galerkin high order finite element method is applied to obtain a semi-discrete system of equations that is usually stiff. A semi-implicit time scheme is used for the time discretization and many linear systems are needed to be solved and previously, preconditioned. The aim of this work is to speed up the convergence of the linear systems solver with a multilevel preconditioner that uses different degrees of the polynomials used in the finite element method. Furthermore, as the matrices that appear in this type of system are very large and sparse, a matrix-free implementation of the preconditioner is developed to avoid the full assembly of the matrices. A benchmark transient tests this methodology. Numerical results show, in comparison with the block Gauss-Seidel preconditioner, an improvement in terms of number of iterations and the necessity of computational resources.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar, D.; Verdú, G. (2022). Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 68-77. https://doi.org/10.4995/YIC2021.2021.12510OCS687

Block strategies to compute the lambda modes associated with the neutron diffusion equation

[EN] Given a configuration of a nuclear reactor core, the neutronic distribution of the power can beapproximated by means of the multigroup neutron diffusion equation. This is an approximationof the neutron transport equation that assumes that the neutron current is proportional to thegradient of the scalar neutron ux with a diffusion coeffcient [1]. This approximation is known asthe Fick's first law. To define the steady-state problem, the criticality of the system must be forced.In this work, the -modes problem is used. That yields a generalized eigenvalue problem whoseeigenvector associated with the dominant eigenvalue represents the distribution of the neutron uxin steady-state.The spatial discretization of the equation is made by a continuous Galerkin high order finite elementmethod is applied [2] to obtain an algebraic eigenvalue problem. Usually, the matrices obtainedfrom the discretization are huge and sparse. Moreover, they have a block structure given by the different number of energy groups. In this work, block strategies are developed to optimize thecomputation of the associated eigenvalue problems.First, different block eigenvalue solvers are studied. On the other hand, the convergence of theseiterative methods mainly depends on the initial guess and the preconditioner used. In this sense,different multilevel techniques to accelerate the rate of convergence are proposed. Finally, the sizeof the problems can be suffciently large to be unfeasible to be solved in personal computers. Thus,a matrix-free methodology that avoids the allocation of the matrices in memory is applied [3].Three-dimensional benchmarks are used to show the effciency of the methodology proposed.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal Ferrándiz, A.; Ginestar Peiró, D.; Verdú, G. (2022). Block strategies to compute the lambda modes associated with the neutron diffusion equation. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 423-430. https://doi.org/10.4995/YIC2021.2021.13470OCS42343

Adaptive modal methods to integrate the neutron diffusion equation

Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2019). Adaptive modal methods to integrate the neutron diffusion equation. R. Company, J. C. Cortés, L. Jódar and E. López-Navarro. 26-31. http://hdl.handle.net/10251/180549S263

Edge-wise perturbations to model vibrating fuel assemblies in the frequency-domain using FEMFFUSION: development and verification

[EN] The mechanical vibrations of fuel assemblies have shown to give high levels of neutron noise, triggering in some circumstances the necessity to operate nuclear reactors at a reduced power level. This behaviour can be modelled using the neutron noise diffusion approximation in the frequency-domain. This work presents an extension of the finite element method code FEMFFUSION, to simulate mechanical vibrations in hexagonal reactors in the frequency domain. This novel strategy in neutron noise simulation is based on introducing perturbations on the edges of the cells associated with the vibrating fuel assemblies, allowing to model the movement of these fuel assemblies accurately and efficiently, without the necessity of using locally refined meshes. Numerical results verify the edge-wise methodology in the frequency-domain against the usual cell-wise frequency-domain model and the time-domain model. The edge-wise frequency-domain methodology has also been compared to other neutronic codes, as CORESIM and PARCS.This project has received funding from the Euratom research and training program 2014-2018 under grant agreement No 754316.Vidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2022). Edge-wise perturbations to model vibrating fuel assemblies in the frequency-domain using FEMFFUSION: development and verification. Annals of Nuclear Energy. 175:1-12. https://doi.org/10.1016/j.anucene.2022.10924611217

A block Arnoldi method for the SPN equations

[EN] The simplified spherical harmonics equations are a useful approximation to the stationary neutron transport equation. The eigenvalue problem associated with them is a challenging problem from the computational point of view. In this work, we take advantage of the block structure of the involved matrices to propose the block inverse-free preconditioned Arnoldi method as an efficient method to solve this eigenvalue problem. For the spatial discretization, a continuous Galerkin finite element method implemented with a matrix-free technique is used to keep reasonable memory demands. A multilevel initialization using linear shape functions in the finite element method is proposed to improve the method convergence. This initialization only takes a small percentage of the total computational time. The proposed eigenvalue solver is compared to the standard power iteration method, the Krylov-Schur method and the generalized Davidson method. The numerical results show that it reduces the computational time to solve the eigenvalue problem.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901. Moreover, it has been supported by the Catedra of the CSN Vicente SerradellVidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2020). A block Arnoldi method for the SPN equations. International Journal of Computer Mathematics. 97(1-2):341-357. https://doi.org/10.1080/00207160.2019.1602768S341357971-

Frequency-domain models in the SPN approximation for neutron noise calculations

[EN] Simulations of the neutron flux fluctuations, known as neutron noise, can be performed by means of the spherical harmonics equations (SPN) approximation with higher accuracy than with the diffusion equation. In this sense, one can solve these equations in the time-domain or in the frequency-domain. This last approach permits solving the neutron noise without performing complete time-dependent simulations for monochromatic perturbations. This work presents two formulations of the SPN equations in the frequency domain, that are obtained by using different treatments of the time derivatives of the field moments. The methodology is verified with several neutron noise problems where the numerical results are compared with the time-domain computations of FEMFFUSION code. The C5G7 noise benchmark compares both SPN formulations, showing the applicability of the diffusive SPN approximation.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2022). Frequency-domain models in the SPN approximation for neutron noise calculations. Progress in Nuclear Energy. 148:1-11. https://doi.org/10.1016/j.pnucene.2022.10423311114

Adaptive time-step control for the modal method to integrate the multigroup neutron diffusion equation

[EN] The distribution of the power inside a reactor core can be described by the time dependent multigroup neutron diffusion equation. One of the approaches to integrate this time-dependent equation is the modal method, that assumes that the solution can be described by the sum of amplitude function multiplied by shape functions of modes. These shape functions can be computed by solving a _-modes problems. The modal method has a great interest when the distribution of the power cannot be well approximated by only one shape function, mainly, when local perturbations are applied during the transient. Usually, the shape functions of the modal methods are updated for the time-dependent equations with a constant time-step size to obtain accurate results. In this work, we propose a modal methodology with an adaptive control time-step to update the eigenfunctions associated with the modes. This algorithm improves efficiency because of time is not spent solving the systems to a level of accuracy beyond relevance and reduces the step size if they detect a numerical instability. Step size controllers require an error estimation. Different error estimations are considered and analyzed in a benchmark problem with a out of phase local perturbation.This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Adaptive time-step control for the modal method to integrate the multigroup neutron diffusion equation. EPJ Web of Conferences (Online). 247:1-8. https://doi.org/10.1051/epjconf/202124707010S1824

Time-dependent simplified spherical harmonics formulations for a nuclear reactor system

[EN] The steady-state simplified spherical harmonics equations (SPNequations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SPN equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full systemThis work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2021). Time-dependent simplified spherical harmonics formulations for a nuclear reactor system. Nuclear Engineering and Technology. 53(12):3861-3878. https://doi.org/10.1016/j.net.2021.06.010S38613878531

Social network analysis by means of Markov chains

[EN] The main aim of this paper is shown a mathematical model about the social networks that enhances the interest of the Mathematics' students that are in rst course. This model is based on Markov chains. Its approach and resolution permit to illustrate the applicability of the discrete equation systems and toemphasize the concepts studied about eigenvalue problems and diagonalization. Furthermore, all results are computed with the Mathematica software.[ES] El objetivo de este trabajo es mostrar un modelo matemático sencillo, aplicado al análisis de las redes sociales, que potencie el interés de los alumnos en la asignatura Matemáticas I. El planteamiento y la resolución de este modelo, basado en las cadenas de Markov, nos permiten ilustrar la aplicabilidad los sistemas de ecuaciones discretas y reforzar los conceptos de autovalores, autovectores y diagonalización. Utilizaremos el software de cálculo Mathematica, al que tienen acceso los estudiantes de la asignatura, para resolver el problema, lo que facilitará trabajar con diversos modelos y dimensiones grandes.Este trabajo ha sido parcialmente subvencionado por el Ministerio de Economía y Competitividad bajo los proyectos BES-2015-072901 y MTM2015-64013-P.Carreño Sanchez, A.; Sanabria Codesal, E. (2019). Análisis de redes sociales mediante cadenas de Markov. Modelling in Science Education and Learning. 12(1):21-30. https://doi.org/10.4995/msel.2019.10938SWORD213012