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[EN] Rainfed agricultural systems in semiarid Mediterranean environments are subject to erratic but often heavy rainfall events. As an agronomic practice, fallow periods can be included even within the existing European Union common policy for crop diversification. This study aimed to quantify the effects of previous mineral fertilization on soil mineral nitrogen (Nmin) content and potential nitrate leaching during no-till fallow periods of crop rotation. The Leaching Estimation and Chemistry Model (LEACHM) was used for soil Nmin prediction after fallow. During fallow periods, Nmin measured in the soil profile increased by an average of 125 kg N ha¿1, whereas the model showed an average increase of 95 kg N ha¿1. The amount of nitrogen (N) leached was estimated to be 11¿38 kg N ha¿1. The N balance simulated using LEACHM might differ from the actual situation. The calibrated LEACHM underestimated soil Nmin after a drought period followed by soil water replenishment, probably due to a pulse in soil C and N mineralization caused by the wetting of dry soil, but overestimated soil Nmin after occasional rainy spells when soil quickly became saturated, probably due to occasional nitrous oxide emissions not being fully accounted for by the model and specific preferential water flow, which might lead to greater nitrate leaching than that simulated by LEACHM using the convection-dispersion equation. The results show that soil Nmin measurements after fallow periods cannot be well predicted by LEACHM. The findings of this study provide a support for fertilization planning for crop rotation systems including a fallow period to avoid overfertilization in the following cropping season and reduce N environmental impacts.

[EN] A discrete fractional order model is proposed to analyze the behaviour of an epidemic process with indirect transmission. This model is based on a discrete version of the GrunwaldLetnikov fractional derivative operator. Some properties of this operator are shown and used to derive a truncated version of the operator, which is used to propose a model with short-term memory. Based on the biological meaning of the problem, some bounds have been obtained to assure the nonnegativity of the model solution. The (alpha, k ) -Basic Reproduction Number has been introduced and used to analyze the stability of the solution around its equilibrium points. Moreover, the influence of the fractional order, alpha, and the memory steps, k , on the behaviour of the solution has been analyzed. Finally, the results obtained have been clarified by means of numerical simulations of a model for the evolution of an infection by Salmonella in a hens farm. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

[ES] En este proyecto se procede a desarrollar un modelo matemático para el despegue de un cuadricóptero y a estudiar la influencia que puedan tener perturbaciones externas como ráfagas de viento. Primero, detallando el marco teórico en el cual se desarrolla y los parámetros de los que depende. A continuación, para el modelo Cheerwing Syma X5SW-V3, se ensayan y caracterizan los parámetros a introducir en el modelo. Posteriormente, se graba un vídeo de la trayectoria que se pretende reproducir, para todos los casos de estudio. Por último, mediante software matemático se analizan los resultados obtenidos por el modelo y se comparan con los datos experimentales, validando el modelo teórico desarrollado para el despegue de un dron., [CA] En aquest projecte es procedeix a desenvolupar un model matem`atic per a l’enlairament d’un quadricopter i a estudiar la influ`encia que puguen tindre pertorbacions externes com r`afegues de vent. Primer, detallant el marc te`oric on es desenvolupa i els par`ametres dels que dependr`a. A continuaci´o, per al model Cheerwing Syma X5SW-V3, s’assagen i es caracteritzen els par`ametres a introduir al model. Posteriorment, es grava la traject`oria que es pret´en reproduir, per tots els casos d’estudi. Finalment, mitjan¸cant software matem`atic s’analitzen els resultat obtenits pel model i es comparen amb les dades experimentals, validant el model te`oric desenvolupat per al enlairament d’un dron., [EN] In this project, a mathematical model for the take-off of a quadcopter is developed and the impact of external disturbances like gusts of wind is checked. First, introducing the theoretical framework in which it is developed and the parameters on which it will depend. Next, for the Cheerwing Syma X5SW-V3 model, the parameters to be entered in the model are tested and characterized. The trajectory it intends to reproduce is filmed for every studied case. By last, using mathematical software, the results obtained by the model are analyzed and compared with the experimental data, validating the theoretical model for the take-off of a drone.

[EN] Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or generators in some developments and applications. This is due to their capacity of being deformed when submitted to an electric field. Some models of their actuation are available, but recently, viscoelastic models have been proposed to give an account of the dissipative behaviour of these materials. Their response to an external mechanical or electrical force field implies a relaxation process towards a new state of thermodynamic equilibrium, which can be described by a relaxation time. However, it is well known that viscoelastic and dielectric materials, as for example, polymers, exhibit a distribution of relaxation times instead of a single relaxation time. In the present approach, a continuous distribution of relaxation times is proposed via the introduction of fractional derivatives of the stress and strain, which gives a better account of the material behaviour. The application of fractional derivatives is described and a comparison with former results is made. Then, a double generalisation is carried out: the first one is referred to the viscoelastic or dielectric models and is addressed to obtain a nonsymmetric spectrum of relaxation times, and the second one is the adoption of the more realistic Mooney-Rivlin equation for the stress-strain relationship of the elastomeric material. A modified Mooney-Rivlin model for the free energy density of a hyperelastic material, VHB 4910 has been used based on experimental results of previous authors. This last proposal ensures the appearance of the bifurcation phenomena which is analysed for equibiaxial dead loads; time-dependent bifurcation phenomena are predicted by the extended Mooney-Rivlin equations.

© 2021 IEEE. Personal use of this material is permitted. Permissíon from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertisíng or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works., [EN] The paper ¿SPICE Model of Photomultiplier Tube Under Different Bias Conditions¿ is commented. We revisit the mathematical formulation to compensate for some ambiguities in the original manuscript, and point out some inconsistencies in the results and reproducibility of the simulations, as well as in the optimized parameters originally obtained with the PSPICE simulation engine. All simulations are recalculated with the NGSPICE software using the corrected parameters and compared against the original figures. The reproducibility of our simulations is independently verified with PSPICE, as well as by numerically solving the analytical system of non-linear equations using Newton¿s method within MATLAB

[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.

[EN] The early detection of anomalies through the analysis of the neutron noise recorded by in-core and ex-core instrumentation gives the possibility to take proper actions before such problems lead to safety concerns or impact plant availability. The study of the neutron fluctuations permits detecting and differentiate anomalies depending on their type and possibly to characterize and localize such anomalies. This method is non-intrusive and does not require any external perturbation of the system. To effectively use the neutron noise for reactor diagnostics it is essential to accurately model the effects of the anomalies on the neutron field. This paper deals with the development and validation of a neutron noise simulator for reactors with different geometries. The neutron noise is obtained by solving the frequency-domain two-group neutron diffusion equation in the first order approximation. In order to solve this partial differential equation a code based on a high order finite element method is developed. The novelty of this simulator resides on the possibility of dealing with rectangular meshes in any kind of geometry, thus allowing for complex domains and any location of the perturbation. The finite element method also permits automatic refinements in the cell size (h-adaptability) and in its polynomial degree (p-adaptability) that lead to a fast convergence. In order to show the possibilities of the neutron noise simulator developed a perturbation in a hexagonal two-dimensional reactor is investigated in this paper.

[EN] The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.

[EN] We develop and validate node homogenization methods, based on one and three-dimensional node equivalent problems, that improve the volume weighted method for the treatment of the rod cusping effect. The numerical implementation of these techniques in a numerical approach based on general spherical harmonics, nodal collocation method, for arbitrary odd order ¿¿, and fully implicit time discretization, is verified on the AECL-7236 PHWR benchmark transient, showing the validity and convergence of the P1 and P3 approximations and the ability of the method to give accurate results for the relative power and scalar flux when a coarse spatial mesh combined with advanced homogenization techniques is used.

[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 system

[EN] The diffusion approximation to the time-dependent Boltzmann transport equation gives accurate results for traditional nuclear reactor designs, but new reactor designs and new fuel elements require neutron transport methods. We develop a numerical approximation to the time-dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics P L equations, for odd L, and on the Backward Euler finite difference discretization of time. The resulting scheme can be written as a stationary form of diffusive second order PL equations. This allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem, using coarse spatial grids to further reduce memory requirements. This scheme is validated with several transient benchmarks, where the convergence properties are established and compared with the simplified PL approximation. A more realistic transient benchmark, based on the two-group C5 MOX problem, is finally introduced, showing the need of high order P L approximation for complex fuel geometries.

[EN] The mechanical vibrations of fuel assemblies have been shown to give rise to high levels of neutron noise, triggering in some circumstances the necessity to operate nuclear reactors at a reduced power level. This work analyses the effect in the neutron field of the oscillation of one single fuel assembly. Results show two different effects in the neutron field caused by the fuel assembly vibration. First, a global slow variation of the total reactor power due to a change in the criticality of the system. Second, an oscillation in the neutron flux in-phase with the assembly vibration. This second effect has a strong spatial dependence that can be used to localize the oscillating assembly. This paper shows a comparison between a time-domain and a frequency-domain analysis of the phenomena to calculate the spatial response of the neutron noise. Numerical results show a really close agreement between these two approaches.

[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 is an Accepted Manuscript of an article published by Taylor & Francis in Nuclear Science and Engineering on 2020, available online: http://www.tandfonline.com/10.1080/00295639.2020.1756617, [EN] The mechanical vibrations of core internals such as fuel assemblies (FAs) cause oscillations in the neutron flux that require in some circumstances nuclear power plants to operate at a reduced power level. This work simulates and analyzes the changes of the neutron flux throughout a nuclear core due to the oscillation of a single FA without considering thermal-hydraulic feedback. The amplitude of the FA vibration is bounded to a few millimeters, and this implies the use of fine meshes and accurate numerical solvers due to the different scales of the problem. The results of the simulations show a main oscillation of the neutron flux with the same frequency as the FA vibration along with other harmonics at multiples of the vibration frequency much smaller in amplitude. Also, this work compares time domain analysis and frequency domain analysis of the mechanical vibrations. Numerical results show a close match between these two approaches for the fundamental frequency.

En el libro se presenta una introducción a los diferentes temas relacionados con la resolución numérica de ecuaciones diferenciales ordinarias y en derivadas parciales. Se comienza introduciendo distintos modelos matemáticos basados en ecuaciones diferenciales, se estudian algunas materias de apoyo como son la interpolación y la resolución de sistemas de ecuaciones lineales y posteriormente se explican algunas técnicas para la resolución numérica de problemas de valores iniciales y de problemas de contorno.En el libro se presentan contenidos teóricos, ejemplos de aplicación y algunas secciones dedicadas a la utilización de Matlab para la resolución de problemas relacionados con la materia presentada. En la presentación de los temas se ha evitado exponer desarrollos abstractos buscando con ello que el libro sea de fácil lectura para los estudiantes de ingeniería. Esta edición contiene nuevos ejercicios resueltos y propuestos,así como un análisis más detallado del cálculo de trayectorias de satélites. La notación utilizada en diferentes partes del texto, así como bastantes de los ejercicios con Matlab han sido modificados para facilitar su comprensión.

[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, [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, [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 ti

[ES] La industria de las aeronaves pilotadas por control remoto (RPAs) y, en concreto, la de los drones, se encuentra en su apogeo. Por ello, surge la necesidad de descubrir modelos matemáticos capaces de definir la trayectoria de estos vehículos aéreos. El presente Trabajo de Fin de Grado (TFG) desarrolla un procedimiento analítico para predecir el itinerario de un cuadricóptero en función de los parámetros geométricos y las actuaciones de sus rotores. Con este fin, se explican los fenómenos presentes en la intervención de un dron, se estudia el comportamiento del aire alrededor de las hélices para definir la tracción generada por estas y se caracterizan los motores del modelo Cheerwing Syma XS5W-V3. A partir de la grabación de dos despegues realizados por este cuadricóptero, se comprueba que es posible pronosticar con un programa de simulación la trayectoria ejecutada. Para acabar, se definen los costes del proyecto y la normativa a tener en cuenta durante su realización., [EN] The industry of remotely piloted aircraft (RPAs) and, specifically, the one that investigates drones, is in full swing. Therefore, there is a need to discover mathematical models capable of defining the trajectory of these aerial vehicles. This Final Degree Project (TFG) develops an analytical procedure to predict the itinerary of a quadcopter based on the geometric parameters and the actions of its rotors. To achieve this, the phenomena present in the intervention of a drone are explained, the behavior of the air around the propellers is studied to define the traction generated by them and the engines of the Cheerwing Syma XS5W-V3 model are characterized. From the recording of two take-offs carried out by this quadcopter, it is verified that it is possible to predict with a simulation program the trajectory executed. Finally, the costs of the project and the regulations to be considered during its implementation are commented.

[EN] Background: The respiratory epithelium is frequently infected by the respiratory syncytial virus, resulting in inflammation, a reduction in cilia activity and an increase in the production of mucus. Methods: In this study, an automatic method has been proposed to characterize the ciliary motility from cell cultures by means of a motility index using a dense optical flow algorithm. This method allows us to determine the ciliary beat frequency (CBF) together with a ciliary motility index of the cells in the cultures. The object of this analysis is to automatically distinguish between normal and infected cells in a culture. Results: The method was applied in 2 stages. It was concluded from the first stage that the CBF is not a good enough indicator to discriminate between the control and infected cultures. However, the ciliary motility index does succeed in discriminating between the control and infected cultures using the t test with a value t ¿ 6.46 and P <.001. In the second stage, it has been shown that the ciliary motility index did not differ significantly between patients, and the analysis of variance test gives a ¿ 0.05, F ¿ 1.61, P ¿.20. A threshold for this index has been determined using a receiver operating characteristics analysis that gives an area under the curve of 0.93. Conclusions: We have obtained a ciliary motility index that is able to discriminate between control and infected cultures after the eighth postinfection day. After infection, there is a rapid cilia loss of the cells and the measured CBF corresponds to the remaining noninfected cells. This is why the CBF does not discriminate between the control and the infected cells.

[EN] Aim of study: The development of a procedure to calibrate the LEACHM and EU-Rotate N models for simulating water and nitrogen dynamics in cauliflower crops. Area of study: Calibration was performed using experimental data obtained from measurements in a cauliflower crop sited in Valencia (Spain) region. Material and methods: A procedure based on generalized sensitivity indices for time-dependent outputs was used to determine the most influencing model parameters, in order to reduce the number of parameters to be calibrated and to avoid overparameterization. The most influencing parameters were introduced in an optimization process that uses the experimental measurements of soil water and nitrate content to determine its optimal value and obtain calibrated models. Main results: After this analysis, the most important hydraulic parameters found were the coefficients of Campbell¿s equation for the LEACHM model and the soil water content at field capacity and drainage coefficient for the EU-Rotate_N model. For the N cycle, the most influencing parameters were those related with the nitrification, humus mineralization rate and residue decomposition for both models. Both calibrated models provided good simulation of soil water content with an error between 5-7%. However, larger errors in soil-nitrate content simulation were found, mainly in the period corresponding to the crop residues incorporation. The prediction of the calibrated models in a different plot gave error values of about 7-9% for soil water content, but for soil nitrate content errors computed were 34% and 58%. Research highlights: After calibration, both models can be used to optimize the farmer water management and fertilization practices in horticultural crops, although in the N case further studies should be performed.

[EN] In nuclear engineering, the lambda-modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite element method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a block generalized Newton method implemented with a matrix-free technique that does not store all matrices explicitly. This technique reduces mainly the computational memory and, in some cases, when the assembly of the matrices is an expensive task, the computational time. The main problem is that the block Newton method requires solving linear systems, which need to be preconditioned. The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not efficient in terms of the memory with the matrix-free implementation. As an alternative, several block preconditioners are studied that only save a few block matrices in comparison with the full problem. To test the performance of these methodologies, different reactor problems are studied.

[EN] The behaviour of the neutrons inside a nuclear reactor core can be modelled by using the time dependent neutron diffusion equation. Different time schemes have been used to integrate this equation. One possibility is to use a modal method, which is based on the expansion of the neutron flux in terms of spatial modes that are the eigenfunctions associated with a given configuration of the reactor core. Several spatial modes can be defined for the neutron diffusion equation such as the ¿, ¿ and ¿-modes. In this work, the ¿, the ¿ and the ¿-modes have been used to develop different modal kinetics equations, using a high order finite element method for the spatial discretization of the neutron diffusion equation. The performance of the different modal kinetic equations has been tested and compared using two 3D transient benchmark problems.

[ES] En el grado de Ingenier´ıa de la Energ´ıa, los alumnos pueden cursar las asignaturas de Tecnolog´ıa Nuclear y Seguridad Nuclear. En ambas asignaturas se estudia el dise˜no de un reactor nuclear para que la central funcione en condiciones seguras. Para conseguir que los alumnos entiendan el funcionamiento de un reactor nuclear es interesante el uso de modelos matem´aticos que reproduzcan tanto la potencia generada como la evoluci´on de otras variables que afectan a la seguridad de la planta. En este trabajo, se presenta un modelo sencillo de reactor de agua a presi´on que permite observar la evoluci´on temporal de la potencia del reactor y las temperaturas de los componentes m´as importantes de la planta. Para obtener la potencia generada se hace uso del modelo de cin´etica puntual, mientras que para el c´alculo de las temperaturas del combustible y del refrigerante se utiliza un modelo sencillo de transmisi´on de calor. De este modo, el problema se formula mediante un sistema de ecuaciones diferenciales ordinarias que se resuelve utilizando m´etodos num´ericos. El desarrollo del modelo permite a los alumnos de Tecnolog´ıa Nuclear y Seguridad Nuclear comprender la evoluci´on temporal de alguna de las variables de un reactor y darse cuenta de los efectos estabilizantes que los par´ametros de la termohidr´aulica tienen sobre la potencia generada., [EN] In the Bachelor’s degree of Energy Engineering, students can attend to Nuclear Technology and Nuclear Safety courses. Both subjects are focused on the reactor design to assure the nuclear power plant operates in safe conditions. The use of mathematical models to reproduce the power generated and other plant safety variables helps the students to understand the nuclear power reactor operation. In this work, a simple model of a Pressurized Water Reactor that allows to represent the temporal evolution of the reactor power and the temperatures of the most important plant components is presented. The point kinetics model is used to obtain the power generated inside the core, while a simple heat transfer model is used to calculate the fuel and coolant temperatures. In this way, the problem is formulated as a system of ordinary differential equations that is solved using numerical methods. The development of the model allows the students of Nuclear Technology and Nuclear Safety to understand the temporal evolution of some of the reactor variables and to become aware of the stabilization effect of the thermohydraulic parameters over the generated power.

[EN] The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization looks for discontinuous solutions through the introduction of discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the Simplified P-N equations using the finite element method. This extension proposes pin discontinuity factors instead of the usual assembly discontinuity factors and the use of the simplified spherical harmonics approximation rather than diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. One dimensional numerical results show that the proposed pin discontinuity factors produce more accurate results than the usual assembly discontinuity factors. The proposed pin discontinuity factors produce precise results for both pin and assembly averaged values without using advanced reconstruction methods. Also, the homogenization methodology is verified against the calculation performed with reference discontinuity factors. (C) 2017 Elsevier B.V. All rights reserved.

La ecuación del transporte neutrónico describe la población de neutrones y las reacciones nucleares dentro de un reactor nuclear. Primero, introducimos esta ecuación y las aproximaciones de la misma. Entonces, estudiamos la ecuación de la difusión neutrónica, la aproximación al transporte más utilizada. Para el caso estacionario, esta aproximación da lugar a un problema diferencial de valores propios. Para resolver la ecuación de la difusión se ha desarrollado un método de elementos finitos h-p. Para mejorar la eficiencia del método se ha implementado un precondicionador del tipo Restricted Additive Schwarz. Una vez hemos obtenido la distribución neutrónica en estado estacionario, usamos esta solución como condición inicial para integrar la ecuación de la difusión. Para probar el comportamiento del método propuesto, hemos simulado numéricamente ejecciones accidentales de barras de control. Sin embargo, cuando una celda tiene parcialmente introducida una barra de control aparece un comportamiento no físico, el efecto rod cusping. Para mitigar este efecto proponemos un esquema de malla móvil, es decir, la malla sigue el movimiento de las barras de control. Los resultados muestran que el efecto rod cusping disminuye con el esquema expuesto. Después, desarrollamos la aproximación de armónicos esféricos simplificados, SPN, para simular el comportamiento del núcleo del reactor el problema en estado estacionario. Esta aproximación extiende los armónicos esféricos en geometrías unidimensionales, PN, a geometrías multidimensionales usando fuertes aproximaciones. Las ecuaciones SPN mejoran la teoría de la difusión pero no convergen cuando N tiende a infinito. Probamos las ventajas y limitaciones de esta aproximación en diversos reactores. Finalmente, estudiamos la homogenización espacial en el contexto de los elementos finitos. La homogenización consiste en cambiar subdominios heterogéneos por homogéneos, de forma que el problema homogeneizado da eficientemente resultados pro, The neutron transport equation describes the neutron population and the nuclear reactions inside a nuclear reactor core. First, this equation is introduced and its assumptions are stated. Then, the stationary neutron diffusion equation which is the most useful approximation of this equation, is studied. This approximation leads to a differential eigenvalue problem. To solve the neutron diffusion equation, a h-p finite element method is investigated. To improve the efficiency of the method a Restricted Additive Schwarz preconditioner is implemented. 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 test the behaviour of the method, rod ejection accidents are numerically simulated. However, a non-physical behaviour appears when a cell is partially rodded: this is, the rod cusping effect, which is solved by using a moving mesh scheme. In other words, the mesh follows the movement of the control rod. Numerical results show that the rod cusping effect is corrected with this scheme. After that, the simplified spherical harmonics approximation, SPN, is developed to solve the steady state problem. This approximation extends the spherical harmonics approximation, PN, in one dimensional geometries to multidimensional geometries with strong assumptions. It improves the diffusion theory results but does not converge as N tends to infinity. The advantages and limitations of this approximation are tested on several one-, two- and three-dimensional reactors. Finally, the spatial homogenization in the context of the finite element method is studied. Homogenization consists in replacing heterogeneous subdomains by homogeneous ones, in such a way that the homogenized problem provides fast and accurate average results. Discontinuous solutions were proposed in the Generalized Equivalence Theory. Here, a discontinuous Galerkin finite element method where the jump conditi, L'equació del transport neutrònic descriu la població de neutrons i les reaccions nuclears dins del nucli d'un reactor nuclear. Primer, introduïm aquesta equació i les seues principals aproximacions. Aleshores, estudiem l'equació de la difusió neutrònica, l'aproximació al transport neutrònic més utilitzada. Aquesta equació genera un problema diferencial de valors propis. Per a resoldre l'equació de la difusió s'ha desenvolupat un mètode d'elements finits h-p. Per millorar l'eficiencia del mètode s'ha implementat un precondicionador del tipus Restricted Additive Schwarz. Una vegada hem obtingut la distribució neutrònica en estat estacionari, usem aquesta solució com a condició inicial per integrar l'equació de la difusió depenent del temps. Amb la voluntat de provar el comportament del mètode proposat, hem simulat numèricament expusions accidentals de barres de control. Però, quan un node té parcialment introduïda una barra de control apareix un comportament no físic, l'efecte rod cusping. Per mitigar aquest efecte proposem un esquema de malla mòbil, és a dir, la malla segueix el moviment de les barres de control. Els resultats numèrics mostren que l'efecte rod cusping disminueix amb l'esquema exposat. Després, desenvolupem l'aproximació d'harmònics esfèrics simplificats, SPN, per a resoldre el problema en estat estacionari. Aquesta equació estén l'aproximació d'harmònics esfèrics en geometries unidimensionals, PN, a geometries multidimensionals usant fortes aproximacions. Les equacions SPN milloren la teoria de la difusió però no convergeixen quan N tendeix a infinit. Provem els avantatges i limitacions d'aquesta aproximació en diversos reactors. Finalment, estudiem l'homogeneïtzació espacial en el context dels elements finits. L'homogeneïtzació consisteix en canviar subdominis heterogenis per homogenis, de forma que el problema homogeneïtzat dóna eficientment resultats mitjos. La Teoria Generalitzada de l'Equivalència per a l'homogeneïtzació proposa factors de disc

[EN] A classical discretization for the angular dependence of the neutron transport equation is based on a truncated spherical harmonics expansion. The resulting system of equations are the PL equations. We review the multi-dimensional P-L equations, for arbitrary odd order L, and then we proceed to the spatial discretization of these equations, for rectangular geometries, using a nodal collocation method based on the expansion of the spatial dependence of the fields in terms of orthonormal Legendre polynomials. The validity of the method to deal with complex reactor problems is then studied with the seven group 2D NEA C5G7 MOX fuel assembly benchmark problem. The solution is computed for different spatial meshes, showing that the PL results are consistent with the reference Monte Carlo solution. Additionally, the first subcritical modes are also computed for the full reactor configuration. (C) 2017 Elsevier Ltd. All rights reserved.

[EN] The dominant lambda-modes associated with a nuclear reactor configuration describe the neutron steady-state distribution and its criticality. Furthermore, they are useful to develop modal methods to study reactor instabilities. Different eigenvalues solvers have been successfully used to obtain such modes, most of them are implemented reducing the original generalized eigenvalue problem to an ordinary one. Thus, it is necessary to solve many linear systems making these methods not very efficient, especially for large problems. In this work, the original generalized eigenvalue problem is considered and two block iterative methods to solve it are studied: the block inverse-free preconditioned Arnoldi method and the modified block Newton method. All of these iterative solvers are initialized using a block multilevel technique. A hybrid multilevel method is also proposed based on the combination of the methods proposed. Two benchmark problems are studied illustrating the convergence and the competitiveness of the methods proposed. A comparison with the Krylov-Schur method and the Generalized Davidson is also included.