Evaluation of preisach model of hysteresis to model superconductors

The Large Hadron Collider is the world’s largest and most powerful particle accelerator. The project is divided in phases. The first one goes from 2009 until 2020. The second phase will consist of the implementation of upgrades. One of the upgrades is to increase the ratio of collision, the luminosity. This objective is the main of one of the most important projects which is carrying out the upgrades: Hi-Lumi LHC project. Increasing luminosity could be done by using a new material in the superconductor magnets placed at the interaction points: Nb3Sn, instead of NbTi, the one being used right now. Before implementing it many aspects should be analysed. One of them is the induction magnetic field quality. The tool used so far has been ROXIE, software developed at CERN by S. Russenschuck. One of the main features of the programme is the time-transient analysis, which is based on three mathematical models. It is quite precise for fields above 1.5 Tesla. However, they are not very accurate for lower fields. Therefore the aim of this project is to evaluate a more accurate model: Classical Preisach Model of Hysteresis, in order to better analyse induced field quality in the new material Nb3Sn. Resumen: El Gran Colisionador de Hadrones es el mayor acelerador de partículas circular del mundo. Se trata de uno de los mayores proyectos de investigación. La primera fase de funcionamiento comprende desde 2009 a 2020, cuando comenzará la siguiente fase. Durante el primer periodo se han pensado mejoras para que puedan ser implementadas en la segunda fase. Una de ellas es el aumento del ratio de las colisiones entre protones por choque. Este es el principal objetivo de uno de los proyectos que está llevando a cabo las mejoras a ser implementadas en 2020: Hi- Lumi LHC. Se cambiarán los imanes superconductores de NbTi de las dos zonas principales de interacción, y se sustituirán por imanes de Nb3Sn. Esta sustituciónn conlleva un profundo estudio previo. Entre otros, uno de los factores a analizar es la calidad del campo magnético. La herramienta utilizada es el software desarrollado por S. Russenschuck en el CERN llamado ROXIE. Está basado en tres modelos de magnetización, los cuales son precisos para campos mayores de 1.5 T. Sin embargo, no lo son tanto para campos menores. Con este proyecto se pretende evaluar la implementación de un cuarto modelo, el modelo clásico de histéresis de Preisach que permita llevar a cabo un mejor análisis de la calidad del campo inducido por el futuro material a utilizar en algunos de los imanes.

Técnicas avanzadas de metrología dimensional indirecta mediante el empleo de máquinas de medición por visión en entornos industriales. Modelo para la estimación de incertidumbres

Esta tesis contiene una investigación detallada sobre las características y funcionamiento de las máquinas de medición por visión. El objetivo fundamental es modelar su comportamiento y dotarlas de trazabilidad metrológica bajo cualquier condición de medida. Al efecto, se ha realizado un exhaustivo análisis de los elementos que conforman su cadena de medición, a saber: sistema de iluminación, estructura, lentes y objetivos, cámara, software de tratamiento de imágenes y software de cálculo. Se han definido modelos físico-matemáticos, de desarrollo propio, capaces de simular con fiabilidad el comportamiento de los elementos citados, agrupados, a efectos de análisis numérico, en dos subsistemas denominados: de visión y mecánico. Se han implementado procedimientos de calibración genuinos para ambos subsistemas mediante el empleo de patrones ópticos. En todos los casos se ha podido determinar la incertidumbre asociada a los diferentes parámetros involucrados, garantizando la trazabilidad metrológica de los resultados. Los distintos modelos desarrollados han sido implementados en Matlab®. Se ha verificado su validez empleando valores sintéticos obtenidos a partir de simulaciones informáticas y también con imágenes reales capturadas en el laboratorio. El estudio experimental y validación definitiva de los resultados se ha realizado en el Laboratorio de Longitud del Centro Español de Metrología y en el Laboratorio de Metrología Dimensional de la ETS de Ingeniería y Diseño Industrial de la UPM. Los modelos desarrollados se han aplicado a dos máquinas de medición por visión de diferentes características constructivas y metrológicas. Empleando dichas máquinas se han medido distintas piezas, pertenecientes a los ámbitos mecánico y oftalmológico. Los resultados obtenidos han permitido la completa caracterización dimensional de dichas piezas y la determinación del cumplimiento de las especificaciones metrológicas en todos los casos, incluyendo longitudes, formas y ángulos. ABSTRACT This PhD thesis contains a detailed investigation of characteristics and performance of the optical coordinate measurement machines. The main goal is to model their behaviour and provide metrological traceability to them under any measurement conditions. In fact, a thorough analysis of the elements which form the measuring chain, i.e.: lighting system, structure, lenses and objectives, camera, image processing software and coordinate metrology software has conducted. Physical-mathematical models, of self-developed, able to simulate with reliability the behavior of the above elements, grouped, for the purpose of numerical analysis, in two subsystems called: “vision subsystem” and “mechanical subsystem”, have been defined. Genuine calibration procedures for both subsystems have been implemented by use of optical standards. In all cases, it has been possible to determine the uncertainty associated with the different parameters involved, ensuring metrological traceability of results. Different developed models have been implemented in Matlab®. Their validity has been verified using synthetic values obtained from computer simulations and also with real images captured in laboratory. The experimental study and final validation of the results was carried out in the Length Laboratory of “Centro Español de Metrología” and Dimensional Metrology Laboratory of the “Escuela Técnica Superior de Ingeniería y Diseño Industrial” of the UPM. The developed models have been applied to two optical coordinate measurement machines with different construction and metrological characteristics. Using such machines, different parts, belonging to the mechanical and ophthalmologist areas, have been measured. The obtained results allow the full dimensional characterization of such parts and determination of compliance with metrological specifications in all cases, including lengths, shapes and angles.

On the Analytical Approach to Present Engineering Problems: Photovoltaic Systems Behavior, Wind Speed Sensors Performance, and High-Speed Train Pressure Wave Effects in Tunnels

At present, engineering problems required quite a sophisticated calculation means. However, analytical models still can prove to be a useful tool for engineers and scientists when dealing with complex physical phenomena. The mathematical models developed to analyze three different engineering problems: photovoltaic devices analysis; cup anemometer performance; and high-speed train pressure wave effects in tunnels are described. In all cases, the results are quite accurate when compared to testing measurements.

Finite element simulation of dispersion in the Bay of Santander

Two mathematical models are used to simulate pollution in the Bay of Santander. The first is the hydrodynamic model that provides the velocity field and height of the water. The second gives the pollutant concentration field as a resultant. Both models are formulated in two-dimensional equations. Linear triangular finite elements are used in the Galerkin procedure for spatial discretization. A finite difference scheme is used for the time integration. At each time step the calculated results of the first model are input to the second model as field data. The efficiency and accuracy of the models are tested by their application to a simple illustrative example. Finally a case study in simulation of pollution evolution in the Bay of Santander is presented

BR3: a biologically inspired fish-like robot actuated by SMA-based artificial muscles

Los peces son animales, donde en la mayoría de los casos, son considerados como nadadores muy eficientes y con una alta capacidad de maniobra. En general los peces se caracterizan por su capacidad de maniobra, locomoción silencioso, giros y partidas rápidas y viajes de larga distancia. Los estudios han identificado varios tipos de locomoción que los peces usan para generar maniobras y natación constante. A bajas velocidades la mayoría de los peces utilizan sus aletas pares y / o impares para su locomoción, que ofrecen una mayor maniobrabilidad y mejor eficiencia de propulsión. A altas velocidades la locomoción implica el cuerpo y / o aleta caudal porque esto puede lograr un mayor empuje y aceleración. Estas características pueden inspirar el diseo y fabricación de una piel muy flexible, una aleta caudal mórfica y una espina dorsal no articulada con una gran capacidad de maniobra. Esta tesis presenta el desarrollo de un novedoso pez robot bio-inspirado y biomimético llamado BR3, inspirado en la capacidad de maniobra y nado constante de los peces vertebrados. Inspirado por la morfología de los peces Micropterus salmoides o también conocido como lubina negra, el robot BR3 utiliza su fundamento biológico para desarrollar modelos y métodos matemáticos precisos que permiten imitar la locomoción de los peces reales. Los peces Largemouth Bass pueden lograr un nivel increíble de maniobrabilidad y eficacia de la propulsión mediante la combinación de los movimientos ondulatorios y aletas morficas. Para imitar la locomoción de los peces reales en una contraparte artificial se necesita del análisis de tecnologías de actuación alternativos, como arreglos de fibras musculares en lugar de servo actuadores o motores DC estándar, así como un material flexible que proporciona una estructura continua sin juntas. Las aleaciones con memoria de forma (SMAs) proveen la posibilidad de construir robots livianos, que no emiten ruido, sin motores, sin juntas y sin engranajes. Asi es como un pez robot submarino se ha desarrollado y cuyos movimientos son generados mediante SMAs. Estos actuadores son los adecuados para doblar la espina dorsal continua del pez robot, que a su vez provoca un cambio en la curvatura del cuerpo. Este tipo de arreglo estructural está inspirado en los músculos rojos del pescado, que son usados principalmente durante la natación constante para la flexión de una estructura flexible pero casi incompresible como lo es la espina dorsal de pescado. Del mismo modo la aleta caudal se basa en SMAs y se modifica para llevar a cabo el trabajo necesario. La estructura flexible proporciona empuje y permite que el BR3 nade. Por otro lado la aleta caudal mórfica proporciona movimientos de balanceo y guiada. Motivado por la versatilidad del BR3 para imitar todos los modos de natación (anguilliforme, carangiforme, subcarangiforme y tunniforme) se propone un controlador de doblado y velocidad. La ley de control de doblado y velocidad incorpora la información del ángulo de curvatura y de la frecuencia para producir el modo de natación deseado y a su vez controlar la velocidad de natación. Así mismo de acuerdo con el hecho biológico de la influencia de la forma de la aleta caudal en la maniobrabilidad durante la natación constante se propone un control de actitud. Esta novedoso robot pescado es el primero de su tipo en incorporar sólo SMAs para doblar una estructura flexible continua y sin juntas y engranajes para producir empuje e imitar todos los modos de natación, así como la aleta caudal que es capaz de cambiar su forma. Este novedoso diseo mecatrónico presenta un futuro muy prometedor para el diseo de vehículos submarinos capaces de modificar su forma y nadar mas eficientemente. La nueva metodología de control propuesto en esta tesis proporcionan una forma totalmente nueva de control de robots basados en SMAs, haciéndolos energéticamente más eficientes y la incorporación de una aleta caudal mórfica permite realizar maniobras más eficientemente. En su conjunto, el proyecto BR3 consta de cinco grandes etapas de desarrollo: • Estudio y análisis biológico del nado de los peces con el propósito de definir criterios de diseño y control. • Formulación de modelos matemáticos que describan la: i) cinemática del cuerpo, ii) dinámica, iii) hidrodinámica iv) análisis de los modos de vibración y v) actuación usando SMA. Estos modelos permiten estimar la influencia de modular la aleta caudal y el doblado del cuerpo en la producción de fuerzas de empuje y fuerzas de rotación necesarias en las maniobras y optimización del consumo de energía. • Diseño y fabricación de BR3: i) estructura esquelética de la columna vertebral y el cuerpo, ii) mecanismo de actuación basado en SMAs para el cuerpo y la aleta caudal, iii) piel artificial, iv) electrónica embebida y v) fusión sensorial. Está dirigido a desarrollar la plataforma de pez robot BR3 que permite probar los métodos propuestos. • Controlador de nado: compuesto por: i) control de las SMA (modulación de la forma de la aleta caudal y regulación de la actitud) y ii) control de nado continuo (modulación de la velocidad y doblado). Está dirigido a la formulación de los métodos de control adecuados que permiten la modulación adecuada de la aleta caudal y el cuerpo del BR3. • Experimentos: está dirigido a la cuantificación de los efectos de: i) la correcta modulación de la aleta caudal en la producción de rotación y su efecto hidrodinámico durante la maniobra, ii) doblado del cuerpo para la producción de empuje y iii) efecto de la flexibilidad de la piel en la habilidad para doblarse del BR3. También tiene como objetivo demostrar y validar la hipótesis de mejora en la eficiencia de la natación y las maniobras gracias a los nuevos métodos de control presentados en esta tesis. A lo largo del desarrollo de cada una de las cinco etapas, se irán presentando los retos, problemáticas y soluciones a abordar. Los experimentos en canales de agua estarán orientados a discutir y demostrar cómo la aleta caudal y el cuerpo pueden afectar considerablemente la dinámica / hidrodinámica de natación / maniobras y cómo tomar ventaja de la modulación de curvatura que la aleta caudal mórfica y el cuerpo permiten para cambiar correctamente la geometría de la aleta caudal y del cuerpo durante la natación constante y maniobras. ABSTRACT Fishes are animals where in most cases are considered as highly manoeuvrable and effortless swimmers. In general fishes are characterized for his manoeuvring skills, noiseless locomotion, rapid turning, fast starting and long distance cruising. Studies have identified several types of locomotion that fish use to generate maneuvering and steady swimming. At low speeds most fishes uses median and/or paired fins for its locomotion, offering greater maneuverability and better propulsive efficiency At high speeds the locomotion involves the body and/or caudal fin because this can achieve greater thrust and accelerations. This can inspire the design and fabrication of a highly deformable soft artificial skins, morphing caudal fins and non articulated backbone with a significant maneuverability capacity. This thesis presents the development of a novel bio-inspired and biomimetic fishlike robot (BR3) inspired by the maneuverability and steady swimming ability of ray-finned fishes (Actinopterygii, bony fishes). Inspired by the morphology of the Largemouth Bass fish, the BR3 uses its biological foundation to develop accurate mathematical models and methods allowing to mimic fish locomotion. The Largemouth Bass fishes can achieve an amazing level of maneuverability and propulsive efficiency by combining undulatory movements and morphing fins. To mimic the locomotion of the real fishes on an artificial counterpart needs the analysis of alternative actuation technologies more likely muscle fiber arrays instead of standard servomotor actuators as well as a bendable material that provides a continuous structure without joins. The Shape Memory Alloys (SMAs) provide the possibility of building lightweight, joint-less, noise-less, motor-less and gear-less robots. Thus a swimming underwater fish-like robot has been developed whose movements are generated using SMAs. These actuators are suitable for bending the continuous backbone of the fish, which in turn causes a change in the curvature of the body. This type of structural arrangement is inspired by fish red muscles, which are mainly recruited during steady swimming for the bending of a flexible but nearly incompressible structure such as the fishbone. Likewise the caudal fin is based on SMAs and is customized to provide the necessary work out. The bendable structure provides thrust and allows the BR3 to swim. On the other hand the morphing caudal fin provides roll and yaw movements. Motivated by the versatility of the BR3 to mimic all the swimming modes (anguilliform, caranguiform, subcaranguiform and thunniform) a bending-speed controller is proposed. The bending-speed control law incorporates bend angle and frequency information to produce desired swimming mode and swimming speed. Likewise according to the biological fact about the influence of caudal fin shape in the maneuverability during steady swimming an attitude control is proposed. This novel fish robot is the first of its kind to incorporate only SMAs to bend a flexible continuous structure without joints and gears to produce thrust and mimic all the swimming modes as well as the caudal fin to be morphing. This novel mechatronic design is a promising way to design more efficient swimming/morphing underwater vehicles. The novel control methodology proposed in this thesis provide a totally new way of controlling robots based on SMAs, making them more energy efficient and the incorporation of a morphing caudal fin allows to perform more efficient maneuvers. As a whole, the BR3 project consists of five major stages of development: • Study and analysis of biological fish swimming data reported in specialized literature aimed at defining design and control criteria. • Formulation of mathematical models for: i) body kinematics, ii) dynamics, iii) hydrodynamics, iv) free vibration analysis and v) SMA muscle-like actuation. It is aimed at modelling the e ects of modulating caudal fin and body bend into the production of thrust forces for swimming, rotational forces for maneuvering and energy consumption optimisation. • Bio-inspired design and fabrication of: i) skeletal structure of backbone and body, ii) SMA muscle-like mechanisms for the body and caudal fin, iii) the artificial skin, iv) electronics onboard and v) sensor fusion. It is aimed at developing the fish-like platform (BR3) that allows for testing the methods proposed. • The swimming controller: i) control of SMA-muscles (morphing-caudal fin modulation and attitude regulation) and ii) steady swimming control (bend modulation and speed modulation). It is aimed at formulating the proper control methods that allow for the proper modulation of BR3’s caudal fin and body. • Experiments: it is aimed at quantifying the effects of: i) properly caudal fin modulation into hydrodynamics and rotation production for maneuvering, ii) body bending into thrust generation and iii) skin flexibility into BR3 bending ability. It is also aimed at demonstrating and validating the hypothesis of improving swimming and maneuvering efficiency thanks to the novel control methods presented in this thesis. This thesis introduces the challenges and methods to address these stages. Waterchannel experiments will be oriented to discuss and demonstrate how the caudal fin and body can considerably affect the dynamics/hydrodynamics of swimming/maneuvering and how to take advantage of bend modulation that the morphing-caudal fin and body enable to properly change caudal fin and body’ geometry during steady swimming and maneuvering.

Aplicación de la técnica de inyección del terreno por fracturación hidráulica a las cimentaciones de puentes en circunstancias de especial dificultad

La implantación de una gran obra de paso facilita en multitud de ocasiones la vida de aquellos que aprovechen las ventajas que ofrece esta estructura. Sin embargo, el camino que lleva a su construcción supone importantes y tortuosos retos cuando el terreno de cimentación presenta condiciones geotécnicas no compatibles con las grandes cargas que el viaducto necesita transmitirle para su buen funcionamiento. El trabajo que desarrolla esta Tesis Doctoral proporciona una herramienta eficaz y económica, por lo reducido de su extensión y medios, que permite allanar el camino que acomete la construcción de una estructura de tal envergadura. Mediante el análisis de la problemática del terreno y de las distintas soluciones de cimentación empleadas en la actualidad, se conduce al lector hacia una técnica de cimentación innovadora que combina la técnica del micropilotaje, para la canalización de las cargas estructurales, junto a la técnica de la inyección de fracturación hidráulica, que mejorará el terreno de implantación de los micropilotes, conformando así un medio capaz de recibir y transmitir grandes cargas en cualquier tipo de terreno. La técnica ya empleada del micropilotaje, por su trabajo esencialmente axil, requiere, aunque sin problema, la constitución de sistemas de fuerzas que equilibren el sistema de cargas provenientes del viaducto. Pero su capacidad resistente viene condicionada por el terreno circundante. Cuando la roca rodea el micropilote, su empleo se realiza sin problema y sin necesidad de mejorar el terreno. Pero sin terreno consistente, el empleo de inyecciones de fracturación hidráulica a través de los propios micropilotes, no sólo mejorará la capacidad resistente de los micropilotes, aumentando la inercia necesaria cuando las cargas son de origen sísmico o estructural ferroviario, sino que resolverá y eliminará los problemas de estabilidad que presentan las laderas que frecuentemente deben recibir las cargas de la obra de paso. Tras recoger el análisis ya realizado en el Trabajo de Investigación, donde se justificaba la alta capacidad resistente de un micropilote con terreno circundante mejorado por la inyección, en la actual Tesis se emplean modelos matemáticos sobre un caso real de viaducto sometido a un gran sismo e implantado en una enorme quebrada (500 m) de Colombia, sujeta a lluvias torrenciales y en la zona de mayor sismicidad del país. Con ello se comprueba la estabilización que se alcanza en el terreno de cimentación con el empleo de esta técnica de transmisión de carga y mejora del terreno. De esta forma se completa un ciclo que justifica las bondades de esta combinación de técnicas de cimentación, pero se abren las puertas a nuevos entornos de aplicación, como edificios antiguos de cualquier tipo que requieran recalces, y no sólo en la implantación de grandes obras de paso. ABSTRACT The establishment of a large bridge represents, in many cases, a better life for those who can take profit of the advantages provided by that structure. Nevertheless, the process of building this structure has to overcome important and difficult circumstances whenever the geotechnical conditions of the bridge site are not adequate to carry the large loads transmitted by the bridge structure. This study develops a method both effective and economical, due to the extension and means necessary for its application, which allows to solve properly the foundation of a structure of that importance. Considering the geotechnical problems inherent to the bridge site, along with the different foundation solutions that are presently used and their limitations, the study leads the reader to an innovative technique which combines the micropile system, for transmission of the structural loads of the bridge, with the technique of hydraulic fracture grouting for improvement of the ground around the micropiles, allowing to both stabilizing and transmitting large loads in any kind of ground. It is well known that the micropiles work axially, and this condition requires an adequate distribution of those units, in order to properly absorb the load system introduced by the viaduct. The resistance of the bridge foundation is, in any case, provided by the ground. When rock is encountered, the micropiles have been successfully used without improving the ground. However, as it is shown in this study, by using the micropiles as sleeve-pipes for hydraulic fracture grouting, not only the micropile resistance can be improved in any ground, but it is possible to develop grouted “solids” in the ground, whose inertia allows to absorb actions of structural and seismic origin. Additionally, as it is shown and analyzed in the Thesis, the ground improvement can give an adequate safety factor to the slopes frequently encountered in bridge sites. In order to properly justify those advantages of combining micropiles with ground improvement through fracture grouting around the micropiles, mathematical models have been developed and applied to a real case of a cable-stayed bridge installed on a very large ravine (500 m) in Colombia, located in the highest seismic zone of the country, and subject to torrential rains. The results of this numerical analysis show the high safety condition provided by the ground improvement to the viaduct site. In conclusion, the Thesis shows the important improvement that can be provided by the combination of micropiles and soil improvement, through fracture grouting, to the problem of founding bridges. However, it can be understood that this technique could be applied successfully to underpinning buildings, specially old buildings of any type, apart from its use in bridge foundations.

Two-dimensional direct rainfall hydraulic models applied to direct calculation of hydrologic events in dams to prevent overtopping

One of the main concerns when conducting a dam test is the acute determination of the hydrograph for a specific flood event. The use of 2D direct rainfall hydraulic mathematical models on a finite elements mesh, combined with the efficiency of vector calculus that provides CUDA (Compute Unified Device Architecture) technology, enables nowadays the simulation of complex hydrological models without the need for terrain subbasin and transit splitting (as in HEC-HMS). Both the Spanish PNOA (National Plan of Aereal Orthophotography) Digital Terrain Model GRID with a 5 x 5 m accuracy and the CORINE GIS Land Cover (Coordination of INformation of the Environment) that allows assessment of the ground roughness, provide enough data to easily build these kind of models

Experimental and statistical investigation of canonical problems in sloshing

En hidrodinámica, el fenómeno de Sloshing se puede definir como el movimiento de la superficie libre de un fluido dentro de un contenedor sometido a fuerzas y perturbaciones externas. El fluido en cuestión experimenta violentos movimientos con importantes deformaciones de su superficie libre. La dinámica del fluido puede llegar a generar cargas hidrodinámicas considerables las cuales pueden afectar la integridad estructural y/o comprometer la estabilidad del vehículo que transporta dicho contenedor. El fenómeno de Sloshing ha sido extensivamente investigado matemática, numérica y experimentalmente, siendo el enfoque experimental el más usado debido a la complejidad del problema, para el cual los modelos matemáticos y de simulación son aun incapaces de predecir con suficiente rapidez y precisión las cargas debidas a dicho fenómeno. El flujo generado por el Sloshing usualmente se caracteriza por la presencia de un fluido multifase (gas-liquido) y turbulencia. Reducir al máximo posible la complejidad del fenómeno de Sloshing sin perder la esencia del problema es el principal reto de esta tesis doctoral, donde un trabajo experimental enfocado en casos canónicos de Sloshing es presentado y documentado con el objetivo de aumentar la comprensión de dicho fenómeno y por tanto intentar proveer información valiosa para validaciones de códigos numéricos. El fenómeno de Sloshing juega un papel importante en la industria del transporte marítimo de gas licuado (LNG). El mercado de LNG en los últimos años ha reportado un crecimiento hasta tres veces mayor al de los mercados de petróleo y gas convencionales. Ingenieros en laboratorios de investigación e ingenieros adscritos a la industria del LNG trabajan continuamente buscando soluciones económicas y seguras para contener, transferir y transportar grandes volúmenes de LNG. Los buques transportadores de LNG (LNGC) han pasado de ser unos pocos buques con capacidad de 75000 m3 hace unos treinta años, a una amplia flota con una capacidad de 140000 m3 actualmente. En creciente número, hoy día se construyen buques con capacidades que oscilan entre 175000 m3 y 250000 m3. Recientemente un nuevo concepto de buque LNG ha salido al mercado y se le conoce como FLNG. Un FLNG es un buque de gran valor añadido que solventa los problemas de extracción, licuefacción y almacenamiento del LNG, ya que cuenta con equipos de extracción y licuefacción a bordo, eliminando por tanto las tareas de transvase de las estaciones de licuefacción en tierra hacia los buques LNGC. EL LNG por tanto puede ser transferido directamente desde el FLNG hacia los buques LNGC en mar abierto. Niveles de llenado intermedios en combinación con oleaje durante las operaciones de trasvase inducen movimientos en los buques que generan por tanto el fenómeno de Sloshing dentro de los tanques de los FLNG y los LNGC. El trabajo de esta tesis doctoral lidia con algunos de los problemas del Sloshing desde un punto de vista experimental y estadístico, para ello una serie de tareas, descritas a continuación, se han llevado a cabo : 1. Un dispositivo experimental de Sloshing ha sido configurado. Dicho dispositivo ha permitido ensayar secciones rectangulares de tanques LNGC a escala con movimientos angulares de un grado de libertad. El dispositivo experimental ha sido instrumentado para realizar mediciones de movimiento, presiones, vibraciones y temperatura, así como la grabación de imágenes y videos. 2. Los impactos de olas generadas dentro de una sección rectangular de un LNGC sujeto a movimientos regulares forzados han sido estudiados mediante la caracterización del fenómeno desde un punto de vista estadístico enfocado en la repetitividad y la ergodicidad del problema. 3. El estudio de los impactos provocados por movimientos regulares ha sido extendido a un escenario más realístico mediante el uso de movimientos irregulares forzados. 4. El acoplamiento del Sloshing generado por el fluido en movimiento dentro del tanque LNGC y la disipación de la energía mecánica de un sistema no forzado de un grado de libertad (movimiento angular) sujeto a una excitación externa ha sido investigado. 5. En la última sección de esta tesis doctoral, la interacción entre el Sloshing generado dentro en una sección rectangular de un tanque LNGC sujeto a una excitación regular y un cuerpo elástico solidario al tanque ha sido estudiado. Dicho estudio corresponde a un problema de interacción fluido-estructura. Abstract In hydrodynamics, we refer to sloshing as the motion of liquids in containers subjected to external forces with large free-surface deformations. The liquid motion dynamics can generate loads which may affect the structural integrity of the container and the stability of the vehicle that carries such container. The prediction of these dynamic loads is a major challenge for engineers around the world working on the design of both the container and the vehicle. The sloshing phenomenon has been extensively investigated mathematically, numerically and experimentally. The latter has been the most fruitful so far, due to the complexity of the problem, for which the numerical and mathematical models are still incapable of accurately predicting the sloshing loads. The sloshing flows are usually characterised by the presence of multiphase interaction and turbulence. Reducing as much as possible the complexity of the sloshing problem without losing its essence is the main challenge of this phd thesis, where experimental work on selected canonical cases are presented and documented in order to better understand the phenomenon and to serve, in some cases, as an useful information for numerical validations. Liquid sloshing plays a key roll in the liquified natural gas (LNG) maritime transportation. The LNG market growth is more than three times the rated growth of the oil and traditional gas markets. Engineers working in research laboratories and companies are continuously looking for efficient and safe ways for containing, transferring and transporting the liquified gas. LNG carrying vessels (LNGC) have evolved from a few 75000 m3 vessels thirty years ago to a huge fleet of ships with a capacity of 140000 m3 nowadays and increasing number of 175000 m3 and 250000 m3 units. The concept of FLNG (Floating Liquified Natural Gas) has appeared recently. A FLNG unit is a high value-added vessel which can solve the problems of production, treatment, liquefaction and storage of the LNG because the vessel is equipped with a extraction and liquefaction facility. The LNG is transferred from the FLNG to the LNGC in open sea. The combination of partial fillings and wave induced motions may generate sloshing flows inside both the LNGC and the FLNG tanks. This work has dealt with sloshing problems from a experimental and statistical point of view. A series of tasks have been carried out: 1. A sloshing rig has been set up. It allows for testing tanks with one degree of freedom angular motion. The rig has been instrumented to measure motions, pressure and conduct video and image recording. 2. Regular motion impacts inside a rectangular section LNGC tank model have been studied, with forced motion tests, in order to characterise the phenomenon from a statistical point of view by assessing the repeatability and practical ergodicity of the problem. 3. The regular motion analysis has been extended to an irregular motion framework in order to reproduce more realistic scenarios. 4. The coupled motion of a single degree of freedom angular motion system excited by an external moment and affected by the fluid moment and the mechanical energy dissipation induced by sloshing inside the tank has been investigated. 5. The last task of the thesis has been to conduct an experimental investigation focused on the strong interaction between a sloshing flow in a rectangular section of a LNGC tank subjected to regular excitation and an elastic body clamped to the tank. It is thus a fluid structure interaction problem.

Estudio de la dinámica no lineal de vórtices magnéticos en nanodots

En el presente trabajo se ha realizado un estudio sobre el comportamiento dinámico de vórtices magnéticos en su aplicación en osciladores de espín-torque. A partir de los modelos matemáticos elaborados en [4] sobre estos osciladores de espín-torque, se han simulado y estudiado usando MATLAB, entre otras herramientas, para tratar de verificar su concordancia con el comportamiento físico observado en el laboratorio. Es importante señalar que el estudio tiene en cuenta la dinámica no lineal, esencial para describir correctamente un sistema oscilante en la práctica. El análisis matemático del sistema que describe la órbita del núcleo de un vórtice magnético sometido a la acción de un campo magnético externo oscilatorio, muestra la posibilidad de la existencia de hasta tres soluciones especiales, según la geometría del dot y el valor del campo magnético de excitación, tanto de su frecuencia como de su amplitud. Estas soluciones corresponden a estados estacionarios del vórtice caracterizados como una órbita circular, aunque en el caso de existir tres equilibrios uno de ellos es inestable. El modelo matemático ha sido estudiado y analizado para reproducir los resultados obtenidos en [4] y adicionalmente se ha implementado el caso de la energía potencial del vórtice con el término no lineal distinto de [4] (con una constante β negativa). Los resultados obtenidos bajo esta suposición muestran una cierta disparidad respecto al caso de β positiva y reflejan la existencia de comportamientos muy diferentes. En la dinámica del vórtice bajo la suposición de β negativa también surgen bifurcaciones de pliegue (fold-over) en la trayectoria del núcleo del vórtice: siempre se obtienen tres equilibrios a bajas frecuencias y aparecen a veces dos intervalos con tres equilibrios en lugar de uno solo, a diferencia del caso de β positiva. El comportamiento del núcleo del vórtice sigue un proceso de histéresis en ambos casos, cuando se dan ciertas condiciones. ABSTRACT. This work is a study about the dynamic behaviour of magnetic vortex in its application in spin-torque oscillators. Starting from mathematical models developed in [4] on this subject, a further analysis has been performed using MATLAB to simulate the behaviour of vortex-based spin-torque oscillators as different relevant parameters vary. The main aim of the study is to check if the obtained results can explain the observed physical behaviour, and an important observation is that fully nonlinear effects are taken into consideration. The mathematical analysis of the system that describes the orbit of the vortex core under the influence of an external oscillatory magnetic field shows that there are up to three possible special solutions depending on the dot geometry and the magnetic driving field value, both on its amplitude as well as its frequency. These solutions correspond to a stable circular orbit of the vortex core, but when the system has three solutions one of them is unstable. This mathematical model has been analyzed and studied to reproduce the results obtained in [4] and, additionally, the effect of a negative value of the nonlinear part of the vortex potential (constant β). Results derived from this assumption exhibit certain differences with respect to the case of a positive constant (β) and reflect the existence of very different patterns. Vortex dynamics under the supposition of a negative β also yield fold-over bifurcations in the trajectory of the vortex core: there are always three solutions at low frequencies and there may be two different intervals with three solutions as opposed to the case of positive β. The vortex core follows a hysteresis process in both cases, when certain conditions are met.

Modelización de procesos de soldadura (GMAW y GTAW) de la aleación de magnesio AZ31B. Estudio y análisis del comportamiento de este tipo de uniones soldadas frente a la corrosión en medios salinos

El magnesio y sus aleaciones representan un interesante campo de investigación dentro de la ingeniería de materiales debido a los retos que plantean tanto su conformabilidad como durabilidad. Las características físicas y mecánicas del magnesio y sus aleaciones los convierten en materiales de gran interés desde el punto de vista industrial al tratarse de uno de los materiales más abundantes y ligeros en un mundo en el que día a día los recursos y materias primas son más escasos por lo que el acceso a materiales abundantes y ligeros que permitan economizar el uso de energía cobrará mayor importancia en el futuro. En la coyuntura actual es por tanto necesario revisar procesos y procedimientos, investigando y tratando de ampliar desde este punto de vista si es posible mejorar los procedimientos de fabricación de los materiales disponibles actualmente o el desarrollo de nuevos, mejores que los anteriores, que permitan ayudar a la sostenibilidad del planeta. El magnesio, pese a ser un material muy abundante y relativamente barato, presenta una serie de inconvenientes que limitan de manera muy seria su aplicación industrial, su alta reactividad en presencia de oxígeno y su mal comportamiento frente a la corrosión así como limitaciones en su conformabilidad han estado limitando su uso y aplicaciones, los investigaciones dentro del campo de la metalurgia física de este material y el desarrollo de nuevas aleaciones han permitido su empleo en múltiples aplicaciones dentro de la industria aeroespacial, militar, automovilística, electrónica, deportiva y médica. La motivación para esta tesis doctoral ha sido tratar de aportar más luz sobre el comportamiento de una de las aleaciones comerciales base magnesio más empleadas, la AZ31B, tratando de modelizar como le afectan los procesos de soldadura y estudiando desde un punto de vista experimental como se ve modificada su microestructura, su comportamiento mecánico y su resistencia frente a la corrosión. Aunque en un principio se pensó en el empleo de métodos electroquímicos para el estudio de la corrosión de estos materiales, rápidamente se decidió prescindir de su uso dada la dificultad observada tanto durante los trabajos de investigación de esta Tesis como los encontrados por otros investigadores. Mediante microdurezas se han caracterizado mecánicamente las soldaduras de aleación de magnesio tipo AZ31 en función de diferentes materiales de aporte, observándose que el empleo de las aleaciones con mayor contenido de aluminio y zinc no contribuye a una mejora significativa de las propiedades mecánicas. Se han podido establecer correlaciones entre los modelos de simulación desarrollados y las microestructuras resultantes de los procesos reales de soldadura que permiten definir a priori que estructuras se van a obtener. De igual forma ha sido posible completar un estudio micrográfico y químico completo de las diferentes fases y microconstituyentes originados durante los procesos de soldadura, gracias a estos resultados se ha propuesto como hipótesis una explicación que justifica el comportamiento frente a la corrosión de estas aleaciones una vez soldadas. Los ensayos de corrosión realizados han permitido determinar correlaciones matemáticas que indican las velocidades de corrosión esperables de este tipo de aleaciones. Desde el punto de vista del diseño, los resultados obtenidos en este trabajo permitirán a otros investigadores y diseñadores tomar decisiones a la hora de decidir qué materiales de aporte emplear junto con las implicaciones que conllevan desde el punto de vista metalúrgico, mecánico o corrosivo las diferentes alternativas. Por último indicar que gracias al trabajo desarrollado se han definido modelos matemáticos para predecir el comportamiento frente a la corrosión de estas aleaciones, se han determinado las posibles causas y mecanismos por las que se gobierna la corrosión en la soldadura de chapas de aleación AZ31B y los motivos por los que se debe considerar el empleo de un material de aporte u otro. Los modelos de simulación desarrollados también han ayudado a comprender mejor la microestructura resultante de los procesos de soldadura y se han determinado que fases y microconstituyentes están presentes en las soldaduras de estas aleaciones. ABSTRACT Magnesium and its alloys represent and interesting research field in the material science due to the challenges of their fabrication and durability. The physical and mechanical properties of magnesium and its alloys make them a very interesting materials from and industrial point of view being one of the most abundant and lightest materials in a world in which day by day the lacking of resources and raw materials is more important, the use of light materials which allow to save energy will become more important in a near future. So that it is necessary to review processes and procedures, investigating and trying to improve current fabrication procedures and developing new ones, better than the former ones, in order to help with the sustainability of the planet. Although magnesium is a very common and relatively cheap material, it shows some inconveniences which limit in a major way their industrial application; its high reactivity in presence of oxygen, its poor corrosion resistance and some manufacturing problems had been limiting their use and applications, metallurgical investigations about this material and the development of new alloys have allowed its use in multiple applications in the aerospacial, military, automobile, electronics, sports and medical industry. The motivation for this thesis has been trying to clarify the behavior of one most used commercial base magnesium alloys, the AZ31, trying to modeling how its affected by thermal cycles of the welding process and studying from an experimental point of view how its microstructure is modified and how these modifications affect its mechanical behavior and corrosion resistance. Although at the beginning of this works it was though about the using of electrochemical techniques to evaluate the corrosion of these materials, rapidly it was decided not to use them because of the difficulty observed by during this research and by other investigators. The results obtained in this thesis have allowed to characterize mechanically AZ31 magnesium welding alloys considering different filler metals, according to this study using filler metals with a high content of aluminum and zinc does not represent an important improve It has been possible to establish correlations between simulation models and the resultant microstructures of the real melting processes originated during welding processes which allow to predict the structures which will be obtained after the welding. In addition to that it is possible to complete a complete micrographic and chemical analysis of the different phases and microconstituents created during welding, due to these results and hypothesis to explain the corrosion behavior of these welded alloys. Corrosion tests carried out have allowed defining mathematical correlations to predict corrosion rates of this kind of alloys. From a designing point of view, the results obtained in this work will let other investigators and designers to make decisions taking into account which implications have the different options from a metallurgical, mechanic and corrosive point of view. Finally we would like to indicate that thanks to this work it has been possible to define mathematical models to predict the corrosion behavior, the causes and the mechanism of this corrosion in the AZ31 welding sheets have been also determined and the reasons for using of one filler metal or another, the developed simulation models have also help to get a better understanding of the result microstructure determining the phases and the microconstituents present in the welding of this alloys.

Co-simulation Methodologies for Hybrid and Electric Vehicle Dynamics

In recent decades, full electric and hybrid electric vehicles have emerged as an alternative to conventional cars due to a range of factors, including environmental and economic aspects. These vehicles are the result of considerable efforts to seek ways of reducing the use of fossil fuel for vehicle propulsion. Sophisticated technologies such as hybrid and electric powertrains require careful study and optimization. Mathematical models play a key role at this point. Currently, many advanced mathematical analysis tools, as well as computer applications have been built for vehicle simulation purposes. Given the great interest of hybrid and electric powertrains, along with the increasing importance of reliable computer-based models, the author decided to integrate both aspects in the research purpose of this work. Furthermore, this is one of the first final degree projects held at the ETSII (Higher Technical School of Industrial Engineers) that covers the study of hybrid and electric propulsion systems. The present project is based on MBS3D 2.0, a specialized software for the dynamic simulation of multibody systems developed at the UPM Institute of Automobile Research (INSIA). Automobiles are a clear example of complex multibody systems, which are present in nearly every field of engineering. The work presented here benefits from the availability of MBS3D software. This program has proven to be a very efficient tool, with a highly developed underlying mathematical formulation. On this basis, the focus of this project is the extension of MBS3D features in order to be able to perform dynamic simulations of hybrid and electric vehicle models. This requires the joint simulation of the mechanical model of the vehicle, together with the model of the hybrid or electric powertrain. These sub-models belong to completely different physical domains. In fact the powertrain consists of energy storage systems, electrical machines and power electronics, connected to purely mechanical components (wheels, suspension, transmission, clutch…). The challenge today is to create a global vehicle model that is valid for computer simulation. Therefore, the main goal of this project is to apply co-simulation methodologies to a comprehensive model of an electric vehicle, where sub-models from different areas of engineering are coupled. The created electric vehicle (EV) model consists of a separately excited DC electric motor, a Li-ion battery pack, a DC/DC chopper converter and a multibody vehicle model. Co-simulation techniques allow car designers to simulate complex vehicle architectures and behaviors, which are usually difficult to implement in a real environment due to safety and/or economic reasons. In addition, multi-domain computational models help to detect the effects of different driving patterns and parameters and improve the models in a fast and effective way. Automotive designers can greatly benefit from a multidisciplinary approach of new hybrid and electric vehicles. In this case, the global electric vehicle model includes an electrical subsystem and a mechanical subsystem. The electrical subsystem consists of three basic components: electric motor, battery pack and power converter. A modular representation is used for building the dynamic model of the vehicle drivetrain. This means that every component of the drivetrain (submodule) is modeled separately and has its own general dynamic model, with clearly defined inputs and outputs. Then, all the particular submodules are assembled according to the drivetrain configuration and, in this way, the power flow across the components is completely determined. Dynamic models of electrical components are often based on equivalent circuits, where Kirchhoff’s voltage and current laws are applied to draw the algebraic and differential equations. Here, Randles circuit is used for dynamic modeling of the battery and the electric motor is modeled through the analysis of the equivalent circuit of a separately excited DC motor, where the power converter is included. The mechanical subsystem is defined by MBS3D equations. These equations consider the position, velocity and acceleration of all the bodies comprising the vehicle multibody system. MBS3D 2.0 is entirely written in MATLAB and the structure of the program has been thoroughly studied and understood by the author. MBS3D software is adapted according to the requirements of the applied co-simulation method. Some of the core functions are modified, such as integrator and graphics, and several auxiliary functions are added in order to compute the mathematical model of the electrical components. By coupling and co-simulating both subsystems, it is possible to evaluate the dynamic interaction among all the components of the drivetrain. ‘Tight-coupling’ method is used to cosimulate the sub-models. This approach integrates all subsystems simultaneously and the results of the integration are exchanged by function-call. This means that the integration is done jointly for the mechanical and the electrical subsystem, under a single integrator and then, the speed of integration is determined by the slower subsystem. Simulations are then used to show the performance of the developed EV model. However, this project focuses more on the validation of the computational and mathematical tool for electric and hybrid vehicle simulation. For this purpose, a detailed study and comparison of different integrators within the MATLAB environment is done. Consequently, the main efforts are directed towards the implementation of co-simulation techniques in MBS3D software. In this regard, it is not intended to create an extremely precise EV model in terms of real vehicle performance, although an acceptable level of accuracy is achieved. The gap between the EV model and the real system is filled, in a way, by introducing the gas and brake pedals input, which reflects the actual driver behavior. This input is included directly in the differential equations of the model, and determines the amount of current provided to the electric motor. For a separately excited DC motor, the rotor current is proportional to the traction torque delivered to the car wheels. Therefore, as it occurs in the case of real vehicle models, the propulsion torque in the mathematical model is controlled through acceleration and brake pedal commands. The designed transmission system also includes a reduction gear that adapts the torque coming for the motor drive and transfers it. The main contribution of this project is, therefore, the implementation of a new calculation path for the wheel torques, based on performance characteristics and outputs of the electric powertrain model. Originally, the wheel traction and braking torques were input to MBS3D through a vector directly computed by the user in a MATLAB script. Now, they are calculated as a function of the motor current which, in turn, depends on the current provided by the battery pack across the DC/DC chopper converter. The motor and battery currents and voltages are the solutions of the electrical ODE (Ordinary Differential Equation) system coupled to the multibody system. Simultaneously, the outputs of MBS3D model are the position, velocity and acceleration of the vehicle at all times. The motor shaft speed is computed from the output vehicle speed considering the wheel radius, the gear reduction ratio and the transmission efficiency. This motor shaft speed, somehow available from MBS3D model, is then introduced in the differential equations corresponding to the electrical subsystem. In this way, MBS3D and the electrical powertrain model are interconnected and both subsystems exchange values resulting as expected with tight-coupling approach.When programming mathematical models of complex systems, code optimization is a key step in the process. A way to improve the overall performance of the integration, making use of C/C++ as an alternative programming language, is described and implemented. Although this entails a higher computational burden, it leads to important advantages regarding cosimulation speed and stability. In order to do this, it is necessary to integrate MATLAB with another integrated development environment (IDE), where C/C++ code can be generated and executed. In this project, C/C++ files are programmed in Microsoft Visual Studio and the interface between both IDEs is created by building C/C++ MEX file functions. These programs contain functions or subroutines that can be dynamically linked and executed from MATLAB. This process achieves reductions in simulation time up to two orders of magnitude. The tests performed with different integrators, also reveal the stiff character of the differential equations corresponding to the electrical subsystem, and allow the improvement of the cosimulation process. When varying the parameters of the integration and/or the initial conditions of the problem, the solutions of the system of equations show better dynamic response and stability, depending on the integrator used. Several integrators, with variable and non-variable step-size, and for stiff and non-stiff problems are applied to the coupled ODE system. Then, the results are analyzed, compared and discussed. From all the above, the project can be divided into four main parts: 1. Creation of the equation-based electric vehicle model; 2. Programming, simulation and adjustment of the electric vehicle model; 3. Application of co-simulation methodologies to MBS3D and the electric powertrain subsystem; and 4. Code optimization and study of different integrators. Additionally, in order to deeply understand the context of the project, the first chapters include an introduction to basic vehicle dynamics, current classification of hybrid and electric vehicles and an explanation of the involved technologies such as brake energy regeneration, electric and non-electric propulsion systems for EVs and HEVs (hybrid electric vehicles) and their control strategies. Later, the problem of dynamic modeling of hybrid and electric vehicles is discussed. The integrated development environment and the simulation tool are also briefly described. The core chapters include an explanation of the major co-simulation methodologies and how they have been programmed and applied to the electric powertrain model together with the multibody system dynamic model. Finally, the last chapters summarize the main results and conclusions of the project and propose further research topics. In conclusion, co-simulation methodologies are applicable within the integrated development environments MATLAB and Visual Studio, and the simulation tool MBS3D 2.0, where equation-based models of multidisciplinary subsystems, consisting of mechanical and electrical components, are coupled and integrated in a very efficient way.

The oculomotor system as a model for the study of neuronal interaction processes

Identified neurons that control eye movements offer an excellent experimental target for the study of Information coding and neuronal interaction processes wíthin the central nervous system. Here are presented some prelimínary results of the motoneuron behaviour during steady eye fíxation, obtained by regressíon and analysis of variance techniques. A flexible information system intended for the systematic acquisitíon and analysis of simultaneous records of neuronal activity and both eyes angular position in a great amount of cells, oriented to the defínition of mathematical models, is also briefly outlíned.

Different population dynamics of human T cell lymphotropic virus type II in intravenous drug users compared with endemically infected tribes

The phylogeny of human T cell lymphotropic virus type II (HTLV-II) was investigated by using strains isolated from Amerindian and Pygmy tribes, in which the virus is maintained primarily through mother-to-child transmission via breast-feeding, and strains from intravenous drug users (IDUs), in which spread is mainly blood-borne via needle sharing. Molecular clock analysis showed that HTLV-II has two different evolutionary rates with the molecular clock for the virus in IDUs ticking 150–350 times faster than the one in endemically infected tribes: 2.7 × 10−4 compared with 1.71/7.31 × 10−7 nucleotide substitutions per site per year in the long terminal repeat region. This dramatic acceleration of the evolutionary rate seems to be related with the mode of transmission. Mathematical models showed the correlation of these two molecular clocks with an endemic spread of HTLV-II in infected tribes compared with the epidemic spread in IDUs. We also noted a sharp increase in the population size of the virus among IDUs during the last decades probably caused by the worldwide increase in intravenous drug use.

Models of immune memory: On the role of cross-reactive stimulation, competition, and homeostasis in maintaining immune memory

There has been much debate on the contribution of processes such as the persistence of antigens, cross-reactive stimulation, homeostasis, competition between different lineages of lymphocytes, and the rate of cell turnover on the duration of immune memory and the maintenance of the immune repertoire. We use simple mathematical models to investigate the contributions of these various processes to the longevity of immune memory (defined as the rate of decline of the population of antigen-specific memory cells). The models we develop incorporate a large repertoire of immune cells, each lineage having distinct antigenic specificities, and describe the dynamics of the individual lineages and total population of cells. Our results suggest that, if homeostatic control regulates the total population of memory cells, then, for a wide range of parameters, immune memory will be long-lived in the absence of persistent antigen (T1/2 > 1 year). We also show that the longevity of memory in this situation will be insensitive to the relative rates of cross-reactive stimulation, the rate of turnover of immune cells, and the functional form of the term for the maintenance of homeostasis.

Antigenic variation and the within-host dynamics of parasites.

Many parasites exhibit antigenic variation within their hosts. We use mathematical models to investigate the dynamical interaction between an antigenically varying parasite and the host's immune system. The models incorporate antigenic variation in the parasite population and the generation of immune responses directed against (i) antigens specific to individual parasite variants and (ii) antigens common to all the parasite variants. Analysis of the models allows us to evaluate the relative importance of variant-specific and cross-reactive immune responses in controlling the parasite. Early in the course of infection within the host, when parasite diversity is below a defined threshold value (the value is determined by the biological properties of the parasite and of the host's immune response), the variant-specific immune responses are predominant. Later, when the parasite diversity is high, the cross-reactive immune response is largely responsible for controlling the parasitemia. It is argued that increasing antigenic diversity leads to a switch from variant-specific to cross-reactive immune responses. These simple models mimic various features of observed infections recorded in the experimental literature, including an initial peak in parasitemia, a long and variable duration of infection with fluctuating parasitemia that ends with either the clearance of the parasite or persistent infection.