Design of a Semi-analytical Method to Describe Plasma-Wall Interactions

Electric probes are objects immersed in the plasma with sharp boundaries which collect of emit charged particles. Consequently, the nearby plasma evolves under abrupt imposed and/or naturally emerging conditions. There could be localized currents, different time scales for plasma species evolution, charge separation and absorbing-emitting walls. The traditional numerical schemes based on differences often transform these disparate boundary conditions into computational singularities. This is the case of models using advection-diffusion differential equations with source-sink terms (also called Fokker-Planck equations). These equations are used in both, fluid and kinetic descriptions, to obtain the distribution functions or the density for each plasma species close to the boundaries. We present a resolution method grounded on an integral advancing scheme by using approximate Green's functions, also called short-time propagators. All the integrals, as a path integration process, are numerically calculated, what states a robust grid-free computational integral method, which is unconditionally stable for any time step. Hence, the sharp boundary conditions, as the current emission from a wall, can be treated during the short-time regime providing solutions that works as if they were known for each time step analytically. The form of the propagator (typically a multivariate Gaussian) is not unique and it can be adjusted during the advancing scheme to preserve the conserved quantities of the problem. The effects of the electric or magnetic fields can be incorporated into the iterative algorithm. The method allows smooth transitions of the evolving solutions even when abrupt discontinuities are present. In this work it is proposed a procedure to incorporate, for the very first time, the boundary conditions in the numerical integral scheme. This numerical scheme is applied to model the plasma bulk interaction with a charge-emitting electrode, dealing with fluid diffusion equations combined with Poisson equation self-consistently. It has been checked the stability of this computational method under any number of iterations, even for advancing in time electrons and ions having different time scales. This work establishes the basis to deal in future work with problems related to plasma thrusters or emissive probes in electromagnetic fields.

El Problema de Lambert en el Contexto de Optimización de Trayectorias Espaciales

Esta tesis se basa en el estudio de la trayectoria que pasa por dos puntos en el problema de los dos cuerpos, inicialmente desarrollado por Lambert, del que toma su nombre. En el pasado, el Problema de Lambert se ha utilizado para la determinación de órbitas a partir de observaciones astronómicas de los cuerpos celestes. Actualmente, se utiliza continuamente en determinación de órbitas, misiones planetaria e interplanetarias, encuentro espacial e interceptación, o incluso en corrección de orbitas. Dada su gran importancia, se decide investigar especialmente sobre su solución y las aplicaciones en las misiones espaciales actuales. El campo de investigación abierto, es muy amplio, así que, es necesario determinar unos objetivos específicos realistas, en el contexto de ejecución de una Tesis, pero que sirvan para mostrar con suficiente claridad el potencial de los resultados aportados en este trabajo, e incluso poder extenderlos a otros campos de aplicación. Como resultado de este análisis, el objetivo principal de la Tesis se enfoca en el desarrollo de algoritmos para resolver el Problema de Lambert, que puedan ser aplicados de forma muy eficiente en las misiones reales donde aparece. En todos los desarrollos, se ha considerado especialmente la eficiencia del cálculo computacional necesario en comparación con los métodos existentes en la actualidad, destacando la forma de evitar la pérdida de precisión inherente a este tipo de algoritmos y la posibilidad de aplicar cualquier método iterativo que implique el uso de derivadas de cualquier orden. En busca de estos objetivos, se desarrollan varias soluciones para resolver el Problema de Lambert, todas ellas basadas en la resolución de ecuaciones transcendentes, con las cuales, se alcanzan las siguientes aportaciones principales de este trabajo: • Una forma genérica completamente diferente de obtener las diversas ecuaciones para resolver el Problema de Lambert, mediante desarrollo analítico, desde cero, a partir de las ecuaciones elementales conocidas de las cónicas (geométricas y temporal), proporcionando en todas ellas fórmulas para el cálculo de derivadas de cualquier orden. • Proporcionar una visión unificada de las ecuaciones más relevantes existentes, mostrando la equivalencia con variantes de las ecuaciones aquí desarrolladas. • Deducción de una nueva variante de ecuación, el mayor logro de esta Tesis, que destaca en eficiencia sobre todas las demás (tanto en coste como en precisión). • Estudio de la sensibilidad de la solución ante variación de los datos iniciales, y como aplicar los resultados a casos reales de optimización de trayectorias. • También, a partir de los resultados, es posible deducir muchas propiedades utilizadas en la literatura para simplificar el problema, en particular la propiedad de invariancia, que conduce al Problema Transformado Simplificado. ABSTRACT This thesis is based on the study of the two-body, two-point boundary-value problem, initially developed by Lambert, from who it takes its name. Since the past, Lambert's Problem has been used for orbit determination from astronomical observations of celestial bodies. Currently, it is continuously used in orbit determinations, for planetary and interplanetary missions, space rendezvous, and interception, or even in orbit corrections. Given its great importance, it is decided to investigate their solution and applications in the current space missions. The open research field is very wide, it is necessary to determine specific and realistic objectives in the execution context of a Thesis, but that these serve to show clearly enough the potential of the results provided in this work, and even to extended them to other areas of application. As a result of this analysis, the main aim of the thesis focuses on the development of algorithms to solve the Lambert’s Problem which can be applied very efficiently in real missions where it appears. In all these developments, it has been specially considered the efficiency of the required computational calculation compared to currently existing methods, highlighting how to avoid the loss of precision inherent in such algorithms and the possibility to apply any iterative method involving the use of derivatives of any order. Looking to meet these objectives, a number of solutions to solve the Lambert’s Problem are developed, all based on the resolution of transcendental equations, with which the following main contributions of this work are reached: • A completely different generic way to get the various equations to solve the Lambert’s Problem by analytical development, from scratch, from the known elementary conic equations (geometrics and temporal), by providing, in all cases, the calculation of derivatives of any order. • Provide a unified view of most existing relevant equations, showing the equivalence with variants of the equations developed here. • Deduction of a new variant of equation, the goal of this Thesis, which emphasizes efficiency (both computational cost and accuracy) over all other. • Estudio de la sensibilidad de la solución ante la variación de las condiciones iniciales, mostrando cómo aprovechar los resultados a casos reales de optimización de trayectorias. • Study of the sensitivity of the solution to the variation of the initial data, and how to use the results to real cases of trajectories’ optimization. • Additionally, from results, it is possible to deduce many properties used in literature to simplify the problem, in particular the invariance property, which leads to a simplified transformed problem.

Evaluación de Ulmus pumila L. y Populus spp. como cultivos energéticos en corta rotación

El presente trabajo estudia el empleo del olmo de Siberia (Ulmus Pumila L.) y el chopo (Populus spp.) en corta rotación y alta densidad para la producción de biomasa con fines energéticos. En el área mediterránea las disponibilidades hídricas son limitadas, por lo que la mayoría de los cultivos energéticos utilizados hasta el momento requieren el aporte de agua de riego; por ello resulta fundamental encontrar especies con bajos requerimientos hídricos, analizar la eficiencia en el uso del agua de diferentes materiales genéticos y optimizar la dosis de riego. Las parcelas experimentales se ubicaron en la provincia de Soria. En el ensayo llevado a cabo con olmo de Siberia se ha analizado el efecto en la producción de la disponibilidad de agua mediante el establecimiento de parcelas en secano y con dos dosis de riego (2000 m3 ha-1 año-1 y 4000 m3 ha-1 año-1 aproximadamente); además, al ser una especie poco estudiada hasta el momento, se ha estudiado también el efecto que tiene sobre el rendimiento la densidad de plantación (3.333 plantas ha-1 y 6.666 plantas ha-1), el tipo de suelo (2 calidades diferentes) y el turno de corta (3 y 4 años). En el caso del chopo, se han evaluado cuatro clones (AF-2, I-214, Monviso y Pegaso) establecidos con una densidad de 20.000 plantas por hectárea. Durante el primer ciclo de tres años se aportó el mismo volumen de riego a todas las parcelas, mientras que durante el segundo ciclo se establecieron 8 regímenes hídricos diferentes. Por otra parte, se ha investigado sobre el uso del potencial hídrico de las plantas para evaluar el estrés hídrico de las mismas y se ha estimado la producción de biomasa foliar y el Índice de Área Foliar (LAI) de ambas especies, relacionando los valores obtenidos con la dosis de riego y la producción de biomasa. Los resultados muestran que los suelos inundados reducen la tasa de supervivencia de los olmos durante el periodo de implantación, sin embargo la mortalidad durante los siguientes periodos vegetativos es baja y muestra buena capacidad de rebrote. La productividad (kg ha-1 año-1) obtenida fue mayor con un turno de corta de cuatro años que con turno de tres años. El área basal y la altura fueron variables eficaces para predecir la producción de biomasa del olmo de Siberia, obteniendo una variabilidad explicada de más del 80%. En cuanto a los parámetros que mayor influencia tuvieron sobre el crecimiento, el tipo de suelo resulto ser el más relevante, obteniéndose en un suelo agrícola considerado de buena calidad una producción en condiciones de secano de unos 8.000 kg ha-1 año-1. En condiciones de regadío el rendimiento del olmo de Siberia fue al menos el doble que en secano, pero la diferencia entre las dos dosis de riego estudiadas fue pequeña. La producción de biomasa fue mayor en la densidad de plantación más alta (6.666 plantas ha–1) en las parcelas de regadío, sin embargo no se encontraron diferencias significativas entre las dos densidades en secano. El clon de chopo que presentó un mayor rendimiento durante el primer ciclo fue AF-2, alcanzando los 14.000 kg ha-1 año-1, sin embargo la producción de este clon bajó sustancialmente durante el segundo ciclo debido a su mala capacidad de rebrote, pasando a ser I-214 el clon más productivo llegando también a los 14.000 kg ha-1 año-1. Un aporte adicional de agua proporcionó un incremento de la biomasa recogida, pero a partir de unos 6.500 m3 ha-1 año-1 de agua la producción se mantiene constante. El potencial hídrico foliar ha resultado ser una herramienta útil para conocer el estrés hídrico de las plantas. Los olmos de regadío apenas sufrieron estrés hídrico, mientras que los implantados en condiciones de secano padecieron un acusado estrés durante buena parte del periodo vegetativo, que se acentuó en la parte final del mismo. Los chopos regados con las dosis más altas no sufrieron estrés hídrico o fue muy pequeño, en las dosis intermedias sufrieron un estrés moderado ocasionalmente y únicamente en las dosis más bajas sufrieron puntualmente un estrés severo. El LAI aumenta con la edad de los brotes y oscila entre 2 y 4 m2 m−2 en los chopos y entre 2 y 7 m2 m−2 en los olmos. Se encontró una buena relación entre este índice y la producción de biomasa del olmo de Siberia. En general, puede decirse que el olmo de Siberia podría ser una buena alternativa para producir biomasa leñosa en condiciones de secano, mientras que el chopo podría emplearse en regadío siempre que se haga una buena elección del clon y de la dosis de riego. ABSTRACT This work explores the possibilities of biomass production, for energy purposes, of Siberian elm (Ulmus Pumila L.) and poplar (Populus spp.) in Spain. Irrigation is required for the viable cultivation of many energy crops in Mediterranean areas because of low water availability, for this reason species with low water requirements should be a good alternative for biomass production. Moreover, the optimal amount of irrigation water and the performance of the different genetic material in terms of production and water use efficiency should be studied in order to use water wisely. The experimental plots were established in the province of Soria in Spain. Given the small amount of information available about Siberian elm, besides studying the influence of water availability (rain-fed and two different irrigation doses) on biomass production, two different plantation densities (3,333 plants ha-1 and 6,666 plants ha-1), two different soil type and two cutting cycles (three years and four years) were assessed. In the case of poplar, four clones belonging to different hybrids (I-214, AF2, Pegaso, and Monviso) were included in a high density plantation (20,000 plants ha-1). During the first cycle, the water supplied in all plots was the same, while 8 different watering regimes were used during the second cycle. The suitability of the use of the leaf water potential to assess the water stress situations has also been investigated. Moreover, leaf biomass production and leaf area index (LAI) were estimated in both species in order to analyze the relationship between these parameters, irrigation dose and biomass production. The results shows that flooded soils have an adverse effect on elm survival in the implantation period, but the percentage of mortality is very low during the following vegetative periods and it shows a good ability of regrowth. The annual yield from a four-year cutting cycle was significantly greater than that from the three-year cutting cycle. Basal diameter and height are effective variables for predicting the production of total biomass; equations with R squared higher than 80% were obtained. The analysis of parameters having an influence on elm growth shows that soil type is the most important factor to obtain a good yield. In soils with enough nutrients and higher waterholding capacity, biomass productions of 8,000 kg ha-1 yr-1 were achieved even under rain-fed conditions. In irrigated plots, Siberian elm production was double than the production of biomass under rain-fed conditions; however, small differences were obtained between the 2 different irrigation doses under study. Biomass yield was greater for the highest planting density (6,666 plants ha–1) in irrigated plots, but significant differences were not found between the 2 densities in rain-fed plots. The clone AF-2 showed the highest production (14,000 kg ha-1 yr-1) during the first cycle, however during the second cycle its growth was lower because of a high mortality rate after regrowth and I-214 achieves the greatest production (14,000 kg ha-1 yr-1). An additional water supply provided a greater amount of biomass, but over about 6500 m3 ha-1 yr-1 of water the production is constant. Leaf water potential has been shown to be a useful tool for finding out plant water status. Irrigated elms hardly suffered water stress, while rain-fed elms suffered a pronounced water stress, which was more marked at the end of the vegetative period. Most of poplars did not show water stress; leaf water potentials only showed an important water stress in the plots irrigated with the lowest doses. LAI increases with shoot age and it ranges from 2 to 4 m2 m−2 in poplars and from 2 to 7 m2 m−2 in elms. A good relationship has been found between this index and Siberian elm production. In general, Siberian elm could be a good alternative to produce woody biomass in rainfed plots, while poplar could be used in irrigated plots if a suitable clone and irrigation dose are chosen.

Combining a Similar Coefficient Identification Algorithm with the Boundary Element Method

The boundary element method (BEM) has been applied successfully to many engineering problems during the last decades. Compared with domain type methods like the finite element method (FEM) or the finite difference method (FDM) the BEM can handle problems where the medium extends to infinity much easier than domain type methods as there is no need to develop special boundary conditions (quiet or absorbing boundaries) or infinite elements at the boundaries introduced to limit the domain studied. The determination of the dynamic stiffness of arbitrarily shaped footings is just one of these fields where the BEM has been the method of choice, especially in the 1980s. With the continuous development of computer technology and the available hardware equipment the size of the problems under study grew and, as the flop count for solving the resulting linear system of equations grows with the third power of the number of equations, there was a need for the development of iterative methods with better performance. In [1] the GMRES algorithm was presented which is now widely used for implementations of the collocation BEM. While the FEM results in sparsely populated coefficient matrices, the BEM leads, in general, to fully or densely populated ones, depending on the number of subregions, posing a serious memory problem even for todays computers. If the geometry of the problem permits the surface of the domain to be meshed with equally shaped elements a lot of the resulting coefficients will be calculated and stored repeatedly. The present paper shows how these unnecessary operations can be avoided reducing the calculation time as well as the storage requirement. To this end a similar coefficient identification algorithm (SCIA), has been developed and implemented in a program written in Fortran 90. The vertical dynamic stiffness of a single pile in layered soil has been chosen to test the performance of the implementation. The results obtained with the 3-d model may be compared with those obtained with an axisymmetric formulation which are considered to be the reference values as the mesh quality is much better. The entire 3D model comprises more than 35000 dofs being a soil region with 21168 dofs the biggest single region. Note that the memory necessary to store all coefficients of this single region is about 6.8 GB, an amount which is usually not available with personal computers. In the problem under study the interface zone between the two adjacent soil regions as well as the surface of the top layer may be meshed with equally sized elements. In this case the application of the SCIA leads to an important reduction in memory requirements. The maximum memory used during the calculation has been reduced to 1.2 GB. The application of the SCIA thus permits problems to be solved on personal computers which otherwise would require much more powerful hardware.

Diffusion Gradient Temporal Difference for Cooperative Reinforcement Learning with Linear Function Approximation

We introduce a diffusion-based algorithm in which multiple agents cooperate to predict a common and global statevalue function by sharing local estimates and local gradient information among neighbors. Our algorithm is a fully distributed implementation of the gradient temporal difference with linear function approximation, to make it applicable to multiagent settings. Simulations illustrate the benefit of cooperation in learning, as made possible by the proposed algorithm.

Extension of the characteristic equation to absorption chillers with adiabatic absorbers

Various researchers have developed models of conventional H2O–LiBr absorption machines with the aim of predicting their performance. In this paper, the methodology of characteristic equations developed by Hellmann et al. (1998) is applied. This model is able to represent the capacity of single effect absorption chillers and heat pumps by means of simple algebraic equations. An extended characteristic equation based on a characteristic temperature difference has been obtained, considering the facility features. As a result, it is concluded that for adiabatic absorbers a subcooling temperature must be specified. The effect of evaporator overflow has been characterized. Its influence on cooling capacity has been included in the extended characteristic equation. Taking into account the particular design and operation features, a good agreement between experimental performance data and those obtained through the extended characteristic equation has been achieved at off-design operation. This allows its use for simulation and control purposes.

Design equations for reinforced concrete members strengthened in shear with external FRP reinforcement formulated in an evolutionary multi-objective framework

Methods for predicting the shear capacity of FRP shear strengthened RC beams assume the traditional approach of superimposing the contribution of the FRP reinforcing to the contributions from the reinforcing steel and the concrete. These methods become the basis for most guides for the design of externally bonded FRP systems for strengthening concrete structures. The variations among them come from the way they account for the effect of basic shear design parameters on shear capacity. This paper presents a simple method for defining improved equations to calculate the shear capacity of reinforced concrete beams externally shear strengthened with FRP. For the first time, the equations are obtained in a multiobjective optimization framework solved by using genetic algorithms, resulting from considering simultaneously the experimental results of beams with and without FRP external reinforcement. The performance of the new proposed equations is compared to the predictions with some of the current shear design guidelines for strengthening concrete structures using FRPs. The proposed procedure is also reformulated as a constrained optimization problem to provide more conservative shear predictions.

Development of AMREM: a Maxwell's equations solver with Adaptive Mesh Refinement

El principal objetivo de la presente tesis es el de desarrollar y probar un código capaz de resolver las ecuaciones de Maxwell en el dominio del tiempo con Malla Refinada Adaptativa (AMR por sus siglas en inglés). AMR es una técnica de cálculo basada en dividir el dominio físico del problema en distintas mallas rectangulares paralelas a las direcciones cartesianas. Cada una de las mallas tendrá distinta resolución y aquellas con mayor resolución se sitúan allí dónde las ondas electromagnéticas se propagan o interaccionan con los materiales, es decir, dónde mayor precisión es requerida. Como las ondas van desplazándose por todo el dominio, las mayas deberán seguirlas. El principal problema al utilizar esta metodología se puede encontrar en las fronteras internas, dónde las distintas mallas se unen. Ya que el método más corrientemente utilizado para resolver las ecuaciones de Maxwell es el de las diferencias finitas en el dominio del tiempo (FDTD por sus siglas en inglés) , el trabajo comenzó tratando de adaptar AMR a FDTD. Tras descubrirse que esta interacción resultaba en problemas de inestabilidades en las fronteras internas antes citadas, se decidió cambiar a un método basado en volúmenes finitos en el dominio del tiempo (FVTD por sus siglas en inglés). Este se basa en considerar la forma en ecuaciones de conservación de las ecuaciones de Maxwell y aplicar a su resolución un esquema de Godunov. Se ha probado que es clave para el correcto funcionamiento del código la elección de un limitador de flujo que proteja los extremos de la onda de la disipación típica de los métodos de este tipo. Otro problema clásico a la hora de resolver las ecuaciones de Maxwell es el de tratar con las condiciones de frontera física cuando se simulan dominios no acotados, es decir, dónde las ondas deben salir del sistema sin producir ninguna reflexión. Normalmente la solución es la de disponer una banda absorbente en las fronteras físicas. En AMREM se ha desarrollado un nuevo método basado en los campos característicos que con menor requisito de CPU funcina suficientemente bien incluso en los casos más desfaborables. El código ha sido contrastado con soluciones analíticas de diferentes problemas y también su velocidad ha sido comparada con la de Meep, uno de los programas más conocidos del ámbito. También algunas aplicaciones han sido simuladas con el fin de demostrar el amplio espectro de campos en los que AMREM puede funcionar como una útil herramienta.