On the free-boundary for an elliptic inverse nonlocal problem arising in the nuclear fusion. A numerical approach

We consider a mathematical model related to the stationary regime of a plasma magnetically confined in a Stellarator device in the nuclear fusion. The mathematical problem may be reduced to an nonlinear elliptic inverse nonlocal two dimensional free{boundary problem. The nonlinear terms involving the unknown functions of the problem and its rearrangement. Our main goal is to determinate the existence and the estimate on the location and size of region where the solution is nonnegative almost everywhere (corresponding to the plasma region in the physical model)

Different approaches to grounding resistance. Calculation for thin wire structures at low frequency energization

The present paper deals with the calculation of grounding resistance of an electrode composed of thin wires, that we consider here as perfect electric conductors (PEC) e.g. with null internal resistance, when buried in a soil of uniform resistivity. The potential profile at the ground surface is also calculated when the electrode is energized with low frequency current. The classic treatment by using leakage currents, called Charge Simulated Method (CSM), is compared with that using a set of steady currents along the axis of the wires, here called the Longitudinal Currents Method (LCM), to solve the Maxwell equations. The method of moments is applied to obtain a numerical approximation of the solution by using rectangular basis functions. Both methods are applied to two types of electrodes and the results are also compared with those obtained using a thirth approach, the Average Potential Method (APM), later described in the text. From the analysis performed, we can estimate a value of the error in the determination of grounding resistance as a function of the number of segments in which the electrodes are divided.

Inverse kinematics of a humanoid robot with non-spherical hip: a hybrid algorithm approach

This paper describes an approach to solve the inverse kinematics problem of humanoid robots whose construction shows a small but non negligible offset at the hip which prevents any purely analytical solution to be developed. Knowing that a purely numerical solution is not feasible due to variable efficiency problems, the proposed one first neglects the offset presence in order to obtain an approximate “solution” by means of an analytical algorithm based on screw theory, and then uses it as the initial condition of a numerical refining procedure based on the Levenberg‐Marquardt algorithm. In this way, few iterations are needed for any specified attitude, making it possible to implement the algorithm for real‐time applications. As a way to show the algorithm’s implementation, one case of study is considered throughout the paper, represented by the SILO2 humanoid robot.

Contribución al análisis cinemático y dinámico de manipuladores paralelos

El documento presentado contiene una aproximación a algunos de los diversos problemas actuales existentes en el campo de la robótica paralela. Primeramente se hace una propuesta para el cálculo de los parámetros estructurales de los robots paralelos, mediante el desarrollo de una metodología que combina las herramientas del estudio de mecanismos con el álgebra lineal; en una segunda sección se propone la solución del problema geométrico directo a partir de la definición de ecuaciones de restricción y su respectiva solución usando métodos numéricos, así como la solución para el problema geométrico inverso; en la tercera parte se aborda el problema dinámico tanto directo como inverso y su solución a partir de una metodología basada en el método de Kane o de trabajos virtuales. Para las propuestas metodológicas expuestas se han desarrollado ejemplos de aplicación tanto teóricos como prácticos (simulaciones y pruebas físicas), donde se demuestra su alcance y desempeño, mediante su utilización en múltiples configuraciones para manipuladores paralelos, entre los que se destacan la plataforma Stewart Gough, y el 3-RRR. Todo con el objetivo de extender su aplicación en futuros trabajos de investigación en el área. ABSTRACT The document presented below provides an approach to some of the many current problems existing in the field of parallel robotics. First is maked a proposal for calculating the structural parameters of the parallel robots, through developing a methodology that combines tools to study mechanisms with the linear algebra; a second section contains a direct geometrical problem solution from the definition of constraint equations and their respective solution using numerical methods, as well as the solution to the inverse geometric problem; in the third part, both, direct and inverse dynamic problem and its solution based on methodology Kane or the method of virtual work are propossed. For each of the exposed methodological proposals they were developed examples of both theoretical and practical application (simulations and physical tests), where its scope and performance is demonstrated by its use in multiple configurations for parallel manipulators, among which stand out the platform Stewart Gough, and 3-RRR. All with the goal of extending its application in future research in the area.

Mixing Snapshots and Fast Time Integration of PDEs

A local proper orthogonal decomposition (POD) plus Galerkin projection method was recently developed to accelerate time dependent numerical solvers of PDEs. This method is based on the combined use of a numerical code (NC) and a Galerkin sys- tem (GS) in a sequence of interspersed time intervals, INC and IGS, respectively. POD is performed on some sets of snapshots calculated by the numerical solver in the INC inter- vals. The governing equations are Galerkin projected onto the most energetic POD modes and the resulting GS is time integrated in the next IGS interval. The major computa- tional e®ort is associated with the snapshots calculation in the ¯rst INC interval, where the POD manifold needs to be completely constructed (it is only updated in subsequent INC intervals, which can thus be quite small). As the POD manifold depends only weakly on the particular values of the parameters of the problem, a suitable library can be con- structed adapting the snapshots calculated in other runs to drastically reduce the size of the ¯rst INC interval and thus the involved computational cost. The strategy is success- fully tested in (i) the one-dimensional complex Ginzburg-Landau equation, including the case in which it exhibits transient chaos, and (ii) the two-dimensional unsteady lid-driven cavity problem

Inverse kinematics of a 6 DoF human upper limb using ANFIS and ANN for anticipatory actuation in ADL-based physical Neurorehabilitation

Objective: This research is focused in the creation and validation of a solution to the inverse kinematics problem for a 6 degrees of freedom human upper limb. This system is intended to work within a realtime dysfunctional motion prediction system that allows anticipatory actuation in physical Neurorehabilitation under the assisted-as-needed paradigm. For this purpose, a multilayer perceptron-based and an ANFIS-based solution to the inverse kinematics problem are evaluated. Materials and methods: Both the multilayer perceptron-based and the ANFIS-based inverse kinematics methods have been trained with three-dimensional Cartesian positions corresponding to the end-effector of healthy human upper limbs that execute two different activities of the daily life: "serving water from a jar" and "picking up a bottle". Validation of the proposed methodologies has been performed by a 10 fold cross-validation procedure. Results: Once trained, the systems are able to map 3D positions of the end-effector to the corresponding healthy biomechanical configurations. A high mean correlation coefficient and a low root mean squared error have been found for both the multilayer perceptron and ANFIS-based methods. Conclusions: The obtained results indicate that both systems effectively solve the inverse kinematics problem, but, due to its low computational load, crucial in real-time applications, along with its high performance, a multilayer perceptron-based solution, consisting in 3 input neurons, 1 hidden layer with 3 neurons and 6 output neurons has been considered the most appropriated for the target application.

An inverse optimization strategy to determine single crystal mechanical behavior from polycrystal tests: Application to AZ31 Mg alloy

An inverse optimization strategy was developed to determine the single crystal properties from experimental results of the mechanical behavior of polycrystals. The polycrystal behavior was obtained by means of the finite element simulation of a representative volume element of the microstructure in which the dominant slip and twinning systems were included in the constitutive equation of each grain. The inverse problem was solved by means of the Levenberg-Marquardt method, which provided an excellent fit to the experimental results. The iterative optimization process followed a hierarchical scheme in which simple representative volume elements were initially used, followed by more realistic ones to reach the final optimum solution, leading to important reductions in computer time. The new strategy was applied to identify the initial and saturation critical resolved shear stresses and the hardening modulus of the active slip systems and extension twinning in a textured AZ31 Mg alloy. The results were in general agreement with the data in the literature but also showed some differences. They were partially explained because of the higher accuracy of the new optimization strategy but it was also shown that the number of independent experimental stress-strain curves used as input is critical to reach an accurate solution to the inverse optimization problem. It was concluded that at least three independent stress-strain curves are necessary to determine the single crystal behavior from polycrystal tests in the case of highly textured Mg alloys.

Analysis of Meshfree Methods for Lagrangian Fluid-Structure Interaction

In this dissertation a new numerical method for solving Fluid-Structure Interaction (FSI) problems in a Lagrangian framework is developed, where solids of different constitutive laws can suffer very large deformations and fluids are considered to be newtonian and incompressible. For that, we first introduce a meshless discretization based on local maximum-entropy interpolants. This allows to discretize a spatial domain with no need of tessellation, avoiding the mesh limitations. Later, the Stokes flow problem is studied. The Galerkin meshless method based on a max-ent scheme for this problem suffers from instabilities, and therefore stabilization techniques are discussed and analyzed. An unconditionally stable method is finally formulated based on a Douglas-Wang stabilization. Then, a Langrangian expression for fluid mechanics is derived. This allows us to establish a common framework for fluid and solid domains, such that interaction can be naturally accounted. The resulting equations are also in the need of stabilization, what is corrected with an analogous technique as for the Stokes problem. The fully Lagrangian framework for fluid/solid interaction is completed with simple point-to-point and point-to-surface contact algorithms. The method is finally validated, and some numerical examples show the potential scope of applications.

Sobre la solución del problema de la radiación electromagnética

En la presente Tesis Doctoral se propone una posible solución al problema de la radiación electromagnética. Este es un problema científico estudiado desde hace mucho tiempo y que hasta el presente no se ha podido resolver de manera exacta. Aquí se redefine nuevamente el problema y se plantea el mismo partiendo un modelo de radiador formado por dos placas planas circulares paralelas, que por sus reducidas dimensiones eléctricas, se considerará para el estudio como un dipolo simétrico corto. Se obtienen las soluciones de las ecuaciones de Maxwell para la radiación electromagnética emitida por dicho sistema y se propone lo que sería el principio de formación de las ondas electromagnéticas. Con ayuda de este se ofrece una posible interpretación de las soluciones obtenidas. Luego a dicho sistema se le realizan correcciones en sus dimensiones a fin de ajustar las soluciones obtenidas al caso real de un dipolo simétrico de media onda. Además se analiza de una manera sencilla la naturaleza material del campo electromagnético y su estructura, lo que hace posible proponer una manera original de comprender las ecuaciones de Maxwell. Finalmente se propone un modelo sencillo mediante el cual se puede demostrar la Ley de Planck para la distribución espectral de la energía electromagnética radiada por los cuerpos calientes, pero desde las soluciones de las ecuaciones de Maxwell para la radiación de las antenas. Con esto podría ser aceptado que la solución del problema de la radiación electromagnética es única tanto para las antenas como para los cuerpos calientes en la naturaleza. ABSTRACT In this PhD thesis a possible solution to the problem of electromagnetic radiation is proposed. This is a scientific problem which has been studied for a long time and to this day has still not exactly been solved. The problem is redefined here again and it is stated using a model formed by two flat parallel circular plates as the radiator system, which due to its reduced electrical dimensions will be considered for the study as a short symmetrical dipole. The solutions of the Maxwell equations for emitted electromagnetic radiation are obtained by said system, and that which could be the principle formation of electromagnetic waves is proposed. Using this principle, a possible interpretation of the solutions obtained is presented. Then the system will perform corrections in size to adjust the solutions obtained to the real case of a symmetrical half-wave dipole. The material nature of the electromagnetic field and its structure is also analyzed in a simple way, making it possible to present an original way to understand the Maxwell equations. Finally, a simple model which can be used to prove Planck's law of spectral distribution of the electromagnetic energy radiated by hot bodies is proposed, but parting from the solutions of the Maxwell equations for the antennas radiation. With this it could be accepted that the solution to the problem of electromagnetic radiation is unique to both antennas and hot bodies on nature.

Reflexión y refracción de la luz en medios anisótropos

En este proyecto se pretende estudiar el comportamiento de la luz al atravesar medios de diversos materiales, tanto isótropos como anisótropos uniáxicos. Para ello se requiere realizar un estudio previo de las condiciones de contorno aplicables a las ecuaciones de Maxwell en la interfase de dos medios que pueden ser isótropos o anisótropos. En el caso de dos materiales isótropos, la solución del problema son los conocidos coeficientes de Fresnel de reflexión y transmisión. En este trabajo se pretende generalizar el estudio al caso del paso de la luz desde un medio isótropo a otro anisótropo uniáxico (con su eje óptico en orientación arbitraria) y viceversa y al caso de dos materiales anisótropos uniáxicos con ejes ópticos en orientaciones arbitrarias. Es de especial interés el caso de un mismo material uniáxico en el que las dos partes tienen el eje óptico con distinta orientación. Una vez planteadas las condiciones de contorno específicas en cada caso, se obtendrá un conjunto de ecuaciones algebraicas cuya resolución permitirá obtener los coeficientes de reflexión y transmisión buscados. Para plantear el sistema de ecuaciones adecuado, será necesario tener una descripción de las características ópticas de los materiales empleados, la orientación de los ejes ópticos en cada caso, y los posibles ángulos de incidencia. Se realizará un tratamiento matricial de modo que el paquete MatLab permite su inversión de manera inmediata. Se desarrollará una interfaz sencilla, realizada con MatLab, que permita al usuario introducir sin dificultad los datos correspondientes a los materiales de los medios incidente y transmitido, la orientación en espacial del o de los ejes ópticos, de la longitud de onda de trabajo y del ángulo de incidencia del haz de luz, con los que la aplicación realizará los cálculos. Los coeficientes de reflexión y refracción obtenidos serán representados gráficamente en función del ángulo de incidencia. Así mismo se representarán los ángulos transmitidos y reflejados en función del de incidencia. Todo ello de esta forma, que resulte sencilla la interpretación de los datos por parte del usuario. ABSTRACT. The reason for this project is to study the behavior of light when light crosses different media of different materials, isotropic materials and uniaxial anisotropic materials. For this, a previous study is necessary where the boundary conditions apply to Maxwell equations at the interface between two media which can be isotropic and anisotropic. If both materials are isotropic, the Fresnel ccoefficients of reflection and refraction are used to solve the problem. The aim of this work is to generalize a study when light crosses from an isotropic media to a uniaxial anisotropic media, where its axis have arbitrary directions, and vicecersa. The system consisting of two materials with axis in arbitrary directions are also being studied. Once the specific boundary conditions are known in each case, a set of algebraic equations are obtained whose solution allows obtaining the reflection coefficients and refraction coefficients. It is necessary to have a description of the optical characteristics of the materials used; of the directions axis in each case and the possible angle of incidence. A matrix is proposed for later treatment in Matlab that allows the immediate inversion. A simple interface will de developed, manufactured with Matlab, that allows the user to enter data easily corresponding to the incident media and transmission media of the different materials, the special axis directions, the wavelength and the angle of incidence of the light beam. This data is used by the application to perform the necessary calculations to solve the problem. When reflection coefficients and refraction coefficients are obtained, the application draws the graphics in function of the angle of incidence. Also transmitted and reflected angles depending on the incidence are represented. This is to perform a data representation which is a simple interpretation of the user data.

Shower approach in the simulation of ion scattering from solids

An efficient approach for the simulation of ion scattering from solids is proposed. For every encountered atom, we take multiple samples of its thermal displacements among those which result in scattering with high probability to finally reach the detector. As a result, the detector is illuminated by intensive “showers,” where each event of detection must be weighted according to the actual probability of the atom displacement. The computational cost of such simulation is orders of magnitude lower than in the direct approach, and a comprehensive analysis of multiple and plural scattering effects becomes possible. We use this method for two purposes. First, the accuracy of the approximate approaches, developed mainly for ion-beam structural analysis, is verified. Second, the possibility to reproduce a wide class of experimental conditions is used to analyze some basic features of ion-solid collisions: the role of double violent collisions in low-energy ion scattering; the origin of the “surface peak” in scattering from amorphous samples; the low-energy tail in the energy spectra of scattered medium-energy ions due to plural scattering; and the degradation of blocking patterns in two-dimensional angular distributions with increasing depth of scattering. As an example of simulation for ions of MeV energies, we verify the time reversibility for channeling and blocking of 1-MeV protons in a W crystal. The possibilities of analysis that our approach offers may be very useful for various applications, in particular, for structural analysis with atomic resolution.

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.

Modelización de contactos termoelastohidrodinámicos

La mejora en la eficiencia energética y la reducción de la tasa de fallos en los contactos lubricados son aspectos que resultan de gran interés en numerosos sectores de la industria, y plantean en estos momentos nuevas dificultades operativas y retos para un futuro próximo. Los avances tecnológicos han incrementado las exigencias técnicas que se requieren a los aceites para cumplir su función al extender sus variables operativas a un mayor espectro de aplicaciones, tanto de condiciones de funcionamiento como a la gran variedad de nuevos materiales constitutivos de los engranajes en los que se tiene que utilizar. Por ello, actualmente se está incentivado el desarrollo de nuevos procedimientos que permitan comprender el comportamiento de este tipo de contactos lubricados, con el fin de lograr mejoras técnicas en su diseño y la correcta selección del aceite. En esta Tesis Doctoral se presenta una metodología de cálculo numérico que permite simular el comportamiento de contactos elastohidrodinámicos (EHD) puntuales, como puede ser el caso de un rodamiento. La resolución de este problema presenta diversas complejidades matemáticas y exige el desarrollo de un elaborado procedimiento de cálculo basado en técnicas multinivel. Para hacer del procedimiento una herramienta válida en un gran número de condiciones de funcionamiento y tipos de lubricantes, se ha tenido en cuenta en el cálculo la posible aparición de comportamientos no-Newtonianos del lubricante y fenómenos de generación y disipación de calor, provocados por el movimiento relativo del fluido y las superficies en contacto. Para la validación del procedimiento, se han contrastado los resultados numéricos obtenidos con nuestro método, con los resultados numéricos y experimentales publicados por otros autores y con valores experimentales propios medidos en un equipo de ensayo de contacto puntual tipo MTM. El desarrollo de este programa ha dotado a la División de Ingeniería de Máquinas de una herramienta que ha permitido, y sobre todo va permitir, evaluar la importancia de cada uno de los parámetros reológicos en los diferentes problemas que va a tener que abordar, evaluación que hasta el momento se realizaba con métodos aproximados que describen la fenomenología con mucha menos precisión. A la hora de emplear nuestro procedimiento numérico para simular situaciones reales, nos hemos encontrado con el obstáculo de que es muy complicado encontrar, en la bibliografía y bases de datos, los parámetros que caracterizan el comportamiento reológico del lubricante en las condiciones de presión, temperatura y grado de cizallamiento habituales en las que se trabaja en este tipo de contactos, y las pocas medidas que existen para estas condiciones de funcionamiento son poco fiables. Por ello como complemento al objetivo principal de esta Tesis hemos desarrollado una metodología para caracterizar los lubricantes en estas condiciones extremas. Dicha metodología está basada en la buena descripción que hace nuestro programa del coeficiente de fricción, lo que nos ha permitido obtener los parámetros reológicos del III lubricante a partir de las medidas experimentales del coeficiente de fricción generado en un equipo MTM lubricado con el lubricante que se desea caracterizar. Madrid, Octubre de 2012 IV Improving energy efficiency and reducing the failure rate in lubricated contacts are issues that are of great interest in many sectors of industry, and right now posed operational difficulties and new challenges for the near future. Technological advances have increased the technical demands required to oils to fulfil its role by extending its operational variables to a wider range of applications, both operating conditions and to the wide variety of new materials which constitute the gear in which must be used. For this reason, has being encouraged currently to develop new procedures to understand the behaviour of this type of lubricated contacts, in order to achieve improvements in design techniques and the correct oil selection. In this Thesis we present a numerical methodology to simulate the puntual elastohydrodynamic contact behaviour (EHD), such as a roller bearing. The resolution of this problem presents various mathematical complexities and requires the development of an elaborate calculation procedure based on multilevel techniques. To make the procedure a valid tool in a wide range of operating conditions and types of lubricants, has been taken into account in calculating the possible occurrence of non-Newtonian behaviour of the lubricant and phenomena of generation and dissipation of heat, caused by the fluid relative motion and contacting surfaces. For the validation of the method, we have compared the numerical results obtained with our method with numerical and experimental results published by other authors also with own experimental values measured on point-contact test equipment MTM. The development of this program has provided the Machines Engineering Division of a tool that has allowed, and especially will allow to evaluate the importance of each of the rheological parameters on the various problems that will have to be addressed, evaluation performed hitherto described methods that phenomenology approximated with much less accuracy. When using our numerical procedure to simulate real situations we have encountered the obstacle that is very difficult to find, in the literature and database, parameters characterizing the rheological behaviour of the lubricant in the usual conditions of pressure, temperature and shear rate in which you work in this type of contact, and the few measures that exist for these operating conditions are unreliable. Thus in addition to the main objective of this thesis, we have developed a methodology to characterize the lubricants in these extreme conditions. This methodology is based on the good description, which makes our program, of the coefficient of friction, that allowed us to obtain the lubricant rheological parameters from experimental measurements of the friction coefficient generated on lubricated MTM equipment with the lubricant to be characterized.

Contribution to the design of waveguide-fed compound-slot arrays by means of equivalent circuit modelling

Los arrays de ranuras son sistemas de antennas conocidos desde los años 40, principalmente destinados a formar parte de sistemas rádar de navíos de combate y grandes estaciones terrenas donde el tamaño y el peso no eran altamente restrictivos. Con el paso de los años y debido sobre todo a importantes avances en materiales y métodos de fabricación, el rango de aplicaciones de este tipo de sistemas radiantes creció en gran medida. Desde nuevas tecnologías biomédicas, sistemas anticolisión en automóviles y navegación en aviones, enlaces de comunicaciones de alta tasa binaria y corta distancia e incluso sistemas embarcados en satélites para la transmisión de señal de televisión. Dentro de esta familia de antennas, existen dos grupos que destacan por ser los más utilizados: las antennas de placas paralelas con las ranuras distribuidas de forma circular o espiral y las agrupaciones de arrays lineales construidos sobre guia de onda. Continuando con las tareas de investigación desarrolladas durante los últimos años en el Instituto de Tecnología de Tokyo y en el Grupo de Radiación de la Universidad Politécnica de Madrid, la totalidad de esta tesis se centra en este último grupo, aunque como se verá se separa en gran medida de las técnicas de diseño y metodologías convencionales. Los arrays de ranuras rectas y paralelas al eje de la guía rectangular que las alimenta son, sin ninguna duda, los modelos más empleados debido a la fiabilidad que presentan a altas frecuencias, su capacidad para gestionar grandes cantidades de potencia y la sencillez de su diseño y fabricación. Sin embargo, también presentan desventajas como estrecho ancho de banda en pérdidas de retorno y rápida degradación del diagrama de radiación con la frecuencia. Éstas son debidas a la naturaleza resonante de sus elementos radiantes: al perder la resonancia, el sistema global se desajusta y sus prestaciones degeneran. En arrays bidimensionales de slots rectos, el campo eléctrico queda polarizado sobre el plano transversal a las ranuras, correspondiéndose con el plano de altos lóbulos secundarios. Esta tesis tiene como objetivo el desarrollo de un método sistemático de diseño de arrays de ranuras inclinadas y desplazadas del centro (en lo sucesivo “ranuras compuestas”), definido en 1971 como uno de los desafíos a superar dentro del mundo del diseño de antennas. La técnica empleada se basa en el Método de los Momentos, la Teoría de Circuitos y la Teoría de Conexión Aleatoria de Matrices de Dispersión. Al tratarse de un método circuital, la primera parte de la tesis se corresponde con el estudio de la aplicabilidad de las redes equivalentes fundamentales, su capacidad para recrear fenómenos físicos de la ranura, las limitaciones y ventajas que presentan para caracterizar las diferentes configuraciones de slot compuesto. Se profundiza en las diferencias entre las redes en T y en ! y se condiciona la selección de una u otra dependiendo del tipo de elemento radiante. Una vez seleccionado el tipo de red a emplear en el diseño del sistema, se ha desarrollado un algoritmo de cascadeo progresivo desde el puerto alimentador hacia el cortocircuito que termina el modelo. Este algoritmo es independiente del número de elementos, la frecuencia central de funcionamiento, del ángulo de inclinación de las ranuras y de la red equivalente seleccionada (en T o en !). Se basa en definir el diseño del array como un Problema de Satisfacción de Condiciones (en inglés, Constraint Satisfaction Problem) que se resuelve por un método de Búsqueda en Retroceso (Backtracking algorithm). Como resultado devuelve un circuito equivalente del array completo adaptado a su entrada y cuyos elementos consumen una potencia acorde a una distribución de amplitud dada para el array. En toda agrupación de antennas, el acoplo mutuo entre elementos a través del campo radiado representa uno de los principales problemas para el ingeniero y sus efectos perjudican a las prestaciones globales del sistema, tanto en adaptación como en capacidad de radiación. El empleo de circuito equivalente se descartó por la dificultad que suponía la caracterización de estos efectos y su inclusión en la etapa de diseño. En esta tesis doctoral el acoplo también se ha modelado como una red equivalente cuyos elementos son transformadores ideales y admitancias, conectada al conjunto de redes equivalentes que representa el array. Al comparar los resultados estimados en términos de pérdidas de retorno y radiación con aquellos obtenidos a partir de programas comerciales populares como CST Microwave Studio se confirma la validez del método aquí propuesto, el primer método de diseño sistemático de arrays de ranuras compuestos alimentados por guía de onda rectangular. Al tratarse de ranuras no resonantes, el ancho de banda en pérdidas de retorno es mucho mas amplio que el que presentan arrays de slots rectos. Para arrays bidimensionales, el ángulo de inclinación puede ajustarse de manera que el campo quede polarizado en los planos de bajos lóbulos secundarios. Además de simulaciones se han diseñado, construido y medido dos prototipos centrados en la frecuencia de 12GHz, de seis y diez elementos. Las medidas de pérdidas de retorno y diagrama de radiación revelan excelentes resultados, certificando la bondad del método genuino Method of Moments - Forward Matching Procedure desarrollado a lo largo de esta tésis. Abstract The slot antenna arrays are well known systems from the decade of 40s, mainly intended to be part of radar systems of large warships and terrestrial stations where size and weight were not highly restrictive. Over the years, mainly due to significant advances in materials and manufacturing methods, the range of applications of this type of radiating systems grew significantly. From new biomedical technologies, collision avoidance systems in cars and aircraft navigation, short communication links with high bit transfer rate and even embedded systems in satellites for television broadcast. Within this family of antennas, two groups stand out as being the most frequent in the literature: parallel plate antennas with slots placed in a circular or spiral distribution and clusters of waveguide linear arrays. To continue the vast research work carried out during the last decades in the Tokyo Institute of Technology and in the Radiation Group at the Universidad Politécnica de Madrid, this thesis focuses on the latter group, although it represents a technique that drastically breaks with traditional design methodologies. The arrays of slots straight and parallel to the axis of the feeding rectangular waveguide are without a doubt the most used models because of the reliability that they present at high frequencies, its ability to handle large amounts of power and their simplicity of design and manufacturing. However, there also exist disadvantages as narrow bandwidth in return loss and rapid degradation of the radiation pattern with frequency. These are due to the resonant nature of radiating elements: away from the resonance status, the overall system performance and radiation pattern diminish. For two-dimensional arrays of straight slots, the electric field is polarized transverse to the radiators, corresponding to the plane of high side-lobe level. This thesis aims to develop a systematic method of designing arrays of angled and displaced slots (hereinafter "compound slots"), defined in 1971 as one of the challenges to overcome in the world of antenna design. The used technique is based on the Method of Moments, Circuit Theory and the Theory of Scattering Matrices Connection. Being a circuitry-based method, the first part of this dissertation corresponds to the study of the applicability of the basic equivalent networks, their ability to recreate the slot physical phenomena, their limitations and advantages presented to characterize different compound slot configurations. It delves into the differences of T and ! and determines the selection of the most suitable one depending on the type of radiating element. Once the type of network to be used in the system design is selected, a progressive algorithm called Forward Matching Procedure has been developed to connect the proper equivalent networks from the feeder port to shorted ending. This algorithm is independent of the number of elements, the central operating frequency, the angle of inclination of the slots and selected equivalent network (T or ! networks). It is based on the definition of the array design as a Constraint Satisfaction Problem, solved by means of a Backtracking Algorithm. As a result, the method returns an equivalent circuit of the whole array which is matched at its input port and whose elements consume a power according to a given amplitude distribution for the array. In any group of antennas, the mutual coupling between elements through the radiated field represents one of the biggest problems that the engineer faces and its effects are detrimental to the overall performance of the system, both in radiation capabilities and return loss. The employment of an equivalent circuit for the array design was discarded by some authors because of the difficulty involved in the characterization of the coupling effects and their inclusion in the design stage. In this thesis the coupling has also been modeled as an equivalent network whose elements are ideal transformers and admittances connected to the set of equivalent networks that represent the antennas of the array. By comparing the estimated results in terms of return loss and radiation with those obtained from popular commercial software as CST Microwave Studio, the validity of the proposed method is fully confirmed, representing the first method of systematic design of compound-slot arrays fed by rectangular waveguide. Since these slots do not work under the resonant status, the bandwidth in return loss is much wider than the longitudinal-slot arrays. For the case of two-dimensional arrays, the angle of inclination can be adjusted so that the field is polarized at the low side-lobe level plane. Besides the performed full-wave simulations two prototypes of six and ten elements for the X-band have been designed, built and measured, revealing excellent results and agreement with the expected results. These facts certify that the genuine technique Method of Moments - Matching Forward Procedure developed along this thesis is valid and trustable.

Simulación en tiempo real de vehículos industriales con modelos multicuerpo de gran complejidad

El objetivo de esta Tesis ha sido la consecución de simulaciones en tiempo real de vehículos industriales modelizados como sistemas multicuerpo complejos formados por sólidos rígidos. Para el desarrollo de un programa de simulación deben considerarse cuatro aspectos fundamentales: la modelización del sistema multicuerpo (tipos de coordenadas, pares ideales o impuestos mediante fuerzas), la formulación a utilizar para plantear las ecuaciones diferenciales del movimiento (coordenadas dependientes o independientes, métodos globales o topológicos, forma de imponer las ecuaciones de restricción), el método de integración numérica para resolver estas ecuaciones en el tiempo (integradores explícitos o implícitos) y finalmente los detalles de la implementación realizada (lenguaje de programación, librerías matemáticas, técnicas de paralelización). Estas cuatro etapas están interrelacionadas entre sí y todas han formado parte de este trabajo. Desde la generación de modelos de una furgoneta y de camión con semirremolque, el uso de tres formulaciones dinámicas diferentes, la integración de las ecuaciones diferenciales del movimiento mediante métodos explícitos e implícitos, hasta el uso de funciones BLAS, de técnicas de matrices sparse y la introducción de paralelización para utilizar los distintos núcleos del procesador. El trabajo presentado en esta Tesis ha sido organizado en 8 capítulos, dedicándose el primero de ellos a la Introducción. En el Capítulo 2 se presentan dos formulaciones semirrecursivas diferentes, de las cuales la primera está basada en una doble transformación de velocidades, obteniéndose las ecuaciones diferenciales del movimiento en función de las aceleraciones relativas independientes. La integración numérica de estas ecuaciones se ha realizado con el método de Runge-Kutta explícito de cuarto orden. La segunda formulación está basada en coordenadas relativas dependientes, imponiendo las restricciones por medio de penalizadores en posición y corrigiendo las velocidades y aceleraciones mediante métodos de proyección. En este segundo caso la integración de las ecuaciones del movimiento se ha llevado a cabo mediante el integrador implícito HHT (Hilber, Hughes and Taylor), perteneciente a la familia de integradores estructurales de Newmark. En el Capítulo 3 se introduce la tercera formulación utilizada en esta Tesis. En este caso las uniones entre los sólidos del sistema se ha realizado mediante uniones flexibles, lo que obliga a imponer los pares por medio de fuerzas. Este tipo de uniones impide trabajar con coordenadas relativas, por lo que la posición del sistema y el planteamiento de las ecuaciones del movimiento se ha realizado utilizando coordenadas Cartesianas y parámetros de Euler. En esta formulación global se introducen las restricciones mediante fuerzas (con un planteamiento similar al de los penalizadores) y la estabilización del proceso de integración numérica se realiza también mediante proyecciones de velocidades y aceleraciones. En el Capítulo 4 se presenta una revisión de las principales herramientas y estrategias utilizadas para aumentar la eficiencia de las implementaciones de los distintos algoritmos. En primer lugar se incluye una serie de consideraciones básicas para aumentar la eficiencia numérica de las implementaciones. A continuación se mencionan las principales características de los analizadores de códigos utilizados y también las librerías matemáticas utilizadas para resolver los problemas de álgebra lineal tanto con matrices densas como sparse. Por último se desarrolla con un cierto detalle el tema de la paralelización en los actuales procesadores de varios núcleos, describiendo para ello el patrón empleado y las características más importantes de las dos herramientas propuestas, OpenMP y las TBB de Intel. Hay que señalar que las características de los sistemas multicuerpo problemas de pequeño tamaño, frecuente uso de la recursividad, y repetición intensiva en el tiempo de los cálculos con fuerte dependencia de los resultados anteriores dificultan extraordinariamente el uso de técnicas de paralelización frente a otras áreas de la mecánica computacional, tales como por ejemplo el cálculo por elementos finitos. Basándose en los conceptos mencionados en el Capítulo 4, el Capítulo 5 está dividido en tres secciones, una para cada formulación propuesta en esta Tesis. En cada una de estas secciones se describen los detalles de cómo se han realizado las distintas implementaciones propuestas para cada algoritmo y qué herramientas se han utilizado para ello. En la primera sección se muestra el uso de librerías numéricas para matrices densas y sparse en la formulación topológica semirrecursiva basada en la doble transformación de velocidades. En la segunda se describe la utilización de paralelización mediante OpenMP y TBB en la formulación semirrecursiva con penalizadores y proyecciones. Por último, se describe el uso de técnicas de matrices sparse y paralelización en la formulación global con uniones flexibles y parámetros de Euler. El Capítulo 6 describe los resultados alcanzados mediante las formulaciones e implementaciones descritas previamente. Este capítulo comienza con una descripción de la modelización y topología de los dos vehículos estudiados. El primer modelo es un vehículo de dos ejes del tipo chasis-cabina o furgoneta, perteneciente a la gama de vehículos de carga medianos. El segundo es un vehículo de cinco ejes que responde al modelo de un camión o cabina con semirremolque, perteneciente a la categoría de vehículos industriales pesados. En este capítulo además se realiza un estudio comparativo entre las simulaciones de estos vehículos con cada una de las formulaciones utilizadas y se presentan de modo cuantitativo los efectos de las mejoras alcanzadas con las distintas estrategias propuestas en esta Tesis. Con objeto de extraer conclusiones más fácilmente y para evaluar de un modo más objetivo las mejoras introducidas en la Tesis, todos los resultados de este capítulo se han obtenido con el mismo computador, que era el top de la gama Intel Xeon en 2007, pero que hoy día está ya algo obsoleto. Por último los Capítulos 7 y 8 están dedicados a las conclusiones finales y las futuras líneas de investigación que pueden derivar del trabajo realizado en esta Tesis. Los objetivos de realizar simulaciones en tiempo real de vehículos industriales de gran complejidad han sido alcanzados con varias de las formulaciones e implementaciones desarrolladas. ABSTRACT The objective of this Dissertation has been the achievement of real time simulations of industrial vehicles modeled as complex multibody systems made up by rigid bodies. For the development of a simulation program, four main aspects must be considered: the modeling of the multibody system (types of coordinates, ideal joints or imposed by means of forces), the formulation to be used to set the differential equations of motion (dependent or independent coordinates, global or topological methods, ways to impose constraints equations), the method of numerical integration to solve these equations in time (explicit or implicit integrators) and the details of the implementation carried out (programming language, mathematical libraries, parallelization techniques). These four stages are interrelated and all of them are part of this work. They involve the generation of models for a van and a semitrailer truck, the use of three different dynamic formulations, the integration of differential equations of motion through explicit and implicit methods, the use of BLAS functions and sparse matrix techniques, and the introduction of parallelization to use the different processor cores. The work presented in this Dissertation has been structured in eight chapters, the first of them being the Introduction. In Chapter 2, two different semi-recursive formulations are shown, of which the first one is based on a double velocity transformation, thus getting the differential equations of motion as a function of the independent relative accelerations. The numerical integration of these equations has been made with the Runge-Kutta explicit method of fourth order. The second formulation is based on dependent relative coordinates, imposing the constraints by means of position penalty coefficients and correcting the velocities and accelerations by projection methods. In this second case, the integration of the motion equations has been carried out by means of the HHT implicit integrator (Hilber, Hughes and Taylor), which belongs to the Newmark structural integrators family. In Chapter 3, the third formulation used in this Dissertation is presented. In this case, the joints between the bodies of the system have been considered as flexible joints, with forces used to impose the joint conditions. This kind of union hinders to work with relative coordinates, so the position of the system bodies and the setting of the equations of motion have been carried out using Cartesian coordinates and Euler parameters. In this global formulation, constraints are introduced through forces (with a similar approach to the penalty coefficients) are presented. The stabilization of the numerical integration is carried out also by velocity and accelerations projections. In Chapter 4, a revision of the main computer tools and strategies used to increase the efficiency of the implementations of the algorithms is presented. First of all, some basic considerations to increase the numerical efficiency of the implementations are included. Then the main characteristics of the code’ analyzers used and also the mathematical libraries used to solve linear algebra problems (both with dense and sparse matrices) are mentioned. Finally, the topic of parallelization in current multicore processors is developed thoroughly. For that, the pattern used and the most important characteristics of the tools proposed, OpenMP and Intel TBB, are described. It needs to be highlighted that the characteristics of multibody systems small size problems, frequent recursion use and intensive repetition along the time of the calculation with high dependencies of the previous results complicate extraordinarily the use of parallelization techniques against other computational mechanics areas, as the finite elements computation. Based on the concepts mentioned in Chapter 4, Chapter 5 is divided into three sections, one for each formulation proposed in this Dissertation. In each one of these sections, the details of how these different proposed implementations have been made for each algorithm and which tools have been used are described. In the first section, it is shown the use of numerical libraries for dense and sparse matrices in the semirecursive topological formulation based in the double velocity transformation. In the second one, the use of parallelization by means OpenMP and TBB is depicted in the semi-recursive formulation with penalization and projections. Lastly, the use of sparse matrices and parallelization techniques is described in the global formulation with flexible joints and Euler parameters. Chapter 6 depicts the achieved results through the formulations and implementations previously described. This chapter starts with a description of the modeling and topology of the two vehicles studied. The first model is a two-axle chassis-cabin or van like vehicle, which belongs to the range of medium charge vehicles. The second one is a five-axle vehicle belonging to the truck or cabin semi-trailer model, belonging to the heavy industrial vehicles category. In this chapter, a comparative study is done between the simulations of these vehicles with each one of the formulations used and the improvements achieved are presented in a quantitative way with the different strategies proposed in this Dissertation. With the aim of deducing the conclusions more easily and to evaluate in a more objective way the improvements introduced in the Dissertation, all the results of this chapter have been obtained with the same computer, which was the top one among the Intel Xeon range in 2007, but which is rather obsolete today. Finally, Chapters 7 and 8 are dedicated to the final conclusions and the future research projects that can be derived from the work presented in this Dissertation. The objectives of doing real time simulations in high complex industrial vehicles have been achieved with the formulations and implementations developed.