En la presente tesis se han estudiado métodos de paso fraccionado para la resolución numérica de la ecuación de Navier-Stokes incompresible mediante el método de los elementos finitos; dicha ecuación rige el movimiento de un fluido incompresible viscoso. Partiendo del análisis del método de proyección clásico, se desarrolla un método para el problema de Stokes (lineal y estacionario) con iguales propiedades en cuanto a discretizacion espacial que aquel, explicando así sus propiedades de estabilización de la presión. Se da también una extensión del nuevo método a la ecuación de Navier-Stokes incompresible estacionaria (no lineal).En la segunda parte de la tesis, se desarrolla un método de paso fraccionado para el problema de evolución que supera un inconveniente del método de proyección relativo a la imposición de las condiciones de contorno.Para todos los métodos desarrollados, se demuestran teoremas de convergencia y estimaciones de error, se proponen implementaciones eficientes y se proporcionan numerosos resultados numéricos.

The study of the electromagnetic phenomena along the last two centuries has brought about outstanding contributions for the human progress. The electromagnetism represents still now, at the beginning of the third millenium, a very important research area. The radiation pattern of particular types of antennas -for example, fractal or microstrip-, the analysis of the effect of the cellular communications on human beings or the detection of buried mines represent specific examples of the wide variety of problems of great interest nowadays. The study of such a variety of problems relies on the application of the Maxwell equations, which rule all the electromagnetic behaviour. Since the analytical solution can only be obtained for very particular cases of canonical forms, to tackle the analysis of an arbitrary problem, one makes use of the numerical methods. The discretization of electromagnetic integral equations by the Method of Moments -MoM- excels as a powerful and reliable tool for analysing bodies composed of locally homogeneous regions -penetrable or perfectly conducting- immerse in a wide and nearly uniform medium -typically the ground or the free-space-. These integral methods result from the surface equivalence theorem, which allows in general two different formulations, the Electric Field Integral Equation (EFIE) and the Magnetic Field Integral Equation (MFIE). For the case of penetrable bodies, the Poggio, Miller, Chang, Harrington and Wu (PMCHW) formulation, that results from the subtraction of the EFIE and MFIE at both sides of the surfaces, can also be employed.The Method of Moments is based on the full expansion of the physical magnitudes, field and current, over the interface surfaces between the regions. In consequence, the solution of the problem is obtained through the inversion of a full-matrix, which, for electrically large problems, requires excessive memory resources and computation time. That is why the MoM is widely considered a brute-force method. The expansion of the magnitudes is carried out through the discretization of the surface; that is, patches spreading over the interface. The first half of this dissertation Thesis tackles the development of the MoM applied to problems with bodies with symmetry of revolution -BoR-. Since in this case the physical magnitudes present an azimuthal periodicity, they can be expressed as a Fourier series. The orthogonality between the different modes enables to obtain separately each azimuthal mode of the solution. It is thus only required to spread the patches along the generating arc of the bodies for each mode, which is very advantageous because the electromagnetic analysis can be carried out indeed for dimensionally large problems. A well-known PeC-EFIE BoR formulation is developed. Accordingly, PeC-MFIE and PMCHW formulations are developed from scratch. Furthermore, it is commented in detail and corrected to some extent the numerical error associated to the fastest-varying part of the PeC-MFIE BoR operator. The BoR-codes are particularly useful in modelling the electromagnetic behaviour of buried mines, which very often show revolution symmetry. The most outstanding contribution of this dissertation Thesis is the study of the appropriate conditions to develop correctly the 3D operators so as to yield accurate results for any structure. Since the discretization implies a break on the continuity properties of the physical magnitudes -field and current- the valid 3D-operators must ensure the physical electromagnetic requirements in the discretized surface. In mathematical terms, these requirements set the rank -field- and domain- -current- spaces, which essentially require the enforcement of the continuity across the edges of either the tangential or the normal component of the expanded magnitudes.For the case of an arbitrary perfectly conducting -PeC- body, it is recommended in this work the use of the divergence-conforming and of the curl-conforming functions respectively in the development of the PeC-EFIE and the PeC-MFIE operators. Low-order sets over triangular facets -RWG and unxRWG- are chosen to develop the PeC-operators. Furthermore, it is reasoned theoretically the inherent misbehaviour of the PeC-MFIE in case the current expansion relies on a divergence-conforming set. A heuristic correction is provided. The better behaviour of PeC-EFIE(RWG) and PeC-MFIE(unxRWG) is confirmed with examples. In view of the results, it is reasoned the suitability of PeC-EFIE(RWG) for the analysis of physical polyhedrons, which makes PeC-MFIE(unxRWG) excel as a more appropriate operator for curved bodies. A procedure for improving the performance of PeC-EFIE(RWG) for coarsely meshed spheres is given.For the case of arbitrary penetrable bodies, the same low-order sets are used to expand the operators EFIE, MFIE and PMCHW. It is shown their compatibility with the combination of the right PeC-operators. In the dielectric case, in addition to the required continuity of the magnitudes across the edges at each region, the fields at both sides of the surface must satisfy the interface continuity, which is ignored in the conducting case -the fields are null inside the conductor-. The impossibility of meeting both continuity requirements at the same time justifies the apparition of inherent and different errors in the dual EFIE-MFIE and in PMCHW. It is thoroughly reasoned and confirmed with examples the suitability of PMCHW for problems with only penetrable regions. It is also shown and discussed in detail the robustness of EFIE-MFIE since its behaviour is appropriate for electrically not too small structures with perfectly conducting or penetrable regions. The analysis of composite structures -very useful to model microstrip antennas- can be considered as a group of disjoint bodies with null distances of separation. For this type of problems, it is recommended in this work the use of EFIE-MFIE since, unlike PMCHW, they can ensure the continuous transition to zero of a distance of separation increasingly small. Finally, efficient methods -IE-MEI and MLFMM- relying on the previous 3D-operators. The development of the PeC 3D IE-MEI cannot maintain the advantages present in the 2D case since the harmonic metrons are not valid in the 3D general case. A new set of metrons that ensures little discontinuity of the current across the edges is presented. It is confirmed with examples how these metrons, so-called quasi-continuous, reduce the number of required coefficients per row for a certain current error. Some examples of penetrable spheres with moderate electrical dimensions analysed under a MLFMM implementation are shown and commented.

A two-dimensional axisymmetric implementation of the smoothed particle hydrodynamics (SPH) technique, called for short AxisSPH, has been described in this thesis, along with a number of basic tests and realistic applications. The main goal of this work was to fill a gap on a topic which has been scarcely addressed in the published literature concerning SPH. Although the application of AxisSPH to the simulation of real problems is restricted to those systems which display the appropriate symmetry there are, however, many interesting examples of physical systems which evolve following the axisymmetric premise. These examples belong to a variety of scientific and technological areas such as, for example, astrophysics, laboratory astrophysics or inertial confinement fusion. Additionally AxisSPH can be also useful in convergence studies of the standard 3D-SPH technique because the higher resolution achieved in 2D can be used to benchmark the three-dimensional codes. The main improvements implemented in AxisSPH with respect existing axisymmetric SPH formulations are summarized as follows: 1) We have derived simple analytical expressions for correction factors which largely improves the calculation of density and velocity in the vicinity of the z-axis. These expressions and their derivatives were given as a function of an adimensional parameter and do not increase the computational load of the scheme. 2) We have obtained the appropriate expression of the fluid Euler equations containing the new correction functions and their derivatives. Far enough from the singular axis, the scheme reduces to the standard formulation discussed by Brookshaw (2003). 3) A novel expression for the heat conduction term, which has to be added to the energy equation was devised and checked. This new term improves the description of the heat flux for those particles located at the axis neighborhoods. 4) Until now axisymmetric SPH hydrocodes handle artificial viscosity using a crude approach because it was treated as a simple restriction of the standard 3D Cartesian viscosity to 2D. Here we propose to calculate the viscous pressure as a combination of two terms, the first one is the (standard) Cartesian part and the second is the axis-converging part of the viscosity respectively. As expected this last term is of special relevance to simulate implosions. 5) We have developed an original method to incorporate gravity into AxisSPH. First the direct ring to ring force was found as a function of the Euclidean distance between the 2D particles. In second place the gravitational force on a given particle was obtained by summing the contributions of all N particles. We have also developed a more efficient scheme to obtain the gravitational force calculating the potential of the ring, instead the force because it involves lesser algebraic operations. The scheme has been checked using a large number of tests cases. These tests range from very specific oriented to check a particular algorithm or a piece of physics, to rather complex ones intended to analyze the behavior of the scheme in potential real applications (ICF, jets, astrophysics). At least in one case, the head on collision of a pair of white dwarfs, the result of the simulations carried out using AxisSPH brings new, unpublished, scientific material. / En esta tesis se ha desarrollado un código, que hemos llamado AxisSPH, en dos dimensiones axisimétrico a partir de la técnica conocida como SPH (“smooothed particle hydrodynamics”). AxisSPH ha sido validado después de realizar una serie de tests básicos y algunas simulaciones de situaciones reales. El objetivo principal de este trabajo ha sido llenar, en parte, el vacío existente al respecto en la literatura sobre SPH. Aunque sólo se puede aplicar AxisSPH en problemas reales que presenten la apropiada simetría, existen muchos ejemplos interesantes de sistemas físicos que presentan la simetría axial demandada. Existen ejemplos en campos de aplicación tanto científica como tecnológica, por ejemplo en astrofísica, en el llamado laboratorio de astrofísica o en fusión por confinamiento inercial (ICF). Otra interesante aplicación de AxisSPH puede ser su utilización en estudios de convergencia con otros códigos 3D-SPH debido a su mayor resolución, al tratarse de un código 2D. Las mejoras implementadas en el código AxisSPH en comparación con otros códigos axisimétricos SPH existentes se pueden resumir en los siguientes puntos: 1) Hemos deducido expresiones analíticas simples para unos factores de corrección que mejoran el cálculo de la densidad y la velocidad en las proximidades del eje z. Dichas expresiones y sus derivadas dependen de un parámetro adimensional que no incrementa mucho el peso computacional del esquema propuesto. 2) Hemos obtenido las expresiones adecuadas de las ecuaciones de Euler que contienen estas nuevas funciones correctoras y sus derivadas. Lejos del eje de singularidad estas ecuaciones se transforman en las de la formulación estándar propuesta por Brookshaw (2003). 3) Una expresión novedosa del término de conducción, que debe de añadirse a la ecuación de la energía, se ha propuesto y validado. Este nuevo término mejora la evolución del flujo de calor de las partículas situadas en las proximidades del eje z. 4) Hasta el momento los códigos hidrodinámicos SPH axisimétricos existentes trabajaban con una aproximación poco elaborada de la viscosidad artificial ya que consistían en una restricción a dos dimensiones de la viscosidad estándar 3D. En este trabajo proponemos el cálculo de la presión debida a la viscosidad como combinación de dos términos, el primero reflejo de la parte cartesiana y la segunda da cuenta de la parte relacionada con la convergencia en el eje. Como era de esperar este último término es de relevante importancia en la simulación de implosiones. 5) Hemos desarrollado un método original para incorporar el cálculo de la gravedad en el código AxisSPH. En primer lugar la fuerza directa de anillo a anillo y en segundo lugar la fuerza de la gravedad que sufre una determinada partícula a partir de la contribución del resto de las N partículas existentes. También hemos desarrollado un esquema más eficiente para calcular la gravedad a partir del cálculo del potencial del anillo en lugar del cálculo directo de la fuerza ya que implica un menor número de operaciones algebraicas. El método ha sido verificado con un gran número de test numéricos. Desde los más específicos orientados a comprobar la validez de un algoritmo particular o la capacidad para simular un fenómeno físico en particular, hasta simulaciones bastante más complejas, con la intención de validar la capacidad de simular aplicaciones potencialmente más reales (ICF, jets, astrofísica). Así, en al menos un caso, en la colisión frontal de dos enanas blancas, los resultados de la simulación utilizando AxisSPH pueden aportar material científico publicable.

La present tesi proposa una metodología per a la simulació probabilística de la fallada de la matriu en materials compòsits reforçats amb fibres de carboni, basant-se en l'anàlisi de la distribució aleatòria de les fibres. En els primers capítols es revisa l'estat de l'art sobre modelització matemàtica de materials aleatoris, càlcul de propietats efectives i criteris de fallada transversal en materials compòsits.El primer pas en la metodologia proposada és la definició de la determinació del tamany mínim d'un Element de Volum Representatiu Estadístic (SRVE) . Aquesta determinació es du a terme analitzant el volum de fibra, les propietats elàstiques efectives, la condició de Hill, els estadístics de les components de tensió i defromació, la funció de densitat de probabilitat i les funcions estadístiques de distància entre fibres de models d'elements de la microestructura, de diferent tamany. Un cop s'ha determinat aquest tamany mínim, es comparen un model periòdic i un model aleatori, per constatar la magnitud de les diferències que s'hi observen.Es defineix, també, una metodologia per a l'anàlisi estadístic de la distribució de la fibra en el compòsit, a partir d'imatges digitals de la secció transversal. Aquest anàlisi s'aplica a quatre materials diferents.Finalment, es proposa un mètode computacional de dues escales per a simular la fallada transversal de làmines unidireccionals, que permet obtenir funcions de densitat de probabilitat per a les variables mecàniques. Es descriuen algunes aplicacions i possibilitats d'aquest mètode i es comparen els resultats obtinguts de la simulació amb valors experimentals. / This thesis proposes a methodology for the probabilistic simulation of the transverse failure of Carbon Fibre Reinforced Polymers (CFRP) by analyzing the random distribution of the fibres within the composite. First chapters are devoted to the State-of-the-art review on the modelization of random materials, the computation of effective properties and the transverse failure of fibre reinforced polymers.The first step in the proposed methodology is the definition of a Statistical Representative Volume Element (SRVE). This SRVE has to satisfy criteria based on the analysis of the volume fraction, the effective properties, the Hill Condition, the statistics of the stress and strain components, the probability density function of the stress and strain components and the inter-fibre distance statistical distributions. Once this SRVE has been achieved, a comparison between a periodic model and a random model is performed to quantitatively analyze the differences between the results they provide.Also a methodology for the statistical analysis of the distribution of the fibre within the composite from digital images of the transverse section. This analysis is performed for four different materials.Finally, a two-scale computational method for the transverse failure of unidirectional laminae is proposed. This method is able to provide probability density functions of the mechanical variables in the composite. Some applications and possibilities of the method are given and the simulation results are compared with experimental tests.

Dynamic optimization methods have become increasingly important over the last years in economics. Within the dynamic optimization techniques employed, optimal control has emerged as the most powerful tool for the theoretical economic analysis. However, there is the need to advance further and take account that many dynamic economic processes are, in addition, dependent on some other parameter different than time. One can think of relaxing the assumption of a representative (homogeneous) agent in macro- and micro-economic applications allowing for heterogeneity among the agents. For instance, the optimal adaptation and diffusion of a new technology over time, may depend on the age of the person that adopted the new technology. Therefore, the economic models must take account of heterogeneity conditions within the dynamic framework.This thesis intends to accomplish two goals. The first goal is to analyze and revise existing environmental policies that focus on defining the optimal management of natural resources over time, by taking account of the heterogeneity of environmental conditions. Thus, the thesis makes a policy orientated contribution in the field of environmental policy by defining the necessary changes to transform an environmental policy based on the assumption of homogeneity into an environmental policy which takes account of heterogeneity. As a result the newly defined environmental policy will be more efficient and likely also politically more acceptable since it is tailored more specifically to the heterogeneous environmental conditions. Additionally to its policy orientated contribution, this thesis aims making a methodological contribution by applying a new optimization technique for solving problems where the control variables depend on two or more arguments --- the so-called two-stage solution approach ---, and by applying a numerical method --- the Escalator Boxcar Train Method --- for solving distributed optimal control problems, i.e., problems where the state variables, in addition to the control variables, depend on two or more arguments.Chapter 2 presents a theoretical framework to determine optimal resource allocation over time for the production of a good by heterogeneous producers, who generate a stock externalit and derives government policies to modify the behavior of competitive producers in order to achieve optimality. Chapter 3 illustrates the method in a more specific context, and integrates the aspects of quality and time, presenting a theoretical model that allows to determine the socially optimal outcome over time and space for the problem of waterlogging in irrigated agricultural production. Chapter 4 of this thesis concentrates on forestry resources and analyses the optimal selective-logging regime of a size-distributed forest.