A general fractional porous medium equation

A general fractional porous medium equation

Fractal Behaviour and Irregularity of Wetting Fronts in Heterogeneous Porous Media

The wetting front is the zone where water invades and advances into an initially dry porous material and it plays a crucial role in solute transport through the unsaturated zone. Water is an essential part of the physiological process of all plants. Through water, necessary minerals are moved from the roots to the parts of the plants that require them. Water moves chemicals from one part of the plant to another. It is also required for photosynthesis, for metabolism and for transpiration. The leaching of chemicals by wetting fronts is influenced by two major factors, namely: the irregularity of the fronts and heterogeneity in the distribution of chemicals, both of which have been described by using fractal techniques. Soil structure can significantly modify infiltration rates and flow pathways in soils. Relations between features of soil structure and features of infiltration could be elucidated from the velocities and the structure of wetting fronts. When rainwater falls onto soil, it doesn?t just pool on surfaces. Water ?or another fluid- acts differently on porous surfaces. If the surface is permeable (porous) it seeps down through layers of soil, filling that layer to capacity. Once that layer is filled, it moves down into the next layer. In sandy soil, water moves quickly, while it moves much slower through clay soil. The movement of water through soil layers is called the the wetting front. Our research concerns the motion of a liquid into an initially dry porous medium. Our work presents a theoretical framework for studying the physical interplay between a stationary wetting front of fractal dimension D with different porous materials. The aim was to model the mass geometry interplay by using the fractal dimension D of a stationary wetting front. The plane corresponding to the image is divided in several squares (the minimum correspond to the pixel size) of size length ". We acknowledge the help of Prof. M. García Velarde and the facilities offered by the Pluri-Disciplinary Institute of the Complutense University of Madrid. We also acknowledge the help of European Community under project Multi-scale complex fluid flows and interfacial phenomena (PITN-GA-2008-214919). Thanks are also due to ERCOFTAC (PELNoT, SIG 14)

Permanent deformation estimates of dynamic equipment foundations: Application to a gas turbine in granular soils

Permanent displacements of a gas turbine founded on a fine, poorly graded, and medium density sand are studied. The amplitudes and modes of vibration are computed using Barkan´s formulation, and the “High-Cycle Accumulation” (HCA) model is employed to account for accumulated deformations due to the high number of cycles. The methodology is simple: it can be easily incorporated into standard mathematical software, and HCA model parameters can be estimated based on granulometry and index properties. Special attention is devoted to ‘transient’ situations at equipment´s start-up, during which a range of frequencies – including frequencies that could be similar to the natural frequencies of the ground – is traversed. Results show that such transient situations could be more restrictive than stationary situations corresponding to normal operation. Therefore, checking the stationary situation only might not be enough, and studying the influence of transient situations on computed permanent displacements is needed to produce a proper foundation design

Classical solutions for a logarithmic fractional diffusion equation

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion ?tu + (?)1/2 log(1 + u) = 0, posed for x ? R, with nonnegative initial data in some function space of LlogL type. The solutions are shown to become bounded and C? smooth in (x, t) for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.

Caracterización de la cerámica para su empleo como material de enfriamiento evaporativo en la envolvente de los edificios = Ceramic characterization towards its application as evaporative cooling material in building envelopes

El presente trabajo de investigación determina las características de la cerámica que más eficientemente se comporta a evaporación y a enfriamiento. Con el objeto de ser empleado como material integrado en la envolvente de los edificios para reducir su carga de refrigeración. La cerámica es un buen material para ser empleado para la refrigeración por evaporación. Es un sólido poroso inerte que, tras ser sometido a cocción a temperaturas por encima de los 900ºC, resulta uno de los materiales que mejor se comportan como contenedor de agua en su red capilar para, posteriormente, ir liberándola por evaporación al mismo tiempo que se enfría su superficie. La metodología general de investigación, se divide en tres etapas: Búsqueda y análisis del estado de la técnica y de la investigación. Estudio teórico de la eficacia del enfriamiento evaporativo como estrategia de enfriamiento pasivo en la arquitectura. Etapa experimental, desarrollada en tres fases: una primera de definición de los parámetros determinantes del Enfriamiento Evaporativo en piezas cerámicas, una segunda de selección cerámica y diseño de ensayos experimentales y una tercera de caracterización de la cerámica bajo criterios de evaporación y de enfriamiento. El recorrido por el estado de la cuestión ha identificado las aplicaciones tecnológicas y las investigaciones científicas que emplean el Enfriamiento Evaporativo con piezas cerámicas como técnica de enfriamiento. Como resultado se ha obtenido una tabla de clasificación de sistemas de enfriamiento evaporativo y se ha constatado que el conjunto de las aplicaciones están centradas en el diseño de piezas o sistemas pero que, sin embargo, no existe una definición de las características de la cerámica para su empleo como material de enfriamiento por evaporación. El estudio teórico de la eficacia del empleo del enfriamiento evaporativo como estrategia de enfriamiento pasivo en la arquitectura se ha realizado mediante cálculos de porcentaje de ampliación de horas en confort con empleo de técnicas de enfriamiento evaporativo directo e indirecto (EED y EEI). Como resultado se obtienen unos mapas para el ámbito español de potencial de aplicación del EED y EEI. Los resultados permiten afirmar que mediante EE se puede llegar a confort en prácticamente la totalidad de las horas de los días más cálidos del año en muchas localidades. La metodología experimental se ha desarrollado en tres fases. En la fase inicial, se han definido los parámetros determinantes del enfriamiento evaporativo en un medio cerámico mediante ensayos experimentales de capacidad de evaporación y de caracterización. Se realizaron un total de 12 ensayos. Se determinó que el material cerámico tiene una gran influencia en la capacidad de evaporación y enfriamiento en las piezas cerámicas, apoyando la hipótesis inicial y la necesidad de caracterizar el material. La primera fase empírica se centró en la selección cerámica y el diseño de los ensayos experimentales de comportamiento hídrico. Se seleccionaron muestras de 5 tipos de cerámica. Se realizaron 4 tipos de ensayos de caracterización y 6 tipos de ensayos experimentales de comportamiento hídrico (total 123 muestras ensayadas). Los resultados obtenidos son de dos tipos, por un lado, se determinó cuál es el tipo de cerámica que más eficientemente se comporta a EE y, por otro, se rediseñaron los ensayos de la última fase experimental. Para la segunda fase experimental se seleccionaron cerámicas de fabricación manual abarcando el mayor número de localidades del ámbito español. Se realizaron ensayos de caracterización de 7 tipos y ensayos de comportamiento hídrico de 5 tipos (total 197 muestras ensayadas). Los resultados de caracterización han permitido aportar unos rangos de las características de la cerámica que más eficientemente se comporta en los ensayos de comportamiento hídrico. Al final de la investigación se ha caracterizado el material cerámico aportando características acerca de su porosidad, capacidad de absorción, color, rugosidad y mineralogía. Así como datos de referencia de su comportamiento hídrico. Además se ha desarrollado una metodología de ensayo específica que permite evaluar la capacidad de enfriamiento eficiente de una pieza cerámica. ABSTRACT The purpose of this research is to determine the characteristics of ceramic materials having the most efficient performance in terms of evaporation and cooling, so that they can be integrated in building envelopes to reduce cooling loads. Ceramics are suitable materials for cooling through passive evaporation. After being fired at temperatures over 900 °C (1,652 °F), the capillary network of this inert porous medium turns to be excellent to retain water, which is progressively liberated by evaporation while the material surface gets colder. Research methodology has involved the following steps: Search and analysis on the state of the art in technology and research. Theoretical study on the efficiency of evaporation as passive cooling strategies in buildings. Experimental stage developed in three phases, namely: definition of parameters determining evaporative cooling in ceramic elements; ceramic selection and design of experimental tests; characterization of ceramic materials under evaporation and cooling criteria. Search and analysis on the state of the art in this field have been useful to identify technology applications and scientific research where ceramics are employed for evaporative cooling. The resulting table shows that applications are wholly focused on the design of pieces and systems. Nonetheless, there is lack of definition of material characteristics in this scope. The theoretical study on efficiency of the passive strategy applied to buildings has been realized by calculation of the percentage increase in comfort hours through direct/indirect evaporative cooling techniques (DEC/IEC). The mapping of their potential application in Spain clearly shows that comfort conditions can be reached in almost all the hours of the hottest days in many towns. In the initial phase of the experimental stage, parameters determining evaporative cooling in ceramic media have been defined. For this purpose, characterization tests and evaporation and cooling rates experiments have been carried out; the number of samples tested amounted to 12. It has been concluded that material characteristics have great influence on these rates, which supports the initial hypothesis and the need for their characterization. The first empirical phase has focused on ceramic selection and design of water behaviour experimental methods. The samples covered five different kinds of ceramic materials. Four different characterization tests and six different water behaviour experiments were carried out; the number of samples tested amounted to 123. The experimental testing procedures served to determine the most efficient types of ceramic materials in terms of evaporative cooling efficiency and, at the same time, made it necessary to change the original designed experimental test for the last phase. In the second phase, a number of varied hand-made ceramic tiles have been selected. Seven different characterization tests and five different water behaviour tests were carried out; the number of samples amounted to 197. The results of characterization served to establish a range of features in ceramic materials according to their efficiency in water behaviour experiments. Finally, ceramic materials have been characterized according to porosity, water absorption, colour, surface roughness and mineralogy. Also, reference data regarding water behaviour have been included. Moreover, an innovative and specific experimental test to evaluate cooling efficiency of ceramic tiles has been developed.

Theoretical study of band alignment in nano-porous ZnO interacting with substituted Phthalocyanines

The aim of this work is the theoretical study of the band alignment between the two components of a hybrid organic-inorganic solar-cell. The working organic molecules are metal tetra-sulphonated phthalocyanines (M-Pc) and the inorganic material is nano-porous ZnO growth in the 001 direction. The theoretical calculations are being made using the density functional theory (DFT) using a GGA functional with the SIESTA code, which projects electron wave functions and density onto a real space grid and uses as basis set a linear combination of numerical, finite-range localized atomic orbitals. We also used the DFT+U method included in the code that allows a semi-empirical inclusion of electronic correlations in the description of electronic spectra for systems such as zinc oxide.

Dynamic analysis of payloads and structures with intermediate modal density

La necesidad de desarrollar técnicas para predecir la respuesta vibroacústica de estructuras espaciales lia ido ganando importancia en los últimos años. Las técnicas numéricas existentes en la actualidad son capaces de predecir de forma fiable el comportamiento vibroacústico de sistemas con altas o bajas densidades modales. Sin embargo, ambos rangos no siempre solapan lo que hace que sea necesario el desarrollo de métodos específicos para este rango, conocido como densidad modal media. Es en este rango, conocido también como media frecuencia, donde se centra la presente Tesis doctoral, debido a la carencia de métodos específicos para el cálculo de la respuesta vibroacústica. Para las estructuras estudiadas en este trabajo, los mencionados rangos de baja y alta densidad modal se corresponden, en general, con los rangos de baja y alta frecuencia, respectivamente. Los métodos numéricos que permiten obtener la respuesta vibroacústica para estos rangos de frecuencia están bien especificados. Para el rango de baja frecuencia se emplean técnicas deterministas, como el método de los Elementos Finitos, mientras que, para el rango de alta frecuencia las técnicas estadísticas son más utilizadas, como el Análisis Estadístico de la Energía. En el rango de medias frecuencias ninguno de estos métodos numéricos puede ser usado con suficiente precisión y, como consecuencia -a falta de propuestas más específicas- se han desarrollado métodos híbridos que combinan el uso de métodos de baja y alta frecuencia, intentando que cada uno supla las deficiencias del otro en este rango medio. Este trabajo propone dos soluciones diferentes para resolver el problema de la media frecuencia. El primero de ellos, denominado SHFL (del inglés Subsystem based High Frequency Limit procedure), propone un procedimiento multihíbrido en el cuál cada subestructura del sistema completo se modela empleando una técnica numérica diferente, dependiendo del rango de frecuencias de estudio. Con este propósito se introduce el concepto de límite de alta frecuencia de una subestructura, que marca el límite a partir del cual dicha subestructura tiene una densidad modal lo suficientemente alta como para ser modelada utilizando Análisis Estadístico de la Energía. Si la frecuencia de análisis es menor que el límite de alta frecuencia de la subestructura, ésta se modela utilizando Elementos Finitos. Mediante este método, el rango de media frecuencia se puede definir de una forma precisa, estando comprendido entre el menor y el mayor de los límites de alta frecuencia de las subestructuras que componen el sistema completo. Los resultados obtenidos mediante la aplicación de este método evidencian una mejora en la continuidad de la respuesta vibroacústica, mostrando una transición suave entre los rangos de baja y alta frecuencia. El segundo método propuesto se denomina HS-CMS (del inglés Hybrid Substructuring method based on Component Mode Synthesis). Este método se basa en la clasificación de la base modal de las subestructuras en conjuntos de modos globales (que afectan a todo o a varias partes del sistema) o locales (que afectan a una única subestructura), utilizando un método de Síntesis Modal de Componentes. De este modo es posible situar espacialmente los modos del sistema completo y estudiar el comportamiento del mismo desde el punto de vista de las subestructuras. De nuevo se emplea el concepto de límite de alta frecuencia de una subestructura para realizar la clasificación global/local de los modos en la misma. Mediante dicha clasificación se derivan las ecuaciones globales del movimiento, gobernadas por los modos globales, y en las que la influencia del conjunto de modos locales se introduce mediante modificaciones en las mismas (en su matriz dinámica de rigidez y en el vector de fuerzas). Las ecuaciones locales se resuelven empleando Análisis Estadístico de Energías. Sin embargo, este último será un modelo híbrido, en el cual se introduce la potencia adicional aportada por la presencia de los modos globales. El método ha sido probado para el cálculo de la respuesta de estructuras sometidas tanto a cargas estructurales como acústicas. Ambos métodos han sido probados inicialmente en estructuras sencillas para establecer las bases e hipótesis de aplicación. Posteriormente, se han aplicado a estructuras espaciales, como satélites y reflectores de antenas, mostrando buenos resultados, como se concluye de la comparación de las simulaciones y los datos experimentales medidos en ensayos, tanto estructurales como acústicos. Este trabajo abre un amplio campo de investigación a partir del cual es posible obtener metodologías precisas y eficientes para reproducir el comportamiento vibroacústico de sistemas en el rango de la media frecuencia. ABSTRACT Over the last years an increasing need of novel prediction techniques for vibroacoustic analysis of space structures has arisen. Current numerical techniques arc able to predict with enough accuracy the vibro-acoustic behaviour of systems with low and high modal densities. However, space structures are, in general, very complex and they present a range of frequencies in which a mixed behaviour exist. In such cases, the full system is composed of some sub-structures which has low modal density, while others present high modal density. This frequency range is known as the mid-frequency range and to develop methods for accurately describe the vibro-acoustic response in this frequency range is the scope of this dissertation. For the structures under study, the aforementioned low and high modal densities correspond with the low and high frequency ranges, respectively. For the low frequency range, deterministic techniques as the Finite Element Method (FEM) are used while, for the high frequency range statistical techniques, as the Statistical Energy Analysis (SEA), arc considered as more appropriate. In the mid-frequency range, where a mixed vibro-acoustic behaviour is expected, any of these numerical method can not be used with enough confidence level. As a consequence, it is usual to obtain an undetermined gap between low and high frequencies in the vibro-acoustic response function. This dissertation proposes two different solutions to the mid-frequency range problem. The first one, named as The Subsystem based High Frequency Limit (SHFL) procedure, proposes a multi-hybrid procedure in which each sub-structure of the full system is modelled with the appropriate modelling technique, depending on the frequency of study. With this purpose, the concept of high frequency limit of a sub-structure is introduced, marking out the limit above which a substructure has enough modal density to be modelled by SEA. For a certain analysis frequency, if it is lower than the high frequency limit of the sub-structure, the sub-structure is modelled through FEM and, if the frequency of analysis is higher than the high frequency limit, the sub-structure is modelled by SEA. The procedure leads to a number of hybrid models required to cover the medium frequency range, which is defined as the frequency range between the lowest substructure high frequency limit and the highest one. Using this procedure, the mid-frequency range can be define specifically so that, as a consequence, an improvement in the continuity of the vibro-acoustic response function is achieved, closing the undetermined gap between the low and high frequency ranges. The second proposed mid-frequency solution is the Hybrid Sub-structuring method based on Component Mode Synthesis (HS-CMS). The method adopts a partition scheme based on classifying the system modal basis into global and local sets of modes. This classification is performed by using a Component Mode Synthesis, in particular a Craig-Bampton transformation, in order to express the system modal base into the modal bases associated with each sub-structure. Then, each sub-structure modal base is classified into global and local set, fist ones associated with the long wavelength motion and second ones with the short wavelength motion. The high frequency limit of each sub-structure is used as frequency frontier between both sets of modes. From this classification, the equations of motion associated with global modes are derived, which include the interaction of local modes by means of corrections in the dynamic stiffness matrix and the force vector of the global problem. The local equations of motion are solved through SEA, where again interactions with global modes arc included through the inclusion of an additional input power into the SEA model. The method has been tested for the calculation of the response function of structures subjected to structural and acoustic loads. Both methods have been firstly tested in simple structures to establish their basis and main characteristics. Methods are also verified in space structures, as satellites and antenna reflectors, providing good results as it is concluded from the comparison with experimental results obtained in both, acoustic and structural load tests. This dissertation opens a wide field of research through which further studies could be performed to obtain efficient and accurate methodologies to appropriately reproduce the vibro-acoustic behaviour of complex systems in the mid-frequency range.