Spectral bounds and basis results for non-self-adjoint pencils, with an application to Hagen-Poiseuille flow

We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil A−λBA−λB in which BB is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances.

Thermal dependence of luminescence lifetimes and radioluminescence in quartz

During time-resolved optical stimulation experiments (TR-OSL), one uses short light pulses to separate the stimulation and emission of luminescence in time. Experimental TR-OSL results show that the luminescence lifetime in quartz of sedimentary origin is independent of annealing temperature below 500 °C, but decreases monotonically thereafter. These results have been interpreted previously empirically on the basis of the existence of two separate luminescence centers LH and LL in quartz, each with its own distinct luminescence lifetime. Additional experimental evidence also supports the presence of a non-luminescent hole reservoir R, which plays a critical role in the predose effect in this material. This paper extends a recently published analytical model for thermal quenching in quartz, to include the two luminescence centers LH and LL, as well as the hole reservoir R. The new extended model involves localized electronic transitions between energy states within the two luminescence centers, and is described by a system of differential equations based on the Mott–Seitz mechanism of thermal quenching. It is shown that by using simplifying physical assumptions, one can obtain analytical solutions for the intensity of the light during a TR-OSL experiment carried out with previously annealed samples. These analytical expressions are found to be in good agreement with the numerical solutions of the equations. The results from the model are shown to be in quantitative agreement with published experimental data for commercially available quartz samples. Specifically the model describes the variation of the luminescence lifetimes with (a) annealing temperatures between room temperature and 900 °C, and (b) with stimulation temperatures between 20 and 200 °C. This paper also reports new radioluminescence (RL) measurements carried out using the same commercially available quartz samples. Gaussian deconvolution of the RL emission spectra was carried out using a total of seven emission bands between 1.5 and 4.5 eV, and the behavior of these bands was examined as a function of the annealing temperature. An emission band at ∼3.44 eV (360 nm) was found to be strongly enhanced when the annealing temperature was increased to 500 °C, and this band underwent a significant reduction in intensity with further increase in temperature. Furthermore, a new emission band at ∼3.73 eV (330 nm) became apparent for annealing temperatures in the range 600–700 °C. These new experimental results are discussed within the context of the model presented in this paper.

The Devil’s Calculus: Mathematical Models of Civil War

In spite of the movement to turn political science into a real science, various mathematical methods that are now the staples of physics, biology, and even economics are thoroughly uncommon in political science, especially the study of civil war. This study seeks to apply such methods - specifically, ordinary differential equations (ODEs) - to model civil war based on what one might dub the capabilities school of thought, which roughly states that civil wars end only when one side’s ability to make war falls far enough to make peace truly attractive. I construct several different ODE-based models and then test them all to see which best predicts the instantaneous capabilities of both sides of the Sri Lankan civil war in the period from 1990 to 1994 given parameters and initial conditions. The model that the tests declare most accurate gives very accurate predictions of state military capabilities and reasonable short term predictions of cumulative deaths. Analysis of the model reveals the scale of the importance of rebel finances to the sustainability of insurgency, most notably that the number of troops required to put down the Tamil Tigers is reduced by nearly a full order of magnitude when Tiger foreign funding is stopped. The study thus demonstrates that accurate foresight may come of relatively simple dynamical models, and implies the great potential of advanced and currently unconventional non-statistical mathematical methods in political science.

The internal structure of hydrogen-air difussion flames

The Internal Structure of Hydrogen-Air Diffusion Flames. Tho purpose of this paper is to study finite rate chemistry effects in diffusion controlled hydrogenair flames undor conditions appearing in some cases in a supersonic combustor. Since for large reaction rates the flame is close to chemical equilibrium, the reaction takes place in a very thin region, so thata "singular perturbation "treatment" of the problem seems appropriate. It has been shown previously that, within the inner or reaction zone, convection effects may be neglocted, the temperature is constant across the flame, and tho mass fraction distributions are given by ordinary differential equations, whore tho only independent variable involved is tho coordinate normal to the flame surface. Tho solution of the outer problom, which is a pure mixing problem with the additional condition that fuol and oxidizer do not coexist in any zone, provides t h e following information: tho flame position, rates of fuel consumption, temperature, concentrators of species, fluid velocity outside of tho flame, and the boundary conditions required to solve the "inner problem." The main contribution of this paper consists in the introduction of a fairly complicated chemical kinetic scheme representing hydrogen-oxygen reaction. The nonlinear equations expressing the conservation of chemical species are approximately integrated by means of an integral method. It has boen found that, in the case considered of a near-equilibrium diffusion flame, tho role played by the dissociation-recombination reactions is purely marginal, and that somo of the second order "shuffling" reactions are close to equilibrium. The method shown here may be applied to compute the distanco from the injector corresponding to a given separation from equilibrium, say ten to twenty percent. For the casos whore this length is a small fraction of the combustion zone length, the equilibrium treatment describes properly tho flame behavior.

Adams methods: implementation as predictor-corrector methods

In this paper, Adams explicit and implicit formulas are obtained in a simple way and a relationship between them is established, allowing for their joint implementation as predictor-corrector methods. It is shown the purposefulness, from a didactic point of view, of Excel spreadsheets for calculations and for the orderly presentation of results in the application of Adams methods to solving initial value problems in ordinary differential equations.

Low-dimensional modeling of streaks in a wedge flow boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) three-dimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate self-similarity (not associated with the boundary layer self-similarity), namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate self-similar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three one-dimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks.

Simulación numérica de un problema de contaminación atmosférica

La ecuación en derivadas parciales de advección difusión con reacción química es la base de los modelos de dispersión de contaminantes en la atmósfera, y los diferentes métodos numéricos empleados para su resolución han sido objeto de amplios estudios a lo largo de su desarrollo. En esta Tesis se presenta la implementación de un nuevo método conservativo para la resolución de la parte advectiva de la ecuación en derivadas parciales que modela la dispersión de contaminantes dentro del modelo mesoescalar de transporte químico CHIMERE. Este método está basado en una técnica de volúmenes finitos junto con una interpolación racional. La ventaja de este método es la conservación exacta de la masa transportada debido al empleo de la ley de conservación de masas. Para ello emplea una formulación de flujo basado en el cálculo de la integral ponderada dentro de cada celda definida para la discretización del espacio en el método de volúmenes finitos. Los resultados numéricos obtenidos en las simulaciones realizadas (implementando el modelo conservativo para la advección en el modelo CHIMERE) se han comparado con los datos observados de concentración de contaminantes registrados en la red de estaciones de seguimiento y medición distribuidas por la Península Ibérica. Los datos estadísticos de medición del error, la media normalizada y la media absoluta normalizada del error, presentan valores que están dentro de los rangos propuestos por la EPA para considerar el modelo preciso. Además, se introduce un nuevo método para resolver la parte advectivadifusiva de la ecuación en derivadas parciales que modeliza la dispersión de contaminantes en la atmósfera. Se ha empleado un método de diferencias finitas de alto orden para resolver la parte difusiva de la ecuación de transporte de contaminantes junto con el método racional conservativo para la parte advectiva en una y dos dimensiones. Los resultados obtenidos de la aplicación del método a diferentes situaciones incluyendo casos académicos y reales han sido comparados con la solución analítica de la ecuación de advección-difusión, demostrando que el nuevo método proporciona un resultado preciso para aproximar la solución. Por último, se ha desarrollado un modelo completo que contempla los fenómenos advectivo y difusivo con reacción química, usando los métodos anteriores junto con una técnica de diferenciación regresiva (BDF2). Esta técnica consiste en un método implícito multipaso de diferenciación regresiva de segundo orden, que nos permite resolver los problemas rígidos típicos de la química atmosférica, modelizados a través de sistemas de ecuaciones diferenciales ordinarias. Este método hace uso de la técnica iterativa Gauss- Seidel para obtener la solución de la parte implícita de la fórmula BDF2. El empleo de la técnica de Gauss-Seidel en lugar de otras técnicas comúnmente empleadas, como la iteración por el método de Newton, nos proporciona rapidez de cálculo y bajo consumo de memoria, ideal para obtener modelos operativos para la resolución de la cinética química atmosférica. ABSTRACT Extensive research has been performed to solve the atmospheric chemicaladvection- diffusion equation and different numerical methods have been proposed. This Thesis presents the implementation of an exactly conservative method for the advection equation in the European scale Eulerian chemistry transport model CHIMERE based on a rational interpolation and a finite volume algorithm. The advantage of the method is that the cell-integrated average is predicted via a flux formulation, thus the mass is exactly conserved. Numerical results are compared with a set of observation registered at some monitoring sites in Spain. The mean normalized bias and the mean normalized absolute error present values that are inside the range to consider an accurate model performance. In addition, it has been introduced a new method to solve the advectiondiffusion equation. It is based on a high-order accurate finite difference method to solve de diffusion equation together with a rational interpolation and a finite volume to solve the advection equation in one dimension and two dimensions. Numerical results obtained from solving several problems include academic and real atmospheric problems have been compared with the analytical solution of the advection-diffusion equation, showing that the new method give an efficient algorithm for solving such problems. Finally, a complete model has been developed to solve the atmospheric chemical-advection-diffusion equation, adding the conservative method for the advection equation, the high-order finite difference method for the diffusion equation and a second-order backward differentiation formula (BDF2) to solve the atmospheric chemical kinetics. The BDF2 is an implicit, second order multistep backward differentiation formula used to solve the stiff systems of ordinary differential equations (ODEs) from atmospheric chemistry. The Gauss-Seidel iteration is used for approximately solving the implicitly defined BDF solution, giving a faster tool than the more commonly used iterative modified Newton technique. This method implies low start-up costs and a low memory demand due to the use of Gauss-Seidel iteration.

Discontinuous solutions and boundary value problems for non-linearly elastic fibre-reinforced materials

En este trabajo se han analizado varios problemas en el contexto de la elasticidad no lineal basándose en modelos constitutivos representativos. En particular, se han analizado problemas relacionados con el fenómeno de perdida de estabilidad asociada con condiciones de contorno en el caso de material reforzados con fibras. Cada problema se ha formulado y se ha analizado por separado en diferentes capítulos. En primer lugar se ha mostrado el análisis del gradiente de deformación discontinuo para un material transversalmente isótropo, en particular, el modelo del material considerado consiste de una base neo-Hookeana isótropa incrustada con fibras de refuerzo direccional caracterizadas con un solo parámetro. La solución de este problema se vincula con instabilidades que dan lugar al mecanismo de fallo conocido como banda de cortante. La perdida de elipticidad de las ecuaciones diferenciales de equilibrio es una condición necesaria para que aparezca este tipo de soluciones y por tanto las inestabilidades asociadas. En segundo lugar se ha analizado una deformación combinada de extensión, inación y torsión de un tubo cilíndrico grueso donde se ha encontrado que la deformación citada anteriormente puede ser controlada solo para determinadas direcciones de las fibras refuerzo. Para entender el comportamiento elástico del tubo considerado se ha ilustrado numéricamente los resultados obtenidos para las direcciones admisibles de las fibras de refuerzo bajo la deformación considerada. En tercer lugar se ha estudiado el caso de un tubo cilíndrico grueso reforzado con dos familias de fibras sometido a cortante en la dirección azimutal para un modelo de refuerzo especial. En este problema se ha encontrado que las inestabilidades que aparecen en el material considerado están asociadas con lo que se llama soluciones múltiples de la ecuación diferencial de equilibrio. Se ha encontrado que el fenómeno de instabilidad ocurre en un estado de deformación previo al estado de deformación donde se pierde la elipticidad de la ecuación diferencial de equilibrio. También se ha demostrado que la condición de perdida de elipticidad y ^W=2 = 0 (la segunda derivada de la función de energía con respecto a la deformación) son dos condiciones necesarias para la existencia de soluciones múltiples. Finalmente, se ha analizado detalladamente en el contexto de elipticidad un problema de un tubo cilíndrico grueso sometido a una deformación combinada en las direcciones helicoidal, axial y radial para distintas geotermias de las fibras de refuerzo . In the present work four main problems have been addressed within the framework of non-linear elasticity based on representative constitutive models. Namely, problems related to the loss of stability phenomena associated with boundary value problems for fibre-reinforced materials. Each of the considered problems is formulated and analysed separately in different chapters. We first start with the analysis of discontinuous deformation gradients for a transversely isotropic material under plane deformation. In particular, the material model is an augmented neo-Hookean base with a simple unidirectional reinforcement characterised by a single parameter. The solution of this problem is related to material instabilities and it is associated with a shear band-type failure mode. The loss of ellipticity of the governing differential equations is a necessary condition for the existence of these material instabilities. The second problem involves a detailed analysis of the combined non-linear extension, inflation and torsion of a thick-walled circular cylindrical tube where it has been found that the aforementioned deformation is controllable only for certain preferred directions of transverse isotropy. Numerical results have been illustrated to understand the elastic behaviour of the tube for the admissible preferred directions under the considered deformation. The third problem deals with the analysis of a doubly fibre-reinforced thickwalled circular cylindrical tube undergoing pure azimuthal shear for a special class of the reinforcing model where multiple non-smooth solutions emerge. The associated instability phenomena are found to occur prior to the point where the nominal stress tensor changes monotonicity in a particular direction. It has been also shown that the loss of ellipticity condition that arises from the equilibrium equation and ^W=2 = 0 (the second derivative of the strain-energy function with respect to the deformation) are equivalent necessary conditions for the emergence of multiple solutions for the considered material. Finally, a detailed analysis in the basis of the loss of ellipticity of the governing differential equations for a combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube is carried out.

Analysis of the static and dynamic behaviour of hydraulic fills

En la actualidad existe un gran conocimiento en la caracterización de rellenos hidráulicos, tanto en su caracterización estática, como dinámica. Sin embargo, son escasos en la literatura estudios más generales y globales de estos materiales, muy relacionados con sus usos y principales problemáticas en obras portuarias y mineras. Los procedimientos semi‐empíricos para la evaluación del efecto silo en las celdas de cajones portuarios, así como para el potencial de licuefacción de estos suelos durantes cargas instantáneas y terremotos, se basan en estudios donde la influencia de los parámetros que los rigen no se conocen en gran medida, dando lugar a resultados con considerable dispersión. Este es el caso, por ejemplo, de los daños notificados por el grupo de investigación del Puerto de Barcelona, la rotura de los cajones portuarios en el Puerto de Barcelona en 2007. Por estos motivos y otros, se ha decidido desarrollar un análisis para la evaluación de estos problemas mediante la propuesta de una metodología teórico‐numérica y empírica. El enfoque teórico‐numérico desarrollado en el presente estudio se centra en la determinación del marco teórico y las herramientas numéricas capaces de solventar los retos que presentan estos problemas. La complejidad del problema procede de varios aspectos fundamentales: el comportamiento no lineal de los suelos poco confinados o flojos en procesos de consolidación por preso propio; su alto potencial de licuefacción; la caracterización hidromecánica de los contactos entre estructuras y suelo (camino preferencial para el flujo de agua y consolidación lateral); el punto de partida de los problemas con un estado de tensiones efectivas prácticamente nulo. En cuanto al enfoque experimental, se ha propuesto una metodología de laboratorio muy sencilla para la caracterización hidromecánica del suelo y las interfaces, sin la necesidad de usar complejos aparatos de laboratorio o procedimientos excesivamente complicados. Este trabajo incluye por tanto un breve repaso a los aspectos relacionados con la ejecución de los rellenos hidráulicos, sus usos principales y los fenómenos relacionados, con el fin de establecer un punto de partida para el presente estudio. Este repaso abarca desde la evolución de las ecuaciones de consolidación tradicionales (Terzaghi, 1943), (Gibson, English & Hussey, 1967) y las metodologías de cálculo (Townsend & McVay, 1990) (Fredlund, Donaldson and Gitirana, 2009) hasta las contribuciones en relación al efecto silo (Ranssen, 1985) (Ravenet, 1977) y sobre el fenómeno de la licuefacción (Casagrande, 1936) (Castro, 1969) (Been & Jefferies, 1985) (Pastor & Zienkiewicz, 1986). Con motivo de este estudio se ha desarrollado exclusivamente un código basado en el método de los elementos finitos (MEF) empleando el programa MATLAB. Para ello, se ha esablecido un marco teórico (Biot, 1941) (Zienkiewicz & Shiomi, 1984) (Segura & Caron, 2004) y numérico (Zienkiewicz & Taylor, 1989) (Huerta & Rodríguez, 1992) (Segura & Carol, 2008) para resolver problemas de consolidación multidimensional con condiciones de contorno friccionales, y los correspondientes modelos constitutivos (Pastor & Zienkiewicz, 1986) (Fiu & Liu, 2011). Asimismo, se ha desarrollado una metodología experimental a través de una serie de ensayos de laboratorio para la calibración de los modelos constitutivos y de la caracterización de parámetros índice y de flujo (Castro, 1969) (Bahda 1997) (Been & Jefferies, 2006). Para ello se han empleado arenas de Hostun como material (relleno hidráulico) de referencia. Como principal aportación se incluyen una serie de nuevos ensayos de corte directo para la caracterización hidromecánica de la interfaz suelo – estructura de hormigón, para diferentes tipos de encofrados y rugosidades. Finalmente, se han diseñado una serie de algoritmos específicos para la resolución del set de ecuaciones diferenciales de gobierno que definen este problema. Estos algoritmos son de gran importancia en este problema para tratar el procesamiento transitorio de la consolidación de los rellenos hidráulicos, y de otros efectos relacionados con su implementación en celdas de cajones, como el efecto silo y la licuefacciones autoinducida. Para ello, se ha establecido un modelo 2D axisimétrico, con formulación acoplada u‐p para elementos continuos y elementos interfaz (de espesor cero), que tratan de simular las condiciones de estos rellenos hidráulicos cuando se colocan en las celdas portuarias. Este caso de estudio hace referencia clara a materiales granulares en estado inicial muy suelto y con escasas tensiones efectivas, es decir, con prácticamente todas las sobrepresiones ocasionadas por el proceso de autoconsolidación (por peso propio). Por todo ello se requiere de algoritmos numéricos específicos, así como de modelos constitutivos particulares, para los elementos del continuo y para los elementos interfaz. En el caso de la simulación de diferentes procedimientos de puesta en obra de los rellenos se ha requerido la modificacion de los algoritmos empleados para poder así representar numéricamente la puesta en obra de estos materiales, además de poder realizar una comparativa de los resultados para los distintos procedimientos. La constante actualización de los parámetros del suelo, hace también de este algoritmo una potente herramienta que permite establecer un interesante juego de perfiles de variables, tales como la densidad, el índice de huecos, la fracción de sólidos, el exceso de presiones, y tensiones y deformaciones. En definitiva, el modelo otorga un mejor entendimiento del efecto silo, término comúnmente usado para definir el fenómeno transitorio del gradiente de presiones laterales en las estructuras de contención en forma de silo. Finalmente se incluyen una serie de comparativas entre los resultados del modelo y de diferentes estudios de la literatura técnica, tanto para el fenómeno de las consolidaciones por preso propio (Fredlund, Donaldson & Gitirana, 2009) como para el estudio del efecto silo (Puertos del Estado, 2006, EuroCódigo (2006), Japan Tech, Stands. (2009), etc.). Para concluir, se propone el diseño de un prototipo de columna de decantación con paredes friccionales, como principal propuesta de futura línea de investigación. Wide research is nowadays available on the characterization of hydraulic fills in terms of either static or dynamic behavior. However, reported comprehensive analyses of these soils when meant for port or mining works are scarce. Moreover, the semi‐empirical procedures for assessing the silo effect on cells in floating caissons, and the liquefaction potential of these soils during sudden loads or earthquakes are based on studies where the underlying influence parameters are not well known, yielding results with significant scatter. This is the case, for instance, of hazards reported by the Barcelona Liquefaction working group, with the failure of harbor walls in 2007. By virtue of this, a complex approach has been undertaken to evaluate the problem by a proposal of numerical and laboratory methodology. Within a theoretical and numerical scope, the study is focused on the numerical tools capable to face the different challenges of this problem. The complexity is manifold; the highly non‐linear behavior of consolidating soft soils; their potentially liquefactable nature, the significance of the hydromechanics of the soil‐structure contact, the discontinuities as preferential paths for water flow, setting “negligible” effective stresses as initial conditions. Within an experimental scope, a straightforward laboratory methodology is introduced for the hydromechanical characterization of the soil and the interface without the need of complex laboratory devices or cumbersome procedures. Therefore, this study includes a brief overview of the hydraulic filling execution, main uses (land reclamation, filled cells, tailing dams, etc.) and the underlying phenomena (self‐weight consolidation, silo effect, liquefaction, etc.). It comprises from the evolution of the traditional consolidation equations (Terzaghi, 1943), (Gibson, English, & Hussey, 1967) and solving methodologies (Townsend & McVay, 1990) (Fredlund, Donaldson and Gitirana, 2009) to the contributions in terms of silo effect (Ranssen, 1895) (Ravenet, 1977) and liquefaction phenomena (Casagrande, 1936) (Castro, 1969) (Been & Jefferies, 1985) (Pastor & Zienkiewicz, 1986). The novelty of the study lies on the development of a Finite Element Method (FEM) code, exclusively formulated for this problem. Subsequently, a theoretical (Biot, 1941) (Zienkiewicz and Shiomi, 1984) (Segura and Carol, 2004) and numerical approach (Zienkiewicz and Taylor, 1989) (Huerta, A. & Rodriguez, A., 1992) (Segura, J.M. & Carol, I., 2008) is introduced for multidimensional consolidation problems with frictional contacts and the corresponding constitutive models (Pastor & Zienkiewicz, 1986) (Fu & Liu, 2011). An experimental methodology is presented for the laboratory test and material characterization (Castro 1969) (Bahda 1997) (Been & Jefferies 2006) using Hostun sands as reference hydraulic fill. A series of singular interaction shear tests for the interface calibration is included. Finally, a specific model algorithm for the solution of the set of differential equations governing the problem is presented. The process of consolidation and settlements involves a comprehensive simulation of the transient process of decantation and the build‐up of the silo effect in cells and certain phenomena related to self‐compaction and liquefaction. For this, an implementation of a 2D axi‐syimmetric coupled model with continuum and interface elements, aimed at simulating conditions and self‐weight consolidation of hydraulic fills once placed into floating caisson cells or close to retaining structures. This basically concerns a loose granular soil with a negligible initial effective stress level at the onset of the process. The implementation requires a specific numerical algorithm as well as specific constitutive models for both the continuum and the interface elements. The simulation of implementation procedures for the fills has required the modification of the algorithm so that a numerical representation of these procedures is carried out. A comparison of the results for the different procedures is interesting for the global analysis. Furthermore, the continuous updating of the model provides an insightful logging of variable profiles such as density, void ratio and solid fraction profiles, total and excess pore pressure, stresses and strains. This will lead to a better understanding of complex phenomena such as the transient gradient in lateral pressures due to silo effect in saturated soils. Interesting model and literature comparisons for the self‐weight consolidation (Fredlund, Donaldson, & Gitirana, 2009) and the silo effect results (Puertos del Estado (2006), EuroCode (2006), Japan Tech, Stands. (2009)). This study closes with the design of a decantation column prototype with frictional walls as the main future line of research.