Differential invariants of second-order ordinary differential equations

The notion of a differential invariant for systems of second-order differential equations on a manifold M with respect to the group of vertical automorphisms of the projection is de?ned and the Chern connection attached to a SODE allows one to determine a basis for second-order differential invariants of a SODE.

Space-time Reconstruction of Oceanic Sea States via Variational Stereo Methods

We present a remote sensing observational method for the measurement of the spatio-temporal dynamics of ocean waves. Variational techniques are used to recover a coherent space-time reconstruction of oceanic sea states given stereo video imagery. The stereoscopic reconstruction problem is expressed in a variational optimization framework. There, we design an energy functional whose minimizer is the desired temporal sequence of wave heights. The functional combines photometric observations as well as spatial and temporal regularizers. A nested iterative scheme is devised to numerically solve, via 3-D multigrid methods, the system of partial differential equations resulting from the optimality condition of the energy functional. The output of our method is the coherent, simultaneous estimation of the wave surface height and radiance at multiple snapshots. We demonstrate our algorithm on real data collected off-shore. Statistical and spectral analysis are performed. Comparison with respect to an existing sequential method is analyzed.

Examples of PDEs all whose points are characteristic

Let π : FM ! M be the bundle of linear frames of a manifold M. A basis Lijk , j < k, of diffeomorphism invariant Lagrangians on J1 (FM) was determined in [J. Muñoz Masqué, M. E. Rosado, Invariant variational problems on linear frame bundles, J. Phys. A35 (2002) 2013-2036]. The notion of a characteristic hypersurface for an arbitrary first-order PDE system on an ar- bitrary bred manifold π : P → M, is introduced and for the systems dened by the Euler-Lagrange equations of Lijk every hypersurface is shown to be characteristic. The Euler-Lagrange equations of the natural basis of Lagrangian densities Lijk on the bundle of linear frames of a manifold M which are invariant under diffeomorphisms, are shown to be an underdetermined PDEs systems such that every hypersurface of M is characteristic for such equations. This explains why these systems cannot be written in the Cauchy-Kowaleska form, although they are known to be formally integrable by using the tools of geometric theory of partial differential equations, see [J. Muñoz Masqué, M. E. Rosado, Integrability of the eld equations of invariant variational problems on linear frame bundles, J. Geom. Phys. 49 (2004), 119-155]

The estimation of truncation error by tau-estimation revisited

The aim of this paper was to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local truncation error using the @t-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error.

A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves

We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents.

An efficient ‘a priori’ model reduction for boundary element models

The Boundary Element Method (BEM) is a discretisation technique for solving partial differential equations, which offers, for certain problems, important advantages over domain techniques. Despite the high CPU time reduction that can be achieved, some 3D problems remain today untreatable because the extremely large number of degrees of freedom—dof—involved in the boundary description. Model reduction seems to be an appealing choice for both, accurate and efficient numerical simulations. However, in the BEM the reduction in the number of degrees of freedom does not imply a significant reduction in the CPU time, because in this technique the more important part of the computing time is spent in the construction of the discrete system of equations. In this way, a reduction also in the number of weighting functions, seems to be a key point to render efficient boundary element simulations.

Bernstein polynomials in element-free Galerkin method

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves,can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

Stabilization in a two-species chemotaxis system with logistic source

We study a system of three partial differential equations modelling the spatiotemporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. For a range of the parameters the system possesses a uniquely determined spatially homogeneous positive equilibrium (u?, v?) globally asymptotically stable within a certain nonempty range of the logistic growth coefficients.

Suitability of artificial neural networks for designing LoC circuits

he simulation of complex LoC (Lab-on-a-Chip) devices is a process that requires solving computationally expensive partial differential equations. An interesting alternative uses artificial neural networks for creating computationally feasible models based on MOR techniques. This paper proposes an approach that uses artificial neural networks for designing LoC components considering the artificial neural network topology as an isomorphism of the LoC device topology. The parameters of the trained neural networks are based on equations for modeling microfluidic circuits, analogous to electronic circuits. The neural networks have been trained to behave like AND, OR, Inverter gates. The parameters of the trained neural networks represent the features of LoC devices that behave as the aforementioned gates. This would mean that LoC devices universally compute.

Mathematical and Numerical Modeling in Maritime Geomechanics

A theoretical and numerical framework to model the foundation of marine offshore structures is presented. The theoretical model is composed by a system of partial differential equations describing coupling between seabed solid skeleton and pore fluids (water, air, oil,…) combined with a system of ordinary differential equations describing the specific constitutive relation of the seabed soil skeleton. Once the theoretical model is described, the finite element numerical procedure to achieve an approximate solution of the governing equations is outlined. In order to validate the proposed theoretical and numerical framework the seaward tilt mechanism induced by the action of breaking waves over a vertical breakwater is numerically reproduced. The results numerically attained are in agreement with the main conclusions drawn from the literature associated with this failure mechanism

Caracterización dinámica del glaciar Hurd combinando observaciones de campo y simulaciones numéricas

El objetivo fundamental de esta tesis es la caracterización de la morfología y del estado de deformaciones y tensiones del Glaciar Hurd (Isla Livingston, Archipiélago de las Shetland del Sur, Antártida), mediante una combinación de observaciones de campo, registros de georradar y simulaciones numéricas. La morfología y el estado de deformaciones y tensiones actuales son la expresión de la evolución dinámica del glaciar desde tiempos pretéritos hasta recientes, y su análisis nos dará las pautas con las cuales ser capaces de predecir, con el apoyo de las simulaciones numéricas, su evolución futura. El primer aspecto que se aborda es el estudio de las estructuras que pueden observarse en la superficie del glaciar. Describimos las distintas técnicas utilizadas (medidas de campo, fotointerpretación de ortofotografías, análisis geoquímico de cenizas volcánicas, etc.) y presentamos el análisis e interpretación de los resultados morfo-estructurales, así como la correlación, mediante análisis geoquímicos (fluorescencia de rayos X), entre las cenizas volcánicas que extruyen en la superficie del Glaciar Hurd y las del volcán Decepción, origen de las cenizas. Esto nos permite realizar una datación de las mismas como Tefra 1, correspondiente a la erupción de 1970, Tefra 2, correspondiente a las erupciones pre-1829, y el conjunto Tefra 3, asociado a las erupciones más antiguas. En segundo lugar nos ocupamos de las estructuras presentes en el interior del glaciar, cuya herramienta de detección fundamental es el georradar. Identificadas estas estructuras internas, las vinculamos con las observadas en la superficie del glaciar. También hemos estudiado la estructura hidrotérmica del glaciar, obteniendo una serie de evidencias adicionales de su carácter politérmico. Entre éstas se contaban, hasta ahora, las basadas en el valor del parámetro de rigidez de la relación constitutiva del hielo determinada por ajuste de modelos dinámicos y observaciones realizados por Otero (2008) y las basadas en las velocidades de las ondas de radar en el hielo determinadas con el método de punto medio común por Navarro y otros (2009). Las evidencias adicionales que aportamos en esta tesis son: 1) la presencia de estructuras típicas de régimen compresivo en la zona terminal del glaciar y de cizalla en los márgenes del mismo, y 2) la presencia de un estrato superficial de hielo frío (por encima de otro templado) en la zona de ablación de los tres lóbulos del Glaciar Hurd –Sally Rocks, Argentina y Las Palmas–, que alcanzan espesores de 70, 50 y 40 m, respectivamente. Este estrato de hielo frío está probablemente congelado al lecho subglaciar en la zona terminal (Molina y otros, 2007; esta tesis). Por último, nos ocupamos de la simulación numérica de la dinámica glaciar. Presentamos el modelo físico-matemático utilizado, discutimos sus condiciones de contorno y cómo éstas se miden en los trabajos de campo, y describimos el procedimiento de resolución numérica del sistema de ecuaciones parciales del modelo. Presentamos los resultados para los campos de velocidades, deformaciones y tensiones, comparando estos resultados con las estructuras observadas. También incluimos el análisis de las elipses de deformación acumulativa, que proporcionan información sobre las estructuras a las que puede dar lugar la evolución del estado de deformaciones y tensiones a las que se ve sometido el hielo según avanza, lentamente, desde la cabecera hasta la zona terminal del glaciar, con tiempos de tránsito de hasta 1.250 años, recogiendo así la historia de deformaciones en el glaciar. Concluyendo, ponemos de manifiesto en esta tesis que las medidas de campo de las estructuras y niveles de cenizas, las medidas de georradar y las simulaciones numéricas de la dinámica glaciar, realizadas de forma combinada, permiten caracterizar el régimen actual de velocidades, deformaciones y tensiones del glaciar, entender su evolución en el pasado y predecir su evolución futura. ABSTRACT The main objective of this thesis is to characterize the morphology and the state of strains and stresses of Hurd Glacier (Livingston Island, South Shetland Islands archipelago, Antarctica) through a combination of field observations, ground-penetrating radar measurements and numerical simulations. The morphology and the current state of strain and stresses are the expression of the dynamic evolution of the glacier from the past to recent times, and their analysis gives us the guidelines to be able to predict, with the support of numerical simulations, its future evolution. The first subject addressed is the study of structures that can be observed on the glacier surface. We describe the different techniques used (field measurements, photointerpretation of orthophotos, geochemical analysis of volcanic ashes, etc.) and we present the analysis and interpretation of the morpho-structural results, as well as the correlation with geochemical analysis (XRF) between the volcanic ashes extruded to the surface of Hurd Glacier and those of Deception Island volcano, from which the ashes originate. This allows us dating the ashes as Tephra 1, corresponding to the 1970 eruption, Tephra 2, corresponding to the pre-1829 eruptions, and the Tephra 3 group, associated with older eruptions. Secondly we focus on the study of the structures present within the glacier, which are detected with the help of ground-penetrating radar. Once identified, we link these internal structures with those observed on the glacier surface. We also study the hydrothermal structure of the glacier, getting a series of additional evidences of its polythermal structure. Among the evidences available so far, we can mention those based on the value of the stiffness parameter of the constitutive relation of ice, determined by fitting dynamic models to observations, as done by Otero (2008), and those based on the velocity of propagation of the radar waves through the glacier ice, measured using the common midpoint method, as done by Navarro et al. (2009). The additional evidences that we provide in this thesis are: 1) the presence of structures typical of compressive regime in the terminal zone of the glacier, together with shear at its margins, and 2) the presence of a surface layer of cold ice (overlying a layer of temperate ice) in the ablation zone of the three lobes of Hurd Glacier –Sally Rocks, Argentina and Las Palmas–, reaching thicknesses of 70, 50 and 40 m, respectively. This cold layer is probably frozen to the subglacial bed in the terminal zone (Molina and others 2007; this thesis). Finally, we deal with the numerical simulation of glacier dynamics. We present the physical-mathematical model, discuss its boundary conditions and how they are measured in the field work, and describe the method of numerical solution of the model’s partial differential equations. We present the results for the velocity, strain and stress fields, comparing these results with the observed structures. We also include an analysis of the ellipses of cumulative deformation, which provide information about the structures that can result from the evolution of the strain and stress regime of the glacier ice as it moves slowly from the head to the snout of the glacier, with transit times of up to 1,250 years, so picking the history of deformation of the glacier. Summarizing, we show in this thesis that field measurements of structures and ash layers, ground-penetrating radar measurements and numerical simulations of glacier dynamics, performed in combination, allow us to characterize the current regime of velocities, strains and stresses of the glacier, to understand its past evolution and to predict its future evolution.

Integración de modelos numéricos de glaciares y procesado de datos de georradar en un sistema de información geográfica

Los modelos de termomecánica glaciar están definidos mediante sistemas de ecuaciones en derivadas parciales que establecen los principios básicos de conservación de masa, momento lineal y energía, acompañados por una ley constitutiva que define la relación entre las tensiones a las que está sometido el hielo glaciar y las deformaciones resultantes de las mismas. La resolución de estas ecuaciones requiere la definición precisa del dominio (la geometría del glaciar, obtenido a partir de medidas topográficas y de georradar), así como contar con un conjunto de condiciones de contorno, que se obtienen a partir de medidas de campo de las variables implicadas y que constituyen un conjunto de datos geoespaciales. El objetivo fundamental de esta tesis es desarrollar una serie de herramientas que nos permitan definir con precisión la geometría del glaciar y disponer de un conjunto adecuado de valores de las variables a utilizar como condiciones de contorno del problema. Para ello, en esta tesis se aborda la recopilación, la integración y el estudio de los datos geoespaciales existentes para la Península Hurd, en la Isla Livingston (Antártida), generados desde el año 1957 hasta la actualidad, en un sistema de información geográfica. Del correcto tratamiento y procesamiento de estos datos se obtienen otra serie de elementos que nos permiten realizar la simulación numérica del régimen termomecánico presente de los glaciares de Península Hurd, así como su evolución futura. Con este objetivo se desarrolla en primer lugar un inventario completo de datos geoespaciales y se realiza un procesado de los datos capturados en campo, para establecer un sistema de referencia común a todos ellos. Se unifican además todos los datos bajo un mismo formato estándar de almacenamiento e intercambio de información, generándose los metadatos correspondientes. Se desarrollan asimismo técnicas para la mejora de los procedimientos de captura y procesado de los datos, de forma que se minimicen los errores y se disponga de estimaciones fiables de los mismos. El hecho de que toda la información se integre en un sistema de información geográfica (una vez producida la normalización e inventariado de la misma) permite su consulta rápida y ágil por terceros. Además, hace posible efectuar sobre ella una serie de operaciones conducentes a la obtención de nuevas capas de información. El análisis de estos nuevos datos permite explicar el comportamiento pasado de los glaciares objeto de estudio y proporciona elementos esenciales para la simulación de su comportamiento futuro. ABSTRACT Glacier thermo-mechanical models are defined by systems of partial differential equations stating the basic principles of conservation of mass, momentum and energy, accompanied by a constitutive principle that defines the relationship between the stresses acting on the ice and the resulting deformations. The solution of these equations requires an accurate definition of the model domain (the geometry of the glacier, obtained from topographical and ground penetrating radar measurements), as well as a set of boundary conditions, which are obtained from measurements of the variables involved and define a set of geospatial data. The main objective of this thesis is to develop tools able to provide an accurate definition of the glacier geometry and getting a proper set of values for the variables to be used as boundary conditions of our problem. With the above aim, this thesis focuses on the collection, compilation and study of the geospatial data existing for the Hurd Peninsula on Livingston Island, Antarctica, generated since 1957 to present, into a geographic information system. The correct handling and processing of these data results on a new collection of elements that allow us to numerically model the present state and the future evolution of Hurd Peninsula glaciers. First, a complete inventory of geospatial data is developed and the captured data are processed, with the aim of establishing a reference system common to all collections of data. All data are stored under a common standard format, and the corresponding metadata are generated to facilitate the information exchange. We also develop techniques for the improvement of the procedures used for capturing and processing the data, such that the errors are minimized and better estimated. All information is integrated into a geographic information system (once produced the standardization and inventory of it). This allows easy and fast viewing and consulting of the data by third parties. Also, it is possible to carry out a series of operations leading to the production of new layers of information. The analysis of these new data allows to explain past glacier behavior, and provides essential elements for explaining its future evolution.

On a competitive system under chemotactic effects with non-local terms

In this paper, we study a system of partial differential equations describing the evolution of a population under chemotactic effects with non-local reaction terms. We consider an external application of chemoattractant in the system and study the cases of one and two populations in competition. By introducing global competitive/cooperative factors in terms of the total mass of the populations, weobtain, forarangeofparameters, thatanysolutionwithpositive and bounded initial data converges to a spatially homogeneous state with positive components. The proofs rely on the maximum principle for spatially homogeneous sub- and super-solutions.

Numerical simulation of group combustion of pulverized coal

A mathematical model for the group combustion of pulverized coal particles was developed in a previous work. It includes the Lagrangian description of the dehumidification, devolatilization and char gasification reactions of the coal particles in the homogenized gaseous environment resulting from the three fuels, CO, H2 and volatiles, supplied by the gasification of the particles and their simultaneous group combustion by the gas phase oxidation reactions, which are considered to be very fast. This model is complemented here with an analysis of the particle dynamics, determined principally by the effects of aerodynamic drag and gravity, and its dispersion based on a stochastic model. It is also extended to include two other simpler models for the gasification of the particles: the first one for particles small enough to extinguish the surrounding diffusion flames, and a second one for particles with small ash content when the porous shell of ashes remaining after gasification of the char, non structurally stable, is disrupted. As an example of the applicability of the models, they are used in the numerical simulation of an experiment of a non-swirling pulverized coal jet with a nearly stagnant air at ambient temperature, with an initial region of interaction with a small annular methane flame. Computational algorithms for solving the different stages undergone by a coal particle during its combustion are proposed. For the partial differential equations modeling the gas phase, a second order finite element method combined with a semi-Lagrangian characteristics method are used. The results obtained with the three versions of the model are compared among them and show how the first of the simpler models fits better the experimental results.

Modelos sin malla en simulación numérica de estructuras aeroespaciales

La presente Tesis Doctoral aborda la introducción de la Partición de Unidad de Bernstein en la forma débil de Galerkin para la resolución de problemas de condiciones de contorno en el ámbito del análisis estructural. La familia de funciones base de Bernstein conforma un sistema generador del espacio de funciones polinómicas que permite construir aproximaciones numéricas para las que no se requiere la existencia de malla: las funciones de forma, de soporte global, dependen únicamente del orden de aproximación elegido y de la parametrización o mapping del dominio, estando las posiciones nodales implícitamente definidas. El desarrollo de la formulación está precedido por una revisión bibliográfica que, con su punto de partida en el Método de Elementos Finitos, recorre las principales técnicas de resolución sin malla de Ecuaciones Diferenciales en Derivadas Parciales, incluyendo los conocidos como Métodos Meshless y los métodos espectrales. En este contexto, en la Tesis se somete la aproximación Bernstein-Galerkin a validación en tests uni y bidimensionales clásicos de la Mecánica Estructural. Se estudian aspectos de la implementación tales como la consistencia, la capacidad de reproducción, la naturaleza no interpolante en la frontera, el planteamiento con refinamiento h-p o el acoplamiento con otras aproximaciones numéricas. Un bloque importante de la investigación se dedica al análisis de estrategias de optimización computacional, especialmente en lo referente a la reducción del tiempo de máquina asociado a la generación y operación con matrices llenas. Finalmente, se realiza aplicación a dos casos de referencia de estructuras aeronáuticas, el análisis de esfuerzos en un angular de material anisotrópico y la evaluación de factores de intensidad de esfuerzos de la Mecánica de Fractura mediante un modelo con Partición de Unidad de Bernstein acoplada a una malla de elementos finitos. ABSTRACT This Doctoral Thesis deals with the introduction of Bernstein Partition of Unity into Galerkin weak form to solve boundary value problems in the field of structural analysis. The family of Bernstein basis functions constitutes a spanning set of the space of polynomial functions that allows the construction of numerical approximations that do not require the presence of a mesh: the shape functions, which are globally-supported, are determined only by the selected approximation order and the parametrization or mapping of the domain, being the nodal positions implicitly defined. The exposition of the formulation is preceded by a revision of bibliography which begins with the review of the Finite Element Method and covers the main techniques to solve Partial Differential Equations without the use of mesh, including the so-called Meshless Methods and the spectral methods. In this context, in the Thesis the Bernstein-Galerkin approximation is subjected to validation in one- and two-dimensional classic benchmarks of Structural Mechanics. Implementation aspects such as consistency, reproduction capability, non-interpolating nature at boundaries, h-p refinement strategy or coupling with other numerical approximations are studied. An important part of the investigation focuses on the analysis and optimization of computational efficiency, mainly regarding the reduction of the CPU cost associated with the generation and handling of full matrices. Finally, application to two reference cases of aeronautic structures is performed: the stress analysis in an anisotropic angle part and the evaluation of stress intensity factors of Fracture Mechanics by means of a coupled Bernstein Partition of Unity - finite element mesh model.