Aplicación de Métodos Meshless al Análisis de Problemas de Autovalores

La presente Tesis Doctoral aborda la aplicación de métodos meshless, o métodos sin malla, a problemas de autovalores, fundamentalmente vibraciones libres y pandeo. En particular, el estudio se centra en aspectos tales como los procedimientos para la resolución numérica del problema de autovalores con estos métodos, el coste computacional y la viabilidad de la utilización de matrices de masa o matrices de rigidez geométrica no consistentes. Además, se acomete en detalle el análisis del error, con el objetivo de determinar sus principales fuentes y obtener claves que permitan la aceleración de la convergencia. Aunque en la actualidad existe una amplia variedad de métodos meshless en apariencia independientes entre sí, se han analizado las diferentes relaciones entre ellos, deduciéndose que el método Element-Free Galerkin Method [Método Galerkin Sin Elementos] (EFGM) es representativo de un amplio grupo de los mismos. Por ello se ha empleado como referencia en este análisis. Muchas de las fuentes de error de un método sin malla provienen de su algoritmo de interpolación o aproximación. En el caso del EFGM ese algoritmo es conocido como Moving Least Squares [Mínimos Cuadrados Móviles] (MLS), caso particular del Generalized Moving Least Squares [Mínimos Cuadrados Móviles Generalizados] (GMLS). La formulación de estos algoritmos indica que la precisión de los mismos se basa en los siguientes factores: orden de la base polinómica p(x), características de la función de peso w(x) y forma y tamaño del soporte de definición de esa función. Se ha analizado la contribución individual de cada factor mediante su reducción a un único parámetro cuantificable, así como las interacciones entre ellos tanto en distribuciones regulares de nodos como en irregulares. El estudio se extiende a una serie de problemas estructurales uni y bidimensionales de referencia, y tiene en cuenta el error no sólo en el cálculo de autovalores (frecuencias propias o carga de pandeo, según el caso), sino también en términos de autovectores. This Doctoral Thesis deals with the application of meshless methods to eigenvalue problems, particularly free vibrations and buckling. The analysis is focused on aspects such as the numerical solving of the problem, computational cost and the feasibility of the use of non-consistent mass or geometric stiffness matrices. Furthermore, the analysis of the error is also considered, with the aim of identifying its main sources and obtaining the key factors that enable a faster convergence of a given problem. Although currently a wide variety of apparently independent meshless methods can be found in the literature, the relationships among them have been analyzed. The outcome of this assessment is that all those methods can be grouped in only a limited amount of categories, and that the Element-Free Galerkin Method (EFGM) is representative of the most important one. Therefore, the EFGM has been selected as a reference for the numerical analyses. Many of the error sources of a meshless method are contributed by its interpolation/approximation algorithm. In the EFGM, such algorithm is known as Moving Least Squares (MLS), a particular case of the Generalized Moving Least Squares (GMLS). The accuracy of the MLS is based on the following factors: order of the polynomial basis p(x), features of the weight function w(x), and shape and size of the support domain of this weight function. The individual contribution of each of these factors, along with the interactions among them, has been studied in both regular and irregular arrangement of nodes, by means of a reduction of each contribution to a one single quantifiable parameter. This assessment is applied to a range of both one- and two-dimensional benchmarking cases, and includes not only the error in terms of eigenvalues (natural frequencies or buckling load), but also of eigenvectors

A robust numerical framework for the analysis of material failure of fibre reinforced soft tissue

In this work, robustness and stability of continuum damage models applied to material failure in soft tissues are addressed. In the implicit damage models equipped with softening, the presence of negative eigenvalues in the tangent elemental matrix degrades the condition number of the global matrix, leading to a reduction of the computational performance of the numerical model. Two strategies have been adapted from literature to improve the aforementioned computational performance degradation: the IMPL-EX integration scheme [Oliver,2006], which renders the elemental matrix contribution definite positive, and arclength-type continuation methods [Carrera,1994], which allow to capture the unstable softening branch in brittle ruptures. The IMPL-EX integration scheme has as a major drawback the need to use small time steps to keep numerical error below an acceptable value. A convergence study, limiting the maximum allowed increment of internal variables in the damage model, is presented. Finally, numerical simulation of failure problems with fibre reinforced materials illustrates the performance of the adopted methodology.

Medidas autosemejantes en el plano, momentos y matrices de Hessenberg

La tesis MEDIDAS AUTOSEMEJANTES EN EL PLANO, MOMENTOS Y MATRICES DE HESSENBERG se enmarca entre las áreas de la teoría geométrica de la medida, la teoría de polinomios ortogonales y la teoría de operadores. La memoria aborda el estudio de medidas con soporte acotado en el plano complejo vistas con la óptica de las matrices infinitas de momentos y de Hessenberg asociadas a estas medidas que en la teoría de los polinomios ortogonales las representan. En particular se centra en el estudio de las medidas autosemejantes que son las medidas de equilibrio definidas por un sistema de funciones iteradas (SFI). Los conjuntos autosemejantes son conjuntos que tienen la propiedad geométrica de descomponerse en unión de piezas semejantes al conjunto total. Estas piezas pueden solaparse o no, cuando el solapamiento es pequeño la teoría de Hutchinson [Hut81] funciona bien, pero cuando no existen restricciones falla. El problema del solapamiento consiste en controlar la medida de este solapamiento. Un ejemplo de la complejidad de este problema se plantea con las convoluciones infinitas de distribuciones de Bernoulli, que han resultado ser un ejemplo de medidas autosemejantes en el caso real. En 1935 Jessen y A. Wintner [JW35] ya se planteaba este problema, lejos de ser sencillo ha sido estudiado durante más de setenta y cinco años y siguen sin resolverse las principales cuestiones planteadas ya por A. Garsia [Gar62] en 1962. El interés que ha despertado este problema así como la complejidad del mismo está demostrado por las numerosas publicaciones que abordan cuestiones relacionadas con este problema ver por ejemplo [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05],[JKS07] [JKS11]. En el primer capítulo comenzamos introduciendo con detalle las medidas autosemejante en el plano complejo y los sistemas de funciones iteradas, así como los conceptos de la teoría de la medida necesarios para describirlos. A continuación se introducen las herramientas necesarias de teoría de polinomios ortogonales, matrices infinitas y operadores que se van a usar. En el segundo y tercer capítulo trasladamos las propiedades geométricas de las medidas autosemejantes a las matrices de momentos y de Hessenberg, respectivamente. A partir de estos resultados se describen algoritmos para calcular estas matrices a partir del SFI correspondiente. Concretamente, se obtienen fórmulas explícitas y algoritmos de aproximación para los momentos y matrices de momentos de medidas fractales, a partir de un teorema del punto fijo para las matrices. Además utilizando técnicas de la teoría de operadores, se han extendido al plano complejo los resultados que G. Mantica [Ma00, Ma96] obtenía en el caso real. Este resultado es la base para definir un algoritmo estable de aproximación de la matriz de Hessenberg asociada a una medida fractal u obtener secciones finitas exactas de matrices Hessenberg asociadas a una suma de medidas. En el último capítulo, se consideran medidas, μ, más generales y se estudia el comportamiento asintótico de los autovalores de una matriz hermitiana de momentos y su impacto en las propiedades de la medida asociada. En el resultado central se demuestra que si los polinomios asociados son densos en L2(μ) entonces necesariamente el autovalor mínimo de las secciones finitas de la matriz de momentos de la medida tiende a cero. ABSTRACT The Thesis work “Self-similar Measures on the Plane, Moments and Hessenberg Matrices” is framed among the geometric measure theory, orthogonal polynomials and operator theory. The work studies measures with compact support on the complex plane from the point of view of the associated infinite moments and Hessenberg matrices representing them in the theory of orthogonal polynomials. More precisely, it concentrates on the study of the self-similar measures that are equilibrium measures in a iterated functions system. Self-similar sets have the geometric property of being decomposable in a union of similar pieces to the complete set. These pieces can overlap. If the overlapping is small, Hutchinson’s theory [Hut81] works well, however, when it has no restrictions, the theory does not hold. The overlapping problem consists in controlling the measure of the overlap. The complexity of this problem is exemplified in the infinite convolutions of Bernoulli’s distributions, that are an example of self-similar measures in the real case. As early as 1935 [JW35], Jessen and Wintner posed this problem, that far from being simple, has been studied during more than 75 years. The main cuestiones posed by Garsia in 1962 [Gar62] remain unsolved. The interest in this problem, together with its complexity, is demonstrated by the number of publications that over the years have dealt with it. See, for example, [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05], [JKS07] [JKS11]. In the first chapter, we will start with a detailed introduction to the self-similar measurements in the complex plane and to the iterated functions systems, also including the concepts of measure theory needed to describe them. Next, we introduce the necessary tools from orthogonal polynomials, infinite matrices and operators. In the second and third chapter we will translate the geometric properties of selfsimilar measures to the moments and Hessenberg matrices. From these results, we will describe algorithms to calculate these matrices from the corresponding iterated functions systems. To be precise, we obtain explicit formulas and approximation algorithms for the moments and moment matrices of fractal measures from a new fixed point theorem for matrices. Moreover, using techniques from operator theory, we extend to the complex plane the real case results obtained by Mantica [Ma00, Ma96]. This result is the base to define a stable algorithm that approximates the Hessenberg matrix associated to a fractal measure and obtains exact finite sections of Hessenberg matrices associated to a sum of measurements. In the last chapter, we consider more general measures, μ, and study the asymptotic behaviour of the eigenvalues of a hermitian matrix of moments, together with its impact on the properties of the associated measure. In the main result we demonstrate that, if the associated polynomials are dense in L2(μ), then necessarily follows that the minimum eigenvalue of the finite sections of the moments matrix goes to zero.

Symmetry considerations in the empirical k.p Hamiltonian for the study of intermediate band solar cells

With the purpose of assessing the absorption coefficients of quantum dot solar cells, symmetry considerations are introduced into a Hamiltonian whose eigenvalues are empirical. In this way, the proper transformation from the Hamiltonian's diagonalized form to the form that relates it with Γ-point exact solutions through k.p envelope functions is built accounting for symmetry. Forbidden transitions are thus determined reducing the calculation burden and permitting a thoughtful discussion of the possible options for this transformation. The agreement of this model with the measured external quantum efficiency of a prototype solar cell is found to be excellent.

Characterising emergent semantics in Twitter lists

Twitter lists organise Twitter users into multiple, often overlapping, sets. We believe that these lists capture some form of emergent semantics, which may be useful to characterise. In this paper we describe an approach for such characterisation, which consists of deriving semantic relations between lists and users by analyzing the cooccurrence of keywords in list names. We use the vector space model and Latent Dirichlet Allocation to obtain similar keywords according to co-occurrence patterns. These results are then compared to similarity measures relying on WordNet and to existing Linked Data sets. Results show that co-occurrence of keywords based on members of the lists produce more synonyms and more correlated results to that of WordNet similarity measures.

A robust numerical framework to the analysis of material failure of fibre reinforced soft tissue

In this work, robustness and stability of continuum damage models applied to material failure in soft tissues are addressed. In the implicit damage models equipped with softening, the presence of negative eigenvalues in the tangent elemental matrix degrades the condition number of the global matrix, leading to a reduction of the computational performance of the numerical model. Two strategies have been adapted from literature to improve the aforementioned computational performance degradation: the IMPL-EX integration scheme [Oliver,2006], which renders the elemental matrix contribution definite positive, and arclength-type continuation methods [Carrera,1994], which allow to capture the unstable softening branch in brittle ruptures. The IMPL-EX integration scheme has as a major drawback the need to use small time steps to keep numerical error below an acceptable value. A convergence study, limiting the maximum allowed increment of internal variables in the damage model, is presented. Finally, numerical simulation of failure problems with fibre reinforced materials illustrates the performance of the adopted methodology.

Boundary elements in potential and elasticity theory

A general theory that describes the B.I.E. linear approximation in potential and elasticity problems, is developed. A method to tread the Dirichlet condition in sharp vertex is presented. Though the study is developed for linear elements, its extension to higher order interpolation is straightforward. A new direct assembling procedure of the global of equations to be solved, is finally showed.

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.

Simulación numérica computacional de fenómenos dinámicos

En este proyecto se trata la simulación numérica de un fenómeno dinámico, basado en el comportamiento de una onda transmitida a lo largo de una cuerda elástica de un instrumento musical, cuyos extremos se encuentran anclados. El fenómeno físico, se desarrolla utilizando una ecuación en derivadas parciales hiperbólicas con variables espacial y temporal, acompañada por unas condiciones de contorno tipo Dirichlet en los extremos y por más condiciones iniciales que dan comienzo al proceso. Posteriormente se han generado algoritmos para el método numérico empleado (Diferencias finitas centrales y progresivas) y la programación del problema aproximado con su consistencia, estabilidad y convergencia, obteniéndose unos resultados acordes con la solución analítica del problema matemático. La programación y salida de resultados se ha realizado con Visual Studio 8.0. y la programación de objetos con Visual Basic .Net In this project the topic is the numerical simulation of a dynamic phenomenon, based on the behavior of a transmitted wave along an elastic string of a musical instrument, whose ends are anchored. The physical phenomenon is developed using a hyperbolic partial differential equation with spatial and temporal variables, accompanied by a Dirichlet boundary conditions at the ends and more initial conditions that start the process. Subsequently generated algorithms for the numerical method used (central and forward finite differences) and the programming of the approximate problem with consistency, stability and convergence, yielding results in line with the analytical solution of the mathematical problem. Programming and output results has been made with Visual Studio 8.0. and object programming with Visual Basic. Net

Dynamic analysis of translational shells

In this paper a dynamic analysis of transnational shells is presented. The general linear shell theory is used in conjunction with additional shallow and curved plate approximations. In order to apply some type of extended Levy solution, the shell is assumed to be limited by a rectangular plan form, with two opposite edges simply supported (gable boundary conditions). First, the shells free vibrations are studied in the usual way, obtaining for each Fourier term the natural frequencies as solutions of a transcendental equation. However, solving these equations arises enormous computational difficulties. This paper deals specifically with this problem, trying to reduce its dimension by a discretization procedure. In the shell dynamic characteristics, namely the mass. The shell mass is lumped along a family of coordinate lines. Therefore, the natural frequencies for each harmonic term can be found from the solution of a typical matrix eigenvalues problem and standard numerical techniques can be applied. The shell response to forced vibrations, particularly to earthquake excitation, can be determined by using conventional procedure either in the time or in the frequency domain. Finally, extending the above procedure, any system of translational shells under dynamic loading can be studied. Then, by using matrix methods, a general computer program is written and applied to some illustrative examples. Numerical results has been obtained in two cases: circular cylindrical shell and box girder bridge.

On the Use of Matrix-Free Shift-Invert Strategies For Global Flow Instability Analysis

A novel time-stepping shift-invert algorithm for linear stability analysis of laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix-free framework. Validations and comparisons to the classical exponential method have been performed in three different cases: (i) stenotic flow, (ii) backward-facing step and (iii) lid-driven swirling flow. Results show that this new approach speeds up the required Krylov subspace iterations and has the capability of converging to specific parts of the global spectrum. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the performance of the present method could be dramatically improved with the use of a preconditioner. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.

Energy-entropy-momentum time integration methods for coupled smooth dissipative problems

Esta tesis aborda la formulación, análisis e implementación de métodos numéricos de integración temporal para la solución de sistemas disipativos suaves de dimensión finita o infinita de manera que su estructura continua sea conservada. Se entiende por dichos sistemas aquellos que involucran acoplamiento termo-mecánico y/o efectos disipativos internos modelados por variables internas que siguen leyes continuas, de modo que su evolución es considerada suave. La dinámica de estos sistemas está gobernada por las leyes de la termodinámica y simetrías, las cuales constituyen la estructura que se pretende conservar de forma discreta. Para ello, los sistemas disipativos se describen geométricamente mediante estructuras metriplécticas que identifican claramente las partes reversible e irreversible de la evolución del sistema. Así, usando una de estas estructuras conocida por las siglas (en inglés) de GENERIC, la estructura disipativa de los sistemas es identificada del mismo modo que lo es la Hamiltoniana para sistemas conservativos. Con esto, métodos (EEM) con precisión de segundo orden que conservan la energía, producen entropía y conservan los impulsos lineal y angular son formulados mediante el uso del operador derivada discreta introducido para asegurar la conservación de la Hamiltoniana y las simetrías de sistemas conservativos. Siguiendo estas directrices, se formulan dos tipos de métodos EEM basados en el uso de la temperatura o de la entropía como variable de estado termodinámica, lo que presenta importantes implicaciones que se discuten a lo largo de esta tesis. Entre las cuales cabe destacar que las condiciones de contorno de Dirichlet son naturalmente impuestas con la formulación basada en la temperatura. Por último, se validan dichos métodos y se comprueban sus mejores prestaciones en términos de la estabilidad y robustez en comparación con métodos estándar. This dissertation is concerned with the formulation, analysis and implementation of structure-preserving time integration methods for the solution of the initial(-boundary) value problems describing the dynamics of smooth dissipative systems, either finite- or infinite-dimensional ones. Such systems are understood as those involving thermo-mechanical coupling and/or internal dissipative effects modeled by internal state variables considered to be smooth in the sense that their evolutions follow continuos laws. The dynamics of such systems are ruled by the laws of thermodynamics and symmetries which constitutes the structure meant to be preserved in the numerical setting. For that, dissipative systems are geometrically described by metriplectic structures which clearly identify the reversible and irreversible parts of their dynamical evolution. In particular, the framework known by the acronym GENERIC is used to reveal the systems' dissipative structure in the same way as the Hamiltonian is for conserving systems. Given that, energy-preserving, entropy-producing and momentum-preserving (EEM) second-order accurate methods are formulated using the discrete derivative operator that enabled the formulation of Energy-Momentum methods ensuring the preservation of the Hamiltonian and symmetries for conservative systems. Following these guidelines, two kind of EEM methods are formulated in terms of entropy and temperature as a thermodynamical state variable, involving important implications discussed throughout the dissertation. Remarkably, the formulation in temperature becomes central to accommodate Dirichlet boundary conditions. EEM methods are finally validated and proved to exhibit enhanced numerical stability and robustness properties compared to standard ones.

Discrete linear stability and sensitivity analysis of fluid flows

Este trabajo presenta un método discreto para el cálculo de estabilidad hidrodinámica y análisis de sensibilidad a perturbaciones externas para ecuaciones diferenciales y en particular para las ecuaciones de Navier-Stokes compressible. Se utiliza una aproximación con variable compleja para obtener una precisión analítica en la evaluación de la matriz Jacobiana. Además, mapas de sensibilidad para la sensibilidad a las modificaciones del flujo de base y a una fuerza constante permiten identificar las regiones del campo fluido donde una modificacin (ej. fuerza puntual) tiene un efecto estabilizador del flujo. Se presentan cuatro casos de prueba: (1) un caso analítico para comprobar la derivación discreta, (2) una cavidad cerrada a bajo Reynolds para mostrar la mayor precisión en el cálculo de los valores propios con la aproximación de paso complejo, (3) flujo 2D en un cilindro circular para validar la metodología, y (4) flujo en un cavidad abierta, presentado para validar el método en casos de inestabilidades convectivamente inestables. Los tres últimos casos mencionados (2-4) se resolvieron con las ecuaciones de Navier-Stokes compresibles, utilizando un método Discontinuous Galerkin Spectral Element Method. Se obtuvo una buena concordancia para el caso de validación (3), cuando se comparó el nuevo método con resultados de la literatura. Además, este trabajo muestra que para el cálculo de los modos propios directos y adjuntos, así como para los mapas de sensibilidad, el uso de variables complejas es de suprema importancia para obtener una predicción precisa. El método descrito es aplicado al análisis para la estabilización de la estela generada por un disco actuador, que representa un modelo sencillo para hélices, rotores de helicópteros o turbinas eólicas. Se explora la primera bifurcación del flujo para un disco actuador, y se sugiere que está asociada a una inestabilidad de tipo Kelvin-Helmholtz, cuya estabilidad se controla con en el número de Reynolds y en la resistencia del disco actuador (o fuerza resistente). En primer lugar, se verifica que la disminución de la resistencia del disco tiene un efecto estabilizador parecido a una disminución del Reynolds. En segundo lugar, el análisis hidrodinmico discreto identifica dos regiones para la colocación de una fuerza puntual que controle las inestabilidades, una cerca del disco y otra en una zona aguas abajo. En tercer lugar, se muestra que la inclusión de un forzamiento localizado cerca del actuador produce una estabilización más eficiente que al forzar aguas abajo. El análisis de los campos de flujo controlados confirma que modificando el gradiente de velocidad cerca del actuador es más eficiente para estabilizar la estela. Estos resultados podrían proporcionar nuevas directrices para la estabilización de la estela de turbinas de viento o de marea cuando estén instaladas en un parque eólico y minimizar las interacciones no estacionarias entre turbinas. ABSTRACT A discrete framework for computing the global stability and sensitivity analysis to external perturbations for any set of partial differential equations is presented. In particular, a complex-step approximation is used to achieve near analytical accuracy for the evaluation of the Jacobian matrix. Sensitivity maps for the sensitivity to base flow modifications and to a steady force are computed to identify regions of the flow field where an input could have a stabilising effect. Four test cases are presented: (1) an analytical test case to prove the theory of the discrete framework, (2) a lid-driven cavity at low Reynolds case to show the improved accuracy in the calculation of the eigenvalues when using the complex-step approximation, (3) the 2D flow past a circular cylinder at just below the critical Reynolds number is used to validate the methodology, and finally, (4) the flow past an open cavity is presented to give an example of the discrete method applied to a convectively unstable case. The latter three (2–4) of the aforementioned cases were solved with the 2D compressible Navier–Stokes equations using a Discontinuous Galerkin Spectral Element Method. Good agreement was obtained for the validation test case, (3), with appropriate results in the literature. Furthermore, it is shown that for the calculation of the direct and adjoint eigenmodes and their sensitivity maps to external perturbations, the use of complex variables is paramount for obtaining an accurate prediction. An analysis for stabilising the wake past an actuator disc, which represents a simple model for propellers, helicopter rotors or wind turbines is also presented. We explore the first flow bifurcation for an actuator disc and it suggests that it is associated to a Kelvin- Helmholtz type instability whose stability relies on the Reynolds number and the flow resistance applied through the disc (or actuator forcing). First, we report that decreasing the disc resistance has a similar stabilising effect to an decrease in the Reynolds number. Second, a discrete sensitivity analysis identifies two regions for suitable placement of flow control forcing, one close to the disc and one far downstream where the instability originates. Third, we show that adding a localised forcing close to the actuator provides more stabilisation that forcing far downstream. The analysis of the controlled flow fields, confirms that modifying the velocity gradient close to the actuator is more efficient to stabilise the wake than controlling the sheared flow far downstream. An interesting application of these results is to provide guidelines for stabilising the wake of wind or tidal turbines when placed in an energy farm to minimise unsteady interactions.