Theory of intermittency applied to classical pathological cases

The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Though it works fine in some model systems, there exist a number of so-called pathological cases characterized by a significant deviation of main characteristics from the values predicted on the basis of the uniform distribution. Recently, we reported on how the reinjection probability density (RPD) can be generalized. Here, we extend this methodology and apply it to different dynamical systems exhibiting anomalous type-II and type-III intermittencies. Estimation of the universal RPD is based on fitting a linear function to experimental data and requires no a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets (dozens of points). Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. We show that the method copes well with dynamical systems exhibiting significantly different statistics reported in the literature. We also derive and classify characteristic relations between the mean laminar length and main controlling parameter in perfect agreement with data provided by numerical simulations

Análisis del estado de tensiones en uniones carpinteras de empalme de llave por el método de los elementos finitos

El objeto del presente trabajo es la determinación de la distribución de tensiones en la unión carpintera de empalme de llave mediante el método de los elementos finitos y su comparación con los valores obtenidos mediante la teoría de Resistencia de Materiales. Se analizan las zonas donde se produce concentración de tensiones y se estudia la influencia del refino de la malla sobre los resultados con el fin de determinar el tamaño que proporciona los valores de tensión más ajustados a la teoría. En las zonas donde la concentración de tensiones es menor, distintos tamaños de la malla muestran valores de tensión similares. Los resultados muestran una simetría central de la distribución de isotensiones donde el centro de simetría se corresponde con el centro geométrico de la unión. En las zonas donde se produce concentración de tensiones, los valores de las mismas aumentan considerablemente con el refino de la malla. La comparación de los niveles de tensiones normales obtenidos por el MEF y la teoría clásica muestran diferencias reducidas, excepto en los puntos de concentración de tensiones. The purpose of this study is to determine the stress distribution in the joint carpenter of halved and tabled joint with the finite element method and its comparison with the values obtained using the theory of Strength of Materials. The stress concentration areas where analyzed and the influence of mesh refinement was studied on the results in order to determine the mesh size that provides the stress values more consistent with the theory. In areas where stress concentration is lower, different mesh sizes show similar stress values. The results show a central symmetry of the isobar lines distribution where the centre of symmetry corresponds to the geometric centre of the joint. In areas where stress concentration occurs, the same values increase considerably with the refinement of the mesh. Comparison of normal stress levels obtained by the FEM and the classical theory shows small differences, except at points of stress concentration.

Analysis of the stress state of a halved and tabled traditional timber scarf joint with the Finite Element Method

The purpose of this study is to determine the stress distribution in the carpentry joint of halved and tabled scarf joint with the finite element method (FEM) and its comparison with the values obtained using the theory of Strength of Materials. The stress concentration areas where analyzed and the influence of mesh refinement was studied on the results in order to determine the mesh size that provides the stress values more consistent with the theory. In areas where stress concentration is lower, different mesh sizes show similar stress values. In areas where stress concentration occurs, the same values increase considerably with the refinement of the mesh. The results show a central symmetry of the isobar lines distribution where the centre of symmetry corresponds to the geometric centre of the joint. Comparison of normal stress levels obtained by the FEM and the classical theory shows small differences, except at points of stress concentration.

Modelización de los Procesos de Cristalización en Poliolefinas

“Por lo tanto, la cristalización de polímeros se supone, y en las teorías se describe a menudo, como un proceso de múltiples pasos con muchos aspectos físico-químicos y estructurales influyendo en él. Debido a la propia estructura de la cadena, es fácil entender que un proceso que es termodinámicamente forzado a aumentar su ordenamiento local, se vea obstaculizado geométricamente y, por tanto, no puede conducirse a un estado de equilibrio final. Como resultado, se forman habitualmente estructuras de no equilibrio con diferentes características dependiendo de la temperatura, presión, cizallamiento y otros parámetros físico-químicos del sistema”. Estas palabras, pronunciadas recientemente por el profesor Bernhard Wunderlich, uno de los mas relevantes fisico-quimicos que han abordado en las ultimas décadas el estudio del estado físico de las macromoléculas, adelantan lo que de alguna manera se explicita en esta memoria y constituyen el “leitmotiv” de este trabajo de tesis. El mecanismo de la cristalización de polímeros esta aun bajo debate en la comunidad de la física de polímeros y la mayoría de los abordajes experimentales se explican a través de la teoría LH. Esta teoría clásica debida a Lauritzen y Hoffman (LH), y que es una generalización de la teoría de cristalización de una molécula pequeña desde la fase de vapor, describe satisfactoriamente muchas observaciones experimentales aunque esta lejos de explicar el complejo fenómeno de la cristalización de polímeros. De hecho, la formulación original de esta teoría en el National Bureau of Standards, a comienzos de la década de los 70, sufrió varias reformulaciones importantes a lo largo de la década de los 80, buscando su adaptación a los hallazgos experimentales. Así nació el régimen III de cristalización que posibilita la creacion de nichos moleculares en la superficie y que dio pie al paradigma ofrecido por Sadler y col., para justificar los experimentos que se obtenian por “scattering” de neutrones y otras técnicas como la técnica de “droplets” o enfriamiento rapido. Por encima de todo, el gran éxito de la teoría radica en que explica la dependencia inversa entre el tamaño del plegado molecular y el subenfriamiento, definido este ultimo como el intervalo de temperatura que media entre la temperatura de equilibrio y la temperatura de cristalización. El problema concreto que aborda esta tesis es el estudio de los procesos de ordenamiento de poliolefinas con distinto grado de ramificacion mediante simulaciones numéricas. Los copolimeros estudiados en esta tesis se consideran materiales modelo de gran homogeneidad molecular desde el punto de vista de la distribución de tamaños y de ramificaciones en la cadena polimérica. Se eligieron estas poliolefinas debido al gran interes experimental en conocer el cambio en las propiedades fisicas de los materiales dependiendo del tipo y cantidad de comonomero utilizado. Además, son modelos sobre los que existen una ingente cantidad de información experimental, que es algo que preocupa siempre al crear una realidad virtual como es la simulación. La experiencia en el grupo Biophym es que los resultados de simulación deben de tener siempre un correlato mas o menos próximo experimental y ese argumento se maneja a lo largo de esta memoria. Empíricamente, se conoce muy bien que las propiedades físicas de las poliolefinas, en suma dependen del tipo y de la cantidad de ramificaciones que presenta el material polimérico. Sin embargo, tal como se ha explicado no existen modelos teóricos adecuados que expliquen los mecanismos subyacentes de los efectos de las ramas. La memoria de este trabajo es amplia por la complejidad del tema. Se inicia con una extensa introducción sobre los conceptos básicos de una macromolecula que son relevantes para entender el contenido del resto de la memoria. Se definen los conceptos de macromolecula flexible, distribuciones y momentos, y su comportamiento en disolución y fundido con los correspondientes parametros caracteristicos. Se pone especial énfasis en el concepto de “entanglement” o enmaranamiento por considerarse clave a la hora de tratar macromoléculas con una longitud superior a la longitud critica de enmaranamiento. Finaliza esta introducción con una reseña sobre el estado del arte en la simulación de los procesos de cristalización. En un segundo capitulo del trabajo se expone detalladamente la metodología usada en cada grupo de casos. En el primer capitulo de resultados, se discuten los estudios de simulación en disolución diluida para sistemas lineales y ramificados de cadena única. Este caso mas simple depende claramente del potencial de torsión elegido tal como se discute a lo largo del texto. La formación de los núcleos “babys” propuestos por Muthukumar parece que son consecuencia del potencial de torsión, ya que este facilita los estados de torsión mas estables. Así que se propone el análisis de otros potenciales que son igualmente utilizados y los resultados obtenidos sobre la cristalización, discutidos en consecuencia. Seguidamente, en un segundo capitulo de resultados se estudian moleculas de alcanos de cadena larga lineales y ramificados en un fundido por simulaciones atomisticas como un modelo de polietileno. Los resultados atomisticos pese a ser de gran detalle no logran captar en su totalidad los efectos experimentales que se observan en los fundidos subenfriados en su etapa previa al estado ordenado. Por esta razon se discuten en los capítulos 3 y 4 de resultados sistemas de cadenas cortas y largas utilizando dos modelos de grano grueso (CG-PVA y CG-PE). El modelo CG-PE se desarrollo durante la tesis. El uso de modelos de grano grueso garantiza una mayor eficiencia computacional con respecto a los modelos atomísticos y son suficientes para mostrar los fenómenos a la escala relevante para la cristalización. En todos estos estudios mencionados se sigue la evolución de los procesos de ordenamiento y de fusión en simulaciones de relajación isoterma y no isoterma. Como resultado de los modelos de simulación, se han evaluado distintas propiedades fisicas como la longitud de segmento ordenado, la cristalinidad, temperaturas de fusion/cristalizacion, etc., lo que permite una comparación con los resultados experimentales. Se demuestra claramente que los sistemas ramificados retrasan y dificultan el orden de la cadena polimérica y por tanto, las regiones cristalinas ordenadas decrecen al crecer las ramas. Como una conclusión general parece mostrarse una tendencia a la formación de estructuras localmente ordenadas que crecen como bloques para completar el espacio de cristalización que puede alcanzarse a una temperatura y a una escala de tiempo determinada. Finalmente hay que señalar que los efectos observados, estan en concordancia con otros resultados tanto teoricos/simulacion como experimentales discutidos a lo largo de esta memoria. Su resumen se muestra en un capitulo de conclusiones y líneas futuras de investigación que se abren como consecuencia de esta memoria. Hay que mencionar que el ritmo de investigación se ha acentuado notablemente en el ultimo ano de trabajo, en parte debido a las ventajas notables obtenidas por el uso de la metodología de grano grueso que pese a ser muy importante para esta memoria no repercute fácilmente en trabajos publicables. Todo ello justifica que gran parte de los resultados esten en fase de publicación. Abstract “Polymer crystallization is therefore assumed, and in theories often described, to be a multi step process with many influencing aspects. Because of the chain structure, it is easy to understand that a process which is thermodynamically forced to increase local ordering but is geometrically hindered cannot proceed into a final equilibrium state. As a result, nonequilibrium structures with different characteristics are usually formed, which depend on temperature, pressure, shearing and other parameters”. These words, recently written by Professor Bernhard Wunderlich, one of the most prominent researchers in polymer physics, put somehow in value the "leitmotiv "of this thesis. The crystallization mechanism of polymers is still under debate in the physics community and most of the experimental findings are still explained by invoking the LH theory. This classical theory, which was initially formulated by Lauritzen and Hoffman (LH), is indeed a generalization of the crystallization theory for small molecules from the vapor phase. Even though it describes satisfactorily many experimental observations, it is far from explaining the complex phenomenon of polymer crystallization. This theory was firstly devised in the early 70s at the National Bureau of Standards. It was successively reformulated along the 80s to fit the experimental findings. Thus, the crystallization regime III was introduced into the theory in order to explain the results found in neutron scattering, droplet or quenching experiments. This concept defines the roughness of the crystallization surface leading to the paradigm proposed by Sadler et al. The great success of this theory is the ability to explain the inverse dependence of the molecular folding size on the supercooling, the latter defined as the temperature interval between the equilibrium temperature and the crystallization temperature. The main scope of this thesis is the study of ordering processes in polyolefins with different degree of branching by using computer simulations. The copolymers studied along this work are considered materials of high molecular homogeneity, from the point of view of both size and branching distributions of the polymer chain. These polyolefins were selected due to the great interest to understand their structure– property relationships. It is important to note that there is a vast amount of experimental data concerning these materials, which is essential to create a virtual reality as is the simulation. The Biophym research group has a wide experience in the correlation between simulation data and experimental results, being this idea highly alive along this work. Empirically, it is well-known that the physical properties of the polyolefins depend on the type and amount of branches presented in the polymeric material. However, there are not suitable models to explain the underlying mechanisms associated to branching. This report is extensive due to the complexity of the topic under study. It begins with a general introduction to the basics concepts of macromolecular physics. This chapter is relevant to understand the content of the present document. Some concepts are defined along this section, among others the flexibility of macromolecules, size distributions and moments, and the behavior in solution and melt along with their corresponding characteristic parameters. Special emphasis is placed on the concept of "entanglement" which is a key item when dealing with macromolecules having a molecular size greater than the critical entanglement length. The introduction finishes with a review of the state of art on the simulation of crystallization processes. The second chapter of the thesis describes, in detail, the computational methodology used in each study. In the first results section, we discuss the simulation studies in dilute solution for linear and short chain branched single chain models. The simplest case is clearly dependent on the selected torsion potential as it is discussed throughout the text. For example, the formation of baby nuclei proposed by Mutukhumar seems to result from the effects of the torsion potential. Thus, we propose the analysis of other torsion potentials that are also used by other research groups. The results obtained on crystallization processes are accordingly discussed. Then, in a second results section, we study linear and branched long-chain alkane molecules in a melt by atomistic simulations as a polyethylene-like model. In spite of the great detail given by atomistic simulations, they are not able to fully capture the experimental facts observed in supercooled melts, in particular the pre-ordered states. For this reason, we discuss short and long chains systems using two coarse-grained models (CG-PVA and CG-PE) in section 3 and 4 of chapter 2. The CG-PE model was developed during the thesis. The use of coarse-grained models ensures greater computational efficiency with respect to atomistic models and is enough to show the relevant scale phenomena for crystallization. In all the analysis we follow the evolution of the ordering and melting processes by both isothermal and non isothermal simulations. During this thesis we have obtained different physical properties such as stem length, crystallinity, melting/crystallization temperatures, and so on. We show that branches in the chains cause a delay in the crystallization and hinder the ordering of the polymer chain. Therefore, crystalline regions decrease in size as branching increases. As a general conclusion, it seems that there is a tendency in the macromolecular systems to form ordered structures, which can grown locally as blocks, occupying the crystallization space at a given temperature and time scale. Finally it should be noted that the observed effects are consistent with both, other theoretical/simulation and experimental results. The summary is provided in the conclusions chapter along with future research lines that open as result of this report. It should be mentioned that the research work has speeded up markedly in the last year, in part because of the remarkable benefits obtained by the use of coarse-grained methodology that despite being very important for this thesis work, is not easily publishable by itself. All this justify that most of the results are still in the publication phase.

Adaptive control system based on lineal control theory for the path-following problem of a car-like mobile robot

The objective of this paper is to design a path following control system for a car-like mobile robot using classical linear control techniques, so that it adapts on-line to varying conditions during the trajectory following task. The main advantages of the proposed control structure is that well known linear control theory can be applied in calculating the PID controllers to full control requirements, while at the same time it is exible to be applied in non-linear changing conditions of the path following task. For this purpose the Frenet frame kinematic model of the robot is linearised at a varying working point that is calculated as a function of the actual velocity, the path curvature and kinematic parameters of the robot, yielding a transfer function that varies during the trajectory. The proposed controller is formed by a combination of an adaptive PID and a feed-forward controller, which varies accordingly with the working conditions and compensates the non-linearity of the system. The good features and exibility of the proposed control structure have been demonstrated through realistic simulations that include both kinematics and dynamics of the car-like robot.

SMS design and aberration theory

The SMS, Simultaneous Multiple Surfaces, design was born to Nonimaging Optics applications and is now being applied also to Imaging Optics. In this paper the wave aberration function of a selected SMS design is studied. It has been found the SMS aberrations can be analyzed with a little set of parameters, sometimes two. The connection of this model with the conventional aberration expansion is also presented. To verify these mathematical model two SMS design systems were raytraced and the data were analyzed with a classical statistical methods: the plot of discrepancies and the quadratic average error. Both the tests show very good agreement with the model for our systems.

Potential theory

In this chapter we will introduce the reader to the techniques of the Boundary Element Method applied to simple Laplacian problems. Most classical applications refer to electrostatic and magnetic fields, but the Laplacian operator also governs problems such as Saint-Venant torsion, irrotational flow, fluid flow through porous media and the added fluid mass in fluidstructure interaction problems. This short list, to which it would be possible to add many other physical problems governed by the same equation, is an indication of the importance of the numerical treatment of the Laplacian operator. Potential theory has pioneered the use of BEM since the papers of Jaswon and Hess. An interesting introduction to the topic is given by Cruse. In the last five years a renaissance of integral methods has been detected. This can be followed in the books by Jaswon and Symm and by Brebbia or Brebbia and Walker.In this chapter we shall maintain an elementary level and follow a classical scheme in order to make the content accessible to the reader who has just started to study the technique. The whole emphasis has been put on the socalled "direct" method because it is the one which appears to offer more advantages. In this section we recall the classical concepts of potential theory and establish the basic equations of the method. Later on we discuss the discretization philosophy, the implementation of different kinds of elements and the advantages of substructuring which is unavoidable when dealing with heterogeneous materials.

Diagnostics of classical and dusty plasmas w ith collecting and emissive langmuir probes = Diagnosis de plasmas clásicos y granulares mediante sondas colectoras y emisoras de Langmuir

Las sondas eléctricas se emplean habitualmente en la diagnosis de plasmas. La presente tesis aborda la operación de las sondas colectoras y emisoras de Langmuir en plasmas fríos de baja densidad. El estudio se ha centrado en la determinación del potencial de plasma, Vsp, mediante el potencial flotante de una sonda emisora. Esta técnica consiste en la medida del potencial de la sonda correspondiente a la condición de corriente neta igual a cero, el cual se denomina potencial flotante, VF. Este potencial se desplaza hacia el potencial del plasma según aumenta la emisión termoiónica de la sonda, hasta que se satura cerca de Vsp. Los experimentos llevados a cabo en la pluma de plasma de un motor iónico y en un plasma de descarga de glow muestran que la corriente de electrones termoiónicos es mayor que la corriente de electrones recogidos para una sonda polarizada por debajo del potencial del plasma, resultado inconsistente con la teoría tradicionalmente aceptada. Para investigar estos resultados se ha introducido el parámetro R, definido como el cociente entre la corriente de electrones emitidos y recogidos por la sonda. Este parámetro, que está relacionado con la diferencia de potencial VF - Vsp, también es útil para la descripción de los modos de operación de la sonda emisora (débil, fuerte y más allá del fuerte). Los resultados experimentales evidencian que, al contrario de lo que indica la teoría, R es mayor que la unidad. Esta discrepancia se puede solucionar introduciendo una población efectiva de electrones. Con dicha población, el nuevo modelo para la corriente total de la sonda reproduce los datos experimentales. El origen de este grupo electrónico es todavía una cuestión abierta, pero podría estar originada por una nueva estructura de potencial cerca de la sonda cuando ésta trabaja en el régimen de emisión fuerte. Para explicar dicha estructura de potencial, se propone un modelo unidimensional compuesto por un mínimo de potencial cerca de la superficie de la sonda. El análisis numérico indica que este pozo de potencial aparece para muy altas temperaturas de la sonda, reduciendo la cantidad de electrones emitidos que alcanzan el plasma y evitando así cualquier posible perturbación de éste. Los aspectos experimentales involucrados en el método del potencial flotante también se han estudiado, incluyendo cuestiones como las diferentes técnicas de obtención del VF, el cociente señal-ruido, el acoplamiento de la señal de los equipos utilizados para la obtención de las curvas I-V o la evidencia experimental de los diferentes modos de operación de la sonda. Estas evidencias empíricas se encuentran en todos los aspectos de operación de la sonda: la recolección de electrones, el potencial flotante, la precisión en las curvas I-V y la emisión electrónica. Ésta última también se estudia en la tesis, debido a que un fenómeno de super emisión tiene lugar en el régimen de emisión fuerte. En este modo de operación, las medidas experimentales indican que las corrientes termoiónicas de electrones son mayores que aquéllas predichas por la ecuación de Richardson-Dushman clásica. Por último, la diagnosis de plasmas usando sondas eléctrica bajo presencia de granos de polvo (plasmas granulares) en plasmas fríos de baja densidad también se ha estudiado, mediante la aplicación numérica de la técnica del potencial flotante de la sonda emisora en un plasma no convencional. Los resultados apuntan a que el potencial flotante de una sonda emisora se vería afectado por altas densidades de polvo o grandes partículas. ABSTRACT Electric probes are widely employed for plasma diagnostics. This dissertation concerns the operation of collecting and emissive Langmuir probes in low density cold plasmas. The study is focused on the determination of the plasma potential, Vsp, by means of the floating potential of emissive probes. This technique consists of the measurement of the probe potential, corresponding to the zero net probe current, which is the so-called floating potential, VF . This potential displaces towards the plasma potential as the thermionic electron emission increases, until it saturates near Vsp. Experiments carried out in the plasma plume of an ion thruster and in a glow discharge plasma show the thermionic electron current of the emissive Langmuir probe is higher than the collected electron current, for a probe with a bias potential below Vsp, which is inconsistent with the traditional accepted theory. To investigate these results, a parameter R is introduced as the ratio between the emitted and the collected electron current. This parameter, which is related to the difference VF - Vsp, is also useful for the description of the operation modes of the emissive Langmuir probe (weak, strong and beyond strong). The experimental results give an inconsistency of R > 1, which is solved by a modification of the theory for emissive probes, with the introduction of an effective electron population. With this new electron group, the new model for the total probe current agrees with the experimental data. The origin of this electron group remains an open question, but it might be originated by a new potential structure near the emissive probe when it operates in the strong emission regime. A simple one-dimension model composed by a minimum of potential near the probe surface is discussed for strongly emitting emissive probes. The results indicate that this complex potential structure appears for very high probe temperatures and the potential well might reduce the emitted electrons population reaching the plasma bulk. The experimental issues involved in the floating potential method are also studied, as the different obtaining techniques of VF, the signal-to-noise ratio, the signal coupling of the I-V curve measurement system or the experimental evidence of the probe operation modes. These empirical proofs concern all the probe operation aspects: the electron collection, the floating potential, the I-V curve accuracy as well as the electron emission. This last issue is also investigated in this dissertation, because a super emission takes place in the strong emission regime. In this operation mode, the experimental results indicate that the thermionic electron currents might be higher than those predicted by the classical Richardson-Dushman equation. Finally, plasma diagnosis using electric probes in the presence of dust grains (dusty plasmas) in low density cold plasmas is also addressed. The application of the floating potential technique of the emissive probe in a non-conventional complex plasma is numerically investigated, whose results point out the floating potential of the emissive probe might be shifted for high dust density or large dust particles.

On sonic anemometer measurement theory

In this paper a model for the measuring process of sonic anemometers (ultrasound pulse based) is presented. The differential equations that describe the travel of ultrasound pulses are solved in the general case of non-steady, non-uniform atmospheric flow field. The concepts of instantaneous line-average and travelling pulse-referenced average are established and employed to explain and calculate the differences between the measured turbulent speed (travelling pulse-referenced average) and the line-averaged one. The limit k1l=1 established by Kaimal in 1968, as the maximum value which permits the neglect of the influence of the sonic measuring process on the measurement of turbulent components is reviewed here. Three particular measurement cases are analysed: A non-steady, uniform flow speed field, a steady, non-uniform flow speed field and finally an atmospheric flow speed field. In the first case, for a harmonic time-dependent flow field, Mach number, M (flow speed to sound speed ratio) and time delay between pulses have revealed themselves to be important parameters in the behaviour of sonic anemometers, within the range of operation. The second case demonstrates how the spatial non-uniformity of the flow speed field leads to an influence of the finite transit time of the pulses (M≠0) even in the absence of non-steady behaviour of the wind speed. In the last case, a model of the influence of the sonic anemometer processes on the measurement of wind speed spectral characteristics is presented. The new solution is compared to the line-averaging models existing in the literature. Mach number and time delay significantly distort the measurement in the normal operational range. Classical line averaging solutions are recovered when Mach number and time delay between pulses go to zero in the new proposed model. The results obtained from the mathematical model have been applied to the calculation of errors in different configurations of practical interest, such as an anemometer located on a meteorological mast and the transfer function of a sensor in an atmospheric wind. The expressions obtained can be also applied to determine the quality requirements of the flow in a wind tunnel used for ultrasonic anemometer calibrations.