Fractais - Aplicações em Engenharia

Num universo despovoado de formas geométricas perfeitas, onde proliferam superfícies irregulares, difíceis de representar e de medir, a geometria fractal revelou-se um instrumento poderoso no tratamento de fenómenos naturais, até agora considerados erráticos, imprevisíveis e aleatórios. Contudo, nem tudo na natureza é fractal, o que significa que a geometria euclidiana continua a ser útil e necessária, o que torna estas geometrias complementares. Este trabalho centra-se no estudo da geometria fractal e na sua aplicação a diversas áreas científicas, nomeadamente, à engenharia. São abordadas noções de auto-similaridade (exata, aproximada), formas, dimensão, área, perímetro, volume, números complexos, semelhança de figuras, sucessão e iterações relacionadas com as figuras fractais. Apresentam-se exemplos de aplicação da geometria fractal em diversas áreas do saber, tais como física, biologia, geologia, medicina, arquitetura, pintura, engenharia eletrotécnica, mercados financeiros, entre outras. Conclui-se que os fractais são uma ferramenta importante para a compreensão de fenómenos nas mais diversas áreas da ciência. A importância do estudo desta nova geometria, é avassaladora graças à sua profunda relação com a natureza e ao avançado desenvolvimento tecnológico dos computadores.

Study of the electromyographic signal dynamic behavior in Amyotrophic Lateral Sclerosis (ALS)

Amyotrophic Lateral Sclerosis (ALS) is a neurodegenerative disease characterized by motor neurons degeneration, which reduces muscular force, being very difficult to diagnose. Mathematical methods are used in order to analyze the surface electromiographic signal’s dynamic behavior (Fractal Dimension (FD) and Multiscale Entropy (MSE)), evaluate different muscle group’s synchronization (Coherence and Phase Locking Factor (PLF)) and to evaluate the signal’s complexity (Lempel-Ziv (LZ) techniques and Detrended Fluctuation Analysis (DFA)). Surface electromiographic signal acquisitions were performed in upper limb muscles, being the analysis executed for instants of contraction for ipsilateral acquisitions for patients and control groups. Results from LZ, DFA and MSE analysis present capability to distinguish between the patient group and the control group, whereas coherence, PLF and FD algorithms present results very similar for both groups. LZ, DFA and MSE algorithms appear then to be a good measure of corticospinal pathways integrity. A classification algorithm was applied to the results in combination with extracted features from the surface electromiographic signal, with an accuracy percentage higher than 70% for 118 combinations for at least one classifier. The classification results demonstrate capability to distinguish members between patients and control groups. These results can demonstrate a major importance in the disease diagnose, once surface electromyography (sEMG) may be used as an auxiliary diagnose method.

Nonlinear behavior of gelatin networks reveals a hierarchical structure

We investigate the strain hardening behavior of various gelatin networks-namely physical gelatin gel, chemically cross-linked gelatin gel, and a hybrid gel made of a combination of the former two-under large shear deformations using the pre-stress, strain ramp, and large amplitude oscillations shear protocols. Further, the internal structures of physical gelatin gels and chemically cross-linked gelatin gels were characterized by small angle neutron scattering (SANS) to enable their internal structures to be correlated with their nonlinear rheology. The Kratky plots of SANS data demonstrate the presence of small cross-linked aggregates within the chemically cross-linked network whereas, in the physical gelatin gels, a relatively homogeneous structure is observed. Through model fitting to the scattering data, we were able to obtain structural parameters, such as the correlation length (ξ), the cross-sectional polymer chain radius (Rc) and the fractal dimension (df) of the gel networks. The fractal dimension df obtained from the SANS data of the physical and chemically cross-linked gels is 1.31 and 1.53, respectively. These values are in excellent agreement with the ones obtained from a generalized nonlinear elastic theory that has been used to fit the stress-strain curves. The chemical cross-linking that generates coils and aggregates hinders the free stretching of the triple helix bundles in the physical gels.

Disseny d'un acoblador direccional amb metamaterials

En aquest treball es descriu el disseny d’un acoblador direccional en anell de 1.5λ, habitualment anomenat rat-race, mitjançant metamaterials. S’utilitza una línia microstrip amb gap capacitiu en sèrie i una cel·la left-handed basada en un ressonador d’anells oberts complementari (CSRR), fent així possible una reducció de la longitud de la línia de 270º en un factor 5. Gràcies a aquesta disminució de longitud, l’àrea del dispositiu és 4 vegades menor que la del rat-race convencional. La mesura mostra que el comportament és el desitjat a la freqüència d’operació.

Multifractal portrayal of the Swiss population

Fractal geometry is a fundamental approach for describing the complex irregularities of the spatial structure of point patterns. The present research characterizes the spatial structure of the Swiss population distribution in the three Swiss geographical regions (Alps, Plateau and Jura) and at the entire country level. These analyses were carried out using fractal and multifractal measures for point patterns, which enabled the estimation of the spatial degree of clustering of a distribution at different scales. The Swiss population dataset is presented on a grid of points and thus it can be modelled as a "point process" where each point is characterized by its spatial location (geometrical support) and a number of inhabitants (measured variable). The fractal characterization was performed by means of the box-counting dimension and the multifractal analysis was conducted through the Renyi's generalized dimensions and the multifractal spectrum. Results showed that the four population patterns are all multifractals and present different clustering behaviours. Applying multifractal and fractal methods at different geographical regions and at different scales allowed us to quantify and describe the dissimilarities between the four structures and their underlying processes. This paper is the first Swiss geodemographic study applying multifractal methods using high resolution data.

Miniaturización de componentes planares de microondas mediante técnicas basadas en metamateriales: aplicación a un desfasador de cuadratura

En este trabajo se describe la miniaturización del diseño de un desfasador de cuadratura mediante líneas de transmisión metamaterial en tecnología microstrip. Para conseguir la miniaturización del diseño se utiliza la topología Spiral Resonator (SR). Los desfasadores de cuadratura han sido diseñados, fabricados y medidos. Mediante este proceso se ha consiguido un dispositivo metamaterial que presenta una reducción de un 77% del tamaño respecto a las líneas convencionales.

GNSS array: diseño de una antena con polarización circular

Aquest projecte es centra en el disseny d’una antena microstrip per a GNSS. Una antena per a GNSS ha de tenir adaptació de impedància d’entrada i polarització circular a dretes, com a principals especificacions, en el rang de 1.15-1.6 GHz. El tipus d’alimentació d’una antena microstrip amb el major ample de banda d’adaptació és l’alimentació mitjançant acoblament per apertura. Si a l’antena s’introdueixen dos apertures de forma ortogonal, alimentades amb un desfasament de 90º entre elles, s’aconsegueix polarització circular. L’opció de separar les apertures redueix la transferència de potència entre elles, i disminueix el guany de polarització creuada. La xarxa d’alimentació dissenyada és un divisor de Wilkinson amb una línia de λ/4 a la freqüència central, encara que el desfasament als extrems de la banda no sigui de 90º. Com a xarxa d’alimentació es va provar un hibrid de 90º, però l’elevat valor del paràmetre S21 de l’antena impossibilita l’adaptació a l’entrada del hibrid.

Implementación de resonadores serie en derivación mediante stubs en derivación de salto de impedancia: análisis y limitaciones

En aquest projecte s'analitzen les limitacions de la implementació de ressonadors sèrie en derivació en tecnologia microstrip mitjançant un stub en derivació de salt d'impedància (SISS). L'esmentada estructura està composada per una línea d'alta impedància carregada amb una línea de baixa impedància acabada en circuit obert. El seu comportament en freqüència és equivalent al d'un ressonador sèrie en derivació sempre i quan les longituds de les seves línies siguin elèctricament petites. Sota aquestes condicions, bàsicament, la línia d'alta impedància sintetitza una inductància, mentre que la línia de baixa impedància una capacitat. La limitació en els valors mínim i màxim de la impedància característica que es poden implementar implica una limitació sobre la inductància i la capacitat que es poden sintetitzar mitjançant el SISS.

Desfasador de dos estats

El projecte es basa en el disseny i la fabricació d’un circuit desfasador, amb una entrada i una sortida, que commuta entre dos estats, un estat amb desfasament de 180º i l’altre sense desfasament. El circuit es dissenya amb microstrip,per una fc=5GHz. S’intenten obtenir les millors característiques per aquest disseny, és a dir, bon ample de banda i pèrdues per inserció baixes, acompanyat d’una bona resposta en les relacions de magnitud i fase. Es segueixen diferents etapes de disseny, on es comença per un model simple i s’avança en complexitat, afegint els nous components que acabaran conformant el circuit final. Després del disseny es passa a la fabricació del circuit, per veure el seu funcionament real. La memòria recull i ordena la informació obtinguda a través d’aquest procés, intentant mostrar-la de manera clara, per tal de seguir el procés de disseny realitzat, i així poder interpretar els resultats obtinguts. L’objectiu final és veure com es comporta el circuit dissenyat i definir les pautes a seguir per millorar-lo en un futur.

Estudio de parámetros circuitales de resonadores

El objetivo de este proyecto consiste en el estudio de los parámetros circuitales (condensadores, bobinas…) de un resonador, realizado con estructuras microstrip, donde permita obtener unos resultados de esos parámetros circuitales cambiando los valores físicos del diseño, tales como la longitud y la anchura del resonador a partir de las medidas de los parámetros S. Para llevar a cabo dicho trabajo, se desarrolla en primer lugar toda la teoría necesaria de resonadores. Empezando por el funcionamiento y la estructura del resonador diseñado, y mostrando especial interés en el modelado de dicho resonador. Seguidamente, se estudia y analiza su comportamiento a través de las simulaciones de los parámetros S. Una vez se ha estudiado y analizado su comportamiento, se procede con las modificaciones de los parámetros físicos y se analiza a través de las simulaciones de los parámetros S cómo afectan estas modificaciones en los parámetros circuitales. Donde se utilizan una serie de herramientas que agilizan la extracción de los valores de los parámetros circuitales del resonador.

Spatial Analysis of Cell Composition Data

A version of Matheron’s discrete Gaussian model is applied to cell composition data.The examples are for map patterns of felsic metavolcanics in two different areas. Q-Qplots of the model for cell values representing proportion of 10 km x 10 km cell areaunderlain by this rock type are approximately linear, and the line of best fit can be usedto estimate the parameters of the model. It is also shown that felsic metavolcanics in theAbitibi area of the Canadian Shield can be modeled as a fractal

Kinematics and dynamic complexity of postural transitions in frail elderly subjects.

The aim of this study was to propose a methodology allowing a detailed characterization of body sit-to-stand/stand-to-sit postural transition. Parameters characterizing the kinematics of the trunk movement during sit-to-stand (Si-St) postural transition were calculated using one initial sensor system fixed on the trunk and a data logger. Dynamic complexity of these postural transitions was estimated by fractal dimension of acceleration-angular velocity plot. We concluded that this method provides a simple and accurate tool for monitoring frail elderly and to objectively evaluate the efficacy of a rehabilitation program.

Multi-parametric evaluation of sit-to-stand and stand-to-sit transitions in elderly people.

The aim of this study was to extract multi-parametric measures characterizing different features of sit-to-stand (Si-St) and stand-to-sit (St-Si) transitions in older persons, using a single inertial sensor attached to the chest. Investigated parameters were transition's duration, range of trunk tilt, smoothness of transition pattern assessed by its fractal dimension, and trunk movement's dynamic described by local wavelet energy. A measurement protocol with a Si-St followed by a St-Si postural transition was performed by two groups of participants: the first group (N=79) included Frail Elderly subjects admitted to a post-acute rehabilitation facility and the second group (N=27) were healthy community-dwelling elderly persons. Subjects were also evaluated with Tinetti's POMA scale. Compared to Healthy Elderly persons, frail group at baseline had significantly longer Si-St (3.85±1.04 vs. 2.60±0.32, p=0.001) and St-Si (4.08±1.21 vs. 2.81±0.36, p=0.001) transition's duration. Frail older persons also had significantly decreased smoothness of Si-St transition pattern (1.36±0.07 vs. 1.21±0.05, p=0.001) and dynamic of trunk movement. Measurements after three weeks of rehabilitation in frail older persons showed that smoothness of transition pattern had the highest improvement effect size (0.4) and discriminative performance. These results demonstrate the potential interest of such parameters to distinguish older subjects with different functional and health conditions.

Dynamical features of reaction-diffusion fronts in fractals

The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration

Speed of reaction-diffusion fronts in spatially heterogeneous media

The front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis, the geometric approach of Hamilton-Jacobi dynamics, and the local speed approach. Exact and perturbed expressions for the front speed are obtained in the limit of large times. For linear and fractal heterogeneities, the analytic results have been compared with numerical results exhibiting a good agreement. Finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities