Analysis of the Van der Pol oscillator containing derivatives of fractional order

In this paper a modified version of the classical Van der Pol oscillator is proposed, introducing fractional-order time derivatives into the state-space model. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, using phase portraits, spectral analysis and bifurcation diagrams. The fractional-order dynamics is illustrated through numerical simulations of the proposed schemes using approximations to fractional-order operators. Finally, the analysis is extended to the forced Van der Pol oscillator.

Experimental and numerical study on the evaluation of ventilation efficiency

Buildings account for 40% of total energy consumption in the European Union. The reduction of energy consumption in the buildings sector constitute an important measure needed to reduce the Union's energy dependency and greenhouse gas emissions. The Portuguese legislation incorporate this principles in order to regulate the energy performance of buildings. This energy performance should be accompanied by good conditions for the occupants of the buildings. According to EN 15251 (2007) the four factors that affect the occupant comfort in the buildings are: Indoor Air Quality (IAQ), thermal comfort, acoustics and lighting. Ventilation directly affects all except the lighting, so it is crucial to understand the performance of it. The ventilation efficiency concept therefore earn significance, because it is an attempt to quantify a parameter that can easily distinguish the different options for air diffusion in the spaces. The two indicators most internationally accepted are the Air Change Efficiency (ACE) and the Contaminant Removal Effectiveness (CRE). Nowadays with the developed of the Computational Fluid Dynamics (CFD) the behaviour of ventilation can be more easily predicted. Thirteen strategies of air diffusion were measured in a test chamber through the application of the tracer gas method, with the objective to validate the calculation by the MicroFlo module of the IES-VE software for this two indicators. The main conclusions from this work were: that the values of the numerical simulations are in agreement with experimental measurements; the value of the CRE is more dependent of the position of the contamination source, that the strategy used for the air diffusion; the ACE indicator is more appropriate for quantifying the quality of the air diffusion; the solutions to be adopted, to maximize the ventilation efficiency should be, the schemes that operate with low speeds of supply air and small differences between supply air temperature and the room temperature.

Numerical solution of a PDE system with non-linear steady state conditions that translates the air stripping pollutants removal

This work deals with the numerical simulation of air stripping process for the pre-treatment of groundwater used in human consumption. The model established in steady state presents an exponential solution that is used, together with the Tau Method, to get a spectral approach of the solution of the system of partial differential equations associated to the model in transient state.

Strap repairs using embedded patches: numerical analysis and experimental results

Adhesively bonded repairs offer an attractive option for repair of aluminium structures, compared to more traditional methods such as fastening or welding. The single-strap (SS) and double-strap (DS) repairs are very straightforward to execute but stresses in the adhesive layer peak at the overlap ends. The DS repair requires both sides of the damaged structures to be reachable for repair, which is often not possible. In strap repairs, with the patches bonded at the outer surfaces, some limitations emerge such as the weight, aerodynamics and aesthetics. To minimize these effects, SS and DS repairs with embedded patches were evaluated in this work, such that the patches are flush with the adherends. For this purpose, in this work standard SS and DS repairs, and also with the patches embedded in the adherends, were tested under tension to allow the optimization of some repair variables such as the overlap length (LO) and type of adhesive, thus allowing the maximization of the repair strength. The effect of embedding the patch/patches on the fracture modes and failure loads was compared with finite elements (FE) analysis. The FE analysis was performed in ABAQUS® and cohesive zone modelling was used for the simulation of damage onset and growth in the adhesive layer. The comparison with the test data revealed an accurate prediction for all kinds of joints and provided some principles regarding this technique.

Numerical and Experimental Analysis of Balanced and Unbalanced Adhesive Single-Lap Joints between Aluminium Adherends

The single-lap joint is the most commonly used, although it endures significant bending due to the non-collinear load path, which negatively affects its load bearing capabilities. The use of material or geometric changes is widely documented in the literature to reduce this handicap, acting by reduction of peel and shear peak stresses or alterations of the failure mechanism emerging from local modifications. In this work, the effect of using different thickness adherends on the tensile strength of single-lap joints, bonded with a ductile and brittle adhesive, was numerically and experimentally evaluated. The joints were tested under tension for different combinations of adherend thickness. The effect of the adherends thickness mismatch on the stress distributions was also investigated by Finite Elements (FE), which explained the experimental results and the strength prediction of the joints. The numerical study was made by FE and Cohesive Zone Modelling (CZM), which allowed characterizing the entire fracture process. For this purpose, a FE analysis was performed in ABAQUS® considering geometric non-linearities. In the end, a detailed comparative evaluation of unbalanced joints, commonly used in engineering applications, is presented to give an understanding on how modifications in the bonded structures thickness can influence the joint performance.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

Desenvolvimento de um procedimento de Numerical Site Calibration para um Parque Eólico usando Dinâmica de Fluidos Computacional

Esta dissertação descreve o desenvolvimento e avaliação de um procedimento de \Numerical Site Calibration" (NSC) para um Parque Eólico, situado a sul de Portugal, usando Dinâmica de Fluídos Computacional (CFD). O NSC encontra-se baseado no \Site Calibration" (SC), sendo este um método de medição padronizado pela Comissão Electrónica Internacional através da norma IEC 61400. Este método tem a finalidade de quantificar e reduzir os efeitos provocados pelo terreno e por possíveis obstáculos, na medição do desempenho energético das turbinas eólicas. Assim, no SC são realizadas medições em dois pontos, no mastro referência e no local da turbina (mastro temporário). No entanto, em Parques Eólicos já construídos, este método não é aplicável visto ser necessária a instalação de um mastro de medição no local da turbina e, por conseguinte, o procedimento adequado para estas circunstâncias é o NSC. O desenvolvimento deste método é feito por um código CFD, desenvolvido por uma equipa de investigação do Instituto Superior de Engenharia do Porto, designado de WINDIETM, usado extensivamente pela empresa Megajoule Inovação, Lda em aplicações de energia eólica em todo mundo. Este código é uma ferramenta para simulação de escoamentos tridimensionais em terrenos complexos. As simulações do escoamento são realizadas no regime transiente utilizando as equações de Navier-Stokes médias de Reynolds com aproximação de Bussinesq e o modelo de turbulência TKE 1.5. As condições fronteira são provenientes dos resultados de uma simulação realizada com Weather Research and Forecasting, WRF. Estas simulações dividem-se em dois grupos, um dos conjuntos de simulações utiliza o esquema convectivo Upwind e o outro utiliza o esquema convectivo de 4aordem. A análise deste método é realizada a partir da comparação dos dados obtidos nas simulações realizadas no código WINDIETM e a coleta de dados medidos durante o processo SC. Em suma, conclui-se que o WINDIETM e as suas configurações reproduzem bons resultados de calibração, ja que produzem erros globais na ordem de dois pontos percentuais em relação ao SC realizado para o mesmo local em estudo.

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.

Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

Fractional Dynamics in the Rayleigh's Piston

This paper studies the dynamics of the Rayleigh piston using the modeling tools of Fractional Calculus. Several numerical experiments examine the effect of distinct values of the parameters. The time responses are transformed into the Fourier domain and approximated by means of power law approximations. The description reveals characteristics usual in Fractional Brownian phenomena.

Introducing numerical analysis tools in engineering: a Scilab user case in electronics course

Software tools in education became popular since the widespread of personal computers. Engineering courses lead the way in this development and these tools became almost a standard. Engineering graduates are familiar with numerical analysis tools but also with simulators (e.g. electronic circuits), computer assisted design tools and others, depending on the degree. One of the main problems with these tools is when and how to start use them so that they can be beneficial to students and not mere substitutes for potentially difficult calculations or design. In this paper a software tool to be used by first year students in electronics/electricity courses is presented. The growing acknowledgement and acceptance of open source software lead to the choice of an open source software tool – Scilab, which is a numerical analysis tool – to develop a toolbox. The toolbox was developed to be used as standalone or integrated in an e-learning platform. The e-learning platform used was Moodle. The first approach was to assess the mathematical skills necessary to solve all the problems related to electronics and electricity courses. Analysing the existing circuit simulators software tools, it is clear that even though they are very helpful by showing the end result they are not so effective in the process of the students studying and self learning since they show results but not intermediate steps which are crucial in problems that involve derivatives or integrals. Also, they are not very effective in obtaining graphical results that could be used to elaborate reports and for an overall better comprehension of the results. The developed tool was based on the numerical analysis software Scilab and is a toolbox that gives their users the opportunity to obtain the end results of a circuit analysis but also the expressions obtained when derivative and integrals calculations, plot signals, obtain vector diagrams, etc. The toolbox runs entirely in the Moodle web platform and provides the same results as the standalone application. The students can use the toolbox through the web platform (in computers where they don't have installation privileges) or in their personal computers by installing both the Scilab software and the toolbox. This approach was designed for first year students from all engineering degrees that have electronics/electricity courses in their curricula.

Improved Numerical Simulation for a Novel Adaptive Control Using Fractional Order Derivatives

A novel control technique is investigated in the adaptive control of a typical paradigm, an approximately and partially modeled cart plus double pendulum system. In contrast to the traditional approaches that try to build up ”complete” and ”permanent” system models it develops ”temporal” and ”partial” ones that are valid only in the actual dynamic environment of the system, that is only within some ”spatio-temporal vicinity” of the actual observations. This technique was investigated for various physical systems via ”preliminary” simulations integrating by the simplest 1st order finite element approach for the time domain. In 2004 INRIA issued its SCILAB 3.0 and its improved numerical simulation tool ”Scicos” making it possible to generate ”professional”, ”convenient”, and accurate simulations. The basic principles of the adaptive control, the typical tools available in Scicos, and others developed by the authors, as well as the improved simulation results and conclusions are presented in the contribution.