Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

A hybrid Laplace transform/finite difference boundary element method for diffusion problems

The solution process for diffusion problems usually involves the time development separately from the space solution. A finite difference algorithm in time requires a sequential time development in which all previous values must be determined prior to the current value. The Stehfest Laplace transform algorithm, however, allows time solutions without the knowledge of prior values. It is of interest to be able to develop a time-domain decomposition suitable for implementation in a parallel environment. One such possibility is to use the Laplace transform to develop coarse-grained solutions which act as the initial values for a set of fine-grained solutions. The independence of the Laplace transform solutions means that we do indeed have a time-domain decomposition process. Any suitable time solver can be used for the fine-grained solution. To illustrate the technique we shall use an Euler solver in time together with the dual reciprocity boundary element method for the space solution

Voltammetry of oxygen in the room-temperature ionic liquids 1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)imide and hexyltriethylammonium bis((trifluoromethyl)sulfonyl)imide: One-electron reduction to form superoxide. Steady-state and transient behavior in the same cyclic voltammogram resulting from widely different diffusion coefficients of oxygen and superoxide

The electrochemical reduction of oxygen in two different room-temperature ionic liquids, 1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)imide ([EMIM][N(Tf)(2)]) and hexyltriethylammonium bis((trifluoromethyl)sulfonyl)imide ([N-6222][N(Tf)(2)]) was investigated by cyclic voltammetry at a gold microdisk electrode. Chronoamperometric measurements were made to determine the diffusion coefficient, D, and concentration, c, of the electroactive oxygen dissolved in the ionic liquid by fitting experimental transients to the Aoki model. [Aoki, K.; et al. J. Electroanal. Chem. 1981, 122, 19]. A theory and simulation designed for cyclic voltammetry at microdisk electrodes was then employed to determine the diffusion coefficient of the electrogenerated superoxide species, O-2(.-), as well as compute theoretical voltammograms to confirm the values of D and c for neutral oxygen obtained from the transients. As expected, the diffusion coefficient of the superoxide species was found to be smaller than that of the oxygen in both ionic liquids. The diffusion coefficients of O-2 and O-2(.-) in [N-6222][N(Tf)(2)], however, differ by more than a factor of 30 (D-O2 = 1.48 x 10(-10) m(2) s(-1), DO2.- = 4.66 x 10(-12) m(2) s(-1)), whereas they fall within the same order of magnitude in [EMIM][N(Tf)(2)] (D-O2 = 7.3 x 10(-10) m(2) s(-1), DO2.- = 2.7 x 10(-10) m(2) s(-1)). This difference in [N-6222][N(Tf)(2)] causes pronounced asymmetry in the concentration distributions of oxygen and superoxide, resulting in significant differences in the heights of the forward and back peaks in the cyclic voltammograms for the reduction of oxygen. This observation is most likely a result of the higher viscosity of [N-6222][N(Tf)(2)] in comparison to [EMIM][N(Tf)(2)], due to the structural differences in cationic component.

Automatic extraction of reduced-order models from CFD simulations for building energy modelling

Accurate modelling of the internal climate of buildings is essential if Building Energy Management Systems (BEMS) are to efficiently maintain adequate thermal comfort. Computational fluid dynamics (CFD) models are usually utilised to predict internal climate. Nevertheless CFD models, although providing the necessary level of accuracy, are highly computationally expensive, and cannot practically be integrated in BEMS. This paper presents and describes validation of a CFD-ROM method for real-time simulations of building thermal performance. The CFD-ROM method involves the automatic extraction and solution of reduced order models (ROMs) from validated CFD simulations. ROMs are shown to be adequately accurate with a total error below 5% and to retain satisfactory representation of the phenomena modelled. Each ROM has a time to solution under 20seconds, which opens the potential of their integration with BEMS, giving real-time physics-based building energy modelling. A parameter study was conducted to investigate the applicability of the extracted ROM to initial boundary conditions different from those from which it was extracted. The results show that the ROMs retained satisfactory total errors when the initial conditions in the room were varied by ±5°C. This allows the production of a finite number of ROMs with the ability to rapidly model many possible scenarios.

Fuzzy, integer and fractional-order control: application on a wind turbine benchmark model

This paper presents a comparison between proportional integral control approaches for variable speed wind turbines. Integer and fractional-order controllers are designed using linearized wind turbine model whilst fuzzy controller also takes into account system nonlinearities. These controllers operate in the full load region and the main objective is to extract maximum power from the wind turbine while ensuring the performance and reliability required to be integrated into an electric grid. The main contribution focuses on the use of fractional-order proportional integral (FOPI) controller which benefits from the introduction of one more tuning parameter, the integral fractional-order, taking advantage over integer order proportional integral (PI) controller. A comparison between proposed control approaches for the variable speed wind turbines is presented using a wind turbine benchmark model in the Matlab/Simulink environment. Results show that FOPI has improved system performance when compared with classical PI and fuzzy PI controller outperforms the integer and fractional-order control due to its capability to deal with system nonlinearities and uncertainties. © 2014 IEEE.

Fractional describing function of systems with Coulomb friction

This paper studies the describing function (DF) of systems constituted by a mass subjected to nonlinear friction. The friction force is decomposed into two components, namely, the viscous and the Coulomb friction. The system dynamics is analyzed in the DF perspective revealing a fractional-order behavior. The reliability of the DF method is evaluated through the signal harmonic contents.

A technology diffusion perspective of Enterprise Resource Planning across European Small and Medium Enterprises: from determinants to Use to Value

Tese apresentada como requisito parcial para obtenção do grau de Doutor em Gestão de Informação

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Homogeneity and heterogeneity of political traditions in the remaking of world order

Deux décennies après la chute de l'URSS (1991), ce mémoire propose une réévaluation de la thèse de Francis Fukuyama sur la Fin de l'Histoire, élaborée en 1989, qui postule qu'avec la chute de l'URSS aucune idéologie ne peut rivaliser avec la démocratie libérale capitaliste; et de la thèse de Samuel P. Huntington sur le Choc des civilisations, élaborée en 1993, qui pose l'existence d'un nombre fini de civilisations homogènes et antagonistes. Pourtant, lorsque confrontées à une étude approfondie des séquences historiques, ces deux théories apparaissent pour le moins relatives. Deux questions ont été traitées: l'interaction entre Idéologie et Conditions historiques, et la thèse de l'homogénéité intracivilisationnelle et de l'hétérogénéité antagoniste intercivilisationnelle. Sans les invalider complètement, cette recherche conclut toutefois que ces deux théories doivent être nuancées; elles se situent aux deux extrémités du spectre des relations internationales. La recherche effectuée a montré que les idéologies et leur poids relatif sont tributaires d'un contexte, contrairement à Fukuyama qui les pose dans l'absolu. De plus, l'étude de la Chine maoïste et particulièrement de la pensée de Mao Zedong montre que les traditions politiques locales sont plus hétérogènes qu'il n'y paraît au premier abord, ce qui relativise la thèse de Huntington. En conclusion, les rapports entre États sont plus dynamiques que ne le laissent penser les thèses de Fukuyama et de Huntington.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Understanding mixing efficiency in the oceans. Do the nonlinearities of the equation of state matter?

There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the turbulent rate of change Wr, turbulent of background gravitational potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to the well known exact equality D(APE)=Wr, turbulent for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=Wr, turbulent/D(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that D(APE) and Wr, turbulent actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE0, and the conversion between GPEr and a different subcomponent of internal energy called "exergy" IEexergy. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency N vertical profile, but different vertical profiles of the parameter Υ=α P/(ρCp), where α is the thermal expansion coefficient, P the hydrostatic pressure, ρ the density, and Cp the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and Wr, turbulent depend critically on the sign and magnitude of dΥ/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.

The exclusive use of integer-order spots for LEED-IV structure analysis of adsorption systems: p(2×2)-O on Ni{111}

Two different ways of performing low-energy electron diffraction (LEED) structure determinations for the p(2 x 2) structure of oxygen on Ni {111} are compared: a conventional LEED-IV structure analysis using integer and fractional-order IV-curves collected at normal incidence and an analysis using only integer-order IV-curves collected at three different angles of incidence. A clear discrimination between different adsorption sites can be achieved by the latter approach as well as the first and the best fit structures of both analyses are within each other's error bars (all less than 0.1 angstrom). The conventional analysis is more sensitive to the adsorbate coordinates and lateral parameters of the substrate atoms whereas the integer-order-based analysis is more sensitive to the vertical coordinates of substrate atoms. Adsorbate-related contributions to the intensities of integer-order diffraction spots are independent of the state of long-range order in the adsorbate layer. These results show, therefore, that for lattice-gas disordered adsorbate layers, for which only integer-order spots are observed, similar accuracy and reliability can be achieved as for ordered adsorbate layers, provided the data set is large enough.

An historical perspective on the development of the thermodynamic equation of seawater - 2010

Oceanography is concerned with understanding the mechanisms controlling the movement of seawater and its contents. A fundamental tool in this process is the characterization of the thermophysical properties of seawater as functions of measured temperature and electrical conductivity, the latter used as a proxy for the concentration of dissolved matter in seawater. For many years a collection of algorithms denoted the Equation of State 1980 (EOS-80) has been the internationally accepted standard for calculating such properties. However, modern measurement technology now allows routine observations of temperature and electrical conductivity to be made to at least one order of magnitude more accurately than the uncertainty in this standard. Recently, a new standard has been developed, the Thermodynamical Equation of Seawater 2010 (TEOS-10). This new standard is thermodynamically consistent, valid over a wider range of temperature and salinity, and includes a mechanism to account for composition variations in seawater. Here we review the scientific development of this standard, and describe the literature involved in its development, which includes many of the articles in this special issue.