Simulación en tiempo real de vehículos industriales con modelos multicuerpo de gran complejidad

El objetivo de esta Tesis ha sido la consecución de simulaciones en tiempo real de vehículos industriales modelizados como sistemas multicuerpo complejos formados por sólidos rígidos. Para el desarrollo de un programa de simulación deben considerarse cuatro aspectos fundamentales: la modelización del sistema multicuerpo (tipos de coordenadas, pares ideales o impuestos mediante fuerzas), la formulación a utilizar para plantear las ecuaciones diferenciales del movimiento (coordenadas dependientes o independientes, métodos globales o topológicos, forma de imponer las ecuaciones de restricción), el método de integración numérica para resolver estas ecuaciones en el tiempo (integradores explícitos o implícitos) y finalmente los detalles de la implementación realizada (lenguaje de programación, librerías matemáticas, técnicas de paralelización). Estas cuatro etapas están interrelacionadas entre sí y todas han formado parte de este trabajo. Desde la generación de modelos de una furgoneta y de camión con semirremolque, el uso de tres formulaciones dinámicas diferentes, la integración de las ecuaciones diferenciales del movimiento mediante métodos explícitos e implícitos, hasta el uso de funciones BLAS, de técnicas de matrices sparse y la introducción de paralelización para utilizar los distintos núcleos del procesador. El trabajo presentado en esta Tesis ha sido organizado en 8 capítulos, dedicándose el primero de ellos a la Introducción. En el Capítulo 2 se presentan dos formulaciones semirrecursivas diferentes, de las cuales la primera está basada en una doble transformación de velocidades, obteniéndose las ecuaciones diferenciales del movimiento en función de las aceleraciones relativas independientes. La integración numérica de estas ecuaciones se ha realizado con el método de Runge-Kutta explícito de cuarto orden. La segunda formulación está basada en coordenadas relativas dependientes, imponiendo las restricciones por medio de penalizadores en posición y corrigiendo las velocidades y aceleraciones mediante métodos de proyección. En este segundo caso la integración de las ecuaciones del movimiento se ha llevado a cabo mediante el integrador implícito HHT (Hilber, Hughes and Taylor), perteneciente a la familia de integradores estructurales de Newmark. En el Capítulo 3 se introduce la tercera formulación utilizada en esta Tesis. En este caso las uniones entre los sólidos del sistema se ha realizado mediante uniones flexibles, lo que obliga a imponer los pares por medio de fuerzas. Este tipo de uniones impide trabajar con coordenadas relativas, por lo que la posición del sistema y el planteamiento de las ecuaciones del movimiento se ha realizado utilizando coordenadas Cartesianas y parámetros de Euler. En esta formulación global se introducen las restricciones mediante fuerzas (con un planteamiento similar al de los penalizadores) y la estabilización del proceso de integración numérica se realiza también mediante proyecciones de velocidades y aceleraciones. En el Capítulo 4 se presenta una revisión de las principales herramientas y estrategias utilizadas para aumentar la eficiencia de las implementaciones de los distintos algoritmos. En primer lugar se incluye una serie de consideraciones básicas para aumentar la eficiencia numérica de las implementaciones. A continuación se mencionan las principales características de los analizadores de códigos utilizados y también las librerías matemáticas utilizadas para resolver los problemas de álgebra lineal tanto con matrices densas como sparse. Por último se desarrolla con un cierto detalle el tema de la paralelización en los actuales procesadores de varios núcleos, describiendo para ello el patrón empleado y las características más importantes de las dos herramientas propuestas, OpenMP y las TBB de Intel. Hay que señalar que las características de los sistemas multicuerpo problemas de pequeño tamaño, frecuente uso de la recursividad, y repetición intensiva en el tiempo de los cálculos con fuerte dependencia de los resultados anteriores dificultan extraordinariamente el uso de técnicas de paralelización frente a otras áreas de la mecánica computacional, tales como por ejemplo el cálculo por elementos finitos. Basándose en los conceptos mencionados en el Capítulo 4, el Capítulo 5 está dividido en tres secciones, una para cada formulación propuesta en esta Tesis. En cada una de estas secciones se describen los detalles de cómo se han realizado las distintas implementaciones propuestas para cada algoritmo y qué herramientas se han utilizado para ello. En la primera sección se muestra el uso de librerías numéricas para matrices densas y sparse en la formulación topológica semirrecursiva basada en la doble transformación de velocidades. En la segunda se describe la utilización de paralelización mediante OpenMP y TBB en la formulación semirrecursiva con penalizadores y proyecciones. Por último, se describe el uso de técnicas de matrices sparse y paralelización en la formulación global con uniones flexibles y parámetros de Euler. El Capítulo 6 describe los resultados alcanzados mediante las formulaciones e implementaciones descritas previamente. Este capítulo comienza con una descripción de la modelización y topología de los dos vehículos estudiados. El primer modelo es un vehículo de dos ejes del tipo chasis-cabina o furgoneta, perteneciente a la gama de vehículos de carga medianos. El segundo es un vehículo de cinco ejes que responde al modelo de un camión o cabina con semirremolque, perteneciente a la categoría de vehículos industriales pesados. En este capítulo además se realiza un estudio comparativo entre las simulaciones de estos vehículos con cada una de las formulaciones utilizadas y se presentan de modo cuantitativo los efectos de las mejoras alcanzadas con las distintas estrategias propuestas en esta Tesis. Con objeto de extraer conclusiones más fácilmente y para evaluar de un modo más objetivo las mejoras introducidas en la Tesis, todos los resultados de este capítulo se han obtenido con el mismo computador, que era el top de la gama Intel Xeon en 2007, pero que hoy día está ya algo obsoleto. Por último los Capítulos 7 y 8 están dedicados a las conclusiones finales y las futuras líneas de investigación que pueden derivar del trabajo realizado en esta Tesis. Los objetivos de realizar simulaciones en tiempo real de vehículos industriales de gran complejidad han sido alcanzados con varias de las formulaciones e implementaciones desarrolladas. ABSTRACT The objective of this Dissertation has been the achievement of real time simulations of industrial vehicles modeled as complex multibody systems made up by rigid bodies. For the development of a simulation program, four main aspects must be considered: the modeling of the multibody system (types of coordinates, ideal joints or imposed by means of forces), the formulation to be used to set the differential equations of motion (dependent or independent coordinates, global or topological methods, ways to impose constraints equations), the method of numerical integration to solve these equations in time (explicit or implicit integrators) and the details of the implementation carried out (programming language, mathematical libraries, parallelization techniques). These four stages are interrelated and all of them are part of this work. They involve the generation of models for a van and a semitrailer truck, the use of three different dynamic formulations, the integration of differential equations of motion through explicit and implicit methods, the use of BLAS functions and sparse matrix techniques, and the introduction of parallelization to use the different processor cores. The work presented in this Dissertation has been structured in eight chapters, the first of them being the Introduction. In Chapter 2, two different semi-recursive formulations are shown, of which the first one is based on a double velocity transformation, thus getting the differential equations of motion as a function of the independent relative accelerations. The numerical integration of these equations has been made with the Runge-Kutta explicit method of fourth order. The second formulation is based on dependent relative coordinates, imposing the constraints by means of position penalty coefficients and correcting the velocities and accelerations by projection methods. In this second case, the integration of the motion equations has been carried out by means of the HHT implicit integrator (Hilber, Hughes and Taylor), which belongs to the Newmark structural integrators family. In Chapter 3, the third formulation used in this Dissertation is presented. In this case, the joints between the bodies of the system have been considered as flexible joints, with forces used to impose the joint conditions. This kind of union hinders to work with relative coordinates, so the position of the system bodies and the setting of the equations of motion have been carried out using Cartesian coordinates and Euler parameters. In this global formulation, constraints are introduced through forces (with a similar approach to the penalty coefficients) are presented. The stabilization of the numerical integration is carried out also by velocity and accelerations projections. In Chapter 4, a revision of the main computer tools and strategies used to increase the efficiency of the implementations of the algorithms is presented. First of all, some basic considerations to increase the numerical efficiency of the implementations are included. Then the main characteristics of the code’ analyzers used and also the mathematical libraries used to solve linear algebra problems (both with dense and sparse matrices) are mentioned. Finally, the topic of parallelization in current multicore processors is developed thoroughly. For that, the pattern used and the most important characteristics of the tools proposed, OpenMP and Intel TBB, are described. It needs to be highlighted that the characteristics of multibody systems small size problems, frequent recursion use and intensive repetition along the time of the calculation with high dependencies of the previous results complicate extraordinarily the use of parallelization techniques against other computational mechanics areas, as the finite elements computation. Based on the concepts mentioned in Chapter 4, Chapter 5 is divided into three sections, one for each formulation proposed in this Dissertation. In each one of these sections, the details of how these different proposed implementations have been made for each algorithm and which tools have been used are described. In the first section, it is shown the use of numerical libraries for dense and sparse matrices in the semirecursive topological formulation based in the double velocity transformation. In the second one, the use of parallelization by means OpenMP and TBB is depicted in the semi-recursive formulation with penalization and projections. Lastly, the use of sparse matrices and parallelization techniques is described in the global formulation with flexible joints and Euler parameters. Chapter 6 depicts the achieved results through the formulations and implementations previously described. This chapter starts with a description of the modeling and topology of the two vehicles studied. The first model is a two-axle chassis-cabin or van like vehicle, which belongs to the range of medium charge vehicles. The second one is a five-axle vehicle belonging to the truck or cabin semi-trailer model, belonging to the heavy industrial vehicles category. In this chapter, a comparative study is done between the simulations of these vehicles with each one of the formulations used and the improvements achieved are presented in a quantitative way with the different strategies proposed in this Dissertation. With the aim of deducing the conclusions more easily and to evaluate in a more objective way the improvements introduced in the Dissertation, all the results of this chapter have been obtained with the same computer, which was the top one among the Intel Xeon range in 2007, but which is rather obsolete today. Finally, Chapters 7 and 8 are dedicated to the final conclusions and the future research projects that can be derived from the work presented in this Dissertation. The objectives of doing real time simulations in high complex industrial vehicles have been achieved with the formulations and implementations developed.

Modelo numérico mecanobiológico para a obtenção da matriz de rigidez estrutural e da densidade mineral na remodelagem óssea

Neste trabalho é proposto um modelo mecanobiológico de remodelagem óssea para a estimativa de variações, provocadas por perturbações mecânicas ou biológicas, na matriz de rigidez estrutural da escala macroscópica e na densidade mineral em uma região do osso. Na cooperação entre as áreas da saúde e da engenharia, como nos estudos estruturais de biomecânica no sistema esquelético, as propriedades mecânicas dos materiais devem ser conhecidas, entretanto os ossos possuem uma constituição material altamente complexa, dinâmica e variante entre indivíduos. Sua dinâmica decorre dos ciclos de absorção e deposição de matriz óssea na remodelagem óssea, a qual ocorre para manter a integridade estrutural do esqueleto e adaptá-lo aos estímulos do ambiente, sejam eles biológicos, químicos ou mecânicos. Como a remodelagem óssea pode provocar alterações no material do osso, espera-se que suas propriedades mecânicas também sejam alteradas. Na literatura científica há modelos matemáticos que preveem a variação da matriz de rigidez estrutural a partir do estímulo mecânico, porém somente os modelos mais recentes incluíram explicitamente processos biológicos e químicos da remodelagem óssea. A densidade mineral óssea é um importante parâmetro utilizado no diagnóstico de doenças ósseas na área médica. Desse modo, para a obtenção da variação da rigidez estrutural e da densidade mineral óssea, propõe-se um modelo numérico mecanobiológico composto por cinco submodelos: da dinâmica da população de células ósseas, da resposta das células ao estímulo mecânico, da porosidade óssea, da densidade mineral óssea e, baseado na Lei de Voigt para materiais compósitos, da rigidez estrutural. Os valores das constantes das equações dos submodelos foram obtidos de literatura. Para a solução das equações do modelo, propõe-se uma implementação numérica e computacional escrita em linguagem C. O método de Runge-Kutta-Dorman-Prince, cuja vantagem consiste no uso de um passo de solução variável, é utilizado no modelo para controlar o erro numérico do resultado do sistema de equações diferenciais. Foi realizada uma avaliação comparativa entre os resultados obtidos com o modelo proposto e os da literatura dos modelos de remodelagem óssea recentes. Conclui-se que o modelo e a implementação propostos são capazes de obter variações da matriz de rigidez estrutural macroscópica e da densidade mineral óssea decorrentes da perturbação nos parâmetros mecânicos ou biológicos do processo de remodelagem óssea.

Nonlinear dynamics of a satellite with deployable solar panel arrays

The multibody dynamics of a satellite in circular orbit, modeled as a central body with two hinge-connected deployable solar panel arrays, is investigated. Typically, the solar panel arrays are deployed in orbit using preloaded torsional springs at the hinges in a near symmetrical accordion manner, to minimize the shock loads at the hinges. There are five degrees of freedom of the interconnected rigid bodies, composed of coupled attitude motions (pitch, yaw and roll) of the central body plus relative rotations of the solar panel arrays. The dynamical equations of motion of the satellite system are derived using Kane's equations. These are then used to investigate the dynamic behavior of the system during solar panel deployment via the 7-8th-order Runge-Kutta integration algorithms and results are compared with approximate analytical solutions. Chaotic attitude motions of the completely deployed satellite in circular orbit under the influence of the gravity-gradient torques are subsequently investigated analytically using Melnikov's method and confirmed via numerical integration. The Hamiltonian equations in terms of Deprit's variables are used to facilitate the analysis. (C) 2003 Published by Elsevier Ltd.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Effective cell-centred time-domain Maxwell's equations numerical solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed. © 2004 Elsevier Inc. All rights reserved.

Modelling of induction motors for system faults and transient stability studies

The research carried out in this thesis was mainly concerned with the effects of large induction motors and their transient performance in power systems. Computer packages using the three phase co-ordinate frame of reference were developed to simulate the induction motor transient performance. A technique using matrix algebra was developed to allow extension of the three phase co-ordinate method to analyse asymmetrical and symmetrical faults on both sides of the three phase delta-star transformer which is usually required when connecting large induction motors to the supply system. System simulation, applying these two techniques, was used to study the transient stability of a power system. The response of a typical system, loaded with a group of large induction motors, two three-phase delta-star transformers, a synchronous generator and an infinite system was analysed. The computer software developed to study this system has the advantage that different types of fault at different locations can be studied by simple changes in input data. The research also involved investigating the possibility of using different integrating routines such as Runge-Kutta-Gill, RungeKutta-Fehlberg and the Predictor-Corrector methods. The investigation enables the reduction of computation time, which is necessary when solving the induction motor equations expressed in terms of the three phase variables. The outcome of this investigation was utilised in analysing an introductory model (containing only minimal control action) of an isolated system having a significant induction motor load compared to the size of the generator energising the system.

Degradação fotocatalítica oxidativa do fenol utilizando carvão obtido da pirólise de diferentes biomassas

The modern industrial progress has been contaminating water with phenolic compounds. These are toxic and carcinogenic substances and it is essential to reduce its concentration in water to a tolerable one, determined by CONAMA, in order to protect the living organisms. In this context, this work focuses on the treatment and characterization of catalysts derived from the bio-coal, by-product of biomass pyrolysis (avelós and wood dust) as well as its evaluation in the phenol photocatalytic degradation reaction. Assays were carried out in a slurry bed reactor, which enables instantaneous measurements of temperature, pH and dissolved oxygen. The experiments were performed in the following operating conditions: temperature of 50 °C, oxygen flow equals to 410 mL min-1 , volume of reagent solution equals to 3.2 L, 400 W UV lamp, at 1 atm pressure, with a 2 hours run. The parameters evaluated were the pH (3.0, 6.9 and 10.7), initial concentration of commercial phenol (250, 500 and 1000 ppm), catalyst concentration (0, 1, 2, and 3 g L-1 ), nature of the catalyst (activated avelós carbon washed with dichloromethane, CAADCM, and CMADCM, activated dust wood carbon washed with dichloromethane). The results of XRF, XRD and BET confirmed the presence of iron and potassium in satisfactory amounts to the CAADCM catalyst and on a reduced amount to CMADCM catalyst, and also the surface area increase of the materials after a chemical and physical activation. The phenol degradation curves indicate that pH has a significant effect on the phenol conversion, showing better results for lowers pH. The optimum concentration of catalyst is observed equals to 1 g L-1 , and the increase of the initial phenol concentration exerts a negative influence in the reaction execution. It was also observed positive effect of the presence of iron and potassium in the catalyst structure: betters conversions were observed for tests conducted with the catalyst CAADCM compared to CMADCM catalyst under the same conditions. The higher conversion was achieved for the test carried out at acid pH (3.0) with an initial concentration of phenol at 250 ppm catalyst in the presence of CAADCM at 1 g L-1 . The liquid samples taken every 15 minutes were analyzed by liquid chromatography identifying and quantifying hydroquinone, p-benzoquinone, catechol and maleic acid. Finally, a reaction mechanism is proposed, cogitating the phenol is transformed into the homogeneous phase and the others react on the catalyst surface. Applying the model of Langmuir-Hinshelwood along with a mass balance it was obtained a system of differential equations that were solved using the Runge-Kutta 4th order method associated with a optimization routine called SWARM (particle swarm) aiming to minimize the least square objective function for obtaining the kinetic and adsorption parameters. Related to the kinetic rate constant, it was obtained a magnitude of 10-3 for the phenol degradation, 10-4 to 10-2 for forming the acids, 10-6 to 10-9 for the mineralization of quinones (hydroquinone, p-benzoquinone and catechol), 10-3 to 10-2 for the mineralization of acids.

Techniques en appui des formats de modulation avancés pour les futurs réseaux optiques

Les systèmes de communication optique avec des formats de modulation avancés sont actuellement l’un des sujets de recherche les plus importants dans le domaine de communication optique. Cette recherche est stimulée par les exigences pour des débits de transmission de donnée plus élevés. Dans cette thèse, on examinera les techniques efficaces pour la modulation avancée avec une détection cohérente, et multiplexage par répartition en fréquence orthogonale (OFDM) et multiples tonalités discrètes (DMT) pour la détection directe et la détection cohérente afin d’améliorer la performance de réseaux optiques. Dans la première partie, nous examinons la rétropropagation avec filtre numérique (DFBP) comme une simple technique d’atténuation de nonlinéarité d’amplificateur optique semiconducteur (SOA) dans le système de détection cohérente. Pour la première fois, nous démontrons expérimentalement l’efficacité de DFBP pour compenser les nonlinéarités générées par SOA dans un système de détection cohérente porteur unique 16-QAM. Nous comparons la performance de DFBP avec la méthode de Runge-Kutta quatrième ordre. Nous examinons la sensibilité de performance de DFBP par rapport à ses paramètres. Par la suite, nous proposons une nouvelle méthode d’estimation de paramètre pour DFBP. Finalement, nous démontrons la transmission de signaux de 16-QAM aux taux de 22 Gbaud sur 80km de fibre optique avec la technique d’estimation de paramètre proposée pour DFBP. Dans la deuxième partie, nous nous concentrons sur les techniques afin d’améliorer la performance des systèmes OFDM optiques en examinent OFDM optiques cohérente (CO-OFDM) ainsi que OFDM optiques détection directe (DDO-OFDM). Premièrement, nous proposons une combinaison de coupure et prédistorsion pour compenser les distorsions nonlinéaires d’émetteur de CO-OFDM. Nous utilisons une interpolation linéaire par morceaux (PLI) pour charactériser la nonlinéarité d’émetteur. Dans l’émetteur nous utilisons l’inverse de l’estimation de PLI pour compenser les nonlinéarités induites à l’émetteur de CO-OFDM. Deuxièmement, nous concevons des constellations irrégulières optimisées pour les systèmes DDO-OFDM courte distance en considérant deux modèles de bruit de canal. Nous démontrons expérimentalement 100Gb/s+ OFDM/DMT avec la détection directe en utilisant les constellations QAM optimisées. Dans la troisième partie, nous proposons une architecture réseaux optiques passifs (PON) avec DDO-OFDM pour la liaison descendante et CO-OFDM pour la liaison montante. Nous examinons deux scénarios pour l’allocations de fréquence et le format de modulation des signaux. Nous identifions la détérioration limitante principale du PON bidirectionnelle et offrons des solutions pour minimiser ses effets.

Adsorption of amyloglucosidase from Aspergillus niger NRRL 3122 using ion exchange resin

Amyloglucosidase enzyme was produced by Aspergillus niger NRRL 3122 from solid-state fermentation, using deffated rice bran as substrate. The effects of process parameters (pH, temperature) in the equilibrium partition coefficient for the system amyloglucosidase - resin DEAE-cellulose were investigated, aiming at obtaining the optimum conditions for a subsequent purification process. The highest partition coefficients were obtained using 0.025M Tris-HCl buffer, pH 8.0 and 25ºC. The conditions that supplied the highest partition coefficient were specified, the isotherm that better described the amyloglucosidase process of adsorption obtained. It was observed that the adsorption could be well described by Langmuir equation and the values of Qm and Kd estimated at 133.0 U mL-1 and 15.4 U mL-1, respectively. From the adjustment of the kinetic curves using the fourth-order Runge-Kutta algorithm, the adsorption (k1) and desorption (k2) constants were obtained through optimization by the least square procedure, and the values calculated were 2.4x10-3 mL U-1 min-1 for k1 and 0.037 min-1 for k2 .

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Korteweg–de Vries solitoniyhtälön diskretointi- ja aikaintegrointimenetelmien vertailua

Solitoni on tunnettu ilmiönä jo 1800-luvun alkupuolelta lähtien. Se on eräänlainen muotonsa säilyttävä ja vakionopeudella etenevä aalto. 1800-luvun loppupuolella esitettiin osittaisdifferentiaaliyhtälön kuvaamaan tällaista matalassa ja kapeassa kanavassa esiintynyttä solitoniaaltoa. Tätä Kortewegin ja de Vries’n mukaan nimettyä osittaisdifferentiaaliyhtälöä tutkivat numeerisesti ensimmäistä kertaa Zabusky ja Kruskal vuonna 1965. Osittaisdifferentiaaliyhtälöiden ratkaisuun tarvitsee usein käyttää numeerisia menetelmiä. Tämän työn alkupuoli käsittelee yleisesti tarvittavia matemaattisia menetelmiä sekä KdV-yhtälön analyyttistä tarkastelua. Loppupuolella tutkitaan KdV-yhtälön mallintamista tietokoneen avulla. Zabuskyn ja Kruskalin käyttämien menetelmien lisäksi kokeillaan montaa muutakin tapaa KdV-yhtälön mallintamiseen. Näistä menetelmistä vertaillaan laskentatehokkuutta sekä menetelmän tarkkuutta. Zabuskyn ja Kruskalin käyttämä paikkadiskretointi todettiin mallinnuksissa tarkimmaksi, mutta ei kuitenkaan mallinnusaikaa tarkastellen tehokkaimmaksi. Aikaintegroinneista Runge-Kutta-menetelmät todettiin parhaiksi. Menetelmien vertailun lisäksi niistä parhaiksi havaittuja sovellettiin muutaman erikoistapauksen, kuten kolmen aallon törmäyksen, mallintamiseen.

MODELAGEM E SIMULAÇÃO DA DESTERPENAÇÃO DO ÓLEO DA CASCA DE LARANJA COM CO2 SUPERCRÍTICO EM MODO SEMI-CONTÍNUO

A desterpenação do óleo da casca de laranja com CO2 supercrítico foi investigada através da modelagem e simulação da separação de uma mistura sintética de limoneno (90 % em peso) e linalol (10 %), em um extrator operando em modo semi-contínuo. A modelagem matemática da extração supercrítica foi realizada por analogia com a destilação de uma mistura binária, em batelada, expressando-se a composição das fases em equilíbrio numa base molar livre de CO2. O cálculo das variáveis do processo foi feito por integração numérica da equação de Rayleigh empregando-se o método de Runge-Kutta de quarta ordem. Para a determinação da relação de equilíbrio entre as fases, adotou-se a equação de Peng-Robinson modificada, com os parâmetros de interação obtidos de dados de ELV dos sistemas binários CO2+limoneno e CO2+linalol e do ternário CO2+limoneno+linalol.

COMPARAÇÃO ENTRE TÉCNICAS NUMÉRICAS PARA A RESOLUÇÃO DO PROBLEMA DE TRANSFERÊNCIA DE CALOR EM ALIMENTOS ENLATADOS

Este trabalho apresenta a comparação das técnicas de resolução numérica de Diferenças Finitas e Runge-Kutta-Gill de 4a ordem com passo de integração variável para a simulação do modelo de transferência de calor em regime transiente bidimensional em coordenadas cilíndricas e unidimensional em coordenadas esféricas, aplicado a alimentos enlatados processados em autoclave, observando-se a rapidez e a precisão de integração comparados aos perfis reais de tempo/temperatura, incluindo desvios de processo. Além disso, uma análise de sensibilidade paramétrica na difusividade térmica foi realizada na tentativa de quantificar a influência que essa propriedade possui na resolução numérica. Para tanto, foram realizados ensaios de penetração de calor em autoclave a vapor, utilizando-se latas cilíndricas de 75x90 e 100x110 contendo simulantes de alimentos com características de aquecimento por condução e convecção.

Numerical Methods for the Unsteady Compressible Navier-Stokes Equations

We consider numerical methods for the compressible time dependent Navier-Stokes equations, discussing the spatial discretization by Finite Volume and Discontinuous Galerkin methods, the time integration by time adaptive implicit Runge-Kutta and Rosenbrock methods and the solution of the appearing nonlinear and linear equations systems by preconditioned Jacobian-Free Newton-Krylov, as well as Multigrid methods. As applications, thermal Fluid structure interaction and other unsteady flow problems are considered. The text is aimed at both mathematicians and engineers.

Numerical analyses of Runge–Kutta implicit–explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models

With the prospect of exascale computing, computational methods requiring only local data become especially attractive. Consequently, the typical domain decomposition of atmospheric models means horizontally-explicit vertically-implicit (HEVI) time-stepping schemes warrant further attention. In this analysis, Runge-Kutta implicit-explicit schemes from the literature are analysed for their stability and accuracy using a von Neumann stability analysis of two linear systems. Attention is paid to the numerical phase to indicate the behaviour of phase and group velocities. Where the analysis is tractable, analytically derived expressions are considered. For more complicated cases, amplification factors have been numerically generated and the associated amplitudes and phase diagnosed. Analysis of a system describing acoustic waves has necessitated attributing the three resultant eigenvalues to the three physical modes of the system. To do so, a series of algorithms has been devised to track the eigenvalues across the frequency space. The result enables analysis of whether the schemes exactly preserve the non-divergent mode; and whether there is evidence of spurious reversal in the direction of group velocities or asymmetry in the damping for the pair of acoustic modes. Frequency ranges that span next-generation high-resolution weather models to coarse-resolution climate models are considered; and a comparison is made of errors accumulated from multiple stability-constrained shorter time-steps from the HEVI scheme with a single integration from a fully implicit scheme over the same time interval. Two schemes, “Trap2(2,3,2)” and “UJ3(1,3,2)”, both already used in atmospheric models, are identified as offering consistently good stability and representation of phase across all the analyses. Furthermore, according to a simple measure of computational cost, “Trap2(2,3,2)” is the least expensive.