Aktuelle technologische Herausforderungen, z.B. in der Werkzeugmaschinenentwicklung, erfordern aufgrund der steigenden Genauigkeitsanforderungen an die thermische Simulation eines zu betrachtenden Systems, dass ebenfalls die Auswirkungen relevanter, translatorischer Relativbewegungen zwischen unterschiedlichen Teilen des Systems berücksichtigt werden. Das Vorgehen, diese Bewegung in den Simulationen durch diskrete Verschiebungen zwischen den Lastschritten einer transienten Analyse umzusetzen, führt bei der Verwendung einer infinit kleinen Zeitschrittweite auf die Abbildung einer kontinuierlichen Bewegung, bringt aber gleichzeitig eine problematische Steigerung des notwendigen Rechenaufwands mit sich. Die Anwendung einer langen Zeitschrittweite bei gleichzeitiger Konservierung der Ergebnisgenauigkeit stellt nun einen Ansatz dar, die Effizienz solcher Analysen über den eingesparten Aufwand der nicht auszuwertenden Lastschritte zu steigern. In dieser Arbeit wurden durch eine gezielte Partitionierung der aus einer Ortsdiskretisierung resultierenden Systemmatrizen zunächst vier qualitativ unterscheidbare Fehlerquellen identifiziert, welche die Verwendung einer groben Zeitdiskretisierung potentiell nach sich ziehen kann. Konkret gehören dazu die Leistungsfähigkeit des zur transienten Auswertung verwendeten Integrationsverfahrens, die diskrete Umsetzung der Bewegung sowie die seltene Aktualisierung der beiden Arten von Kontaktlasten. Für die einzelnen Fehler werden die möglichen Auswirkungen jeweils allgemein quantifiziert. Für zwei, dabei als relevant identifizierte Fehlerquellen werden mit der BD- und der RUMHI-Korrektur zugehörige Verfahren entwickelt, mit denen die Ergebnisgenauigkeit trotz grober Zeitdiskretisierung aufwandsarm bewahrt werden kann. Dass ein strukturvariables, thermisches Problem durch die kombinierte dieser Korrekturverfahren deutlich effizienter berechnet werden kann, wird in der Arbeit abschließend an zwei stellvertretenden Problemen beispielhaft gezeigt.

Collision avoidance is an essential component of autonomous shipping. As ships begin to advance towards autonomy, developing an advisory system is one of the first steps. An advisory system with a strong collision avoidance component can help the crew act more quickly and accurately in dangerous situations. One way to avoid colission is to make the vessel stop as fast as possible. In this work, two scenarios are studied, firstly, stopping along a predefined path, and secondly, stopping within a safe area defined by surrounding obstacles. The first scenario was further worked with to formulate a real-time solution. Movements of a vessel, described in three degrees of freedom with continuous dynamics, were simulated using mathematical models of the forces acting on the ship. Nonlinear optimal control problems were formulated for each scenario and solved numerically using discretization and a direct multiple shooting method. The results for the first problem showed that the vessel could stop without much deviation from the path. Paths with different curvatures were tested, and it was shown that a slightly longer distance was traveled when the curvature of the path was greater. The results for the second problem showed that the vessel stays within the safe area and chooses a relatively straight path as the optimal way of stoping. This results in a shorter distance traveled compared to the solution of the first problem. Two different real-time approaches were formulated, firstly a receding-horizon approach and secondly a lookup-based approach. Both approaches were solved with real-time feasibility, where the receding-horizon approach gave a better solution while lookup-based approach had a shorter computational time.

With steadily increasing computational power, computational fluid dynamics (CFD) can be applied to unsteady problems such as seakeeping simulations. Therefore, a good balance between accuracy and computational speed is required. This thesis investigates the application of CFD to seakeeping performance predictions and aims to propose a best-practice procedure for efficient seakeeping simulations. The widely used KVLCC2 research vessel serves as a test case for this thesis and FINEŠ/Marine software package is used for CFD computations. In order to validate the simulations, results are compared to recent experimental data from SSPA as well as predictions with potential ˛ow code SHIPFLOW® Motions. As for the calm water simulations, both inviscid and viscous ˛ow computations are performed in combination with three mesh refinement levels. Seakeeping simulations with regular head waves of different wavelengths are set-up correspondingly. Furthermore, different strategies for time discretization are investigated. With the given computational resources, it is not feasible to complete seakeeping simulations with a ˝ne mesh. However, already the coarse meshes give good agreement to experiments and SHIPFLOW® Motions' predictions. Viscous ˛ow simulations turn out to be more robust than Euler ˛ow computations and thus should be preferred. Regarding the time discretization, a fixed time discretization of 150 steps per wave period has shown the best balance between accuracy and speed. Based on these findings, a best-practice procedure for seakeeping performance predictions in FINEŠ/Marine is established. Taking the most efficient settings obtained from head wave simulations, the vessel is subjected to oblique waves with 160° encounter angle. Under similar wave conditions, CFD predictions of a similar thesis show close agreement in terms of added wave resistance. Compared to the previous head wave conditions of this study, added resistance in 160° oblique waves is found to be significantly higher. This underlines that oblique bow quartering waves represent a relevant case for determining the maximum required power of a ship. CFD and potential ˛ow show similar accuracy with respect to ship motions and added wave resistance, albeit potential ˛ow outperforms CFD in terms of computational speed. Hence, CFD should be applied in cases where viscous effects are known to have large influence on a vessel's seakeeping behavior. This can be the case if motion control and damping devices are to be evaluated, for instance. / Tack vare den stadigt ökande beräkningskraften kan beräkningsuiddynamik (CFD) idag användas på beräkningsintensiva problem som sjöegenskapssimulationer. Den här rapporten undersöker användning av CFD på sjöegenskapsprestanda och syftar till att foreslå ett best-practice förfaringssätt för effektiv sjöegenskapssimulationer. Forskningsskrovet KVLCC2 fungerar som ett testfall för denna rapport och FINE—/Marine-mjukvarupaketet används för CFD-beräkningar. Viktiga parametrar, såsom ödestyp, beräkningsnät och tidssteg varierars systematiskt. Resultaten jämförs med experiment gjorda vid SSPA. Baserat på resultaten förelås en best-practice. Den föreslagna best-practice användas vidare för berökningar av sjöegenskaper i sneda vågor. Jämförelse av resultaten med liknande studier visar god överensstämmelse. Genom att använda det föreslagna förfarandet för best-practice kan CFD-sjöegenskapssimulationer användas på fall där viskösa krafter måste beaktas, till exempel rörelseregleringsanordningar.

This paper explores the noise shaping and noise producing qualities of Delta-Sigma Modulators (DSM) and Pulse-Width Modulators (PWM). DSM has long been dominant in the Delta Sigma Analog-to-Digital Converter (DSADC) as a noise-shaped quantizer and time discretizer, while PWM, with a similar self oscillating structure, has seen use in Class D Power Amplifiers, performing a similar function. It has been shown that the PWM in Class D Amplifiers outperforms the DSM [1], but could this advantage be used in DSADC use-cases? LTSpice simulation and printed circuit board implementation and test are used to present data on four variations of these modulators: The DSM, PWM, the out-of-loop discretized PWM (OOLDP), and the cascaded modulator. A generic form of an Nth order loop filter is presented, where three orders of this generic topology are analyzed in simulation for each modulator, and two orders are used in physical testing.

No description available.

We consider a family of variational time discretizations that generalizes discontinuous Galerkin (dG) and continuous Galerkin-Petrov (cGP) methods. In addition to variational conditions the methods also contain collocation conditions in the time mesh points. The single family members are characterized by two parameters that represent the local polynomial ansatz order and the number of non-variational conditions, which is also related to the global temporal regularity of the numerical solution. Moreover, with respect to Dahlquist’s stability problem the variational time discretization (VTD) methods either share their stability properties with the dG or the cGP method and, hence, are at least A-stable. With this thesis, we present the first comprehensive theoretical study of the family of VTD methods in the context of non-stiff and stiff initial value problems as well as, in combination with a finite element method for spatial approximation, in the context of parabolic problems. Here, we mainly focus on the error analysis for the discretizations. More concrete, for initial value problems the pointwise error is bounded, while for parabolic problems we rather derive error estimates in various typical integral-based (semi-)norms. Furthermore, we show superconvergence results in the time mesh points. In addition, some important concepts and key properties of the VTD methods are discussed and often exploited in the error analysis. These include, in particular, the associated quadrature formulas, a beneficial postprocessing, the idea of cascadic interpolation, connections between the different VTD schemes, and connections to other classes of methods (collocation methods, Runge-Kutta-like methods). Numerical experiments for simple academic test examples are used to highlight various properties of the methods and to verify the optimality of the proven convergence orders.:List of Symbols and Abbreviations Introduction I Variational Time Discretization Methods for Initial Value Problems 1 Formulation, Analysis for Non-Stiff Systems, and Further Properties 1.1 Formulation of the methods 1.1.1 Global formulation 1.1.2 Another formulation 1.2 Existence, uniqueness, and error estimates 1.2.1 Unique solvability 1.2.2 Pointwise error estimates 1.2.3 Superconvergence in time mesh points 1.2.4 Numerical results 1.3 Associated quadrature formulas and their advantages 1.3.1 Special quadrature formulas 1.3.2 Postprocessing 1.3.3 Connections to collocation methods 1.3.4 Shortcut to error estimates 1.3.5 Numerical results 1.4 Results for affine linear problems 1.4.1 A slight modification of the method 1.4.2 Postprocessing for the modified method 1.4.3 Interpolation cascade 1.4.4 Derivatives of solutions 1.4.5 Numerical results 2 Error Analysis for Stiff Systems 2.1 Runge-Kutta-like discretization framework 2.1.1 Connection between collocation and Runge-Kutta methods and its extension 2.1.2 A Runge-Kutta-like scheme 2.1.3 Existence and uniqueness 2.1.4 Stability properties 2.2 VTD methods as Runge-Kutta-like discretizations 2.2.1 Block structure of A VTD 2.2.2 Eigenvalue structure of A VTD 2.2.3 Solvability and stability 2.3 (Stiff) Error analysis 2.3.1 Recursion scheme for the global error 2.3.2 Error estimates 2.3.3 Numerical results II Variational Time Discretization Methods for Parabolic Problems 3 Introduction to Parabolic Problems 3.1 Regularity of solutions 3.2 Semi-discretization in space 3.2.1 Reformulation as ode system 3.2.2 Differentiability with respect to time 3.2.3 Error estimates for the semi-discrete approximation 3.3 Full discretization in space and time 3.3.1 Formulation of the methods 3.3.2 Reformulation and solvability 4 Error Analysis for VTD Methods 4.1 Error estimates for the l th derivative 4.1.1 Projection operators 4.1.2 Global L2-error in the H-norm 4.1.3 Global L2-error in the V-norm 4.1.4 Global (locally weighted) L2-error of the time derivative in the H-norm 4.1.5 Pointwise error in the H-norm 4.1.6 Supercloseness and its consequences 4.2 Error estimates in the time (mesh) points 4.2.1 Exploiting the collocation conditions 4.2.2 What about superconvergence!? 4.2.3 Satisfactory order convergence avoiding superconvergence 4.3 Final error estimate 4.4 Numerical results Summary and Outlook Appendix A Miscellaneous Results A.1 Discrete Gronwall inequality A.2 Something about Jacobi-polynomials B Abstract Projection Operators for Banach Space-Valued Functions B.1 Abstract definition and commutation properties B.2 Projection error estimates B.3 Literature references on basics of Banach space-valued functions C Operators for Interpolation and Projection in Time C.1 Interpolation operators C.2 Projection operators C.3 Some commutation properties C.4 Some stability results D Norm Equivalences for Hilbert Space-Valued Polynomials D.1 Norm equivalence used for the cGP-like case D.2 Norm equivalence used for final error estimate Bibliography

La simulation numérique est aujourd'hui un outils majeur dans la conception des objets aérodynamiques, que ce soit dans l'aéronautique, l'automobile, l'industrie navale, etc... Un des défis majeurs pour repousser les limites des codes de simulation est d'améliorer leur précision, tout en utilisant une quantité fixe de ressources (puissance et/ou temps de calcul). Cet objectif peut être atteint par deux approches différentes, soit en construisant une discrétisation fournissant sur un maillage donné une solution d'ordre très élevé, soit en construisant un schéma compact et massivement parallèlisable, de manière à minimiser le temps de calcul en distribuant le problème sur un grand nombre de processeurs. Dans cette thèse, nous tentons de rassembler ces deux approches par le développement et l'implémentation de Schéma Distribuant le Résidu (RDS) d'ordre très élevé et de compacité maximale. Ce manuscrit commence par un rappel des principaux résultats mathématiques concernant les Lois de Conservation hyperboliques (CLs). Le but de cette première partie est de mettre en évidence les propriétés des solutions analytiques que nous cherchons à approcher, de manière à injecter ces propriétés dans celles de la solution discrète recherchée. Nous décrivons ensuite les trois étapes principales de la construction d'un schéma RD d'ordre très élevé : - la représentation polynomiale d'ordre très élevé de la solution sur des polygones et des polyèdres; - la description de méthodes distribuant le résidu de faible ordre, compactes et conservatives, consistantes avec une représentation polynomiale des données de très haut degré. Parmi elles, une attention particulière est donnée à la plus simple, issue d'une généralisation du schéma de Lax-Friedrichs (\LxF); - la mise en place d'une procédure préservant la positivité qui transforme tout schéma stable et linéaire, en un schéma non linéaire d'ordre très élevé, capturant les chocs de manière non oscillante. Dans le manuscrit, nous montrons que les schémas obtenus par cette procédure sont consistants avec la CL considérée, qu'ils sont stables en norme $\L^{\infty}$ et qu'ils ont la bonne erreur de troncature. Même si tous ces développements théoriques ne sont démontrés que dans le cas de CLs scalaires, des remarques au sujet des problèmes vectoriels sont faites dès que cela est possible. Malheureusement, lorsqu'on considère le schéma \LxF, le problème algébrique non linéaire associé à la recherche de la solution stationnaire est en général mal posé. En particulier, on observe l'apparition de modes parasites de haute fréquence dans les régions de faible gradient. Ceux-ci sont éliminés grâce à un terme supplémentaire de stabilisation dont les effets et l'évaluation numérique sont précisément détaillés. Enfin, nous nous intéressons à une discrétisation correcte des conditions limites pour le schéma d'ordre élevé proposé. Cette théorie est ensuite illustrée sur des cas test scalaires bidimensionnels simples. Afin de montrer la généralité de notre approche, des maillages composés uniquement de triangles et des maillages hybrides, composés de triangles et de quandrangles, sont utilisés. Les résultats obtenus par ces tests confirment ce qui est attendu par la théorie et mettent en avant certains avantages des maillages hybrides. Nous considérons ensuite des solutions bidimensionnelles des équations d'Euler de la dynamique des gaz. Les résultats sont assez bons, mais on perd les pentes de convergence attendues dès que des conditions limite de paroi sont utilisées. Ce problème nécessite encore d'être étudié. Nous présentons alors l'implémentation parallèle du schéma. Celle-ci est analysée et illustrée à travers des cas test tridimensionnel de grande taille. / Numerical simulations are nowadays a major tool in aerodynamic design in aeronautic, automotive, naval industry etc... One of the main challenges to push further the limits of the simulation codes is to increase their accuracy within a fixed set of resources (computational power and/or time). Two possible approaches to deal with this issue are either to contruct discretizations yielding, on a given mesh, very high order accurate solutions, or to construct compact, massively parallelizable schemes to minimize the computational time by means of a high performance parallel implementation. In this thesis, we try to combine both approaches by investigating the contruction and implementation of very high order Residual Distribution Schemes (RDS) with the most possible compact stencil. The manuscript starts with a review of the mathematical theory of hyperbolic Conservation Laws (CLs). The aim of this initial part is to highlight the properties of the analytical solutions we are trying to approximate, in order to be able to link these properties with the ones of the sought discrete solutions. Next, we describe the three main steps toward the construction of a very high order RDS: - The definition of higher order polynomial representations of the solution over polygons and polyhedra; - The design of low order compact conservative RD schemes consistent with a given (high degree) polynomial representation. Among these, particular accest is put on the simplest, given by a generalization of the Lax-Friedrich's (\LxF) scheme; - The design of a positivity preserving nonlinear transformation, mapping first-order linear schemes onto nonlinear very high order schemes. In the manuscript, we show formally that the schemes obtained following this procedure are consistent with the initial CL, that they are stable in $L^{\infty}$ norm, and that they have the proper truncation error. Even though all the theoretical developments are carried out for scalar CLs, remarks on the extension to systems are given whenever possible. Unortunately, when employing the first order \LxF scheme as a basis for the construction of the nonlinear discretization, the final nonlinear algebraic equation is not well-posed in general. In particular, for smoothly varying solutions one observes the appearance of high frequency spurious modes. In order to kill these modes, a streamline dissipation term is added to the scheme. The analytical implications of this modifications, as well as its practical computation, are thouroughly studied. Lastly, we focus on a correct discretization of the boundary conditions for the very high order RDS proposed. The theory is then extensively verified on a variety of scalar two dimensional test cases. Both triangular, and hybrid triangular-quadrilateral meshes are used to show the generality of the approach. The results obtained in these tests confirm all the theoretical expectations in terms of accuracy and stability and underline some advantages of the hybrid grids. Next, we consider solutions of the two dimensional Euler equations of gas dynamics. The results obtained are quite satisfactory and yet, we are not able to obtain the desired convergence rates on problems involving solid wall boundaries. Further investigation of this problem is under way. We then discuss the parallel implementation of the schemes, and analyze and illustrate the performance of this implementation on large three dimensional problems. Due to the preliminary character and the complexity of these three dimensional problems, a rather qualitative discussion is made for these tests cases: the overall behavior seems to be the correct one, but more work is necessary to assess the properties of the schemes in three dimensions.

No presente trabalho desenvolve-se um estudo de modelagem matemática e simulação numérica do escoamento bifásico gás-sólido na coluna ascendente de um leito fluidizado circulante. Utiliza-se o modelo euleriano de duas fases separadas considerando dois procedimentos diferentes para a modelagem do tensor das tensões da fase sólida: modelo tradicional e a teoria cinética dos escoamentos granulares (TCEG). As simulações numéricas são conduzidas com a utilização do código MFIX que é um software livre e disponível na rede (Internet). Os resultados da simulação numérica são avaliados por meio da análise da influência dos seguintes parâmetros: malha computacional, correlações para o computo do tensor das tensões da fase sólida e esquemas de discretização dos termos advectivos. Também se desenvolve estudo de caracterização de estruturas coerentes - \"clusters\". De forma complementar foram realizadas duas análises teóricas compreendendo: uma análise da influência das diversas correlações utilizadas na TCEG para o computo da viscosidade dinâmica do sólido; e uma análise enfocando o emprego de diversos esquemas de discretização para os termos advectivos presentes nas equações de conservação (Foup, Muscl, Van Leer, Minmod e Superbee). De todos os estudos e resultados apresentados no trabalho conclui-se que os escoamentos gás-sólido em leitos fluidizados circulantes são muito complexos, sendo necessário a realização de futuras pesquisas para uma melhor compreensão dos fenômenos físicos inerentes a esses escoamentos. / In the present work is described a mathematical model and numerical study simulation of the gas-solid flow in the riser of a circulating fluidized bed. It is used the two fluids eulerian model considering two different procedures for the solid phase stress tensor modeling: the traditional model and the kinetic theory of granular flows (KTGF). The numerical simulation results are evaluated through the influence analysis of the following parameters: computational mesh, correlations for computing the solid phase stress tensor and the discretization of the advective terms. It is also presented a study concerning the characterization coherent structures - \"clusters\". Complementing the above studies were accomplished two theoretical analyses comprehending: an influence analysis of several correlations used in the KTGF for computing the dynamic viscosity of the solid phase; and an analysis concerning several discretization schemes for the advective terms present in the conservative equations. Considering the developed studies and the obtained results it is concluded that the gas-solid flows in circulating fluidized beds are very complex, being necessary future research works for a better comprehension of the inherent physical phenomena to these flows.

No presente trabalho desenvolve-se um estudo de modelagem matemática e simulação numérica do escoamento bifásico gás-sólido na coluna ascendente de um leito fluidizado circulante. Utiliza-se o modelo euleriano de duas fases separadas considerando dois procedimentos diferentes para a modelagem do tensor das tensões da fase sólida: modelo tradicional e a teoria cinética dos escoamentos granulares (TCEG). As simulações numéricas são conduzidas com a utilização do código MFIX que é um software livre e disponível na rede (Internet). Os resultados da simulação numérica são avaliados por meio da análise da influência dos seguintes parâmetros: malha computacional, correlações para o computo do tensor das tensões da fase sólida e esquemas de discretização dos termos advectivos. Também se desenvolve estudo de caracterização de estruturas coerentes - \"clusters\". De forma complementar foram realizadas duas análises teóricas compreendendo: uma análise da influência das diversas correlações utilizadas na TCEG para o computo da viscosidade dinâmica do sólido; e uma análise enfocando o emprego de diversos esquemas de discretização para os termos advectivos presentes nas equações de conservação (Foup, Muscl, Van Leer, Minmod e Superbee). De todos os estudos e resultados apresentados no trabalho conclui-se que os escoamentos gás-sólido em leitos fluidizados circulantes são muito complexos, sendo necessário a realização de futuras pesquisas para uma melhor compreensão dos fenômenos físicos inerentes a esses escoamentos. / In the present work is described a mathematical model and numerical study simulation of the gas-solid flow in the riser of a circulating fluidized bed. It is used the two fluids eulerian model considering two different procedures for the solid phase stress tensor modeling: the traditional model and the kinetic theory of granular flows (KTGF). The numerical simulation results are evaluated through the influence analysis of the following parameters: computational mesh, correlations for computing the solid phase stress tensor and the discretization of the advective terms. It is also presented a study concerning the characterization coherent structures - \"clusters\". Complementing the above studies were accomplished two theoretical analyses comprehending: an influence analysis of several correlations used in the KTGF for computing the dynamic viscosity of the solid phase; and an analysis concerning several discretization schemes for the advective terms present in the conservative equations. Considering the developed studies and the obtained results it is concluded that the gas-solid flows in circulating fluidized beds are very complex, being necessary future research works for a better comprehension of the inherent physical phenomena to these flows.

La modélisation du vivant, en particulier la modélisation de l'activité cardiaque, est devenue un défi scientifique majeur. Le but de cette thématique est de mieux comprendre les phénomènes physiologiques et donc d'apporter des solutions à des problèmes cliniques. Nous nous intéressons dans cette thèse à la modélisation et à l'étude numérique de l'activité électrique du cœur, en particulier l'étude des électrocardiogrammes (ECGs). L'onde électrique dans le cœur est gouvernée par un système d'équations de réaction-diffusion appelé modèle bidomaine ce système est couplé à une EDO représentant l'activité cellulaire. Afin simuler des ECGs, nous tenons en compte la propagation de l'onde électrique dans le thorax qui est décrite par une équation de diffusion. Nous commençons par une démonstrer l'existence d'une solution faible du système couplé cœur-thorax pour une classe de modèles ioniques phénoménologiques. Nous prouvons ensuite l'unicité de cette solution sous certaines conditions. Le plus grand apport de cette thèse est l'étude et la simulation numérique du couplage électrique cœur-thorax. Les résultats de simulations sont représentés à l'aide des ECGs. Dans une première partie, nous produisons des simulations pour un cas normal et pour des cas pathologiques (blocs de branche gauche et droit et des arhythmies). Nous étudions également l'impact de certaines hypothèses de modélisation sur les ECGs (couplage faible, utilisation du modèle monodomaine, isotropie, homogénéité cellulaire, comportement résistance-condensateur du péricarde,. . . ). Nous étudions à la fin de cette partie la sensibilité des ECGs par apport aux paramètres du modèle. En deuxième partie, nous effectuons l'analyse numérique de schémas du premier ordre en temps découplant les calculs du potentiel d'action et du potentiel extérieur. Puis, nous combinons ces schémas en temps avec un traîtement explicite du type Robin-Robin des conditions de couplage entre le cœur et le thorax. Nous proposons une analyse de stabilité de ces schémas et nous illustrons les résultats avec des simulations numériques d'ECGs. La dernière partie est consacrée à trois applications. Nous commençons par l'estimation de certains paramètres du modèle (conductivité du thorax et paramètres ioniques). Dans la deuxième application, qui est d'originie industrielle, nous utilisons des méthodes d'apprentissage statistique pour reconstruire des ECGs à partir de mesures ('électrogrammes). Enfin, nous présentons des simulations électro-mécaniques du coeur sur une géométrie réelle dans diverses situations physiologiques et pathologiques. Les indicateurs cliniques, électriques et mécaniques, calculés à partir de ces simulations sont très similaires à ceux observés en réalité.