The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of fluids, from water moving around a naval vessel to blood flowing through the arteries of the cardiovascular system. Furthermore, the secrets of turbulence are widely believed to be locked within the Navier-Stokes equations. Despite the enormous applicability of the Navier-Stokes equations, the underlying behavior of solutions to the partial differential system remains little understood. Indeed, one of the Clay Mathematics Institute's famed Millenium Prize Problems involves the establishment of existence and smoothness results for Navier-Stokes solutions, and turbulence is considered, in the words of famous physicist Richard Feynman, to be "the last great unsolved problem of classical physics." Numerical simulation has proven to be a very useful tool in the analysis of the Navier-Stokes equations. Simulation of incompressible flows now plays a major role in the industrial design of automobiles and naval ships, and simulation has even been utilized to study the Navier-Stokes existence and smoothness problem. In spite of these successes, state-of-the-art incompressible flow solvers are not without their drawbacks. For example, standard turbulence models which rely on the existence of an energy spectrum often fail in non-trivial settings such as rotating flows. More concerning is the fact that most numerical methods do not respect the fundamental geometric properties of the Navier-Stokes equations. These methods only satisfy the incompressibility constraint in an approximate sense. While this may seem practically harmless, conservative semi-discretizations are typically guaranteed to balance energy if and only if incompressibility is satisfied pointwise. This is especially alarming as both momentum conservation and energy balance play a critical role in flow structure development. Moreover, energy balance is inherently linked to the numerical stability of a method. In this dissertation, novel B-spline discretizations for the generalized Stokes and Navier-Stokes equations are developed. The cornerstone of this development is the construction of smooth generalizations of Raviart-Thomas-Nedelec elements based on the new theory of isogeometric discrete differential forms. The discretizations are (at least) patch-wise continuous and hence can be directly utilized in the Galerkin solution of viscous flows for single-patch configurations. When applied to incompressible flows, the discretizations produce pointwise divergence-free velocity fields. This results in methods which properly balance both momentum and energy at the semi-discrete level. In the presence of multi-patch geometries or no-slip walls, the discontinuous Galerkin framework can be invoked to enforce tangential continuity without upsetting the conservation and stability properties of the method across patch boundaries. This also allows our method to default to a compatible discretization of Darcy or Euler flow in the limit of vanishing viscosity. These attributes in conjunction with the local stability properties and resolution power of B-splines make these discretizations an attractive candidate for reliable numerical simulation of viscous incompressible flows. / text

Neste trabalho, construímos duas generalizações de uma classe de modelos discretos bidimensionais, assim chamados \"Quantum Double Models\", definidos em variedades orientáveis, compactas e sem fronteiras. Na primeira generalização, introduzimos campos de matéria aos vértices e, na segunda, às faces. Além das propriedades básicas dos modelos, estudamos como se comporta a sua ordem topológica sob a hipótese de que os estados de base são indexados por grupos Abelianos. Na primeira generalização, surge um novo fenômeno de confinamento. Como consequência, a degenerescência do estado fundamental se torna independente do grupo fundamental sobre o qual o modelo está definido, dependendo da ação do grupo de calibre e do segundo grupo de homologia. A segunda generalização pode ser vista como o dual algébrico da primeira. Nela, as mesmas propriedades de confinamento de quasipartículas está presente, mas a degenerescência do estado fundamental continua dependendo do grupo fundamental. Além disso, degenerescências adicionais aparecem, relacionadas ao homomorfismo de coação entre os grupos de matéria e de calibre. / In this work, we constructed two generalizations of a class of discrete bidimensional models, the so called Quantum Double Models, defined in orientable, compact and boundaryless manifolds. In the first generalization we introduced matter fields to the vertices and, in the second one, to the faces. Beside the basic model properties, we studied its topological order behaviour under the hypothesis that the basic states be indexed by Abelian groups. In the first generalization, appears a new phenomenon of quasiparticle confinement. As a consequence, the ground state degeneracy becomes independent of the fundamental group of the manifold on which the model is defined, depending on the action of the gauge group and on the second group of homology. The second generalization can be seen as the algebraic dual of the first one. In it, the same quasiparticle confinement properties are present, but the ground state degeneracy stay dependent on the fundamental group. Besides, additional degeneracies appear, related to a coaction homomorphism between matter and gauge groups.

Neste trabalho, construímos duas generalizações de uma classe de modelos discretos bidimensionais, assim chamados \"Quantum Double Models\", definidos em variedades orientáveis, compactas e sem fronteiras. Na primeira generalização, introduzimos campos de matéria aos vértices e, na segunda, às faces. Além das propriedades básicas dos modelos, estudamos como se comporta a sua ordem topológica sob a hipótese de que os estados de base são indexados por grupos Abelianos. Na primeira generalização, surge um novo fenômeno de confinamento. Como consequência, a degenerescência do estado fundamental se torna independente do grupo fundamental sobre o qual o modelo está definido, dependendo da ação do grupo de calibre e do segundo grupo de homologia. A segunda generalização pode ser vista como o dual algébrico da primeira. Nela, as mesmas propriedades de confinamento de quasipartículas está presente, mas a degenerescência do estado fundamental continua dependendo do grupo fundamental. Além disso, degenerescências adicionais aparecem, relacionadas ao homomorfismo de coação entre os grupos de matéria e de calibre. / In this work, we constructed two generalizations of a class of discrete bidimensional models, the so called Quantum Double Models, defined in orientable, compact and boundaryless manifolds. In the first generalization we introduced matter fields to the vertices and, in the second one, to the faces. Beside the basic model properties, we studied its topological order behaviour under the hypothesis that the basic states be indexed by Abelian groups. In the first generalization, appears a new phenomenon of quasiparticle confinement. As a consequence, the ground state degeneracy becomes independent of the fundamental group of the manifold on which the model is defined, depending on the action of the gauge group and on the second group of homology. The second generalization can be seen as the algebraic dual of the first one. In it, the same quasiparticle confinement properties are present, but the ground state degeneracy stay dependent on the fundamental group. Besides, additional degeneracies appear, related to a coaction homomorphism between matter and gauge groups.

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.

In the paper we construct exact three-point discretizations of linear and nonlinear two-point boundary value problems with boundary conditions of the first kind. The finite element approach uses basis functions defined by the coefficients of the differential equations. All the discretized boundary value problems are of inverse isotone type and so are its exact discretizations which involve tridiagonal M-matrices in the linear case and M-functions in the nonlinear case.

In the paper we construct exact three-point discretizations of linear and nonlinear two-point boundary value problems with boundary conditions of the first kind. The finite element approach uses basis functions defined by the coefficients of the differential equations. All the discretized boundary value problems are of inverse isotone type and so are its exact discretizations which involve tridiagonal M-matrices in the linear case and M-functions in the nonlinear case.

This dissertation presents results of the study on symplectic and multisymplectic numerical methods for solving linear and nonlinear Hamiltonian wave equations. The emphasis is put on the second order space and time discretizations of the linear wave, the Klein-Gordon and the sine-Gordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and non-symplectic numerical methods. Tools used in the linear analysis are related to the Fourier transform and consist of the dispersion relationship and the power spectrum of the numerical solution. Nonlinear analysis, in turn, is closely connected to the temporal evolution of the total energy (Hamiltonian) and can be viewed from the topological perspective as preservation of the phase space structures. Using both linear and nonlinear diagnostics we find qualitative differences between MS and non-MS methods. The first difference can be noted in simulations of the linear wave equation solved for broad spectrum Gaussian initial data. Initial wave profiles of this type immediately split into an oscillatory wave-train with the high modes traveling faster (MS schemes), or slower (non-MS methods), than the analytic group velocity. This result is confirmed by an analysis of the dispersion relationship, which also indicates improved qualitative agreement of the dispersive curves for MS methods over non-MS ones. Moreover, observations of the convergence patterns in the wave profile obtained for the sine-Gordon equation for the initial data corresponding to the double-pole soliton and the temporal evolution of the Hamiltonian functional computed for solutions obtained from different discretizations suggest a change of the geometry of the phase space. Finally, we present some theoretical considerations concerning wave action. Lagrangian formulation of linear partial differential equations (PDEs) with slowly varying solutions is capable of linking the wave action conservation law with the dispersion relationship thus suggesting the possibility to extend this connection to multisymplectic PDEs.

The goal of the current study was to develop a computational framework for modelling the coupled fluid-structure interaction problem of squeeze films often encountered in MEMS devices. Vibratory MEMS devices such as gyroscopes, RF switches, and 2D resonators often have a thin plate like structure vibrating transversely to a Fixed substrate, and are generally not perfectly vacuum packed. This results in a thin air film being trapped between the vibrating plate and the fixed substrate which behaves like a squeeze film offering both stiffness and damping to the vibrating plate. For accurate modelling of the squeeze film effect, one must account for the coupled fluid-structure interaction. The majority of prior works attempting to address the coupled problem either approximate the mode shape of the vibrating plate or resort to cumbersome iterative solution strategies to address the problem in an indirect way. In the current work, we discuss the development of a fully coupled finite element based numerical scheme to solve the 2D Reynolds equation coupled with the 3D plate elasticity equation in a single step. The squeeze film solver so developed has been implemented into a commercial FEA package NISA as part of its Micro-Systems module. Further, extending on a prior analytical work, the effect of variable ow boundaries for an all sides clamped plate on squeeze film parameters has been thoroughly investigated. The developed FEM based numerical scheme has been used to validate the results of the prior analytical study. The developed numerical scheme models the 2D Reynolds equation thus limiting the model to account for the effects of the fluid volume strictly confined between the structure and the substrate. To study the effect of surrounding fluid volume ANSYS FLOTRAN simulations have been performed by numerically solving the full 3D Navier Stokes equation in the extended fluid domain for the different flow boundary scenarios. Cut-off frequencies are established beyond which one can consider a 2D fluid domain without considerable loss of accuracy. First, a displacement based finite element formulation is presented for the 2D Reynolds equation coupled with the 3D elasticity equation. Both lower order 8 node and higher order 27 node 3D elements are developed. Only a single type of 3D element is used for modelling along with a 2D fluid layer represented by the \wet" face of the 3D structural domain. The results from our numerical model are compared with experimental data from literature for a MEMS cantilever. The results from the 27 node displacement based elements show good agreement with published experimental data. The results from the lower order 8 node displacement based elements however show huge errors even for relatively fine meshes due to locking issues in modelling high aspect ratio structures. This limits the implementation of the displacement based solver in commercial FE packages where the available mesh generators are generally restricted to lower order 3D elements. In order to overcome the limitations faced by lower order elements (primarily locking issues) in modelling high aspect ratio MEMS geometries, a coupled hybrid formulation is developed next. A thorough performance study is presented considering both the hybrid and displacement based elements for lower order 8 node and higher order 27 node ele- ments. The optimal element choice for modelling squeeze film geometries is determined based on the comparative studies. The effect of element aspect ratio for hybrid and displacement based elements are studied and the superiority of hybrid formulation over displacement based formulations is established for lower order 8 node elements. The coupled hybrid nite element formulation developed for lower order elements is implemented in the commercial FEA package NISA. The implementation scheme to integrate the developed coupled hybrid 8 node squeeze film solver into the commercial FEA package is discussed. The pre-integration analysis and subsequent requirement gaps are first investigated. Based on the gap analysis, certain GUI modifications are undertaken and parser programs are developed to re-format data according to NISA input requirements. Certain special features are included in the package to aid in post processing data analysis by MEMS designers such as \frequency sweep" and \node of interest" selection. As a case study for validation, we also present the modelling of a MEMS cantilever and show that the simulation results from our software are in good agreement with experimental data reported in the literature. Finally as a case study, an extension of a prior analytical work, which studies the effect of varying flow boundaries on squeeze film parameters, is discussed. Explanations are provided for the findings reported in the prior analytical work. The concept of using variation in flow boundaries as a frequency tuning tool is introduced. The analytical results are validated with the coupled numerical scheme discussed before, by considering imposed mode shape for an all sides clamped plate as prescribed displacement to the fluid domain. The simulated results are used to study the intricacies in squeeze film damping and stiffness variations with respect to spatial changes in the fluid flow boundary conditions. In particular, it has been shown that the boundary venting conditions can be used effectively to tune the dynamic response of a micromechanical structure over a fairly large range of frequencies and somewhat smaller range of squeeze film damping. Next, the effect of the surrounding fluid volume for various venting conditions is studied. ANSYS FLOTRAN is used to solve for the full 3D Navier Stokes equation over the extended fluid domain. Results from the extended domain study are used to determine cut-off frequencies beyond which one need not resort to an extended mesh study, and yet be within 5% accuracy of the full extended mesh model.

Les discrétisations adaptatives sont importantes dans les problèmes de fluxcompressible/incompressible puisqu'il est souvent nécessaire de résoudre desdétails sur plusieurs niveaux, en permettant de modéliser de grandes régionsd'espace en utilisant un nombre réduit de degrés de liberté (et en réduisant letemps de calcul).Il existe une grande variété de méthodes de discrétisation adaptative, maisles grilles cartésiennes sont les plus efficaces, grâce à leurs stencilsnumériques simples et précis et à leurs performances parallèles supérieures.Et telles performance et simplicité sont généralement obtenues en appliquant unschéma de différences finies pour la résolution des problèmes, mais cetteapproche de discrétisation ne présente pas, au contraire, un chemin faciled'adaptation.Dans un schéma de volumes finis, en revanche, nous pouvons incorporer différentstypes de maillages, plus appropriées aux raffinements adaptatifs, en augmentantla complexité sur les stencils et en obtenant une plus grande flexibilité.L'opérateur de Laplace est un élément essentiel des équations de Navier-Stokes,un modèle qui gouverne les écoulements de fluides, mais il se produit égalementdans des équations différentielles qui décrivent de nombreux autres phénomènesphysiques, tels que les potentiels électriques et gravitationnels. Il s'agitdonc d'un opérateur différentiel très important, et toutes les études qui ontété effectuées sur celui-ci, prouvent sa pertinence.Dans ce travail seront présentés des approches de différences finies et devolumes finis 2D pour résoudre l'opérateur laplacien, en appliquant des patchsde grilles superposées où un niveau plus fin est nécessaire, en laissant desmaillages plus grossiers dans le reste du domaine de calcul.Ces grilles superposées auront des formes quadrilatérales génériques.Plus précisément, les sujets abordés seront les suivants:1) introduction à la méthode des différences finies, méthode des volumes finis,partitionnement des domaines, approximation de la solution;2) récapitulatif des différents types de maillages pour représenter de façondiscrète la géométrie impliquée dans un problème, avec un focussur la structure de données octree, présentant PABLO et PABLitO. Le premier estune bibliothèque externe utilisée pour gérer la création de chaque grille,l'équilibrage de charge et les communications internes, tandis que la secondeest l'API Python de cette bibliothèque, écrite ad hoc pour le projet en cours;3) la présentation de l'algorithme utilisé pour communiquer les données entreles maillages (en ignorant chacune l'existence de l'autre) en utilisant lesintercommunicateurs MPI et la clarification de l'approche monolithique appliquéeà la construction finale de la matrice pour résoudre le système, en tenantcompte des blocs diagonaux, de restriction et de prolongement;4) la présentation de certains résultats; conclusions, références.Il est important de souligner que tout est fait sous Python comme framework deprogrammation, en utilisant Cython pour l'écriture de PABLitO, MPI4Py pour lescommunications entre grilles, PETSc4py pour les parties assemblage et résolutiondu système d'inconnues, NumPy pour les objets à mémoire continue.Le choix de ce langage de programmation a été fait car Python, facile àapprendre et à comprendre, est aujourd'hui un concurrent significatif pourl'informatique numérique et l'écosystème HPC, grâce à son style épuré, sespackages, ses compilateurs et pourquoi pas ses versions optimisées pour desarchitectures spécifiques. / Adaptive discretizations are important in compressible/incompressible flow problems since it is often necessary to resolve details on multiple levels,allowing large regions of space to be modeled using a reduced number of degrees of freedom (reducing the computational time).There are a wide variety of methods for adaptively discretizing space, but Cartesian grids have often outperformed them even at high resolutions due totheir simple and accurate numerical stencils and their superior parallel performances.Such performance and simplicity are in general obtained applying afinite-difference scheme for the resolution of the problems involved, but this discretization approach does not present, by contrast, an easy adapting path.In a finite-volume scheme, instead, we can incorporate different types of grids,more suitable for adaptive refinements, increasing the complexity on thestencils and getting a greater flexibility.The Laplace operator is an essential building block of the Navier-Stokes equations, a model that governs fluid flows, but it occurs also in differential equations that describe many other physical phenomena, such as electric and gravitational potentials, and quantum mechanics. So, it is a very importantdifferential operator, and all the studies carried out on it, prove itsrelevance.In this work will be presented 2D finite-difference and finite-volume approaches to solve the Laplacian operator, applying patches of overlapping grids where amore fined level is needed, leaving coarser meshes in the rest of the computational domain.These overlapping grids will have generic quadrilateral shapes.Specifically, the topics covered will be:1) introduction to the finite difference method, finite volume method, domainpartitioning, solution approximation;2) overview of different types of meshes to represent in a discrete way thegeometry involved in a problem, with a focuson the octree data structure, presenting PABLO and PABLitO. The first one is anexternal library used to manage each single grid’s creation, load balancing and internal communications, while the second one is the Python API ofthat library written ad hoc for the current project;3) presentation of the algorithm used to communicate data between meshes (beingall of them unaware of each other’s existence) using MPI inter-communicators and clarification of the monolithic approach applied building the finalmatrix for the system to solve, taking into account diagonal, restriction and prolongation blocks;4) presentation of some results; conclusions, references.It is important to underline that everything is done under Python as programmingframework, using Cython for the writing of PABLitO, MPI4Py for the communications between grids, PETSc4py for the assembling and resolution partsof the system of unknowns, NumPy for contiguous memory buffer objects.The choice of this programming language has been made because Python, easy to learn and understand, is today a significant contender for the numerical computing and HPC ecosystem, thanks to its clean style, its packages, its compilers and, why not, its specific architecture optimized versions.

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.