Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

Over the last three decades, CFD simulations have become commonplace as a tool in the engineering and design of high-speed aircraft. Experiments are often complemented by computational simulations, and CFD technologies have proved very useful in both the reduction of aircraft development cycles, and in the simulation of conditions difficult to reproduce experimentally. Great advances have been made in the field since its introduction, especially in areas of meshing, computer architecture, and solution strategies. Despite this, there still exist many computational limitations in existing CFD methods; in particular, reliable higher order and hp-adaptive methods for the Navier-Stokes equations that govern viscous compressible flow. Solutions to the equations of viscous flow can display shocks and boundary layers, which are characterized by localized regions of rapid change and high gradients. The use of adaptive meshes is crucial in such settings -- good resolution for such problems under uniform meshes is computationally prohibitive and impractical for most physical regimes of interest. However, the construction of "good" meshes is a difficult task, usually requiring a-priori knowledge of the form of the solution. An alternative to such is the construction of automatically adaptive schemes; such methods begin with a coarse mesh and refine based on the minimization of error. However, this task is difficult, as the convergence of numerical methods for problems in CFD is notoriously sensitive to mesh quality. Additionally, the use of adaptivity becomes more difficult in the context of higher order and hp methods. Many of the above issues are tied to the notion of robustness, which we define loosely for CFD applications as the degradation of the quality of numerical solutions on a coarse mesh with respect to the Reynolds number, or nondimensional viscosity. For typical physical conditions of interest for the compressible Navier-Stokes equations, the Reynolds number dictates the scale of shock and boundary layer phenomena, and can be extremely high -- on the order of 10⁷ in a unit domain. For an under-resolved mesh, the Galerkin finite element method develops large oscillations which prevent convergence and pollute the solution. The issue of robustness for finite element methods was addressed early on by Brooks and Hughes in the SUPG method, which introduced the idea of residual-based stabilization to combat such oscillations. Residual-based stabilizations can alternatively be viewed as modifying the standard finite element test space, and consequently the norm in which the finite element method converges. Demkowicz and Gopalakrishnan generalized this idea in 2009 by introducing the Discontinous Petrov-Galerkin (DPG) method with optimal test functions, where test functions are determined such that they minimize the discrete linear residual in a dual space. Under the ultra-weak variational formulation, these test functions can be computed locally to yield a symmetric, positive-definite system. The main theoretical thrust of this research is to develop a DPG method that is provably robust for singular perturbation problems in CFD, but does not suffer from discretization error in the approximation of test functions. Such a method is developed for the prototypical singular perturbation problem of convection-diffusion, where it is demonstrated that the method does not suffer from error in the approximation of test functions, and that the L² error is robustly bounded by the energy error in which DPG is optimal -- in other words, as the energy error decreases, the L² error of the solution is guaranteed to decrease as well. The method is then extended to the linearized Navier-Stokes equations, and applied to the solution of the nonlinear compressible Navier-Stokes equations. The numerical work in this dissertation has focused on the development of a 2D compressible flow code under the Camellia library, developed and maintained by Nathan Roberts at ICES. In particular, we have developed a framework allowing for rapid implementation of problems and the easy application of higher order and hp-adaptive schemes based on a natural error representation function that stems from the DPG residual. Finally, the DPG method is applied to several convection diffusion problems which mimic difficult problems in compressible flow simulations, including problems exhibiting both boundary layers and singularities in stresses. A viscous Burgers' equation is solved as an extension of DPG to nonlinear problems, and the effectiveness of DPG as a numerical method for compressible flow is assessed with the application of DPG to two benchmark problems in supersonic flow. In particular, DPG is used to solve the Carter flat plate problem and the Holden compression corner problem over a range of Mach numbers and laminar Reynolds numbers using automatically adaptive schemes, beginning with very under-resolved/coarse initial meshes. / text

This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

This study deals with the use of high-order spectral/hp approximation functions in the finite element models of various nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order p greater than or equal to 4) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., p < 3) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements for beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. The higher-order spectral/hp basis functions avoid the interpolation error in the numerical schemes, thereby making them accurate and stable. Furthermore, for fluid flows, when combined with least-squares variational principles, such technology allows us to develop efficient finite element models, that always yield a symmetric positive-definite (SPD) coefficient matrix, and thereby robust direct or iterative solvers can be used. The least-squares formulation avoids ad-hoc stabilization methods employed with traditional low-order weak-form Galerkin formulations. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities (e.g., dilatation, volume, and mass) and stable evolution of variables with time in the case of unsteady flows. The present study uses spectral/hp approximations in the (1) weak-form Galerkin finite element models of viscoelastic beams, (2) weak-form Galerkin displacement finite element models of shear-deformable elastic shell structures under thermal and mechanical loads, and (3) least-squares formulations for the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical simulations using the developed technology of several non-trivial benchmark problems are presented to illustrate the robustness of the higher-order spectral/hp based finite element technology.

Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases. In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method. An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T. In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.

This thesis is motivated by problems studied as part of the Diavik Waste Rock Pile Project. Located at the Diavik Diamond Mine in the Northwest Territories, with academic support from the University of Waterloo, the University of Alberta, and the University of British Columbia, this project focuses on constructing mine waste rock piles and studying their physical and chemical properties and the transport processes within them. One of the main reasons for this investigation is to determine the effect of environmental factors on acid mine drainage (AMD) due to sulfide oxidation and the potential environmental impact of AMD. This research is concerned with modeling pressure driven flow through waste rock piles. Unfortunately, because of the irregular shape of the piles, very little data for fluid flow about such an obstacle exists, and the numerical techniques available to work with this domain are limited. Since this restricts the study of the mathematics behind the flow, this thesis focuses on a cylindrical domain, since flow past a solid cylinder has been subjected to many years of study. The cylindrical domain also facilitates the implementation of a pseudo-spectral method. This thesis examines a pressure driven flow through a cylinder of variable permeability subject to turbulent forcing. An equation for the steady flow of an incompressible fluid through a variable permeability porous medium is derived based on Darcy's law, and a pseudo-spectral model is designed to solve the problem. An unsteady time-dependent model for a slightly compressible fluid is then presented, and the unsteady flow through a constant permeability cylinder is examined. The steady results are compared with a finite element model on a trapezoidal domain, which provides a better depiction of a waste rock pile cross section.

Este trabalho apresenta o desenvolvimento de um gerador de malha de elementos finitos baseado no Algoritmo de Refinamento de Delaunay. O pacote é versátil e pode ser aplicado às malhas seriais e adaptativas ou à decomposição de uma malha inicial grossa ou pré-refinada usando processamento paralelo. O algoritmo desenvolvido trabalha com uma entrada de dados na forma de um gráfico de linhas retas planas. A construção do algoritmo de Delaunay foi baseada na técnica de Watson para a triangulação fronteiriça e nos métodos seqüenciais de Ruppert e Shewchuk para o refinamento com paralelismo. A técnica elaborada produz malhas que mantêm as propriedades de uma triangulação de Delaunay. A metodologia apresentada foi implementada utilizando os conceitos de Programação Orientada a Objetos com o auxílio de bibliotecas de código livre. Aproveitando a flexibilidade de algumas dessas bibliotecas acopladas foi possível parametrizar a dimensão do problema, permitindo gerar malhas seqüenciais bidimensionais e tridimensionais. Os resultados das aplicações em malhas seriais, adaptativas e com programação paralela mostram a eficácia desta ferramenta. Uma versão acadêmica do algoritmo de refinamento de Delaunay bidimensional para o Ambiente Mathematica também foi desenvolvido. / This work presents the development of a finite elements mesh generation based on Delaunay Triangulation Algorithm. The package is versatile and applicable to the serial and adaptable meshes or to either the coarse or pre-refined initial mesh decomposition using parallel computing. The developed algorithm works with data input in the form of Planar Straight Line Graphics. The building of the Delaunay Algorithm was based on the Watson\'s technique for the boundary triangulation and in both Ruppert and Shewchuk sequential methods for the parallel refinement. The proposed technique produces meshes maintaining the properties of the Delaunay triangulation. The presented methodology was implemented using the Programming Object-Oriented concepts, which is supported by open source libraries. Taking advantage of the flexibility of some of those coupled libraries the parametrization of the problem dimension was possible, allowing to generate both two and three-dimensional sequential meshes. The results obtained with the applications in serial, adaptive and in parallel meshes have shown the effectiveness of this tool. An academic version of the twodimensional Delaunay refinement algorithm for the Mathematica Environment was also developed.

A maioria das espécies marinhas possui um ciclo de vida complexo em que a fase adulta é precedida por uma fase larval pelágica. Os adultos produzem larvas que são soltas na coluna d'água onde elas são influenciadas por processos hidrodinâmicos como advecção e difusão turbulenta. Para os organismos que na maturidade são fixos a um substrato, como as cracas em particular, estudos mostram que estes processos podem afetar diretamente o assentamento das larvas e a dinâmica das populações de adultos. Neste trabalho apresentamos um modelo em Elementos Finitos que permite estudar as interações entre os processos biológicos e oceanográficos na dinâmica de um organismo marinho (craca Balanus glandula) com uma fase adulta séssil restrita à linha costeira e uma fase larval planctônica. Alguns cenários são investigados no estudo da variação, ao longo do tempo e do espaço, do número de indivíduos e da distribuição desta população, com o objetivo de entender, explicar e predizer como os fatores abióticos e bióticos afetam tais variações. O modelo considerou a dinâmica da população estágio-estruturada tanto de uma única espécie quanto de duas espécies, diferentes padrões idealizados de correntes marinhas, qualidade do habitat, influência da temperatura e reconstrução do campo de velocidades a partir de dados locais. / Most marine benthic invertebrates have a complex life cycle in which the adult phase is preceded by a planktonic larval phase. The passive nature of larval dispersal of these species results in a larval transport governed by hydrodynamic processes like advection and eddy diffusion. The recruitment depends on a variety of physical and biological factors, which include spawning, larval dispersal and survival, larval settlement, metamorphosis, post-settlement events, inter-specific and intra-specific competitions. In order to assess interactions between oceanographic and biological processes that determine the population dynamics of marine organisms with a sessile adult phase restricted to the coastline and a planktonic larval phase, we develop a stage-structured finite element model for the barnacle Balanus glandula that inhabits the rocky intertidal zone of central California, USA. As the larval dispersal depends on knowing the flow pattern, we also develop a numerical procedure to couple physical and biological models in a very simple way when the velocity flow field is known of some discrete points of the domain of interest and at a given time. We investigate the effects of different flow patterns and velocity speeds on the abundance and the distribution of this organism as well as the influence of other abiotic interactions such as temperature and habitat quality. The interplay of different intra- and inter-specific competitions is also addressed.

Orientador: Vicente Luiz Scalon / Banca: Ivan De Domenico Valarelli / Banca: Carlos Alberto Carrasco Altemani / Resumo: Com a crise energética recente, houve uma nova conscientização da necessidade de utilização mais racional da energia. Desta feita, uma série de pesquisas com fontes alternativas de energia, que vinham sendo preteridos em função da impressão que a crise energética do início da década de 1970 havia passado, tem ganho nova força. Dentre todas as alternativas para aproveitamento de energia solar, uma das mais utilizadas são os chamados "sistemas domésticos de aquecimento de água". Este tipo de sistema, entretanto, ainda é complexo, constituído de uma série de dutos e conexões entre coletor e tanque armazenador, que contribuem para o elevado custo destes dispositivos. Uma alternativa para otimizar o custo final é o uso dos sistemas solar integrados coletor-tanque (ICS). Neste trabalho, foi avaliado o processo de movimentação natural do fluido em uma das geometrias mais comuns de sistemas deste tipo: a trapezoidal. Foi aplicada a condição de fluxo de calor constante na face inclinada para avaliação... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The recent energy crisis has developed a new conscience for necessity of rational energy use. Several works treating about renewable energy was stopped in past based in a false idea that the 1970's energy crisis was finished. Nowadays, these works have been retaken with the large use of solar energy in the Solar Domestic Hot Water Systems. However, this device is quite complex and has several components like pipes and fittings coupling solar collectors and storage tanks. This characteristic makes it an expensive system and bring difficulties for his large use. An alternative to turn it cheaper is the construction of a device with solar collector and storage tank integrated in one single component (ICS). In this work was done an evaluation of free convection process in a common geometry of this device: the trapezoidal shape. For this analysis, a constant heat flux condition was applied to the inclined face for evaluation of free convection process. Numerical results... (Complete abstract click electronic access below) / Mestre