[pt] O emprego de materiais compósitos em estruturas tem ganhado importância na prática da engenharia devido às suas características de alta resistência mecânica, baixa densidade e boa estabilidade a efeitos térmicos. Uma das classes de compósitos, a de laminados fibrosos, pode ser utilizada em tubulações sujeitas às diversas formas de carregamentos, como pressão interna e/ou externa, tração longitudinal, torção, temperatura, etc. O presente Trabalho tem por objetivo propor, implementar e testar a formulação de um modelo de elemento finito axissimétrico, para a representação do comportamento de um tubo laminado por camadas de materiais compósitos fibrosos. A modelagem consiste em representar a seção geratriz de um tubo cilíndrico por um elemento quadrilateral de quatro nós, com três graus de liberdade por nó, com os deslocamentos nodais tomados em relação aos eixos de um sistema cilíndrico de coordenadas. Considera-se a perfeita adesão das camadas, garantindo a continuidade do campo de deslocamentos. Modelos constitutivos de materiais com o comportamento ortotrópico e/ou o transversalmente isotrópico foram implementados, obtendo-se respostas para os campos de deslocamentos, de deformações e de tensões atuantes. Na validação do modelo numérico, considerou-se a comparação de seus resultados com os de soluções analíticas, disponíveis na literatura, e aqueles fornecidos por um programa comercial de elementos finitos, empregando o modelo com elementos sólidos. Foram propostos, para os testes em ambos os casos, exemplos de laminados com uma a quatro camadas, com fibras orientadas em diferentes ângulos. Destas comparações, verifica-se uma boa convergência das soluções numéricas obtidas com o presente modelo, representativo das principais características cinemáticas da classe de problemas representada. / [en] The use of composite materials in structures has grown in the engineering practice due to its characteristics, of high strength, low density and a good stability to thermal effects. A class of composites, the fibrous laminates, is generally used in tubes subjected to many types of loadings as internal and/or external pressure, traction, torsion, temperatures, etc. This work has the objectives to propose, implement and test an axisymmetric finite element model formulation that represents the mechanical behavior of a fibrous laminated composite tube. Modeling consists in representing the cylindrical tube generating section by a quadrilateral element with four nodes and three degrees-of-freedom per node, with three nodal displacements defined in a cylindrical coordinate system. Layers are considered perfectly bonded together, assuring continuity between elements on the displacement fields. Orthotropic and/or transverse isotropic constitutive material models were implemented, allowing solutions for displacement, strain and stress fields. In the element numerical model validation, result comparisons with those from analytical solutions available on literature and those from the use of layered solid elements in a commercial finite element program were considered. Some examples, considering one to four layers, with different fiber angles, were proposed for model testing. It is noted a good numerical convergence for the presenting model solutions which represent the main kinematic characteristics for this class of problems.

The low-level interpretation of images provides constraints on 3D surface shape at multiple resolutions, but typically only at scattered locations over the visual field. Subsequent visual processing can be facilitated substantially if the scattered shape constraints are immediately transformed into visible-surface representations that unambiguously specify surface shape at every image point. The required transformation is shown to lead to an ill-posed surface reconstruction problem. A well-posed variational principle formulation is obtained by invoking 'controlled continuity,' a physically nonrestrictive (generic) assumption about surfaces which is nonetheless strong enough to guarantee unique solutions. The variational principle, which admits an appealing physical interpretation, is locally discretized by applying the finite element method to a piecewise, finite element representation of surfaces. This forms the mathematical basis of a unified and general framework for computing visible-surface representations. The computational framework unifies formal solutions to the key problems of (i) integrating multiscale constraints on surface depth and orientation from multiple visual sources, (ii) interpolating these scattered constraints into dense, piecewise smooth surfaces, (iii) discovering surface depth and orientation discontinuities and allowing them to restrict interpolation appropriately, and (iv) overcoming the immense computational burden of fine resolution surface reconstruction. An efficient surface reconstruction algorithm is developed. It exploits multiresolution hierarchies of cooperative relaxation processes and is suitable for implementation on massively parallel networks of simple, locally interconnected processors. The algorithm is evaluated empirically in a diversity of applications.

Computational modules early in the human vision system typically generate sparse information about the shapes of visible surfaces in the scene. Moreover, visual processes such as stereopsis can provide such information at a number of levels spanning a range of resolutions. In this paper, we extend this multi-level structure to encompass the subsequent task of reconstructing full surface descriptions from the sparse information. The mathematical development proceeds in three steps. First, the surface most consistent with the sparse constraints is characterized as the solution to an equilibrium state of a thin flexible plate. Second, local, finite element representations of surfaces are introduced and, by applying the finite element method, the continuous variational principle is transformed into a discrete problem in the form of a large system of linear algebraic equations whose solution is computable by local-support, cooperative mechanisms. Third, to exploit the information available at each level of resolution, a hierarchy of discrete problems is formulated and a highly efficient multi-level algorithm, involving both intra-level relaxation processes and bi-directional inter-level algorithm, involving both intra-level relaxation processes and bidirectional inter-level local interpolation processes is applied to their simultaneous solution.. Examples of the generation of hierarchies of surface representations from stereo constraints are given. Finally, the basic surface approximation problem is revisited in a broader mathematical context whose implications are of relevance to vision.

Proppants are a specific application of granular materials used in oil/gas well stimulation. Employment of hard and soft particle mixtures is one of the many approaches availed by the industry to improve fracture resistance and the stability of the granular pack in the hydraulic fracture. Current industrial practices of proppant characterization involve long term and expensive conductivity tests. However, the mechanics governing the proppant pack response, in particular the effects due to material, shape and size of particles on the pack porosity, stiffness and particle fragmentation are not understood clearly. The present research embodies analytical and experimental approach to model hard (ceramic) and soft (walnut shell and/or pure aluminum) proppant mixtures by taking into account polydispersity in size, shape and material type of individual particles. The hydraulic fracture condition is represented through confined compression and flowback loads. The particle interactions clearly illustrate changes in pore space as a function of pressure, mixture composition and friction. Single particle compression tests on individual particles are carried out to obtain mechanical properties which are incorporated into the finite element models and are further correlated with the compression/crush response of the mixture. The proppant pack stiffness and particle fragmentation depends strongly on the mixture composition as illustrated in the models and experiments. The flowback models demonstrated that the formation of a stable arch is essential to pack stability. Additional variables that enhance flowback resistance are identified as: addition of softer particles to a pack, softer rock surfaces and higher inter-particle friction. The computational studies also led to the discovery of better, and more efficient pack compositions such as - short and thin pure Al needles/ceramic and the pistachio shells/ceramic mixtures. These analytical results have generated great interest and are engaged in the design of experiments to formulate future proppant pack mixtures at Baker Hughes Pressure Pumping, Tomball, TX.

This thesis is devoted to the numerical analysis of mesh-free methods and, in particular, to the study of the possible advantages of the EFG (Element Free Galerkin) mesh-free method against the well-known FE (Finite Element) method. More precisely, the EFG method and the FE method behavior are compared in two particular interesting problems: (1) analysis of volumetric locking in mechanical problems and (2) accurate resolution of transient convection dominated problems. In both cases the good properties and possibilities of mesh-free methods become apparent. However, in several situations the FE method is still more competitive: for instance, the computation of the FE shape functions and its integrals are less costly, and essential boundary conditions can be easily imposed. Thus, in order to take advantage of the good properties of both methods, a mixed interpolation combining FE and EFG is proposed. This formulation can be applied in two useful situations: (i) enrichment of finite elements with EFG, and (ii) coupling of FE and EFG. An a priori error estimate for the first one is presented and proved. Several examples show the applicability of the mixed interpolation in adaptive computations. / Aquesta tesi està dedicada a l'anàlisi numèrica dels mètodes sense malla i, en particular, a l'estudi dels possibles avantatges del mètode EFG (Element Free Galerkin) davant del ben conegut MEF (Mètode dels Elements Finits). Concretament, es comparen el mètode EFG i el MEF en dos problemes concrets d'interès: (1) l'anàlisi del bloqueig volumètric en problemes mecànics i (2) la resolució precisa de problemes transitoris amb convecció dominant. Les bones propietats i possibilitats dels mètodes sense malla es fan evidents en tots dos casos.Tot i així, en varis aspectes el MEF resulta més competitiu: per exemple, el càlcul de les funcions de forma i de les seves integrals es menys costós, i les condicions de contorn essencials es poden imposar fàcilment. Amb l'objectiu d'aprofitar les bones qualitats dels dos mètodes, es proposa una interpolació mixta combinant elements finits y EFG, aplicable en dues situacions: (i) enriquiment d'elements finits amb EFG i (ii) acoblament d'elements finits i EFG. Per al primer cas, es presenta i demostra una cota a priori de l'error. L'aplicabilitat d'aquesta interpolació mixta en processos adaptatius es mostra amb varis exemples. / Esta tesis está dedicada al análisis numérico de los métodos sin malla y, en particular, al estudio de las posibles ventajas del método EFG (Element Free Galerkin) frente al bien conocido MEF (Método de los Elementos Finitos). Concretamente, se comparan el método EFG y el MEF en dos problemas concretos de interés: (1) el análisis del bloqueo volumétrico en problemas mecánicos y (2) la resolución precisa de problemas transitorios con convección dominante. Las buenas propiedades y posibilidades de los métodos sin malla se hacen evidentes en ambos casos.Sin embargo, en varios aspectos el MEF resulta más competitivo: por ejemplo, el cálculo de las funciones de forma y sus integrales es menos costoso, y las condiciones de contorno esenciales se pueden imponer fácilmente. Con el objetivo de aprovechar las buenas cualidades de ambos métodos, se propone una interpolación mixta combinando elementos finitos y EFG, aplicable en dos situaciones: (i) enriquecimiento de elementos finitos con EFG, y (ii) acoplamiento de elementos finitos y EFG. Para el primer caso, se presenta y demuestra una cota a priori del error. La aplicabilidad de esta interpolación mixta en procesos adaptativos se muestra con varios ejemplos.

For centuries physicians, scientists, engineers, mathematicians, and many others have been asking: 'what are the forces that drive tissues in an embryo to their final geometric forms?' At the tissue and whole embryo level, a multitude of very different morphogenetic processes, such as gastrulation and neurulation are involved. However, at the cellular level, virtually all of these processes are evidently driven by a relatively small number of internal structures all of whose forces can be resolved into equivalent interfacial tensions γ. Measuring the cell-level forces that drive specific morphogenetic events remains one of the great unsolved problems of biomechanics. Here I present a novel approach that allows these forces to be estimated from time lapse images. In this approach, the motions of all visible triple junctions formed between trios of cells adjacent to each other in epithelia (2D cell sheets) are tracked in time-lapse images. An existing cell-based Finite Element (FE) model is then used to calculate the viscous forces needed to deform each cell in the observed way. A recursive least squares technique with variable forgetting factors is then used to estimate the interfacial tensions that would have to be present along each cell-cell interface to provide those forces, along with the attendant pressures in each cell. The algorithm is tested extensively using synthetic data from an FE model. Emphasis is placed on features likely to be encountered in data from live tissues during morphogenesis and wound healing. Those features include algorithm stability and tracking despite input noise, interfacial tensions that could change slowly or suddenly, and complications from imaging small regions of a larger epithelial tissue (the frayed boundary problem). Although the basic algorithm is highly sensitive to input noise due to the ill-conditioned nature of the system of equations that must be solved to obtain the interfacial tensions, methods are introduced to improve the resulting force and pressure estimates. The final algorithm returns very good estimates for interfacial tensions and intracellular cellular pressures when used with synthetic data, and it holds great promise for calculating the forces that remodel live tissue.

For centuries physicians, scientists, engineers, mathematicians, and many others have been asking: 'what are the forces that drive tissues in an embryo to their final geometric forms?' At the tissue and whole embryo level, a multitude of very different morphogenetic processes, such as gastrulation and neurulation are involved. However, at the cellular level, virtually all of these processes are evidently driven by a relatively small number of internal structures all of whose forces can be resolved into equivalent interfacial tensions γ. Measuring the cell-level forces that drive specific morphogenetic events remains one of the great unsolved problems of biomechanics. Here I present a novel approach that allows these forces to be estimated from time lapse images. In this approach, the motions of all visible triple junctions formed between trios of cells adjacent to each other in epithelia (2D cell sheets) are tracked in time-lapse images. An existing cell-based Finite Element (FE) model is then used to calculate the viscous forces needed to deform each cell in the observed way. A recursive least squares technique with variable forgetting factors is then used to estimate the interfacial tensions that would have to be present along each cell-cell interface to provide those forces, along with the attendant pressures in each cell. The algorithm is tested extensively using synthetic data from an FE model. Emphasis is placed on features likely to be encountered in data from live tissues during morphogenesis and wound healing. Those features include algorithm stability and tracking despite input noise, interfacial tensions that could change slowly or suddenly, and complications from imaging small regions of a larger epithelial tissue (the frayed boundary problem). Although the basic algorithm is highly sensitive to input noise due to the ill-conditioned nature of the system of equations that must be solved to obtain the interfacial tensions, methods are introduced to improve the resulting force and pressure estimates. The final algorithm returns very good estimates for interfacial tensions and intracellular cellular pressures when used with synthetic data, and it holds great promise for calculating the forces that remodel live tissue.

The use of B-spline interpolation functions in the finite element method (FEM) is not a new subject. B-splines have been utilized in finite elements for many reasons. One reason is the higher continuity of derivatives and smoothness of B-splines. Another reason is the possibility of reducing the required number of degrees of freedom compared to a conventional finite element analysis. Furthermore, if B-splines are utilized to represent the geometry of a finite element model, interfacing a finite element analysis program with existing computer aided design programs (which make extensive use of B-splines) is possible. While B-splines have been used in finite element analysis due to the aforementioned goals, it is difficult to find resources that describe the process of implementing B-splines into an existing finite element framework. Therefore, it is necessary to document this methodology. This implementation should conform to the structure of conventional finite elements and only require exceptions in methodology where absolutely necessary. One goal is to implement B-spline interpolation functions in a finite element framework such that it appears very similar to conventional finite elements and is easily understandable by those with a finite element background. The use of B-spline functions in finite element analysis has been studied for advantages and disadvantages. Two-dimensional B-spline and standard FEM have been compared. This comparison has addressed the accuracy as well as the computational efficiency of B-spline FEM. Results show that for a given number of degrees of freedom, B-spline FEM can produce solutions with lower error than standard FEM. Furthermore, for a given solution time and total analysis time B-spline FEM will typically produce solutions with lower error than standard FEM. However, due to a more coupled system of equations and larger elemental stiffness matrix, B-spline FEM will take longer per degree of freedom for solution and assembly times than standard FEM. Three-dimensional B-spline FEM has also been validated by the comparison of a three-dimensional model with plane-strain boundary conditions to an equivalent two-dimensional model using plane strain conditions.

Presence of cultural refuse has long posed a serious challenge to meaningful geological interpretation of near surface controlled–source electromagnetic data (CSEM). Cultural refuse, such as buried pipes, underground storage tanks, unexploded ordnance, is often highly conductive and magnetically permeable. Interpretation of the CSEM response in the presence of cultural noise requires an understanding of electromagnetic field diffusion and the effects of anomalous highly conductive and permeable structures embedded in geologic media. While many numerical techniques have been used to evaluate the response of three dimensional subsurface conductivity distributions, there is a lack of approaches for modeling the EM response incorporating variations in both subsurface conductivity σ and relative permeability μr. In this dissertation, I present a new three dimensional edge–based finite element (FE) algorithm capable of modeling the CSEM response of buried conductive and permeable targets. A coupled potential formulation for variable μ using the vector magnetic potential A and scalar electric potential V gives rise to an ungauged curl–curl equation. Using reluctivity (v=1/mu ), a new term in geophysical applications instead of traditional magnetic susceptibility, facilitates a separation of primary and secondary potentials. The resulting differential equation is solved using the finite element method (FEM) on a tetrahedral mesh with local refinement capabilities. The secondary A and V potentials are expressed in terms of the vector edge basis vectors and the scalar nodal basis functions respectively. The finite element matrix is solved using a Jacobi preconditioned QMR solver. Post processing steps to interpolate the vector potentials on the nodes of the mesh are described. The algorithm is validated against a number of analytic and multi dimensional numeric solutions. The code has been deployed to estimate the influence of magnetic permeability on the mutual coupling between multiple geological and cultural targets. Some limitations of the code with regards to speed and performance at high frequency, conductivity and permeability values have been noted. Directions for further improvement and expanding the range of applicability have been proposed.

Linear discontinuous (LD) spatial discretization of the transport operator can generate negative angular flux solutions. In slab geometry, negativities are limited to optically thick cells. However, in multi-dimension problems, negativities can even occur in voids. Past attempts to eliminate the negativities associated with LD have focused on inherently positive solution shapes and ad-hoc fixups. We present a new, strictly non-negative finite element method that reduces to the LD method whenever the LD solution is everywhere positive. The new method assumes an angular flux distribution, e , that is a linear function in space, but with all negativities set-to- zero. Our new scheme always conserves the zeroth and linear spatial moments of the transport equation. For these reasons, we call our method the consistent set-to-zero (CSZ) scheme. CSZ can be thought of as a nonlinear modification of the LD scheme. When the LD solution is everywhere positive within a cell, psi csz = psi LD. If psi LD < 0 somewhere within a cell, psi csz is a linear function psi csz with all negativities set to zero. Applying CSZ to the transport moment equations creates a nonlinear system of equations which is solved to obtain a non-negative solution that preserves the moments of the transport equation. These properties make CSZ unique; it encompasses the desirable properties of both strictly positive nonlinear solution representations and ad-hoc fixups. Our test problems indicate that CSZ avoids the slow spatial convergence properties of past inherently positive solutions representations, is more accurate than ad-hoc fixups, and does not require significantly more computational work to solve a problem than using an ad-hoc fixup. Overall, CSZ is easy to implement and a valuable addition to existing transport codes, particularly for shielding applications. CSZ is presented here in slab and rect- angular geometries, but is readily extensible to three-dimensional Cartesian (brick) geometries. To be applicable to other simulations, particularly radiative transfer, additional research will need to be conducted, focusing on the diffusion limit in multi-dimension geometries and solution acceleration techniques.