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.

Cette thèse a porté sur la construction de densités d'énergie de déformation permettant de décrire le comportement non linéaire de matériaux anisotropes tels que les tissus biologiques souples (ligaments, tendons, parois artérielles etc.) ou les caoutchoucs renforcés par des fibres. Les densités que nous avons proposées ont été élaborées en se basant sur la théorie mathématique des polynômes invariants et notamment sur le théorème de Noether et l'opérateur de Reynolds. Notre travail a concerné deux types de matériaux anisotropes, le premier avec une seule famille de fibre et le second avec quatre familles. Le concept de polyconvexité a également été étudié car il est notoire qu'il joue un rôle important pour s'assurer de l'existence de solutions. Dans le cas d'un matériau comportant une seule famille de fibre, nous avons démontré qu'il était impossible qu'une densité polynomiale de degré quelconque puisse prédire des essais de cisaillement avec un chargement parallèle puis perpendiculaire à la direction des fibres. Une densité polynomiale linéaire combinée avec une fonction puissance a permis de contourner cet obstacle. Dans le cas d'un matériau comportant quatre familles de fibre, une densité polynomiale a permis de prédire correctement des résultats d'essai en traction bi-axiale extraits de la littérature. Les deux densités proposées ont été implémentées avec la méthode des éléments finis et en langage C++ dans le code de calcul universitaire FER. Pour se faire, une formulation lagrangienne totale a été adoptée. L'implémentation a été validée par des comparaisons avec des solutions analytiques de référence que nous avons exhibée dans le cas de chargements simples conduisant à des déformations homogènes. Des exemples tridimensionnels plus complexes, impliquant des déformations non-homogènes, ont également été étudiés. / This thesis has focused on the construction of strain energy densities for describing the non-linear behavior of anisotropic materials such as biological soft tissues (ligaments, tendons, arterial walls, etc.) or fiber-reinforced rubbers. The densities we have proposed have been developed with the mathematical theory of invariant polynomials, particularly the Noether theorem and the Reynolds operator. Our work involved two types of anisotropic materials, the first with a single fiber family and the second with a four-fiber family. The concept of polyconvexity has also been studied because it is well known that it plays an important role for ensuring the existence of solutions. In the case of a single fiber family, we have demonstrated that it is impossible for a polynomial density of any degree to predict shear tests with a loading parallel and then perpendicular to the direction of the fibers. A linear polynomial density combined with a power-law function allowed to overcome this problem. In the case of a material made of a four-fiber family, a polynomial density allowed to correctly predict bi-axial tensile test data extracted from the literature. The two proposed densities were implemented in C++ language in the university finite element software FER by adopting a total Lagrangian formulation. This implementation has been validated by comparisons with reference analytical solutions exhibited in the case of simple loads leading to homogeneous deformations. More complex three-dimensional examples, involving non-homogeneous deformations, have also been studied.

To simulate real world problems modeled by differential equations, it is often not sufficient to  consider and tackle a single equation. Rather, complex phenomena are modeled by several partial dierential equations that are coupled to each other. For example, a heart beat involve electric activity, mechanics of the movement of the walls and valves, as well as blood fow - a true multiphysics problem. There may also be ordinary differential equations modeling the reactions on a cellular level, and these may act on a much finer scale in both space and time. Determining efficient and accurate simulation tools for such multiscalar multiphysics problems is a challenge. The five scientific papers constituting this thesis investigate and present solutions to issues regarding accurate and efficient simulation using adaptive finite element methods. These include handling local accuracy through submodeling, analyzing error propagation in time-dependent  multiphysics problems, developing efficient algorithms for adaptivity in time and space, and deriving error analysis for coupled PDE-ODE systems. In all these examples, the error is analyzed and controlled using the framework of dual-weighted residuals, and the spatial meshes are handled using octree based data structures. However, few realistic geometries fit such grid and to address this issue a discontinuous Galerkin Nitsche method is presented and analyzed.

In this study, a heterogeneous flow model is proposed based on a non-overlapping domain decomposition method. The model combines potential flow and incompressible viscous flow. Both flow domains contain a free surface boundary. The heterogeneous domain decomposition method is formulated following the Dirichlet-Neumann method. Both an implicit scheme and an explicit scheme are proposed. The algebraic form of the implicit scheme is of the same form of the Dirichlet--Neumann method, whereas the explicit scheme can be interpreted as the classical staggered scheme using the splitting of the Dirichlet-Neumann method. The explicit scheme is implemented based on two numerical solvers, a Boundary element method (BEM) solver for the potential flow model, and a finite element method (FEM) solver for the Navier-Stokes equations (NSE). The implementation based on the two solvers is validated using numerical examples. / Graduation date: 2013

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.

This dissertation addresses design and analysis aspects of stochastic dynamical systems using Fokker-Planck equation (FPE). A new numerical methodology based on the partition of unity meshless paradigm is developed to tackle the greatest hurdle in successful numerical solution of FPE, namely the curse of dimensionality. A local variational form of the Fokker-Planck operator is developed with provision for h- and p- refinement. The resulting high dimensional weak form integrals are evaluated using quasi Monte-Carlo techniques. Spectral analysis of the discretized Fokker- Planck operator, followed by spurious mode rejection is employed to construct a new semi-analytical algorithm to obtain near real-time approximations of transient FPE response of high dimensional nonlinear dynamical systems in terms of a reduced subset of admissible modes. Numerical evidence is provided showing that the curse of dimensionality associated with FPE is broken by the proposed technique, while providing problem size reduction of several orders of magnitude. In addition, a simple modification of norm in the variational formulation is shown to improve quality of approximation significantly while keeping the problem size fixed. Norm modification is also employed as part of a recursive methodology for tracking the optimal finite domain to solve FPE numerically. The basic tools developed to solve FPE are applied to solving problems in nonlinear stochastic optimal control and nonlinear filtering. A policy iteration algorithm for stochastic dynamical systems is implemented in which successive approximations of a forced backward Kolmogorov equation (BKE) is shown to converge to the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Several examples, including a four-state missile autopilot design for pitch control, are considered. Application of the FPE solver to nonlinear filtering is considered with special emphasis on situations involving long durations of propagation in between measurement updates, which is implemented as a weak form of the Bayes rule. A nonlinear filter is formulated that provides complete probabilistic state information conditioned on measurements. Examples with long propagation times are considered to demonstrate benefits of using the FPE based approach to filtering.

In the dissertation we discuss the rectangular elements, Adini's elements and $p-$order Lagrange elements, which were constructed in the rectangular finite spaces. The special rectangular partitions enable the finite element solutions $u_h$ more efficient in interpolation of the true solution for Elliptic equation $u_I$. The convergence rates of $|u_h-u_I|_1$ are one or two orders higher than the optimal convergence rates. For post-processings we construct higher order interpolation operation $Pi_p$ to reach superconvergence $|u-Pi_p u_h|_1$. To our best knowledge, we at the first time provided the a posteriori interpolant formulas of Adini's elements and biquadratic Lagrange elements to obtain the global superconvergence, and at the first time reported the numerical verification for supercloseness $O(h^4)-O(h^5) $, global superconvergence $O(h^5)$ in $H^1$-norm and the high rates $O(h^6|ln h|)$ in the infinity norm for Poisson's equation(i.e., $-Delta u = f$). Since the finite element methods is fail to deal with the singularity problems, in the dissertation, the combinations of the Ritz-Galerkin method and the finite element methods are used for the singularity problem, i.e., Motz's problem. To couple two methods along their common boundary, we adopt the simplified hybrid, penalty, and penalty plus hybrid techniques. The analysis are made in the dissertation to derive the almost best global superconvergence $O(h^{p+2-delta})$ in $H^1$-norm, $0<delta << 1$, for the combination using $p(geq 2)$-rectangles in the smooth subdomain, and the best global superconvergence $O(h^{3.5})$ in $H^1$-norm for combinations of Adini's elements in the smooth subdomain. The numerical experiments have been carried out for the combinations of the Ritz-Galerkin method and Adini's elements, to verify the theoretical superconvergence derived.

In this study the relationship between the dynamic response of the flexible substrate and the power generation for energy harvesting system is proposed. High electro-mechanical transformation of piezoelectric materials, high efficient energy transfer of mechanical structure and controlled circuit make the piezoelectric generator a high performance. The devices of cantilevers with lump structures on the flexible substrate and piezoelectric film (ZnO) are designed. Then some individual layers of power generator are stocked in parallel to form a multi-layer system with a broad resonant band width. When the generator is operated in a wide frequency range vibration environment, the multi-layer piezoelectric films in the form of cantilever structures can induce current. First the finite element method for the piezoelectric cantilever beam is constructed by using ANSYS software. Both modal analysis and harmonic response analysis are performed to obtain the structural modal parameters and frequency response functions, respectively. Besides, the beam structure is modeled by 3D coupled field piezoelectric element. This research will apply Taguchi¡¦s method to design including variations of dimensions and material properties for energy harvesting system. The flexible substrate is polymeric film (PET). Imprinting process is applied to transfer the simulated geometric configuration onto a flexible substrate to obtain a maximum power output. The results show the single devices can improve efficiently by using lump structures on the flexible substrate, the generator could achieve maximum OCV of 2.25V which is 0.276£gW every centimeter squared when attached to a stable source of vibration. The multi-layer system can be used in 50~500Hz of low frequency environment. Furthermore, the output voltage (OCV) is upward when the flexible substrate with low Young¡¦s modulus.

<p>The work in this thesis focuses on the use of explicit finite element analysis (FEA) in the modelling of fluid-structure interaction of panel-water impacts. Paper A, considers modelling of a two-dimensional rigid wedge impacting a calm water surface. From analytical methods and results of a systematic parameter study a generalised approach for determination of fluid discretization and contact parameters in the modelling of arbitrary hull-water impact situations is developed and presented. In paper B the finite element modelling methodology suggested in paper A is evaluated for elastic structures by a convergence study of structural response and hydrodynamic load. The structural hydroelastic response is systematically studied by a number of FE-simulations of different impact situations concerning panel deadrise, impact velocity and boundary conditions. In paper B a tentative method for dynamic characterization is also derived. The results are compared with other published results concerning hydroelasticity in panel water impacts. The long-term goal of this work is to develop design criteria, by which it can be determined whether the loading situation of a certain vessel type should be regarded as quasi-static or dynamic, and which consequence on the design a dynamic loading has.</p>

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.