O Método dos Elementos Finitos Generalizados, MEFG, compartilha importantes características dos métodos sem malha. As funções de aproximação do MEFG, atreladas aos pontos nodais, são enriquecidas de modo análogo ao refinamento p realizado no Método das Nuvens hp. Por outro lado, por empregar uma malha de elementos para construir as funções partição da unidade, ele também pode ser entendido como uma forma não convencional do Método dos Elementos Finitos. Neste trabalho, ambas as interpretações são consideradas. Os métodos sem malha, particularmente o Método de Galerkin Livre de Elementos e o Método das Nuvens hp, são introduzidos com o propósito de estabelecer os conceitos fundamentais para a descrição do MEFG. Na seqüência, apresentam-se aplicações numéricas em análise linear e evidenciam-se características que tornam o MEFG interessante para a simulação da propagação de descontinuidades. Após discutir os modelos de dano adotados para representar o comportamento não-linear do material, são introduzidos exemplos de aplicação, inicialmente do Método das Nuvens hp e depois do MEFG, na análise de estruturas de concreto. Os resultados obtidos servem de argumento para a implementação de um procedimento p-adaptativo, particularmente com o MEFG. Propõe-se, então a adaptação do Método dos Resíduos em Elementos Equilibrados à formulação do MEFG. Com vistas ao seu emprego em problemas não-lineares, algumas modificações são introduzidas à formulação do estimador. Mostra-se que a medida obtida para representar o erro, apesar de fundamentada em diversas hipóteses nem sempre possíveis de serem satisfeitas, ainda assim viabiliza a análise não-linear p-adaptativa. Ao final, são enumeradas propostas para a aplicação do MEFG em problemas caracterizados pela propagação de defeitos / The Generalized Finite Element Method, GFEM, shares several features with the so called meshless methods. The approximation functions used in the GFEM are associated with nodal points like in meshless methods. In addition, the enrichment of the approximation spaces can be done in the same fashion as in the meshless hp-Cloud method. On the other hand, the partition of unity used in the GFEM is provided by Lagrangian finite element shape functions. Therefore, this method can also be understood as a variation of the Finite Element Method. Indeed, both interpretations of the GFEM are valid and give unique insights into the method. The meshless character of the GFEM justified the investigation of meshless methods in this work. Among them, the Element Free Galerkin Method and the hp-Cloud Method are described aiming to introduce key concepts of the GFEM formulation. Following that, several linear problems are solved using these three methods. Such linear analysis demonstrates several features of the GFEM and its suitability to simulate propagating discontinuities. Next, damage models employed to model the nonlinear behavior of concrete structures are discussed and numerical analysis using the hp-Cloud Method and the GFEM are presented. The results motivate the implementation of a p-adaptive procedure tailored to the GFEM. The technique adopted is the Equilibrated Element Residual Method. The estimator is modified to take into account nonlinear peculiarities of the problems considered. The hypotheses assumed in the definition of the error measure are sometimes violated. Nonetheless, it is shown that the proposed error indicator is effective for the class of p-adaptive nonlinear analysis investigated. Finally, several suggestions are enumerated considering future applications of the GFEM, specially for the simulation of damage and crack propagation

This project presents the development and implementation of a GPU-accelerated meshless two-phase incompressible fluid flow solver. The solver uses a variant of the Generalized Finite Difference Meshless Method presented by Gerace et al. [1]. The Level Set Method [2] is used for capturing the fluid interface. The Compute Unified Device Architecture (CUDA) language for general-purpose computing on the graphics-processing-unit is used to implement the GPU-accelerated portions of the solver. CUDA allows the programmer to take advantage of the massive parallelism offered by the GPU at a cost that is significantly lower than other parallel computing options. Through the combined use of GPU-acceleration and a radial-basis function (RBF) collocation meshless method, this project seeks to address the issue of speed in computational fluid dynamics. Traditional mesh-based methods require a large amount of user input in the generation and verification of a computational mesh, which is quite time consuming. The RBF meshless method seeks to rectify this issue through the use of a grid of data centers that need not meet stringent geometric requirements like those required by finite-volume and finite-element methods. Further, the use of the GPU to accelerate the method has been shown to provide a 16-fold increase in speed for the solver subroutines that have been accelerated.

Le but de cette thèse est de développer une méthode numérique simple, robuste, efficace et précise pour résoudre des problèmes d'ingénierie de grande taille à partir de la méthode Taylor Meshless (TMM) et fournir de nouvelles idées principales de TMM est d'utiliser comme fonctions de forme des polynômes d'ordre élevé qui sont des solutions approchées de l'EDP. Ainsi la discrétisation ne concerne que la frontière. Les coefficients de ces fonctions de forme sont obtenus en discrétisant les conditions aux limites par des procédures de collocation associées à la méthode des moindres carrés. TMM est alors une véritable méthode sans maillage sans processus d'intégration, les conditions aux limites étant obtenues par collocation. Les principales contributions de cette thèse sont les suivantes: 1) Basé sur TMM, un algorithme général et efficace a été développé pour résoudre des EDP elliptiques tridimensionnelles; 2) Trois techniques de couplage pour des résolutions par morceaux ont été discutées dans des cas de problèmes à grande échelle: la méthode de collocation par les moindres carrés et deux méthodes de couplage basées sur les multiplicateurs de Lagrange; 3) Une méthode numérique générale pour résoudre les EDP non-linéaires a été proposée en combinant la méthode de Newton, la TMM et la technique de différentiation automatique. 4) Pour résoudre des problèmes avec un bord non régulier, des solutions singulières satisfaisant l'équation de contrôle sont introduites comme des fonctions de forme complémentaires, ce qui fournit une base théorique pour la résolution de problèmes singuliers / Based on Taylor Meshless Method (TMM), the aim of this thesis is to develop a simple, robust, efficient and accurate numerical method which is capable of solving large scale engineering problems and to provide a new idea for the follow-up study on meshless methods. To this end, the influence of the key factors in TMM has been studied by solving three-dimensional and non-linear Partial Differential Equations (PDEs). The main idea of TMM is to use high order polynomials as shape functions which are approximated solutions of the PDE and the discretization concerns only the boundary. To solve the unknown coefficients, boundary conditions are accounted by collocation procedures associated with least-square method. TMM that needs only boundary collocation without integration process, is a true meshless method. The main contributions of this thesis are as following: 1) Based on TMM, a general and efficient algorithm has been developed for solving three-dimensional PDEs; 2) Three coupling techniques in piecewise resolutions have been discussed and tested in cases of large-scale problems, including least-square collocation method and two coupling methods based on Lagrange multipliers; 3) A general numerical method for solving non-linear PDEs has been proposed by combining Newton Method, TMM and Automatic Differentiation technique; 4) To apply TMM for solving problems with singularities, the singular solutions satisfying the control equation are introduced as complementary shape functions, which provides a theoretical basis for solving singular problems