Tratamento de descontinuidade de material no método dos elementos finitos generalizado

Abstract

The study of Non-Conventional Element Method Formulations is based on alternative approaches to the Finite Element Method including the Meshless Methods and the Generalized Finite Element Method. The Generalized Finite Element Method has important characteristics in common with Meshless Methods. Its approximating functions linked to the nodal points are enriched in a similar manner to the p refinement performed in the hp-clouds. The Finite Element Method, gives good results for wave propagation problems with small electric domainand problems with discontinuity, however it generally demands high computational cost to achieve a better accuracy in its results, for example, solving wave propagation problems with bigger electric domain. Then, it becomes clear that there is a demand for methodologies and tools for analysis and verification of these cases of electromagnetism problems. In this thesis, electromagnetic wave propagation problems modeled by bi two dimensional scalar Helmholtz equation are numerically solved by the Generalized Finite Element Method. In order to generate the Generalized Finite Element Method space, plane wave functions were used to enrichthe Finite Element space. The enrichment of the approximation space through these functions allows reducing the effects of error pollution generated by the high wave number. It makes possible to obtain a solution with high accuracy for the problem using a reduced number ofdegrees of freedom in comparison to the classic Finite Element method.In problems where the domain is composed of regions with different media, the function space of Generalized Finite Element Method presents discontinuity. The usual form to treat this discontinuity is to use the Lagrange Multipliers to enforce the interface conditions. However, thisapproach leads to ill conditioned and non positive definite matrix systems, that impose severe restrictions over what method should be used to solve the system. The main contributions presented in this thesis are related to the treatment of this discontinuity for the Generalized Finite Element Method. The novelty presented in this thesis is the replacement of the Lagrange Multipliers by an approach based in the Mortar Element Method, this procedure has the advantage of generating a matrix that preserve the sparsity of the system with a much smaller dimensions than the systems obtained by Lagrange Multipliers. Addtionaly, the resulting matrix is and positive definite. This method showed to be as precise as the Lagrange Multipliers with less computational cost. Another important contribution of this work, also connected to the treatment of discontinuity,was to extend the formulation of Generalized Finite Element Method with enrichment by plane waves to incorporate nonconforming subdomains separated by linear or curve piecewise interfaces. This analysis is made decomposing the global domain of the problem into subdomains and then it is performed an analysis in each of these subdomain. In order to ensure the conditions of continuity between these subdomains, it is proposed an approach performed by Lagrange Multipliers and another by Mortar Element Method. Two special integration schemes were proposed. To ensure the continuity between the subdomains one for the linear case and other for the curve case. The numerical results demonstrates the efficiency of the proposed techniques. Problems where the analytical solution is known or where the solution is obtained by the Finite Element Method are presented in order de show the efficiency of each proposed technique. Finally, the convergence of the method is also presented as a function of the number of the plane wave directions.