Método de mallado y algoritmos adaptativos de dos y tres dimensiones para la resolución de problemas electromagnéticos cerrados mediante el método de los elementos finitos

Abstract

[SPA] La primera parte de esta tesis, desarrollada en los capítulos 1, 2, 3 y 4, está dedicada al diseño de nuevos métodos de mallado bidimensional, superficial y volumétrico que sigan estas premisas. Aunque esos métodos han sido desarrollados en el contexto del análisis de guiado de ondas y el diseño de cavidades resonantes de microondas, su aplicación puede abarcar cualquier campo de la física, pues la fase de discretización del MEF presenta una clara independencia del problema tratado. La segunda parte de esta tesis, presentada en el capítulo 5, está dedicada al estudio de este tipo de métodos. En ella se describen los distintos indicadores de error y estrategias de refinamiento h desarrolladas, y se presentan y analizan los resultados obtenidos con ellos en distintas estructuras de guiado de ondas.

[ENG] In its first part, the dissertation develops a multiblock methodology of surface and volumetric mesh generation from the discretization of the problem boundary, in order to apply the finite element method. Different structured and unstructured meshing techniques in 2D (interpolation and generalized fast advancing front), 3D surface and volumetric (advancing front) domains are presented. Moreover, some a posteriori techniques for improvement of quality mesh are described. The second part of this dissertation deals with adaptive meshing within an adaptive finite element method. This technique is an iterative variant of the finite element method where, in a first step, an initial mesh with few and low order elements is generated, the corresponding algebraic problem is solved and the error in the solution is estimated in order to add degrees of freedom in those regions of the domain with the biggest error estimation. This process is repeated until an ending condition is reached. The two basic stages in this method are the error indication and the mesh enrichment. In this dissertation, within the analysis of waveguiding structures, three kinds of error indicator have been developed: (1) Error indicators based on the residual of the vector wave equation and the boundary conditions at the edges of each element. (2) Error indicators based on the comparison of the solution curl with a smoothed or recovered curl, obtained from the solution curl. (3) Error indicators based on the flux (electric or magnetic) continuity through the inner edges in the mesh. In addition, an overview on refinement techniques is presented, and the h-refinement employed in this work is in depth described. Results obtained with the different error indicators and refinement strategies are discussed and compared with the classical, non-adaptive finite element method.