Métodos sem malha aplicados ao eletromagnetismo: formas fracas enfraquecidas e funções de forma vetoriais

Abstract

This thesis presents the study and application of meshless methods in electromagnetic problems.vIt can be said that scalar problems are well consolidated with the existing methods.vHowever, there is a need to develop new meshless techniques to overcome the difficultiesinvolved in vector problems such as not satisfying the divergence free condition and the appearancevof numerical spurious solutions.One of the contributions of this work is the application of the Point Interpolation Methodv(PIM) using weakened weak forms. Weakened weak forms arrised in order to eliminate incompatibilityvissues present in PIM shape functions. The method is initially applied in scalarvelectromagnetic problems. Then a restricted formulation is proposed with the penalty methodvfor application in vector problems.Another contribution to solve vector problems is developed as an extension of the vectorvRadial Basis Function (RBF), but using weak forms. The vector RBFs are based in nodesvand yet generate approximations with the divergence free condition. Therefore, they can beused with weak forms without the need of adding constraints, unlike the PIM methods withvweakened weak forms.vA third contribution for vector problems was the development of vector shape functionsvconstructed from a set of edges rather than a set of nodes. This technique allows the approximationsvto satisfy the divergence free condition without needing to use restricted formulations,vby the apropriate choice of vector basis functions. The degrees of freedom are associated withvthe edges and the imposition of Dirichlet boundary conditions is done in a simplified manner.vAll the aforementioned techniques are tested in time-harmonic vector electromagnetic problemsvand the results presented along with the mathematical formulations.