Método sem malha nodal utilizando funções de forma vetoriais baseadas em H(curl)

Abstract

Unlike mesh-based methods, meshless methods are characterized by the use of a set of nodes spread across the problem domain, instead of a mesh or grid. It can be said that boundary value problems formulated in terms of scalar functions are satisfactorily solved with the existing nodal meshless methods. However, for vector problems, there is still a need to develop new meshless techniques to overcome some difficulties, such as the presence of spurious solutions, the imposition of boundary and interface conditions between different materials. In this work, the Vector Nodal Meshless Method is presented, which spreads a series of nodes across the problem domain and its boundary to build the approximations. Each node is assigned a unit vector, a vector direction. Using these nodes and these directions, vector shape functions are constructed with the following characteristics: (i) they are based on H(curl) spaces; (ii) its projection in the vector direction of its node is equal to 1 and equal to zero in the direction of the vectors of the other support nodes; (iii) do not generate spurious modes; (iv) they simply impose the boundary and interface conditions between different materials. In this way, a method without meshes based on nodes is created, but with characteristics that allow its use in vector problems where the unknown is the electric field or the magnetic field, that is, problems that are normally solved using Edge Finite Elements. The method is applied in the interpolation of vector fields and in the solution of the problem of eigenvalues of several waveguides. The performed tests use node distributions and vector directions regular and irregular. The method solves the proposed applications, but some problems are detected, such as the low convergence rate for higher order shape functions and the sensitivity to perturbations, especially in vector directions associated with nodes. To overcome these problems, it is proposed to generate a planar subdivision based on the nodes and their vector directions. Thus, it is possible to correctly select the support nodes and obtain better results.