Construção das funções de forma 3-D para o método sem malha nodal vetorial

Abstract

Traditional meshless methods use scalar-based functions and therefore, present difficulties in approximating vector fields. The Vector Nodal Meshless Method (VNMM) constructs its approximations using shape functions based on H(curl) spaces and Nédélec’s first type elements polynomial space. For the application of the method, a set of nodes is distributed in the domain and on its boundary. Each node has an associated unit vector. For the direct imposition of the boundary conditions, the nodes located on the domain boundary and on the interfaces must have the vector direction tangent to the respective boundary or interface. The VNMM has been applied to solve two-dimensional electromagnetic problems. This thesis presents the construction of VNMM shape functions for use in three-dimensional vector problems. The interpolation capability of the shape function and the convergence rates of the method are proven. The problem of eigenvalues of a rectangular cavity is solved with the VNMM and the results are compared with those obtained using the Edge Finite Element Method (EFEM). The numerical solution for the eigenvalues approximated by VNMM is not corrupted by spurious modes. Three magnetostatic problems are solved: a magnetic shielding problem, a coaxial cable problem, and a problem with periodic structure and nonlinear material. In all cases, the numerical solution is validated, there are no difficulties regarding the imposition of boundary conditions, sources and interface conditions. Periodic boundary conditions are introduced into VNMM and validated. Nonlinearity is solved by successive approximations. An algorithm for the arbitrary selection of nodes in the support domain of the VNMM shape function is developed in order to eliminate the dependence on a mesh or an spatial subdivision in the choice of support nodes. Shape function interpolation tests with arbitrary support are presented.