Vector Nodal Meshless Method for 3-D Applications

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 the
$H\mathrm {(curl)}$
spaces and Nédélec’s first type elements polynomial space. In this sense, a set of nodes is distributed in the domain and on its boundary, and each node has an associated unit vector. The VNMM has been applied to solve two-dimensional electromagnetic problems. This article presents the construction of the VNMM shape functions for use in 3-D vectorial problems. The interpolation capacity of the shape function and the method convergence rates are shown. Then, the VNMM is applied to an eigenvalue problem and compared to the traditional edge finite element method (EFEM). The numerical solution for eigenvalues is not corrupted by spurious modes. Finally, a nonlinear magnetostatic problem is solved. A good performance of the method is observed.