THE VARIATIONAL MESHLESS METHOD FOR THE SOLUTION OF ELECTROMAGNETIC EIGENPROBLEMS

The meshless methods are encountered in in various kind of applications. In particular they are used in the solution of mechanical problems, imaging, fluid dynamics, solution of partial differential equations in 2D. In the last decade this tool started to gain the attention also of the electromagnetic community and has been applied to various kind of problems such as time domain simulations, microwave imaging, and 2D bounded problems. The interest about the Meshless Method is increasing because this technique presents some advantages compared to the most traditional ones like finite element (FEM) or finite difference method. The mesh generation is in fact a time and memory consuming process and this is unwanted in particular in real time simulations when frequent re-meshing steps are required. Particularly in the case in which the basis functions are defined over triangular element pairs like the Rao-Wilton-Glisson (RWG). Another important feature of the meshless method is that they are based on a particular kind of basis functions that can approximate the solution in a more accurate way than the low order polynomials generally encountered in the FEM. In general the problems encountered in the microwave engineering can be divided in two families: those which bring to an inversion problem (e.g. imaging, inverse scattering, boundary problems) and those which bring to an eigen problem (e.g. propagation inside a waveguide or resonant modes inside a cavity). The aim of this thesis is the application of the meshless method to the second family. In particular various cases are taken into account: (1) the 2D scalar problem of finding the modes inside a shielded waveguide filled with an homogeneous material, (2) the 2D vector problem of finding the dispersion diagram of the modes propagating inside a shielded waveguide filled with an inhomogeneous material, and (3) the 3D vector problem of finding the resonant modes inside shielded cavity filled with an inhomogeneous material. However the meshless method presents some numerical limitations when is applied to the electromagnetic eigenporoblems in its original form applied to eigenproblems. These limitations are, in brief, the low accuracy on the calculations of the first TE modes of a waveguide filled with a homogeneous material and the strong dependence of the solutions on the position of the collocation points, the non symmetric nature of the built matrices that brings to the ill-conditioning of the problem. To overcome all these issue the Variational technique has been applied in conjunction with the meshless method permitting to develop a new numerical technique that can handle all the cases listed before in a reliable way obtaining in all the reported simulations an high number of modes with a relative low number of unknowns. The method has been called Variational Meshless Method (VMM).