A VECTOR MULTI-RESOLUTION MOM FOR COMPLEX ANTENNAS ON TRIANGULAR MESHES

A Multi-Resolution (MR) approach to the Electric Field Integral Equation - Method of Moments (EFIE-MoM) analysis is presented; it keeps the different scales of the problem directly into the basis functions representation: the obtained MoM matrix is stable against clipping sparsification, and a considerable improvement is obtained in the conver- gence of iterative solvers. THE MULTI-RESOLUTION APPROACH The EFIE-MoM approach is widely used in the simulation of large and complex antennas that are electrically large, yet with fine geometrical details that require discretizations much smaller than the wavelength: the well known limitations are the matrix size and condition number. In these problems, the structure of the solution exhibits very different scales of variation. In the proposed Multi-Resolution (MR) approach, we focus on keeping these different scales directly into the formulation and solution process: this leads to a good control of the conditioning of the matrix, a key in the convergence of iterative solvers, as well as the gateway to the sparsification of the matrix. The present work extends the theory and applications of the approach in Pirinoli et al.(4). The starting point of the proposed Multi-Resolution (MR) approach is the generation of a nested set of hierar- chical meshes over the conductor surface. The first such mesh, called "coarse-level" mesh, is labeled j =1 : this level can be generated with any commercial "mesher" (software to generate meshes from a CAD surface). Then subsequent, nested, domain decomposition steps are performed, that are equivalent to the generation of hierarchically nested grids of increasing detail; each of these meshes is identified as j-level; the finest such mesh, with j = L, is called the "pixel" mesh and it essentially corresponds to the cell size that would be employed in a standard MoM. In Fig.1 the mesh generation scheme is applied to a Bow-tie antenna example. Fig.1: Single Bow-tie antenna: (a) coarse mesh (j=1); (b) subsequent level mesh (j=2). The shaded area shows the subdivision of a cell. The j =1 (initial) and j ≥ 2 levels are treated differently, and MR functions in the strict sense are defined only the latter; on the j =1 mesh we define "connecting functions" that link the various sub-domains: they will be discussed later on. The basis generation is based on the splitting of the unknown surface current into its solenoidal (TE) and quasi-irrotational (quasi-TM, qTM) components Vecchi(1), in a manner similar to the loop-star decomposition. This allows: 1. separating the singular, near-field behaviours, to avoid the "low-frequency catastrophe" associated with the extreme bad conditioning of the MoM matrix when the electric length of the cell size approaches static or quasi-static conditions Vecchi(1); 2. the mapping of the vector MR basis functions to scalar quantities that possess the same degree of regu- larity in both (local) spatial directions, and on which the introduction of wavelet-like constructs is easier. The qTM part is related to the charge density, defined over the cells of the mesh, and the TE part is derived from a "solenoidal potential", defined over the (inner) nodes of the mesh. For the qTM component, for each level