# On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation
## 2 Materials and Methods
### Triangle-Grid
#### 2.3.1 Surface Patch Approach: Overview
The extensive research on developing such surfaces expanded the applications that may employ them to include the design of controlled-wave electromagnetics devices [210; 211], reduced scattering-wave ones [212], antennas with novel properties such as radiation toward angles that would otherwise be shadowed [213], and even waveguides [214; 215]. As mentioned above, these surfaces used a wide range of patch sizes and shapes. For example, to design an anisotropic impedance surface, slices in a lattice of square patches were used in [216], and circular ones were used in [217]. Moreover, a considerable number of works have been devoted to investigating plane [218] and curved [7; 219] polygons in regular mesh and irregular computational grid generating [220]. One conclusion can be made from this works, that no matter which patch is used to generate the mesh of arbitrarily shaped objects: triangular [221; 222], rectangular [223; 224], triangular and rectangular [225], quadrilateral with roof-top vector basis functions and the Galerkin's method [226], polygonal [220; 227], or even combinations of different patches [228], or special ones developed for bodies with complete [229] or not complete [230] symmetry of revolution or translation, the analysis is usually done by the MoM surface formulations. Furthermore, the MoM volumetric formulations have also been used to analyze bodies meshed by triangular [231], parallelepipeds [232; 233], and tetrahedrons patches [234; 235; 236], and even volumetric rooftop functions with parallelepiped ones [237]. Moreover, the researchers developed a new set of triangular patch basis functions capable of representing any linear current distribution over each triangle in contrast to RWG basis functions, and they were much less sensitive to the meshing scheme. In addition, these basic functions can represent the actual current distribution better than RWG with the same computational complexity [238]. However, if the body to be analyzed has a large geometric, like the sea surface for instance, the low order MoM with a lot of unknowns is not efficient. For this reason, the higher order hierarchical basis functions were developed and the higher order MoM (HO-MoM) was achieved [239; 240; 241]. HO-MoM used different types of patches, for ex ample, the bilinear ones with higher order interpolation curvilinear quadrilaterals [242], NURBS patches [243; 244; 245], Bezier patches [246; 247], and combination of NURBS and Bezier patches [248]. Modeling using such patches is simple, and requires few basic functions to obtain a precise representation of complex bodies. In addition, it allows avoiding the errors that might occur when meshing by simple elements. Nevertheless, it cannot accurately approximate a high curvature surface with only one patch, and it is hard to be adopted in most of CAD geometric design tools. Among all patch types mentioned above, the triangular surface patch demonstrated its merits in solving general three-dimensional electromagnetic problems [249] and even the junction problem [250]. Triangular vector basis functions were proposed, for the first time, by [PERSON], [PERSON], and [PERSON] in their paper [251]. In original formulation, these functions were developed to obtain solutions of field scattering problems involving complex practical structures such as aircrafts, vehicles, ships, and other objects not amenable to exact mathematical solution [252]. Subsequently, this type of basic functions came to be called RWG-functions in honor of their creators and became widely used in various electromagnetic problems. The accuracy of the analysis that uses RWG functions was verified in several works. The results obtained by RWG functions were compared for example in [253] with those by the rooftops basic functions, which are piecewise linear functions defined on rectangular subdomains.