# 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  
By virtue of the thin-wire codes ability to model structures of arbitrary shapes, the representation of any structure with its equivalent conducting wire-grid model became not only possible but also efficient in solving radiation and scattering problems [51]. In spite of all the advantages of this modeling approach, it is not always an optimal choice, since for complex structures, direct calculations of their surface currents using the surface patch approach [186] might be more accurate than calculating them in their wire-grid model [187; 188]. The surface patch approach is used with several methods, especially with MoM, and it even forms the base of some of them. For example, the electric surface current model (ESCM) [189] that uses prediction to determine the current density distribution on the antenna upper surface by the same way the dipoles may be analyzed. Then, antenna characteristics can be calculated using these currents and an appropriate Green's function [190]. Unlike complicated methods based on calculating the true current distribution using integral equation like in [191], ESCM enables achieving a closed form solution, which reduces  
Figure 5: Workflow of the Wire-Grid antenna modeling process. the analysis time even less than MoM. This approach can be used to analyze any microstrip structure since it gives a clear physical picture of the microstrip radiation mechanism. However, it is particularly used to analyze of microstrip antennas on cylindrical metallic bodies [192], and with the presence of superstrate made of one [193] or different [194] lossy dielectric materials. However, both ESCM with rectangular-pulse bases [85] and MoM-based wire-grid share similar limitations such as the restrictions on the maximum cell width (about \(\lambda/10\)) and the conducting body total surface area (no more than \(\lambda 2\)) except for symmetrical or revolution bodies. Another method that uses the benefits of the surface patch approach but not subject to such restrictions is the reaction technique. This technique is also an EFIE-MoM-based and its concept has been discussed in [195; 196; 197]. It uses the zero-reaction concept [198] with [PERSON]'s method to reduce the formulated EFIE to a matrix equation. It differs in that the surface-current distribution is expanded in overlapping sinusoidal bases in the direction of current flow, and rectangular-pulse bases in the transverse direction, which satisfies [PERSON]'s current law without introducing line charges. The most famous reaction formulation is the piecewise-sinusoidal reaction technique for scattering and radiation from perfectly conducting bodies of an arbitrary shape [199], derived from that for thin-wire antennas [200; 201]. RWG and rooftop basis functions are examples of low-order divergence-conforming sets, which enforce the normal continuity of the current across the edges arising from the discretization. However, there are some other numerical schemes of EFIE discretization that overcome this drawback, for example, the point-based (collocation) Nystrom [202; 203] and the patch-based monopolar-RWG [204; 205]. One more method that depends on the surface patch approach is the patterning used to generate highly anisotropic and smoothly varying impedance surfaces between square, rectangular, hexagonal, and other shapes with a wide range of aspect ratios [206]. The impedance surfaces can be assumed as a metasurface [207] that is fabricated with periodic metallic patches on a grounded dielectric substrate [208]. The first impedance surfaces were generated using rectangular cells in spite of the original shape of the unit cell, but with the same size and impedance [209].