### I. Introduction

### II. Theoretical Background

### 1. Two-Dimensional MOM

*L*is an integro-differential operator,

*f*is the unknown function, and

*g*is a known excitation source.

*f*can be expanded into a sum of

*N**

*M*weighted basis functions, as follows:

*L*, (1) can be expressed as

*α*

*)*

_{mn}*denote the unknown weighting coefficients.*

_{mn}*f*

*and (3), the entire problem can be reduced to*

_{pq}*Z*

*should be computed and then explicitly stored in memory. To accelerate the solution of the MOM linear system, Heldring et al. [18] proposed a sparsified adaptive cross approximation (S-ACA) algorithm by substituting sub-blocks of the impedance matrix Z using “compressed” approximations, which allowed for reduced storage and accelerated iterative solution. The obtained numerical experiment reveals a computational complexity close to*

_{mn, pq}*N log N*.

### 2. Two-Dimensional Transverse Wave Approach

*E*and

*H*can be expressed as:

*T*refers to the tangential components.

*E*

*) and transverse magnetic field (*

_{T}*H*

*)generates both incident and reflected waves from the discontinuity interface (see Fig. 1).*

_{T}*Z*

_{0}

*stands for the wave impedance of the homogeneous isotropic region*

_{i}*i*∈{1,2} (Fig. 1), which is given by:

*η*

_{0}is the intrinsic impedance of free space defined as

*ɛ*

_{ri},

*μ*

_{ri}, are the relative permittivity (capacitivity) and relative permeability (inductivity) of the medium

*i*;

*J*

*=*

_{T}*H*

*×*

_{T}*v*, with

*v*denoting the outgoing normal vector oriented towards region

*i*; and τ is a Boolean parameter referring to the wave nature (i.e.,

*τ*= 0 for incident wave ([

*W*=

*A*] and τ = 1 for reflected wave [

*W*=

*B*]).

*B*_{0}is wave excitation source;(n) is the iteration order and the underlined waves are presented in the spatial domain, the others in the modal domain;

**Γ̂**and**Ŝ**大发体育 are two linear operators in Hilbert space;**Γ̂**denotes the reflection operator in the modal domain; and**Ŝ**defines the diffraction operator in the spatial domain, describing the boundary conditions from the discontinuity surface Ω.

*W*

^{(}

^{n}^{)}be the difference in terms of waves between two successive waves,

*W*

^{(}

^{n}^{)}and

*W*

^{(}

^{n}^{−1)}, as follows:

*Ŝ*that is less than unity, and the unitarity of

**demonstrates that the spectral radii of ||**

*Γ̂***|| and ||**

*Γ̂Ŝ***||are less than unity. Consequently, the norms||Δ**

*ŜΓ̂**B*

^{(}

^{n}^{)}|| → 0 and ||Δ

*A*

^{(}

^{n}^{)}|| → 0 and the convergence of (16) is reached. This proves the stability of our TWA approach.

*N log N*where

*N*represents the total number of pixels chosen for the EM simulation of the studied structure.

### III. Proposed Structures for EM Investigation

### 1. First Planar Structure: Printed Trapezoidal Monopole Antenna

### 2. Second Planar Structure: A Strip-Loaded Coplanar Waveguide Fed Pentagonal Antenna

### IV. Simulation Results and Discussion

*S*

_{11}, and the results were then compared against accurate measurements obtained inside an anechoic chamber. These methods are the finite integration technique (FIT)-based time domain solver (TDS), finite element method (FEM)-based frequency domain solver (FDS) of CST microwave technology, and the FEM-based High-Frequency Structure Simulator (HFSS) of ANSYS. For our simulation results, the well-known state-of-the-art simulation tool, the so-called Advanced Design System (ADS) Momentum, and our EM simulation tool, well-developed in [25], were used for the EM investigation. These tools were developed based on the MOM and TWA methods, respectively. Table 2 displays the different geometric and modeling simulation parameters for EM analysis of the aforementioned planar antennas presented in Figs. 2 and 3. These parameters have been chosen judiciously to be much closer to the one chosen in [24].

*S*

_{11}, of the two reference antennas underlines the EM validity in the context of EM-modeling of complex RF/microwave structures referring to the specific data mentioned above. Indeed, on the one hand, the simulation results for the trapezoidal antenna depicted in Fig. 4 show two bands using MOM and TWA. The first MOM band appears from 2.79 GHz to 4.88 GHz, while the TWA band appears from 3.26 GHz to 4.73 GHz; the second band is detected from 5.84 GHz to7.43 GHz for MOM and from 5.65 GHz to 6.12 GHz using TWA. On the other hand, for the strip-loaded CPW-fed pentagonal antenna, the results prove the presence of the different resonant bands with small shifting. These obtained simulation results, based either on the MOM or TWA method and as depicted in Figs. 4 and 5, respectively, are nearly identical to the results in [24]. The satisfactory agreement of these results demonstrates the potentialities of these approaches; nevertheless, TWA remains more flexible and efficient than MOM owing to the minimum complexity effort pertaining to the fast iterative process of TWA. The fluctuations observed in Fig. 5 are due to the presence of the periodic walls, which can be circumvented by setting the antenna as far as possible from the walls. This can be performed by increasing the input resolution, which can adversely affect the total CPU time for EM simulation. In this case, the anisotropic mesh technique (AMT) developed by our research team and already implemented in the advanced TWA approach can be a good solution for tackling this problem.