## 4 Analysis of Numerical Results  
In this section, we analyze the results of numerical simulations conducted to design resonators using a combination of fifteen geometric shapes and three materials, as detailed in Section 3, to enhance the performance of a probe-fed patch antenna. Following the decoupled method, the probe-fed patch antenna is designed first, and the resonator is subsequently introduced to improve its performance. For making the comparisons clear, the \(|S_{11}|\) and radiation pattern plots include results from the probe-fed patch antenna obtained with and without the resonator. The numerical simulations were conducted using the Microwave & RF package of Computer Simulation Technology (CST) software [39] release Version 2023.0. The computational setup consisted of an Intel Core i7 processor (2.10 GHz clock speed) Intel Corporation, Santa Clara, USA with 32 GB of RAM. The simulation time for each antenna geometry ranged from 8 to 72 h, depending on the complexity of the structure. Regarding the simulation, we point out the following factors that influence the simulation time:  
* Solver choice: The frequency-domain (FD) and time-domain (TD) solvers have different computational requirements. The TD solver generally require more memory; however, it enable faster broadband analysis. This solver is preferred for simulating the interaction between an antenna and a resonator because it provides a broadband response in a single simulation, making it more efficient for complex geometries. It facilitates the analysis of coupling, S-parameters, and transient response. The FD solver, conversely, is better for specific frequencies and dispersive materials. Therefore, the TD solver was chosen for this work, as it is well-suited for simulating antennas and resonators, providing efficient broadband analysis in a single simulation. * Mesh refinement: A finer mesh improves accuracy but significantly increases memory usage and CPU time. In many cases, a mesh convergence study is recommended to find a suitable balance between accuracy and efficiency. In the simulations with graphite, it was necessary to gradually reduce the mesh due to long simulation times and memory errors. The reduction was made carefully to maintain the accuracy of the results, while balancing computational efficiency. This approach allowed the simulations to run without compromising the quality of the results. * Boundary conditions: For simulating antennas and resonators in CST, the ideal boundary conditions depend on the simulation setup. An electric perfect conductor is best for metal structures, as it establishes a perfect conductor boundary. A magnetic perfect conductor can be used in specific cases involving magnetic resonance or properties. For symmetrical structures, electric/magnetic symmetry can reduce computation time. Open (Add Space) with perfectly matched Layers (PML) is recommended for free-space simulations, as it absorbs outgoing waves, preventing reflections. Generally, Open (Add Space) is the preferred condition for antenna and resonator simulations in open environments. Therefore, in our simulations, we opted for the Open (Add Space) boundary condition. * Material properties: Complex materials, such as dispersive or lossy media, require additional computations to accurately model their behavior. This increases processing time, as the solver must handle complex equations to account for absorption and dispersion effects. In this study, DW and glycerin moderately extend simulation time due to their high permittivity, while graphite, being highly conductive, has a much greater impact on computation time. * Resonator complexity: Complex resonator shapes and material combinations directly impact computational load, as these geometries require finer meshing to maintain simulation accuracy. Additionally, solver convergence may take longer, as intricate structures often need more iterations to reach a stable solution.