# Enhancing Efficiency of Electromagnetic Simulation in Time Domain with Transformation Optics
## 5 Results from 1D and 2D Non-Uniform Grids and Their Efficiency  
The results from the mapping function \(x=f(u)\) to the 1D non-uniform grids are demonstrated, and the numerical cell size in all direction are set to be \(\Delta u=\Delta v=\Delta w=\Delta=2.5\) nm. For the multiple-linear mapping function, we let \(m=4\), and \(\Delta x_{j}=2.5\times 2^{j-1}\) nm for \(j=1\) to \(m\). The end points of the divided regions {\(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\)} are 80 nm, 120 nm, 160 nm, and 4000 nm, respectively. The mesh of the non-uniform grids around the aperture is shown Figure 5a. The cell size in the \(x\) direction increases as \(x\) increases. For the linear-nonlinear mapping function, the cell sizes and end points in the first three regions are identical to those used in the multiple-linear function while \(\gamma\) and \(\alpha\) used in Equation (8) are 1.85 and 0.79, respectively, when the units of \(x\) and \(u\) are in micrometers. The mesh of the non-uniform grids is shown Figure 5b. For \(\mid x\mid\leq 160\) nm, the mesh is same as that shown in Figure 5a. For \(160\) nm \(<\mid x\mid<200\) nm, the slope of Equation (7) in this range is about 0.68. The cell size in the \(x\) direction is therefore about \(\Delta x=(0.68)^{-1}\Delta u=3.68\) nm. The slope will decrease with \(\mid x\mid\) such that the cell size will increase. The film is simulated with the PEC-like material (see Section 3). In these \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \multicolumn{3}{c}{**Uniform Grids**} & \multicolumn{3}{c}{**1D Non-Uniform Grids**} & \multicolumn{3}{c}{**2D Non-Uniform Grids**} \\ \hline  & **10** & **5** & **2.5** & **Multiple-Linear** & **Linear-Nonlinear** & **Multiple-Linear** & **Multiple-Linear** \\  & **nm** & **nm** & **nm** & **(2.5 nm)** & **(2.5 nm)** & **(2.5 nm)** & **(0.625 nm)** \\ \hline \(\lambda_{p}\) & 727 & 678 & 657 & 665 & 665 & 665 & 665 & 642 \\ \(T(\lambda_{p})\) & 4.14 & 3.87 & 3.64 & 3.83 & 3.80 & 3.82 & 3.68 \\ \hline \end{tabular}
\end{table} 
Table 1: Wavelength at peak transmittance \(\lambda_{p}\) and corresponding peak \(T(\lambda_{p})\) for the cases of uniform grids, 1D non-uniform grids, and 2D non-uniform grids. Figure 4: Normalized transmittance spectrums from: the cases of uniform grids when \(\Delta=10\) nm (green curve), 5 nm (cyan curve), and 2.5 nm (blue curve), respectively, the converged profile (purple curve) from the linear curve fitting (see text), theoretical prediction (black curve) from [33], the cases of 1D non-uniform grids with \(\Delta x_{1}=2.5\) nm when the mapping functions are multiple-linear (long dashed brown curve) and linear-nonlinear (long and short dashed golden curve), respectively, and the cases of 2D non-uniform grids with \(\Delta x_{1}=\Delta z_{1}=2.5\) nm (dashed red curve) and 0.625 nm ( long dashed black curve), respectively, when both of the mapping functions are multiple-linear. cases, the stability factor is set to be \(S\cong 0.45\), slightly smaller than the default 0.5, due to the stability requirement of the PEC-like material. The transmittance spectrums are shown in Figure 4. Both spectrums coincide with each other. The wavelength at peak transmittance \(\lambda_{p}\) and the corresponding peak \(T(\lambda_{p})\) are listed in Table 1. Between these two cases, we obtain that \(\lambda_{p}\) is the same and \(T(\lambda_{p})\) has the less than 1% difference. Moreover, they are in good agreement with those from the uniform 2.5-nm case. Therefore, the results are accurate. We verify that the cell size resolution in the central area is important to the accuracy. It should be noted that the linear-nonlinear case implies the flexibility of the mapping functions for complex configurations. To quantify the efficiency, we define the computational cost for each simulation:  
\[C=n_{c}\times t_{s}, \tag{16}\]  
where \(n_{c}\) is the number of parallel-computing cores and \(t_{s}\) is the total hours of simulation.