Iteration-Based Temporal Subgridding Method for the Finite-Difference Time-Domain Algorithm

Abstract

A novel temporal subgridding technique is proposed for the finite-difference time-domain (FDTD) method to solve two-dimensional Maxwell’s equations of electrodynamics in the $T{E}_z$ mode. Based on the subgridding FDTD algorithm with a separated spatial and temporal interface, our method focuses on the temporal subgridding region, as it is the main source of late-time instability. Different from other subgridding algorithms that work on the interpolation between coarse and fine meshes, our method stabilizes the solution by using iterative updating equations on the temporal coarse–fine mesh interface. This new method presents an alternative approach aimed at improving the stability of the subgridding technique without modifying the interpolation formulas. We numerically study the stability of the proposed algorithm via eigenvalue tests and by performing long-term simulations. We employ a refinement ratio of 2:1 in our study. Our findings indicate the stability of the conventional temporal subgridding FDTD algorithm with a magnetic field ($H_z$) interpolation. However, when electric fields ($E_x$ and $E_y$) are utilized in interpolation, late-time instability occurs. In contrast, the proposed iteration-based method with an electric field interpolation appears to be stable. We further employ our method as the forward problem solver in the Through-the-Wall Radar (TWR) imaging application.

1. Introduction

The finite-difference time-domain (FDTD) method (Yee’s algorithm) is a widely used numerical technique for solving time-domain Maxwell’s equations of electrodynamics [1,2,3]. Maxwell’s equations are discretized on a space and time-staggered rectangular grid (Yee’s lattice) using centered difference approximations. Each field component is sampled and evaluated at a particular space position, and the magnetic and electric fields are obtained at different instants of time delayed by half the sampling time step. As an explicit numerical method, the FDTD method is subject to the Courant–Friedrichs–Lewy (CFL) stability condition [2,3].

The accuracy and efficiency of the FDTD method can be significantly improved through a subgridding technique, where the grid is selectively refined only in certain regions of the computational domain. In the last few decades, there have been numerous investigations of subgridding and adaptive mesh refinement (AMR) algorithms [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Subgridding methods reduce the computational cost and increase the computational efficiency over methods using uniform meshes when the region of interest occupies a small portion of the domain. The Huygens subgridding (HSG) technique [26,27,28,29] can reach an arbitrarily large ratio of spatial steps as large as 99. A main weakness of the subgridding technique is the late-time instability problem. Numerical studies of this issue performed in the past, such as [4,9,29], indicated that the stability of the subgridding algorithm is sensitive to the interpolation method and the choice of the interpolated fields. In [23], the spatial subgridding and temporal subgridding are handled separately in order to obtain a better stability. It has been pointed out in [30] that the interpolation of fields in the temporal subgridding region is the main source of the late-time instability.

In this paper, we propose a new iteration-based method to stabilize the temporal subgridding FDTD algorithm. We apply separated spatial and temporal mesh interfaces so that the stability of spatial subgridding and temporal subgridding can be investigated separately. Different from other subgridding methods that focus on the interpolation or special interface treatment between coarse and fine meshes, we iterate the updating equations on the coarse–fine mesh interface a few times to obtain a stable solution for a long-term simulation. Our method is an alternative approach to improve the stability without modifying the interpolation formulas. To test the stability, we computed the eigenvalues of the system of updating equations. The maximum magnitude of the eigenvalues corresponds to the amplification factor of the system. The amplification factor of 1 or less corresponds to a stable solution. We also perform long-term simulations for a million steps to verify the stability of the solution.

The paper is organized as follows: In Section 2, we review a couple of conventional 2D subgridding FDTD methods. In Section 3, we present the proposed iteration-based temporal subgridding algorithm. Numerical examples to verify the stability of the proposed method and an application to the Through-the-Wall Radar (TWR) imaging are presented in Section 4.

2. The FDTD Method and Subgridding Techniques

Consider the two-dimensional Maxwell’s equations (Faraday’s and Ampère’s laws) in the transverse-electric mode with respect to z ($T{E}_z$) [3]

$$\begin{array}{cc}\hfill & {ϵ}_0\frac{\partial E_x}{\partial t}=\frac{\partial H_z}{\partial y},\hfill \\ \hfill & {ϵ}_0\frac{\partial E_y}{\partial t}=-\frac{\partial H_z}{\partial x},\hfill \\ \hfill {\mu }_0& \frac{\partial H_z}{\partial t}=\frac{\partial E_x}{\partial y}-\frac{\partial E_y}{\partial x},\hfill \end{array}$$

(1)

where E and H represent the electric and magnetic fields, respectively. ${ϵ}_0=8.854\times {10}^{-12}$ F/m and ${\mu }_0=4\pi \times {10}^{-7}$ H/m are the vacuum permittivity and permeability, respectively. The discretized updating equations using the Yee FDTD method can be written as [3]

$$E_{xi+\frac{1}{2},j}^{n+1}=E_{xi+\frac{1}{2},j}^n+\left(\frac{\Delta t}{{ϵ}_0\Delta y}\right)(H_{zi+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{zi+\frac{1}{2},j-\frac{1}{2}}^{n+\frac{1}{2}}),$$

(2)

$$E_{yi,j+\frac{1}{2}}^{n+1}=E_{yi,j+\frac{1}{2}}^n-\left(\frac{\Delta t}{{ϵ}_0\Delta x}\right)(H_{zi+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}-H_{zi-\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}),$$

(3)

$$\begin{array}{c}{H}_{zi+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}=H_{zi+\frac{1}{2},j+\frac{1}{2}}^{n-\frac{1}{2}}+\left(\frac{\Delta t}{{\mu }_0\Delta y}\right)(E_{xi+\frac{1}{2},j+1}^n-E_{xi+\frac{1}{2},j}^n)\hfill \\ \hfill -\left(\frac{\Delta t}{{\mu }_0\Delta x}\right)(E_{yi+1,j+\frac{1}{2}}^n-E_{yi,j+\frac{1}{2}}^n),\end{array}$$

(4)

where $\Delta x$, $\Delta y$ and $\Delta t$ are the grid cell sizes in space and time, respectively.

There are two ways to handle the interface between coarse and fine meshes in the subgridding method, based on the choice of ghost values (either magnetic or electric fields). Here, we consider a coarse–fine mesh ratio of 2:1, as shown in Figure 1.

2.1. Temporal Subgridding Algorithm Using Ghost H Fields

Referring to Figure 2, the temporal subgridding algorithm with ghost $H_z$ fields in a 2D $T{E}_z$ mode consists of the following steps:

• Update the magnetic field $H_{zc}$ on the coarse mesh from time $n-\frac{1}{2}$ to $n+\frac{1}{2}$;
• Update the electric fields $E_{xc}$ and $E_{yc}$ on the coarse mesh from time n to $n+1$;
• Interpolate the ghost magnetic field ${\widehat{H}}_{zf}^{n-\frac{1}{2}}$ using nearby values on the coarse mesh;
• Update the magnetic field $H_{zf}$ on the fine mesh from time $n-\frac{1}{2}$ to n;
• Update the electric fields $E_{xf}$ and $E_{yf}$ on the fine mesh from time $n-\frac{1}{4}$ to $n+\frac{1}{4}$;
• Interpolate the ghost magnetic field ${\widehat{H}}_{zf}^n$ using nearby values on the coarse mesh;
• Update the magnetic field $H_{zf}$ on the fine mesh from time n to $n+\frac{1}{2}$;
• Update the electric fields $E_{xf}$ and $E_{yf}$ on the fine mesh from time $n+\frac{1}{4}$ to $n+\frac{3}{4}$;
• Replace the magnetic field $H_{zc}$ on the coarse mesh with the fine mesh magnetic field $H_{zf}$ in the region where fine mesh overlaps coarse mesh: $H_{zci+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}=H_{zfi+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}$;
• Recalculate the electric fields $E_{xc}$ and $E_{yc}$ on the coarse mesh, as in step 2.

The above steps are applied recursively at each refinement level. The ghost magnetic fields are obtained by interpolation using neighboring values. All electric fields in the fine-mesh are computed using the standard Yee FDTD algorithm. On the coarse grid, the $H_{zc}$ field is updated from time $n-1/2$ to $n+1/2$, followed by the $E_{xc}$ and $E_{yc}$ fields, which update from time n to $n+1$. In the meantime, the fine mesh solutions are updated twice: the first update of $H_{zf}$ is from $n-1/2$ to n, followed by the updates of $E_{xf}$ and $E_{yf}$ from $n-1/4$ to $n+1/4$; the second update of $H_{zf}$ is from n to $n+1/2$, followed by the updates of $E_{xf}$ and $E_{yf}$ from $n+1/4$ to $n+3/4$. On the coarse mesh, some magnetic fields have been updated in step 9, so we recalculate the electric fields affected by this change on the coarse–fine mesh interface in step 10.

In order to calculate the electric field on the boundary of the fine mesh, we need the coarse magnetic field to recover the missing fine region magnetic fields that are ghost values. In step 3, the ghost magnetic fields are borrowed from the coarse mesh, similar to the method in [24]:

$${\widehat{H}}_{zfi+\frac{1}{2},j+\frac{1}{2}}^{n-\frac{1}{2}}=H_{zci+\frac{1}{2},j+\frac{1}{2}}^{n-\frac{1}{2}}.$$

(5)

In step 6, the ghost magnetic fields are interpolated from the coarse mesh [24]:

$${\widehat{H}}_{zfi+\frac{1}{2},j+\frac{1}{2}}^n=\frac{1}{2}(H_{zci+\frac{1}{2},j+\frac{1}{2}}^{n-\frac{1}{2}}+H_{zci+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{2}}).$$

(6)

2.2. Temporal Subgridding Algorithm Using Ghost E Fields

The temporal subgridding algorithm using ghost E fields is similar to the case using ghost H fields, but slightly more complicated since there are two electric field components ${\widehat{E}}_{xf}$ and ${\widehat{E}}_{yf}$ in the $T{E}_z$ mode. In this case, the magnetic fields are defined at step n and the electric fields are located at step $n+1/2$. Referring to Figure 3, the algorithm can be summarized as follows:

• Update the electric fields $E_{xc}$ and $E_{yc}$ on the coarse mesh from time $n-\frac{1}{2}$ to $n+\frac{1}{2}$;
• Update the magnetic field $H_{zc}$ on the coarse mesh from time n to $n+1$;
• Interpolate the ghost electric fields ${\widehat{E}}_{xf}^{n-\frac{1}{2}}$ and ${\widehat{E}}_{yf}^{n-\frac{1}{2}}$ using nearby values on the coarse mesh;
• Update the electric fields $E_{xf}$ and $E_{yf}$ on the fine mesh from time $n-\frac{1}{2}$ to n;
• Update the magnetic field $H_{zf}$ on the fine mesh from time $n-\frac{1}{4}$ to $n+\frac{1}{4}$;
• Interpolate the ghost electric fields (${\widehat{E}}_{xf}^n$ and ${\widehat{E}}_{yf}^n$) using nearby values on the coarse mesh;
• Update the electric fields $E_{xf}$ and $E_{yf}$ on the fine mesh from time n to $n+\frac{1}{2}$;
• Update the magnetic field $H_{zf}$ on the fine mesh from time $n+\frac{1}{4}$ to $n+\frac{3}{4}$;
• Replace the electric fields $E_{xc}$ and $E_{yc}$ with the corresponding electric fields $E_{xf}$ and $E_{yf}$ in the region where the fine grid overlaps a coarse grid: $E_{xci+\frac{1}{2},j}^{n+\frac{1}{2}}=E_{xfi+\frac{1}{2},j}^{n+\frac{1}{2}}$ and $E_{yci,j+\frac{1}{2}}^{n+\frac{1}{2}}=E_{yfi,j+\frac{1}{2}}^{n+\frac{1}{2}}$;
• Recalculate the magnetic field $H_{zc}$ on the coarse mesh, as in step 2.

The above steps are applied recursively at each refinement level. In this case, the coarse grid electric fields $E_{xc}$ and $E_{yc}$ are updated from $n-1/2$ to $n+1/2$, followed by $H_{zc}$ updates from n to $n+1$. The fine mesh solutions are updated as follows: $E_{xf}$ and $E_{yf}$ update from $n-1/2$ to n, followed by the $H_{zf}$ update from $n-1/4$ to $n+1/4$, then $E_{xf}$ and $E_{yf}$ update from n to $n+1/2$, followed by the $H_{zf}$ update from $n+1/4$ to $n+3/4$. In step 9, we replace the electric field on the coarse grid by the time-averaged electric fields on the fine grid at the coarse–fine mesh interface and recalculate the corresponding magnetic fields on the coarse mesh in step 10. The last step is to update the magnetic field by using the newly updated electric fields in step 9. In step 3, we interpolate the ghost electric fields from coarse mesh, similar to the interpolation in [24]:

$${\widehat{E}}_{xfi+\frac{1}{2},j}^n=\frac{1}{2}(E_{xci+\frac{1}{2},j}^{n-\frac{1}{2}}+E_{xci+\frac{1}{2},j}^{n+\frac{1}{2}}),$$

(7)

$${\widehat{E}}_{yfi,j+\frac{1}{2}}^n=\frac{1}{2}(E_{yci,j+\frac{1}{2}}^{n-\frac{1}{2}}+E_{yci,j+\frac{1}{2}}^{n+\frac{1}{2}}).$$

(8)

In step 6, we interpolate the ghost electric field by using the corresponding values on the coarse mesh [24]:

$${\widehat{E}}_{xfi+\frac{1}{2},j}^{n+\frac{1}{2}}=E_{xci+\frac{1}{2},j}^{n+\frac{1}{2}},$$

(9)

$${\widehat{E}}_{yfi,j+\frac{1}{2}}^{n+\frac{1}{2}}=E_{yci,j+\frac{1}{2}}^{n+\frac{1}{2}}.$$

(10)

3. Iteration-Based Temporal Subgridding FDTD Algorithm

In this section, we present a new temporal subgridding FDTD algorithm based on the use of an iteration, aiming at overcoming the late-time instability problem of the conventional subgridding method. As we have observed from the previous section, the conventional temporal subgridding with a coarse–fine mesh ratio of 2:1 consists of the update of the solution on the coarse mesh by one $\Delta t$ time step and two consecutive $\Delta t/2$ updates of the fine mesh solutions. The conventional temporal subgridding method is illustrated in the flow chart (a) of Figure 4, and the iteration-based subgridding technique is illustrated in the flow chart (b) of Figure 4.

The iteration-based subgridding FDTD algorithm with a ghost E field is summarized in the following steps:

• Update $E_{xc}$ and $E_{yc}$ on the coarse mesh from time $n-\frac{1}{2}$ to $n+\frac{1}{2}$;
• Update $H_{zc}$ on the coarse mesh from time n to $n+1$;
• Interpolate the ghost electric fields ${\widehat{E}}_{xf}^{n-\frac{1}{2}}$ and ${\widehat{E}}_{yf}^{n-\frac{1}{2}}$ using the nearby fields on the coarse mesh;
• Update $E_{xf}$ and $E_{yf}$ on the fine mesh from time $n-\frac{1}{2}$ to n;
• Update $H_{zf}$ on the fine mesh from time $n-\frac{1}{4}$ to $n+\frac{1}{4}$;
• Recalculate $H_{zc}$ on the coarse–fine mesh interface at time n using the fine mesh values $H_{zf}$ at time $n-\frac{1}{4}$ and $n+\frac{1}{4}$ [24]

  $$H_{zci+\frac{1}{2},j+\frac{1}{2}}^n=\frac{1}{2}(H_{zfi+\frac{1}{2},j+\frac{1}{2}}^{n-\frac{1}{4}}+H_{zfi+\frac{1}{2},j+\frac{1}{2}}^{n+\frac{1}{4}});$$

  (11)
• Repeat steps 1 to 6 several times on the coarse–fine mesh interface;
• Interpolate ghost electric fields ${\widehat{E}}_{xf}^n$ and ${\widehat{E}}_{yf}^n$ using the nearby coarse mesh values;
• Update $E_{xf}$ and $E_{yf}$ on the fine mesh from time n to $n+\frac{1}{2}$;
• Update $H_{zf}$ on the fine mesh from time $n+\frac{1}{4}$ to $n+\frac{3}{4}$;
• Replace the coarse grid electric fields $E_{xc}$ and $E_{yc}$ with the corresponding fine grid values $E_{xf}$ and $E_{yf}$ in the region where the fine grid overlaps the coarse grid: $E_{xci+\frac{1}{2},j}^{n+\frac{1}{2}}=E_{xfi+\frac{1}{2},j}^{n+\frac{1}{2}}$ and $E_{yci,j+\frac{1}{2}}^{n+\frac{1}{2}}=E_{yfi,j+\frac{1}{2}}^{n+\frac{1}{2}}$;
• Recalculate $H_{zc}$ on the coarse mesh, as in step 2.

In step 6, the magnetic field on the coarse mesh is recalculated so it can be used to update the electric field on the coarse mesh, as in step 1. Therefore, the update steps 1–6 can be viewed as an implicit system of updating equations. A classic algorithm to solve implicit problems is by using an iterative method, and it motivates us to develop the iteration-based method.

A key step of the iteration-based subgridding algorithm is step 7, where the iteration is carried out for N times. Through numerical eigenvalue tests, we study the correlation between N and the amplification factor (the largest magnitude of the eigenvalues of the system of updating equations). We found that when the number of iterations increases, the amplification factor decreases. When $N\ge 4$, the amplification factor becomes 1, up to the machine round-off error.

Note that the iteration is conducted only for the electric fields on the coarse–fine mesh interface and its neighboring magnetic fields, so it does not significantly increase the computational cost.

Using the proposed iteration-based temporal subgridding together with a regular spatial subgridding algorithm, we can obtain an iteration-based spatial–temporal subgridding method.

4. Numerical Tests

4.1. Tests for Temporal Subgridding Method

We first test the temporal subgridding methods. In the first example, we simulate a cylindrical wave in the free space. The incident source of a continuous wave with a wavelength of $0.05\mathsf{\mu }\mathrm{m}$ is placed in the center of the domain. The domain is surrounded by PML boundaries [3]. The temporal fine mesh is a $80\times 80$ square zone from grid points $(80,80)$ to $(160,160)$. The numerical result is shown in Figure 5. The small rectangular box shows the boundary of the fine mesh. We can observe that there is no significant reflection or scattering after the wave crosses the interface of coarse and fine meshes.

The second test is a convergent test of the proposed temporal subgridding method, and the results are shown in

Table 1. Convergence test of the iteration-based temporal subgridding method.

Mesh Size	${\mathit{L}}_2$ Norm	Order of Convergence
$10\times 10$	$2.84\times {10}^{-2}$	-
$20\times 20$	$6.77\times {10}^{-3}$	$2.07$
$40\times 40$	$1.68\times {10}^{-3}$	$2.01$
$80\times 80$	$4.01\times {10}^{-4}$	$2.07$
$160\times 160$	$8.68\times {10}^{-5}$	$2.21$

. The $L_2$ norm is calculated by taking the differences between the solutions of the standard FDTD method (without subgridding) and the proposed temporal subgridding method. As observed from this table, the rate of convergence of the iteration-based subgridding algorithm maintains a second order.

In the next test, we assess the stability of the iteration-based subgridding algorithms by performing long-term simulations. We provide a random magnetic field as an initial condition and run the simulation for one million steps. The computation for this model was performed using a coarse mesh size of $200\times 200$. A box from grid points $(40,40)$ to $(80,80)$ is the region consisting of the fine mesh. We conduct a comparison between the conventional ghost-E subgridding method and the iteration-based approach. Figure 6 illustrates the time history of the solutions obtained from both methods at a selected location. For the conventional method, the solution starts to grow exponentially after about 800,000 steps, so it demonstrates a late-time instability. In contrast, the iteration-based approach maintains a bounded solution throughout the entire one-million-step simulation, thus avoiding late-time instability.

In the last test of this subsection, we carried out eigenvalue tests using small mesh sizes. Figure 7 shows the plots of all eigenvalues in the complex plane on a $20\times 20$ coarse mesh embedded with a $6\times 6$ fine mesh. For the ghost-E subgridding method, there are some eigenvalues outside the unit circles, so their largest magnitude is greater than 1, which results in weak growth of the solution and late-time instability. For the ghost-H and the iteration-based subgridding methods, all eigenvalues lie on or inside the unit circle (enough close to one up to the machine round-off error of ${10}^{-14}$), leading to stable solutions.

Furthermore, the largest eigenvalues (in terms of their magnitudes) are computed with various coarse grid sizes of $10\times 10$, $20\times 20$, $30\times 30$, and $40\times 40$. The fine mesh is a square region fixed at a region with $6\times 6$ cells. Results are shown in

Table 2. List of largest eigenvalues (in terms of magnitude) for three temporal subgridding methods in various mesh sizes.

Method	Mesh Size	Largest Eigenvalue (Round-Off Error $\sim {10}^{-14}$)
Conventional Ghost-H	$10\times 10$	$1+1.3\times {10}^{-14}$
	$20\times 20$	$1+1.6\times {10}^{-14}$
	$30\times 30$	$1+1.7\times {10}^{-14}$
	$40\times 40$	$1+3.2\times {10}^{-14}$
Conventional Ghost-E	$10\times 10$	$1+1.37\times {10}^{-8}$
	$20\times 20$	$1+1.29\times {10}^{-8}$
	$30\times 30$	$1+1.43\times {10}^{-8}$
	$40\times 40$	$1+1.28\times {10}^{-8}$
Iteration-based Ghost-E	$10\times 10$	$1+0.9\times {10}^{-14}$
	$20\times 20$	$1+1.5\times {10}^{-14}$
	$30\times 30$	$1+1.9\times {10}^{-14}$
	$40\times 40$	$1+3.8\times {10}^{-14}$

. We observe that the largest eigenvalue of the ghost-E method is larger than 1. In contrast, the largest eigenvalues of the ghost-H and the iteration-based ghost-E subgridding methods are 1 (accurate to the machine round-off error of ${10}^{-14}$).

Note that the ghost-H method works only for a 2D case so only the ghost-E method is suitable when extending the subgridding method to 3D. A future work is to extend the iteration-based method to three dimensions.

4.2. Tests for Spatial-Temporal Subgridding Method

In this subsection, we test the iteration-based spatial and temporal subgridding algorithm. Similar to the previous examples, we first test the cylindrical wave propagation where the incident source is placed at the center of the domain. The coarse mesh contains $400\times 400$ cells. The spatial fine mesh is an $80\times 80$ square zone from grid points $(80,80)$ to $(160,160)$. The temporal fine mesh is $84\times 84$, two cells larger than the spatial fine mesh in all directions. On the coarse grid, $\Delta x=\Delta y=2$ nm and the time step $\Delta t$ is computed using a Courant number of 0.5. In the spatial–temporal fine region, the grid cells are halved, so they are $\Delta x/2,\Delta y/2,\Delta t/2$. Between the spatial and temporal fine mesh, the grid cell sizes are $\Delta x,\Delta y,\Delta t/2$. As shown in Figure 8, the numerical results show a good agreement between the iteration-based subgridding method and the standard FDTD method.

Similar to the previous examples, we test the stability with random initial data and run the simulation for one million steps. The computation for this model was performed using a coarse grid of $200\times 200$ cells embedded with a $40\times 40$ fine mesh. As shown in Figure 9, a stable solution is obtained for the proposed method after one million steps.

Figure 10 and

Table 3. List of the largest eigenvalues (in terms of magnitude) of different grid sizes for the iteration-based subgridding FDTD method.

Mesh Size	Largest Eigenvalue (Round-Off Error $\sim {10}^{-14}$)
$10\times 10$	$1+0.4\times {10}^{-14}$
$20\times 20$	$1+0.8\times {10}^{-14}$
$30\times 30$	$1+2.0\times {10}^{-14}$
$40\times 40$	$1+2.3\times {10}^{-14}$

show the result of the eigenvalue test. Similar to the previous tests, the coarse mesh contains $20\times 20$ cells, and the fine grid contains $6\times 6$ cells. Figure 10 shows that all the eigenvalues are located inside or on the unit circle. Table 3 confirms that the largest eigenvalue (in terms of magnitude) is 1.0 (up to the machine round-off error of ${10}^{-14}$) for all grid sizes, indicating the stability of the method.

4.3. Through-the-Wall Radar Imaging

In this example, we implement the proposed iteration-based subgridding FDTD method as the forward problem solver in the Through-the-Wall Radar (TWR) imaging application. We consider a two-dimensional rectangular room with dimensions $L_x$ by $L_y$, where an object is placed in the room. The relative permittivities of the room and the object are 1 and ${ϵ}_{obj}$, respectively. We ignore the influence of the walls by setting their relative permittivity to 1. A radar is located outside the room. It sends out an electromagnetic pulse through a transmitting antenna (TX). When the signal hits an object, the reflected wave propagates back and is recorded by a receiving antenna (RX). As shown in Figure 11, by letting the radar go around the room’s perimeter, we obtain a set of transmitted and received signals. In our simulation, we select three locations on each side of the room, so in total, we collect 12 data sets. Using these recorded data, we can reconstruct the shape of the object using the algorithm given in [31].

In our simulations, we set the room size to be $4\mathrm{m}\times 4\mathrm{m}$. The object in the center of the room is a circle (diameter is $0.5\mathrm{m}$) or a square (side length $0.5\mathrm{m}$) with a permittivity of ${ϵ}_{obj}=80$. The electromagnetic incident source is a Gaussian pulse with a carrier frequency of $f=600{\mathrm{MH}}_{\mathrm{z}}$.

For the standard FDTD method, the mesh size is $200\times 200$, so $\Delta x=\Delta y=2\mathrm{cm}$. For the iteration-based subgridding method, we use a coarse mesh of $100\times 100$ ($\Delta x=\Delta y=4\mathrm{cm}$), embedded with a small box (1 m by 1 m) surrounding the object using a fine mesh with half grid size $\Delta x=\Delta y=2\mathrm{cm}$.

Figure 12 shows the simulation results. Green curves represent the boundaries of the objects. The open circles are the reconstructed boundary points of the object using the standard FDTD method. The red markers indicate the reconstructed points by the iteration-based subgridding method. The results of the two methods are similar for the circular object. For the squared object, the reconstructed points by the subgridding method are slightly closer to the actual boundary. In terms of computational time, the FDTD method runs for approximately 5 min, while the subgridding method takes only about 2 min. Thus, from this example, we observe that the subgridding method is more efficient.

5. Conclusions

In this paper, we develop a temporal subgridding algorithm appropriate for the FDTD method for Maxwell’s equations in two dimensions and the $T{E}_z$ mode. Different from other subgridding methods, our method iterates the updating equations on the temporal interface to obtain stable solutions. It is an alternative approach to enhance the stability of the subgridding algorithm without modifying the interpolation formulas. The refinement ratio is 2:1, and the ghost value interpolation technique is applied when updating the solutions on the coarse–fine mesh interface. We numerically test the stability of the algorithm by eigenvalue tests and by performing long-term simulations. Our study shows that conventional temporal subgridding with magnetic field interpolation is stable, while if an electric field interpolation is used, it is late-time unstable since some eigenvalues have magnitudes larger than 1. The proposed iteration-based method with an electric field interpolation is stable since all eigenvalues are located on or inside the unit circle. We further use the proposed subgridding FDTD algorithm to serve as the forward problem solver in through-the-wall radar (TWR) imaging. Future work includes the extension to different refinement ratios and three dimensions, along with comparison to other subgridding methods.