## 3 Simulation Setups and Results  
### Simulation Setups  
The most commonly used numerical techniques to calculate electromagnetic radiation are finite different time domain (FDTD), finite element method (FEM), and finite volume method (FVM) [21]. These techniques divide a system domain with a finite number of grids, elements, volumes in appropriate radiation boundary conditions such as Neuman, Dirichlet, and mixed boundary conditions [21]. The concepts are well suited for problems with inhomogeneous or cluttered media. However, the methods are highly computationally intensive and not ideally suited for thin shell shielded enclosures with a low surface area to volume ratio, because a large number of grid cells are required in the free space region inside the shield [7]. To reduce heavily computational time, a surface integral equation (SIE) can be derived. An advantage of the SIE method is that it is explicitly meshed and calculated on the surface of objects, not in the free-space region. For example, equivalent currents are only placed on the boundaries separating material regions. To avoid low-frequency instability for applications of the SIE, an augmented Muller formulation of [PERSON]'s equations was suggested for stable very low frequency and very high contrast materials with the locally corrected Nystrom (LCN) method [7; 22]. Recently, the LCN method of the SIE was shown to match the experimental data of nondestructive shielding effectiveness with 3 dB [7].  
In this paper, we employ a commercial electromagnetic software, FEKO [23]. We apply a method of moments (MoM) that is a full-wave solution of [PERSON]'s surface integral equations (SIE) in the frequency domain.  
Figure 3: \(H\) field and surface current with a different relative permeability of (**a**) 1 and (**b**) 1000 for cylinder materials at 50 Hz case. All cylinder materials have \(10^{6}\) S/m conductivity with a 0.2 m radius and height. The center of the cylinder at the bottom (\(z=0\)) is \(x=y=0\).  
Figure 4 shows an illustration of FEKO simulation setup with a hidden box inside a thin-shelled cubic shield. The system is illuminated by an infinitesimal point magnetic dipole that is oriented in the negative z direction with amplitude 1 Am\({}^{2}\). The side lengths of the outside shell are set at 1.2 m, and the hidden box is located at \(x=y=0.6\) m. The thickness of the outside shell is 2.7 mm. The size and location of the outside shell are fixed in all cases. The outside shell was utilized and designed in recent previous experiments [7,8]. The transmitting magnetic dipole is located at \(x=y=0.6\) m and \(z=1.2\) m. The frequency of the dipole is 50 Hz, which is on the low end of the ELF/VLF band. Fifty and 60 Hz are commonly used by much of the world as power line operational frequencies. Utilizing 50 Hz allows us to indirectly investigate the potential of using background power line signals for ELF imaging. Furthermore, the 50 Hz frequency most clearly shows the hidden objects because of the inherently high degree of penetration [8]. The 200 Hz and 1 kHz cases also have similar qualitative field profiles due to distortions from the hidden objects; however, the incident signal-to-noise ratio (SNR) is notably higher. All electromagnetic quantities are assumed to be time harmonic (\(e^{jwt}\)) and all data are calculated when the source field has zero phase (\(t=0\)). The receivers are assumed to be located on the light blue plane below the box and parallel planes below it. The locations of the transmitter, center of the inside box, and outside shell are fixed in all cases. The simulation utilizes 8770 surface elements on the box and the observation has 1 cm grid resolution.