# Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry
## 5 Numerical Results  
### Multilayered Sphere  
The multilayered sphere is exposed to the magnetic flux density generated by a coil with a radius of 5 cm that is located 13 cm above the center of the sphere, as shown in Figure 3. The axis of the coil is the \(z\)-axis and the origin of the reference system corresponds to the center of the sphere. The operating frequency is 50 Hz and the current flowing through the coil is 1 kA.  
The induced electric field is computed using the numerical method described in Section 2 for both tetrahedral and voxel-based mesh. In the tetrahedral mesh, the number of nodes is about 323,600 and of tetrahedra is 1,883,300. The corresponding voxel mesh is generated using 1 mm for the voxel side. This results in a model with about 2,205,000 nodes and 2,144,100 voxels. It is crucial to observe that the symmetry of the exposure scenario forces the induced currents to circulate within a single tissue. Consequently, hot spots due to field singularity and/or high contrast between conductivities of contiguous tissues are avoided. Therefore, the only source of numerical artifacts in this case is the stair-casing approximation (for the voxel mesh).  
Table 2 shows the maximum value of the electric field in each tissue for the analytical solution. The common post-processing method of the numerical results coming from a voxel model involves the application of specific metrics to remove numerical artifacts. The analysis and comparison of the different filtering metrics are beyond the scope of this paper. The underlying idea behind all techniques is to remove the hot spots of the electric field due to numerical artifacts. To this aim, the ICNIRP guidelines propose to consider the 99 th percentile [5]. However, many authors point out that in the case of localized exposure, considering the 99 th percentile can underestimate the actual maximum induced field [22; 33].  
Figure 5: Manual control of the tetrahedral mesh: the boundaries were smoothed to avoid singularities. Tissue boundaries before local smooth edits (**a**) and the related mesh (**b**). Tissue boundaries after local smooth edits (**c**) and the related mesh (**d**).  
To overcome this issue, the 99.9 th percentile is sometimes preferred [3; 15]. Other techniques, based on statistical considerations rather than on arbitrary thresholds, remove the outliers in the distribution of the induced electric fields [6; 7] in a post-processing operation, or try to prevent them by smoothing the contrast of conductivities between adjacent tissues [6; 22]. In this paper, the maximum value obtained in the raw data is presented together with the 99.9 th and the 99 th percentile. For the tetrahedral mesh only the maximum value is presented. To better appreciate the deviation from the analytical solution, each value obtained in the numerical simulations is divided by the related analytical reference value. Due to the properties of the exposure scenario, the maximum exposure is registered in the outermost layer (skin). In this tissue, in the voxel mesh the maximum value overestimates the exposure by 15%, the 99.9 th percentile overestimates the exposure by 5% and the 99 th percentile underestimates the exposure by 2%. Using the tetrahedral mesh, without any filtering metric, the maximum deviation is 0.5%.