## 3 Applying FEM-DBCI to Computation of Coupling Factors  
The FEM-DBCI and FEM-SDBCI methods described in the previous section appear very suitable for the computation of coupling factors, as can be understood by considering the following points.  
1. The finite element mesh is composed of tetrahedra which are well suited to describe complex bodies, even the most realistic and complex human models, with several types of tissues.  
2. The tetrahedral mesh can be minimized, since there is no need to discretize any free space in the FEM-SDBCI by selecting the integration surface \(\Gamma\) as the skin of the human body and the fictitious boundary \(\Gamma_{\rm F}\) to coincide with it. In the case of FEM-DBCI, it is sufficient to mesh only a thin layer of air by selecting \(\Gamma\) as before and \(\Gamma_{\rm F}\) at a small distance from the human skin. 3. These reduced meshes allow us to place the source coils outside the analysis domain, thus vector \(\mathbf{b_{0}}\) in (6) vanishes and the various matrices \(\mathbf{A}\), \(\mathbf{A_{\rm F}}\), \(\mathbf{H}\), and \(\mathbf{G}\) do not change from one numerical analysis to another by modifying the external coil. Only the \(\mathbf{e_{\rm ext}}\) vector depends on the actual coil type and position; the entries of this array are computed by means of \[\overline{\rm E}(\rm P_{\rm F})=-j\omega\overline{\Lambda}(\rm P_{\rm F})\] (16) where \(\overline{\Lambda}(\rm P_{\rm F})\) is the magnetic vector potential, which can be computed analytically if one considers simple coil geometries. 4. Note that the \(\mathbf{A}\) and \(\mathbf{A_{\rm F}}\) matrices in the FEM equation are complex, while \(\mathbf{H}\) and \(\mathbf{G}\) in the integral equation are real; in fact, these matrices depend only on the geometry of the surfaces \(\Gamma_{\rm F}\) and \(\Gamma\) and on the mesh; hence the matrices \(\mathbf{H}\) and \(\mathbf{G}\) do not change with the (low) frequency of the source currents. Due to the low conductivity of the human model, the skin depth is very large, so it is possible to use a tetrahedral finite element mesh with an average edge size of about 3 cm, without modifications up to frequencies of 10 megahertz. 5. In post-processing, the magnitudes of the electrical field \(\rm E\), and hence of the current density \(\rm J=\sigma E\), are computed at the barycenters of the various tetrahedra in the mesh and also at the barycenters of the various triangular faces on the skin of the human model: \[\rm E=\left|\overline{\rm E}\right|=\left|\sum_{m=1}^{6}\rm E_{m}\overline{ \alpha}_{m}(\rm P_{n})\right|\] (17) where \(\rm P_{n}\) denotes the generic barycenter. Note that the heating power density (in \(\rm W/m^{3}\)) is obtained as \(\rm\gamma\alpha E^{2}\). 6. The computation of the flux density \(\rm B\), if needed, can be performed in post-processing on a per-element basis: \[\overline{\rm B}=-\frac{1}{j\omega}\rm curl\,\overline{\rm E}=-\frac{1}{j \omega}\sum_{m=1}^{6}\rm E_{m}curl\,\overline{\alpha}_{m}\] (18) Since the vector shape functions \(\alpha_{m}\) vary linearly in space, the flux density in (18) is constant in each tetrahedron. As will be explained in the next section, the magnitude of the magnetic field \(\rm B_{0}\) must be computed in the tetrahydro of the human model nearest to the coil. 7. For the solution of the hybrid global systems, the use of the GMRES, as described in the previous section, is more suitable since its convergence characteristics appear more robust than that of the simple iteration. The restating parameter has been set to 10.