## 4 Numerical Computations  
A simple 3-D human model (head and torso) is used for the computations as suggested in the IEC 62311 document [8]. The electrical conductivity \(\sigma\) is set to \(0.2\,\rm S/m\), while free space permeability \(\mu_{0}\) is assumed. The lowest point of the model is located at the origin, while the \(z\)-axis is its rotational symmetry axis. The height of the human model is set to \(\rm h_{0}=152.8\) cm. An ideal circular coil is considered with its axis parallel to the \(z\)-axis, radius \(\rm R_{c}=16\) cm, placed at \(\rm z_{c}=100\) cm from the soil (the xy plane). Its distance d from the model varies from 10 to 30 cm. A time-harmonic source current is assumed to flow in the coil, with intensity I = 1 A and frequency f = 50 Hz. Due to symmetry reasons, only half the human model is meshed by assuming a homogeneous Dirichlet boundary condition on the xz symmetry plane. Figure 2 shows the adopted tetrahedral mesh (3067 nodes, 12,882 tetrahedra, and 17,764 edges) as seen frontally from a viewpoint in the negative \(y\)-axis. The average length of the edges is about 3 cm. Indicating with \(\rm C\equiv(x_{c}\), 0, \(\rm z_{c})\) the center of the circular coil, the three Cartesian components of the magnetic vector potential at a point P = (x, y, z) in free space are given by  
\[\rm A_{x}(P)=-A_{\varphi}\sin(\varphi),\quad A_{y}(P)=A_{\varphi}\cos(\varphi ),\quad A_{z}(P)=0 \tag{19}\]  
where  
\[\rm A_{\varphi}=\frac{\mu_{0}I}{\pi k}\sqrt{\frac{R}{\rho}}\Big{[}\Big{(}1-k^{ 2}/2\Big{)}E(k)-F(k)\Big{]} \tag{20}\]  
\[\rm\rho=\sqrt{\left(x-x_{c}\right)^{2}+y^{2}} \tag{21}\]  
\[\rm k=\sqrt{\frac{4R\rho}{\left(R+\rho\right)^{2}+\left(z-z_{c}\right)^{2}}} \tag{22}\]  
and E(k) and F(k) are the complete elliptic integrals of the first and second kinds, respectively, of modulus k.  
In order to compute the generic entry \(\rm E_{m}\) of the vector \(\rm e_{F}\), the Gauss quadrature is used for integration along the edge \(\rm e_{m}\) lying on \(\rm\Gamma_{F}\):  
\[\rm E_{m}=\frac{1}{L_{m}}\int_{e_{m}}\overline{E}\cdot\hat{t}_{m}ds=-\frac{i \omega}{L_{m}}\sum_{i=1}^{N_{S}}w_{i}\big{(}A_{x}(P_{i})\,e_{x}+A_{y}(P_{i})\, e_{y}\big{)} \tag{23}\]  
where \(\rm N_{g}\) is the number of Gauss points \(\rm P_{i}\) (i = 1,\(\ldots\),\(\rm N_{g}\)) on the edge \(\rm e_{m}\), \(\rm w_{i}\) (i = 1,\(\ldots\),\(\rm N_{g}\)) are their weights, and \(\rm e_{x}\) and \(\rm e_{y}\) are the x- and y-components of the edge. The number \(\rm N_{g}\) is selected as follows: if \(\rm d_{m}\) is the distance of the barycenter \(\rm B_{m}\) of edge \(\rm e_{m}\) from the coil point \(\rm P_{0}\)\(\equiv\)(\(x_{C}-R_{c}\),0,\(z_{C}\)) nearest to the model, then for \(\rm L_{m}/d\leq 0.2\) a one-point quadrature  
Figure 2: Tetrahedral mesh of the human model and its enlargement in the head part. is used; for \(0.2<\mathrm{L_{m}}/\mathrm{d}\leq 1.1\), two Gauss points are used; otherwise, three points are used on the edge. The FEM-SDBCI method was used to analyze this system, so that the human skin is at the same time both the fictitious boundary \(\Gamma_{\mathrm{F}}\) and the integration surface \(\Gamma\), and the various coils are left outside the domain of the FEM analysis. The solution has been obtained by means of the GMRES solver as described in Section 2; only seven iterations were performed to reach convergence with an end-iteration tolerance of \(\eta_{0}=0.01\) percent. From the solution, the modulus of the electrical field was evaluated at each barycenter of the tetrahedra inside the body and of the triangles on the skin, and the maximum value \(\mathrm{E_{max}}\) is obtained. From this, we obtain the maximum value \(\mathrm{J_{max}}=\sigma\mathrm{E_{max}}\) of the current density in order to compute the coupling factor. \[\mathrm{K}=\frac{\mathrm{J_{max}}}{\mathrm{J_{max}^{(u)}}} \tag{24}\]  
Also, the maximum value \(\mathrm{J_{max}^{(u)}}\) of the current density needs to be computed.