## 2 The FEM-DBCI Method  
Figure 1 depicts a typical system studied in an eddy current problem: a set of coils carry given time-harmonic source currents near some massive conductors. By assuming a quasi-static time-harmonic steady-state behavior, the problem can be mathematically formulated in terms of the following partial differential equation:  
\[\
abla\times\left(\mu^{-1}\,\
abla\times\overline{\mathbb{E}}\right)+\mathrm{ j}\omega\,\sigma\,\overline{\mathbb{E}}=-\mathrm{j}\omega\,\overline{\mathrm{J}}_{ \mathrm{s}} \tag{1}\]  
where \(\overline{\mathbb{E}}\) is the phasor vector of the unknown electric field, \(\mu\) is the magnetic permeability, \(\sigma\) is the electric conductivity, \(\omega\) is the angular frequency, and \(\overline{\mathrm{J}}_{\mathrm{s}}\) is the phasor vector of the assigned source current density in the coils, assumed to be skin-effect-free. Due to the unbounded nature of the free space around the system and in order to apply FEM, the free space is truncated by means of a closed fictitious boundary \(\Gamma_{\rm F}\), enclosing all the massive conductors. Optionally, some coils may be placed internally to \(\Gamma_{\rm F}\), but others may be left outside. On the truncation boundary \(\Gamma_{\rm F}\), an unknown non-homogeneous vector, the Dirichlet boundary condition, is assumed to hold:  
\[-\hat{\rm n}_{\rm F}\times\hat{\rm n}_{\rm F}\times\overline{\rm E}=\overline{ \rm E}_{\rm F} \tag{2}\]  
where \(\hat{\rm n}_{\rm F}\) is the outward unit vector normal to \(\Gamma_{\rm F}\) and \(\overline{\rm E}_{\rm F}\) is the unknown component of the electric field tangent to the fictitious boundary. The bounded domain inside \(\Gamma_{\rm F}\) is discretized by means of tetrahedral edge elements [21]. In each tetrahedron, the electric field is developed as  
\[\overline{\rm E}=\sum_{\rm m=1}^{6}{\rm E}_{\rm m}\overline{\alpha}_{\rm m} \tag{3}\]  
where \(\overline{\alpha}_{\rm m}\) are the non-dimensional vector shape functions:  
\[\overline{\alpha}_{\rm m}={\rm L}_{\rm m}(\xi_{\rm 1m}\
abla\xi_{\rm 2m}-\xi_{ \rm 2m}\
abla\xi_{\rm 1m}) \tag{4}\]  
with \(\xi_{\rm 1m}\) and \(\xi_{\rm 2m}\) local coordinates (in the tetrahedron) associated with the first and second node, respectively, of the m-th edge \({\rm e}_{\rm m}\), whose length is \({\rm L}_{\rm m}\); the scalar quantity \({\rm E}_{\rm m}\) (in \({\rm V}/{\rm m}\)) is the m-th expansion coefficient of the electric field and is given by the mean value of the tangent component of the field along the edge:  
\[{\rm E}_{\rm m}=\frac{1}{{\rm L}_{\rm m}}\int_{{\rm e}_{\rm m}}\overline{\rm E }\cdot\hat{\rm t}_{\rm m}{\rm d}s \tag{5}\]  
where \(\hat{\rm t}_{\rm m}\) is the unit vector along the m-th edge. By using the vector shape functions \(\overline{\alpha}_{\rm m}\) as weighting functions, the [PERSON] method leads to the following matrix equation:  
\[{\bf A}\,{\bf e}={\bf b}_{0}-{\bf A}_{\rm F}{\bf e}_{\rm F} \tag{6}\]  
where \({\bf A}\) and \({\bf A}_{\rm F}\) are sparse matrices depending on geometry and materials, \({\bf e}\) and \({\bf e}_{\rm F}\) are the unknown vectors of the field expansion coefficients \({\rm E}_{\rm m}\) for the internal and boundary edges, respectively, and \({\bf b}_{0}\) is a vector due to the internal source currents, if there are any. Figure 1: Massive eddy current conductors (characterized by conductivity \(\sigma\) and permeability \(\mu\)) and distributed internal source coils (characterized by the current density J) are enclosed by the fictitious truncation boundary \(\Gamma_{\rm F}\). Other lumped and distributed coils are left outside (external coils).