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29) BI j with the 2D Green’s function given by G 2D (ρ, ρ ) = 4 H0(2) (k 0 |ρ − ρ |). Here H0(2) is the zerothorder Hankel function of the second kind. Noting that the total field everywhere outside BI must be the sum of the scattered and ˜ inc + H ˜ scat ) on BI . 27) gives the boundary integral equations (BIEs) 1 γ× Ms − γt (Ms ) + γt (J˜s ) = γt Einc , 2 1˜ ˜ inc . 28) is now a principal value integral excluding the observation point ρ. 31) being the magnetic field integral equation (MFIE). 31) lies in the space H−1/2 (Div; ∂ ).
Nearly 10 years after the first full 3D implementation, the FE–BI methods are now among the mainstream methods for EM analysis. The FE–BI technique exploits the surface equivalence principle  to decouple the interior and boundary fields of the FEM domain. In other words, the fields are expressed everywhere on the surface of the FEM domain using a boundary integral characterized by equivalent sources representing both the tangential electric and magnetic field components. On the other hand, the fields inside the FEM domain are formulated using a discretization of the PDE equation.
11) as nˆ × EIt = nˆ × EII t , ˜ II . cls 30 November 8, 2006 18:28 FREQUENCY DOMAIN HYBRID FINITE ELEMENT METHODS On surfaces of electrically conducting materials pec , the PEC boundary condition is characterized by the elimination of the tangential electric fields, namely, γt Et | pec = 0. 13) Similarly, on surfaces of magnetically conducting materials pmc , the PMC boundary condition is characterized by the vanishing tangential magnetic fields, namely, ˜ t| γt H pmc = 0. 14) Impedance Boundary Condition The impedance boundary condition (IBC) is often used to model a thin coating as a simple single surface or simply to model an imperfect conductor.