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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 [28] 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.

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