By Alexander L. Fetter

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We proceeded then to introduce appropriate basis/expansion functions for 2D and 3D applications. These were based on quadrilateral and hexahedral element discretization, and the reader is referred to other references and texts for tetrahedrals and triangular discretizations [32, 37–40]. cls November 8, 2006 18:28 25 CHAPTER 2 Two-Dimensional Hybrid FE–BI The finite element method (FEM) has been demonstrated as one of the most efficient and versatile computational tools [37], especially when considering closed-domain structures such as waveguides [44].

39) BI ˜ inc d w · nˆ × H + ∀w| BI ∈ WE . 38), Eq. 39) has testing function space dual to V J as required by Theorem 3 in Chapter 1. 40) ˜ inc d . 38) can be combined under certain circumstances to produce a symmetric matrix upon discretization. This approach was used recently [22] and allows for a more robust implementation of the standard FE–BI [37]. 38), we assumed that imp ∩ BI = {∅} and res ∩ BI = {∅}. 15). Also, if res ∩ BI = {∅}, then the only change to the FE–BI system is to assume that Et | BI = EIt and Et | = EII t on the surface res .

Fig. 3 represents one such vector basis function defined on the support of two adjacent hexahedral elements. 21) into the variational Eqs. 22) Z Z ⎣Z ⎦ ⎣E ⎦ = ⎣fM ⎦ , ¯0 ¯Z J M ¯Z J J J g # N with # = {E, M} and where the unknown vectors represent columns, namely, E# = {χk# }k=1 J N XY ¯ J = {υk }k=1 . 3: Illustration of the variation in second-order electric field basis functions used in the context of 3D FE–BI. 22) take the explicit forms Z¯ E E = A¯ E E , Z¯ EM = A¯ EM , Z¯ ME = A¯ ME , Z¯ MM = A¯ MM , Z¯ iMJ j = jk 0 α Z¯ iJj M = 2 vi · wjM d + α vi · i BI 1 Z¯ iJj J = (1 − α) 2 i BI wiM · vj d i BI nˆ × wjM vi · vj d − (1 − α) i BI d + (1 − α) (nˆ × vi ) · i BI (vj ) d + α (nˆ × vi ) · i BI vj · i BI nˆ × wjM d (vj ) d .

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