ABSTRACT

Following implementation of low-and high-order SIBCs for the boundary integral equations in Chapters 6 and 7 we now turn our attention to implementation of SIBC for volume discretization methods. We start with the FDTD and implement the SIBCs developed in Chapter 3 for high-frequency applications. The contour-path method, based on the integral forms of Faraday and Ampere’s laws, is used for the implementation because it allows the grid to conform to the boundary and by doing so eliminates one of the main issues in use of the FDTD method-the use of regular grids. The fundamentals of the contour-path FDTD method are outlined separately in Appendix 8.A.1. This approach reduces the errors caused by discretization of boundaries and is well suited for incorporation of SIBCs. The conditions of applicability of the coupling between the FDTD and SIBC are discussed and examples to its use given to demonstrate the implementation. The ﬁnite integration technique (FIT) is another volume discretization

method related to the FDTD method. Unlike the FDTD method which solves for the electric and magnetic ﬁelds, the FIT method operates on circulations of the electric ﬁeld (voltages), circulations of the magnetic ﬁeld (magnetomotive forces) and ﬂuxes of the electric and magnetic ﬁeld. In that sense it may be viewed as a generalized FDTD method for the solution of Maxwell’s equations in their integral form. As with the FDTD method, the FIT method is implemented for high-frequency problems using low-and high-order SIBCs in Cartesian and tetrahedral grids. The relations between the FIT and FDTD grids are discussed as well as the limits of applicability and the performance of the method is evaluated through an example. Perhaps the best-known volumetric discretization method is the ﬁnite-

element method (FEM) because of its extensive use in all engineering disciplines. It is however fundamentally different than the previous two methods in that Maxwell’s equations must be transformed and adapted to the method before discretization and solution can commence. This in turn has implications on the implementation of the SIBCs. Also, the FEM has a diversity of element types and methods of formulation. For the sake of brevity we will discuss here only the formulation in the ‘‘weak’’ sense using tetrahedral edge

elements although the method and approach are general enough to be useful for other types of elements and formulations. Numerical examples complete this chapter as well.