In 1983, Holland published the first FDTD algorithm for generalized non-orthogonal coordinates [71]. In this approach, the fields are expressed in terms of their covariant (flow along a coordinate direction) and contravariant (flux through a constant coordinate surface) components and an integral formulation is used to obtain the update equations. In an orthogonal coordinate system the covariant and contravarient components are co-linear, while in a non-orthogonal one they are not and auxiliary equations must be obtained to express one form in terms of the other. Holland's formulation was later revisited by Fusco [72,73], Lee et al. [74,75], and Harms et al. [76,77]. Fusco applied the algorithm to cylindrical scatterers while Lee et al. applied it to waveguide problems and derived stability equations. Navarro et al. [26,78] have studied aspects of non-orthogonal grids, including numerical dispersion and non-zero divergence, and have adapted the Berenger PML for use in such grids.
Mei et al. [79] presented a conformal technique that employed second-order polynomials and six grid points (ten grid points for 3-D). In order to take advantage of all the work that had been done using Cartesian grids and yet gain the benefit of a conformal approach, Yee et al. [80] developed an FDTD overlapping-grid technique which used a conformal grid in the neighborhood of material boundaries and a Cartesian grid elsewhere. Jurgens et al. [81] and Jurgens and Taflove [82] extended the technique previously used to model sub-cellular structures so that it was suitable for modeling curved surfaces. This ``contour path'' method used a Cartesian grid everywhere except in the vicinity of the material boundaries where a distorted grid was used to conform to the boundary.
Conformal modeling is an area of increasing interest and several new techniques have recently been proposed. Many of these conformal methods, such as the finite-volume time-domain (FVTD) method [83] and the discrete surface integral (DSI) method [84], are only loosely related to the original FDTD method. Although these techniques fall outside the scope of this paper, several of the publications that describe them are listed in the on-line database and the interested reader is directed there.