Conformal Mapping and Hexagonal Nodal Methods —I: Mathematical Foundation

The conventional transverse integration method of deriving nodal diffusion equations does not satisfactorily apply to hexagonal nodes. The transversely integrated nodal diffusion equation contains nonphysical singular terms, and the features that appear in the nodal equations for rectangular nodes cannot be retained for hexagonal ones. A method is presented that conformally maps a hexagonal node to a rectangular node before the transverse integration is applied so that the resulting nodal equations are formally analogous to the ones for rectangular nodes without the appearance of additional singular terms. Utilizing the invariance of the Laplacian diffusion operator under conformal mappings, it is shown that the diffusion equation for a homogeneous hexagonal node can be transformed to the diffusion equation for an inhomogeneous rectangular node. The inhomogeneity comes in through a smoothly varying mapping scale function, which depends only on the geometry. The steps of conformal mapping from a hexagonal node to a rectangular node are given, and the mapping scale function is derived, evaluated, and applied to nodal equation derivations.