CONFORMAL MAPPING AND HEXAGONAL NODAL METHODS .1. MATHEMATICAL FOUNDATION

The conventional transverse integration method of deriving nodal diffu sion equations does not satisfactorily apply to hexagonal nodes. The t ransversely integrated nodal diffusion equation contains nonphysical s ingular 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 no dal equations are formally analogous to the ones for rectangular nodes without the appearance of additional singular terms. Utilizing the in variance of the Laplacian diffusion operator under conformal mappings, it is shown that the diffusion equation for a homogeneous hexagonal n ode can be transformed to the diffusion equation for an inhomogeneous rectangular node. The inhomogeneity comes in through a smoothly varyin g mapping scale function, which depends only on the geometry. The step s of conformal mapping from a hexagonal node to a rectangular node are given, and the mapping scale function is derived, evaluated and appli ed to nodal equation derivations.