Optimal spatial grids for efficient numerical simulation of time-domain electromagnetic phenomena by finite methods

This research develops a mathematical scheme to construct finite method based discrete models of transient electromagnetic wave propagation and guidance problems. The proposed algorithm generates a special sequence of grid steps such that a standard finite-difference discretization that uses these grid steps produces an accurate approximation to the solution along a finely sampled equidistant grid at predetermined receiver locations along the wave-guiding segment. The generated grid steps are optimal in the sense that the resulting spatial sampling approaches the Nyquist limit of two points-per-wavelength, resulting in a highly efficient spatial sampling of the computational domain.;Two primary advantages render the algorithm appropriate for time domain electromagnetic modeling. First, a methodology has been described to allow application of the algorithm to problems involving arbitrary lumped or distributed terminations of the computational domain. However, the optimal spatial grids need only be designed for problems with Dirichlet and Neumann terminations, so the optimal grids can be constructed just once and used repeatedly for problems with similar material characteristics but different boundary conditions.;The second advantage is that the presented algorithm facilitates a procedure that, subsequent to the finite method based time integration along the constructed optimal grids, allows one to construct an approximation of the solution along a fine equidistant grid from the solution along the coarse optimal grid. Hence, the reduced spatial sampling enforced by the optimal grid does not preclude solution sampling along a standard, finely discretized, equidistant spatial grid.;The primary benefits of the derived optimal grids are that the number of segments is minimized to just over the Nyquist limit over a broad frequency range without an increase in the size of the second order finite-difference stencil, that the grids can be used with the circuit solver SPICE for transmission line problems by multiplying per-unit-length series impedance and shunt admittance parameters whilst ensuring passivity by construction, and that the grids are robust enough to be utilized in problems involving frequency dependent losses, inhomogeneities, and anisotropies even if they are only derived assuming a lossless, homogeneous medium.;In particular, a novel application of optimal grids to perfectly matched layer absorbing boundary conditions is presented. In this development, a single optimal grid combining the interior computational and exterior absorbing regions (with Dirichlet or Neumann boundary) is constructed, and the perfectly matched layer loss parameters are defined along this grid.