Local Space–Time Dirichlet–Neumann Nonreflecting Boundary Values: Half-Space Spatial Orthogonal Mode Projections

Abstract

This paper follows the time-honored analyses to provide a formal bridge connecting the PML and the well-understood (classical) Dirichlet and Neumann boundary value problems. The connection offers many theoretical and practical advantages not immediately obvious from any PML reconstructions. For example, the analysis shows that the reflection-free of PML boundaries is composed of the superposition of one Dirichlet and one Neumann solution acting out of phase constituents. In fact, it uncovers that the existing PML solutions are readily obtained from the projection of the well-understood 1-D Robin impedance-matched wave equations. The projection hence not only explains how the PML split fields fundamentally come about but also that the PML implicitly uses only an orthogonal projection. Naturally, we further extend the construction to include nonorthogonal decomposition. Local-to-global coordinate transformation is introduced to give a general nonreflecting boundary value solution valid for planar orientations not aligned with the Cartesian coordinates. Nonorthogonal generalizations also make it geometrically explicit as to what causes an incorrect PML problem that can arise from mediums lacking orthogonal symmetry invariant terminations.