A simple implementation of PEC structures in the FDTD technique

Abstract

An off-grid boundary condition within Yee's finite- difference time-domain (FDTD) is presented. The boundary fields are extrapolated from the internal computational domain using the Theory of Image. The method enhances the flexibility of the FDTD method with respect to complex geometrical domains in a uniform standard FDTD mesh, using the standard FDTD code. Curved surfaces are introduced without reducing the time step and maintaining the stability of the simulation. I. INTRODUCTION In many structures, using the Finite-Difference Time- Domain (FDTD) method, it may not be possible to position the boundaries as integer multiples of the chosen spatial step. In the last years, many papers were written on this topic, and the solutions may be still improved. The first way to handle such structures was a reduction of the spatial step, but the CPU time and memory requested can increase too much. On the other hand the use of a nonuniform grid (1) can lower the accuracy, it usually requires a smaller time step, and can lead in instabilities after many time iterations. Offset curved metal boundaries can be treated by few FDTD modifications. These include locally conformal methods (2) such as the conformal finite-difference time-domain (C-FDTD) (3) and the contour- path finite-difference time-domain (CP-FDTD) (4) methods, as well as methods using new FDTD formulations (5) or sub- cell models (6). All of these methods require big changes in the existing FDTD code, as well as changes in the grid and time step. Recently appeared an alternative method (7) that does not disturb the original uniform spatial grid. The adjacent field values are extrapolated from the internal computational domain to obtain exterior (auxiliary) field values. With respect to the standard FDTD method, the modifications concern only the outer boundary values of the field and the existing conventional FDTD code and the grid remain unchanged. Time-step remain the same and the memory requirements are practically unchanged. In this contribution we introduce a new way to evaluate the exterior fields. When a Perfect Electric Conductor (PEC) must be modeled, instead of Dirichlet or Neumann boundary conditions as in (7) we evaluated the exterior field applying the Theory of Image. With this method the field prediction is more accurate: curved and oblique surfaces can be simulated without any staircase approximation. As example, in the case of a 45 ◦ surface and a squared grid, the solution exactly matches the standard FDTD solution, requiring the same time and substantially the same memory. Standard proofs to validate the method were reported.