Conformal FDTD
Why Staircasing Limits the Yee Grid
The classic Yee algorithm samples the electric and magnetic fields on a staggered Cartesian lattice, which is elegant for rectangular structures but blunt at any curved or slanted surface. A perfect electric conductor that cuts a cell diagonally is forced onto the nearest grid lines, so a smooth circular waveguide wall becomes a jagged sawtooth of full-cell steps. That geometric error is first order in the cell size Δ, meaning the resonant frequencies, surface currents, and quality factor converge only linearly as the mesh is refined. Doubling resolution to halve the error costs eight times the memory and roughly sixteen times the run time in 3D, so brute-force refinement is rarely affordable at millimeter-wave frequencies where features are already small relative to a wavelength.
Conformal FDTD attacks the problem locally rather than globally. Only the cells that the boundary actually intersects are modified; everywhere else the ordinary leapfrog update runs unchanged, preserving the explicit, matrix-free character that makes FDTD attractive. The most widely deployed formulation is the Dey-Mittra scheme, which recasts the magnetic-field update as a discrete contour integral of Faraday's law around the deformed cell face. The integration loop is shortened to the portion of the cell edge that lies in free space, and the enclosed area is reduced to the partial cell area outside the conductor. The result is a single modified coefficient per cut cell, trivially added to an existing solver.
The same partial-cell idea extends to dielectric interfaces through an effective-medium average. When a curved material boundary slices a cell, the local permittivity used in the electric-field update is volume-weighted between the two media by the fill fraction, which removes the abrupt index jumps that otherwise scatter spurious numerical energy. Together these conformal corrections recover the second-order accuracy of the underlying scheme for arbitrary smooth geometry without leaving the comfort of a single structured grid.
Conformal Update and Stability
Hzn+1 = Hzn − (Δt / μΔ) [ (Ey)i+1 − (Ey)i − (Ex)j+1 + (Ex)j ]
Dey-Mittra conformal H-update (cut cell):
Hzn+1 = Hzn − (Δt / μAz) × ∑ Eℓ · Δℓ
where Az = free-space area of the deformed face, Δℓ = length of each free edge segment
Dielectric fill-fraction average:
εeff = f × ε1 + (1 − f) × ε2, f = fraction of cell volume in medium 1
Naive cut-cell stability (the problem):
Δtcut ≈ √(Az / Afull) × ΔtCFL, ΔtCFL = 1 / [ c √(1/Δx² + 1/Δy² + 1/Δz²) ]
Stabilized cut-cell update (enlarged-cell / BCK fix):
Az,eff = max(Az, fmin Afull) by redistributing flux into neighbor faces ⇒ Δt = ΔtCFL
The naive scheme lets a sliver of area fraction Az/Afull force Δt down by roughly its square root (a 1% sliver costs about 10×). The stabilized formulations raise the effective face area to a floor fmin (typically 0.1 to 0.5) without altering the modeled geometry, so the solver runs back at the standard Courant limit ΔtCFL.
Conformal FDTD Versus Other Boundary Treatments
| Method | Boundary accuracy | Time-step penalty | Implementation effort | Best application |
|---|---|---|---|---|
| Staircased Yee | 1st order (curved) | None | Trivial | Rectangular / axis-aligned |
| Dey-Mittra (stabilized) | 2nd order | 1 to 3% | Low (coefficient edit) | Smooth curved conductors |
| BCK conformal | 2nd order | < 1% | Moderate | Sliver-heavy meshes |
| Subgridding | 2nd order (local) | Interface dependent | High (field interp) | Embedded fine features |
| FEM / DGTD | High (p-refinable) | Implicit solve cost | Very high | Arbitrary unstructured shapes |
Frequently Asked Questions
Does conformal FDTD force a smaller time step than standard FDTD?
The naive Dey-Mittra scheme keeps the explicit leapfrog update but shrinks the magnetic-loop area in cut cells, which lowers the locally stable time step; a 1% area sliver would drag the Courant limit down by roughly its square root. Stabilized Dey-Mittra and BCK variants clamp the smallest allowed area to a floor of about 0.1 to 0.5 of a full cell, so the model runs within 1 to 3% of the uniform-grid Δt while still recovering second-order accuracy.
How much does conformal FDTD improve resonant-frequency accuracy over a staircased mesh?
Staircasing gives first-order boundary error, so frequency and Q converge only linearly. A cylindrical cavity meshed at 20 cells across can show 1 to 3% frequency error and spurious mode splitting on a staircased grid; a conformal model on the same grid restores second-order convergence and typically cuts error below 0.1 to 0.3%. That lets you reach a target accuracy with two to four times coarser cells, roughly an order of magnitude fewer total unknowns in 3D.
When should I use conformal FDTD instead of subgridding or FEM?
Reach for conformal FDTD when the geometry is mostly smooth curves and you want one global Cartesian grid: rounded waveguide bends, circular horn apertures, vias, radomes, tissue surfaces. It removes staircase error in place without subgridding's interface bookkeeping or FEM's matrix solve. Use subgridding for small features needing local resolution, and FEM or DGTD for truly unstructured geometry, sharp re-entrant corners, or strongly inhomogeneous media.