Cylindrical FDTD

Besides the standard Cartesian mesh, openEMS can solve the same EC-FDTD equations in cylindrical \((\rho, \alpha, z)\) coordinates instead of Cartesian \((x, y, z)\). Support for cylindrical coordinates is one of openEMS’s distinguishing features among open-source FDTD tools, originally developed for simulating RF coils in traveling-wave Magnetic Resonance Imaging (MRI), where the antennas and the surrounding structure are naturally axisymmetric. This page covers why it exists and its trade-offs; for the practical steps to build a cylindrical mesh, see Mesh.

Why Cylindrical Coordinates?

A Cartesian mesh approximates any boundary that isn’t axis-aligned as a staircase of small steps. For round or axisymmetric structures — circular waveguides and resonators, disk or ring antennas, coaxial lines, MRI coils — this “staircasing” error only shrinks as the mesh is refined; it never vanishes, and figures of merit that are sensitive to the exact radius (a resonant or cutoff frequency, for instance) inherit that error.

A cylindrical mesh sidesteps the problem for the geometry that actually benefits from it: because mesh lines run exactly along \(\rho\) and \(\alpha\), a boundary at constant radius is represented exactly, independent of resolution. The gain is specific to curved or rotationally symmetric features — it doesn’t help (and isn’t meant to help) with axis-misaligned rectilinear geometry, which stair-steps in a cylindrical mesh just as it would in a Cartesian one.

The Origin Singularity

Cylindrical coordinates introduce a problem that has no Cartesian counterpart: the origin (\(\rho = 0\)) is a coordinate singularity. At a fixed angular resolution \(\Delta\alpha\), a cell’s arc length (\(\rho \cdot \Delta\alpha\)) shrinks toward zero as \(\rho\) decreases, so cells near the axis become extremely thin wedges.

Because the FDTD timestep is bounded by the smallest cell anywhere in the mesh (the CFL limit, see Mesh), a naively-generated cylindrical mesh forces a timestep far smaller than the structure’s electrical size would otherwise call for — the tiny cells near the axis end up dominating the runtime of the whole simulation, not just the region near the axis.

Subgridding

openEMS mitigates this with subgridding: at one or more chosen radii, half of the azimuthal mesh lines are dropped, halving the angular resolution. Repeating this while moving inward keeps a cell’s arc length roughly constant instead of shrinking linearly with radius, which keeps the timestep close to what the outer, physically relevant mesh would require rather than what the innermost ring would otherwise force.

This is a coarsening applied at specific radii the user chooses, not a fully automatic or continuously graded refinement — getting the trade-off right between angular resolution, subgridding radii, and runtime still takes some manual tuning, similar in spirit to (but a separate mechanism from) the non-uniform Cartesian mesh described in Mesh.

Note

Each subgridding radius requires interpolating field values between the staggered sub-grids on either side of it. This interpolation is mostly stable, but it is one of the few known exceptions to FDTD’s otherwise inherent numerical stability (see Numerical Method): inhomogeneous material crossing a subgridding interface can destabilize the simulation. If a simulation using subgridding becomes unstable, consider moving the affected material boundary away from the subgridding radius, or moving the radius itself.

Practical Trade-offs and Rough Edges

Cylindrical support isn’t a drop-in replacement for Cartesian everywhere in openEMS; a few areas need extra care:

Primitives. Not every primitive is equally well-suited to a cylindrical mesh — for example, a full cylinder is usually better represented with AddBox() over the radial/angular/z ranges than with AddCylindricalShell(), which can mesh incorrectly in some cases. See the note in Primitives.
Disc-material files. Voxel-based material files (CT/MRI phantoms) are inherently Cartesian; openEMS re-samples them at each cylindrical cell’s Cartesian position, which works but is an added conversion step per cell. See Discrete (Voxelized) Material.
Excitations and mode profiles. Radial or azimuthal field patterns (coax lines, circular waveguide modes) need explicit \(\rho, \alpha\) expressions rather than reusable Cartesian mode profiles, see Symbolic Expression Parser: fparser.
Ports. Only some port types are cylindrical-mesh aware; the circular waveguide port is one, see the circular waveguide tutorial and Ports.
Structure. The mesh is still fundamentally a 2D radial/axial profile extruded around the axis — it doesn’t provide arbitrary local (h-adaptive) refinement the way an unstructured mesh in FEM might for an isolated hotspot away from the axis.

Cylindrical vs. a Refined Cartesian Mesh

Cylindrical coordinates are worth the extra setup effort when a structure has genuine rotational symmetry (or close to it) at a scale where staircasing would otherwise perturb the result that matters — small resonators, curved traces, RF coils. For structures that are only loosely round, or where a circular feature is incidental to an otherwise rectilinear design (e.g. one arc trace on a PCB), a locally-refined Cartesian mesh is usually the simpler and sufficiently accurate choice — consistent with the general “importing is easy, meshing is hard” caution that applies to openEMS modeling overall, see Modeling Basics.