Mesh generation and conformal geometry#

Geometry kernel and material filling#

Solid geometry is authored through a CSG layer (geometry/) backed by the Open CASCADE kernel via pythonocc-core (DD-003, DD-016). Material assignment on the grid uses exact boundary-representation queries (solid classification, 3D face–solid intersection, planar cross-sections) rather than voxel sampling. This is engineering infrastructure on top of a third-party kernel, not a numerical-methods contribution.

The planar cross-sections that feed the conformal fractions are taken three ways. Planar faces are sectioned exactly by an in-house engine (intersection points on straight edges, segments stitched through shared edges). Free-form faces — parametric surfaces, lofts, imported B-spline patches — are sectioned on a triangulation of the body at the section deflection: the crossing points are lifted back onto the exact surface (one Newton step from the interpolated surface parameters, which removes the chord error that a shallow cut would otherwise amplify), and each segment is refined in-plane until its sagitta is a tenth of the deflection. Analytic curved faces — cylinders, cones, spheres, tori — go through the kernel’s Boolean section, whose exact curves are tessellated to the same chord budget, a tenth of the deflection. The deflection is a hundredth of the smallest cell for the sub-cell fractions and a tenth for the cell classification. The chord budget is what sets the residual geometry error of a curved face: a polygon of chords books (2/3) of the sagitta budget too little area per unit boundary length, which is a radius short by about a 1500th of the smallest cell — first order in the cell size, so it is kept an order below the deflection rather than at it (at the full deflection it was a 150th of a cell, and it did not shrink faster than the mesh).

The three paths meet inside one body, and which one a face takes depends on its neighbours. A solid that carries even a single free-form face is sectioned on its triangulation as a whole, and that reaches further than it looks: a Boolean fuses the operands, so a free-form body subtracted from an enclosure puts the enclosure on the same path, including faces nowhere near it. Analytic faces are exempt from the triangulation’s chords whatever else the solid carries: planar faces are exact as facets; cylinders are projected onto their own quadric in closed form and their conic traces laid down as exact conics, so a family of parallel planes cutting a bore books identical areas and a waveguide port on such a cross-section keeps its exact transparent boundary; spheres, cones and tori are projected onto their implicit surface, so every section vertex lies on the surface to rounding, the poles of a sphere included. All three paths hold the same chord budget, so a body books the same masses to within that budget whichever path answers it: a cylinder sectioned across its axis by the exact engine, the facet path and the kernel’s Boolean section agrees to rounding, and a sphere, cone or torus agrees within the budget (their vertices are placed differently but all lie on the surface). A parameter sweep that carries a body from one path to another — the first non-zero fillet radius, a loft joining a solid — therefore sees no step from the change of path. Section contours are wound by nesting parity (holes against their outer boundary) before the area kernels sum them, whatever path produced them. The kernel’s section runs over the faces whose extent reaches the plane rather than over the whole body — the same curves, without the kernel preparing a thousand faces the plane never meets — and a planar face that carries the outline of every feature on it (the floor of an air body under a row of posts) is cut once into tiles, so that a plane crossing one tile is sectioned over that tile alone; a plane on a cut line or across several tiles takes the whole face. A grid plane that coincides with a face of the model is never sectioned in place: the fractions there come from sections a small step to either side, and the step clears the kernel’s tolerance by a margin (four times the largest tolerance in the model, or the section deflection if that is larger), because a section within the tolerance of a face reports that face on both sides.

The free length of a grid edge — the part outside any conductor — comes from the crossings of its carrier line with the conductor’s faces. Every edge of a structured grid lies on one of comparatively few such lines; the lines are collected once and intersected in bulk, planar faces at their exactly known crossing points and curved faces through the kernel, and each edge reads the crossings inside its own window. An edge whose crossings cannot anchor the inside/outside bookkeeping — a crossing on a face outline, as on every edge that runs along a conductor’s corner — classifies its sub-segment midpoints on probe lines of their own, and a midpoint within tolerance of a face counts as conductor, as the kernel’s classifier would have it.

Graded Cartesian mesh#

The mesh generator (mesh/mesher.py) produces a graded (non-uniform) Cartesian tensor-product grid: geometry-derived fixpoints (“anchors”, plane clustering, DD-059…DD-062) plus feature-based two-scale refinement (h_fine near features, h_coarse in bulk, geometric grading between them, DD-028). Graded Cartesian meshes and the accuracy trade-offs of local grading are standard FDTD/FIT practice [5]; the specific fixpoint, plane-clustering and thin-sheet heuristics are in-house engineering.

Which wavelength sets the bulk cell size#

The bulk cell size is λ / min_nodes_per_wavelength, and on a tensor-product grid the question is which λ. A grid line spans the whole domain, so the finest sensible resolution is per slab: each interval between two grid planes on an axis is a slab of the domain, and the densest material whose bounding box reaches into that slab sets the slab’s wavelength (DD-192, the default MeshControl(wavelength_rule="local")). The air box around a small ceramic is meshed at the air wavelength on every axis interval the ceramic does not reach; the slabs the ceramic occupies — and every slab of an axis the ceramic spans entirely, such as the in-plane axes of a full-width substrate — stay at the ceramic’s wavelength. The background material fills whatever no solid covers and counts in every slab. The bounding box is exact for bricks and conservative for curved or rotated bodies, so a slab is never meshed coarser than the material in it. wavelength_rule="global" restores the older rule — the densest material anywhere sets one bulk size for the whole domain.

The two rules differ only far from material interfaces. Feature refinement (min_cells_per_feature), the geometric grading from an interface into the bulk, the DD-107 domain-face buffer and the edge floor below are the same under both; the edge floor keeps the densest material’s wavelength as its reference, because it bounds the time step, and the time step follows the smallest cell anywhere. This is the rule hex-mesh generators apply as per-material mesh settings; the slab-wise form is its consequence on a tensor grid.

Which geometry gets a grid plane#

Grid planes come from two passes over the CAD model. The face pass places a plane on every planar face with an axis-parallel normal and on the axis-normal tangent positions of cylinders and spheres — the material boundaries. The edge pass (DD-191) places a plane wherever a B-rep edge lies flat in an axis-normal plane: the circle where a chamfer cone meets a cylinder, the straight line where a fillet leaves a box face, the section curves of a loft, the iris and equator circles of a revolved profile. These are the positions where a body’s cross-section changes character along an axis, and they are invisible to the face pass (a chamfer is a cone; a fillet is a quarter cylinder whose tangent positions lie outside its trimmed extent). Only sharp edges count: seam edges, degenerated edges and the split lines a Boolean leaves between two faces of one surface contribute nothing — a cylinder’s seam is a straight line through its axis and would put a phantom plane there.

Edge planes are a soft class. They never move or outrank a material plane, and they ask for one cell across the interval they bound — enough for the cell’s midplane to see the feature — rather than the min_cells_per_feature a material gap gets. They are floored: an edge plane whose cell would be smaller than h_max / max_edge_refinement (default 4), or than min_cell_size, is dropped and reported by a warning that names the coarsest dropped position, the cell it would have created and the ratio that keeps it. The ratio bounds the time-step cost of resolving small edges (the explicit loop takes one step bounded by the smallest cell anywhere); max_edge_refinement=0 switches the edge pass off.

Refining the conductor edges#

Where a conductor forms a wedge of less than 180° — the edges of a strip, a patch, an iris — the field and the surface current are singular: \(r^{-1/3}\) at a 90° edge, \(r^{-1/2}\) at a knife edge. A grid cannot represent that, and everything that integrates the edge field — the line impedance, the effective permittivity and with them the phase of \(S_{21}\) and the resonant frequency of a patch — converges only about first order in the cell that holds the edge, however fine the bulk is. Measured on a 50 Ω microstrip: the impedance the port solver reads off the grid is 51.5, 52.2 and 52.6 Ω for edge cells of 50, 25 and 12.5 µm (limit about 52.9 Ω), and three grids with the same 12.5 µm edge cell agree within 0.1 Ω although their cell counts differ by almost two.

MeshControl(singularity_refinement=k) starts the grading at the planes that hold such an edge at h_fine / k instead of h_fine, on both sides of the plane, and grows by growth_factor from there. Which edges count is read from the CAD model: a convex edge of a metal body (PEC or lossy metal), or a concave edge of a non-metal body whose surroundings at the edge are metal — a vacuum body cut out of a PEC background, where the iris rim is the sharp metal wedge. The corners of a cavity are concave metal edges and regular; a fillet’s onset is tangential; dielectric edges are much weaker and not refined. The refinement never adds a plane (the edge’s plane is a material face or an edge plane already), never touches the domain’s own end planes, and stops at min_cell_size. Coupled lines with a tight gap are the case that needs it most: the odd mode of a 25 µm gap between 5 µm fingers on 254 µm alumina lives within a few tens of micrometres of the surface, and its impedance moves from 22 Ω at the default grading to 42 Ω once the cells next to the metal are below 10 µm (factor 8 with a 6 µm floor), while the even mode moves by a tenth of that — the Lange coupler how-to designs on exactly that grid.

The factor is off by default, and the reason is the time step. The edge cell bounds it, so a factor of 2 halves the step and adds the cells of the two ramps: on the mesh-convergence how-to’s structure the factors 1, 2 and 3 lie on one cost-versus-error curve — the factor moves resolution from the bulk to the edges, it does not buy accuracy for free. It pays where the impedance or the effective permittivity is the quantity of interest, where the time step is bound by a min_cell_size floor or by another axis anyway (then the edge cells are free), and where memory rather than time is the limit: at equal edge cell the refined grids of the microstrip need a third fewer cells. Model-wide refinement is still what min_nodes_per_wavelength is for.

Why a feature below the grid is silent rather than approximately represented is a property of the conformal material matrices, below.

Inspecting the grid#

A grid plane is a decision, and the mesh keeps the record of every one of them: mesh.planes lists, per axis, the position of each plane, the rule that asked for it and the shape it came from. The vocabulary:

  • face — an analytic material face: a brick’s side, a cylinder’s tangent plane, the plane of a sheet;

  • extent — the bounding box of a shape, which coincides with a face for bricks and is a conservative envelope for everything else;

  • edge — a geometry edge lying flat in the plane: the onset of a chamfer or a fillet, a loft section, the circle of an iris;

  • sheet — the single plane of a thin metallisation;

  • wire — a vertex of a thin wire;

  • symmetry — a symmetry face with a position;

  • forced — a position from MeshControl(forced_planes=...).

A plane usually has several sources (a brick face is a face, an extent and the edge of the brick’s own sides at once); domain end marks the bounding box, singular a plane that holds a conductor edge and grades from the refined fine size. Requested planes that did not make it into the grid are listed as well: dropped for edge planes below the edge floor, absorbed for material planes merged by the hard min_cell_size floor — and unplaced for anything the mesher discarded without recording it, which is empty in a consistent build and otherwise the first place to look.

print(mesh.planes) prints the record. mesh.planes.as_dict() is the same in a form that can be pinned, and the test suite keeps such a pin for every example model: a change of the meshing rules then shows up as a readable diff of planes, not as a physics test that moved for reasons unknown. plots.plot_mesh_section(mesh, "z", z0, geometry=model) (or mesh.plot_section(...)) draws the section from the mesh’s side: each grid line in the style of its highest-ranking source, the graded fill as hairlines, absorber cells hatched, the exact section outline on top — and the cells shaded by one of three quantities:

  • fill="coverage" (default) — the exact area share of every cell that lies inside a conductor, as measured by the sub-cell classifier on the primal faces of the node plane nearest to the cut; each cell in the colour of its material, blended towards conductor grey by that share. A round conductor is a round disc here.

  • fill="material" — the classification: the material whose volume contains the cell centre. This is the staircase baseline that the sub-cell values override on every cut cell, not the accuracy of the discretisation; thin sheets do not appear in it.

  • fill="conformal" — the permittivity \(\bar\varepsilon\) the electric material matrix holds for the field component normal to the cut, on the dual cells around the nodes (bounded by the cell midpoints, so half a cell off the grid lines; 0 = conductor). Edges that run along a conductor surface are held at its potential, so around a conductor the masked cells reach one node beyond the contour — coarser than the coverage, and the honest picture for a field along the cut normal.

edges=True adds the primal edges of that node plane: edges held at conductor potential, edges only partly inside a conductor (with their free-length share \(f_L\) setting the opacity) and the short ones lent to a longer neighbour by the enlarged-cell rule. Reading the coverage against the contour is the quickest plausibility check a mesh gets before the solver runs; the vocabulary of the shares is explained in the next section.

The record exists for meshes built by Mesh.from_geometry; a mesh assembled from a grid has none. Positions are the merged geometry coordinates: absorber cells are appended outside them, and on a PMC face the end node is pulled a third of a cell inwards, which the record shows as -> node at.

Conformal sub-cell material matrices (partially filled cells)#

Material boundaries that cut through grid cells are represented by area/length-weighted averaging in the mass matrices instead of staircasing: per primal edge the classifier stores an averaged \(\bar\varepsilon\), a free (non-PEC) length fraction and a free dual-face area fraction; per dual face a corresponding \(\bar\mu\) and free-area data (unified per-edge/per-face sub-cell classification, DD-051). This family of techniques — retaining the standard leapfrog update and encoding sub-cell geometry purely in the material matrices — was introduced for FIT by Krietenstein, Schuhmann, Thoma and Weiland [7]. The conformal fill of plot_mesh_section shows exactly this \(\bar\varepsilon\), and its coverage fill the conductor area share of the primal faces.

The average is taken over the dual face transverse to the edge; it does not integrate along the edge. A boundary that crosses the grid edges is therefore resolved continuously — moving a cylinder’s radius by a twentieth of a cell moves a resonance by a proportionate, linearly scaling amount — while a feature that varies along the edges inside one cell layer has no lever at all until it reaches the layer’s midplane. A chamfer or a shallow recess on a face that is smaller than half a cell contributes exactly nothing, and then switches on in one step when it crosses the midplane. This is the reason for the edge pass above: with a grid plane at the chamfer’s onset the chamfer occupies a cell layer of its own, whose dual faces see it. Where the edge floor drops that plane, the mesher says so.

For perfectly conducting boundaries the classifier additionally shortens partially-PEC edges (free-length weighting), which is the conformal-PEC idea of Dey and Mittra [8] (DD-036, since generalised into the unified classifier of DD-051).

Two refinements are in-house:

  • LC-consistent pair coupling (DD-053, couple_face_material_pairs): on dual faces with a locally translation-invariant ladder direction, the averaged \(\bar\mu\) is replaced by the value that makes the co-located product \(M_\varepsilon M_\mu\) equal the exact transmission-line value \(\varepsilon_0\mu_0\,\varepsilon\mu\,d\tilde d\), so a discrete travelling wave on a uniform line is exact (derivation in design-decisions.md DD-053).

  • Enlarged-cell donor (DD-058, implemented but dormant — measured neutral): stabilising strongly cut cells by borrowing area from the uncut neighbour. The published antecedent is the family of uniformly stable conformal schemes / enlarged-cell techniques, e.g. Zagorodnov, Schuhmann and Weiland [9].

Thin conducting sheets#

Zero-thickness or sub-cell metallisation is detected before gridding (DD-035, DD-059) and represented as PEC edge masks on the primal grid (apply_thin_pec_sheet, DD-017) — the standard thin-sheet treatment in Cartesian time-domain solvers [5] (subcell thin-sheet models are ch. 10 there; the detection pipeline itself is in-house). The footprint a sheet paints is exact, not its bounding box: one section of the metal at mid-thickness gives the outline, and every tangential edge of the sheet plane is tested against it by the even-odd rule — holes stay open, and edges on the outline count as metal — the same path the cell classifier uses, so a copper layer of thousands of faces costs one section, not one solid classification per edge. Each contour of the section is tested only on the edges inside its own bounding box, so a layer of many small pads costs the sum of their boxes, not pads × plane.

Thin-wire sub-cell model#

ThinWire(curve, radius) embeds a conductor thinner than a cell as a PEC edge chain with corrected surrounding material matrices (mesh/thin_wire.py, DD-080). The model is the classic thin-wire sub-cell treatment of Holland and Simpson [10], realised in the paired \((m, 1/m)\) encoding of Noda and Yokoyama [11]: the four encircling dual faces scale \(M_\mu\) by \(m = \ln(\delta/a)/\ln(\delta/r_0)\) and the co-located radial edges scale \(M_\varepsilon\) by \(1/m\), so the wire presents the physical per-length inductance \(L' = (\mu/2\pi)\ln(\delta/a)\) while the pair product — and hence the wave speed and the CFL bound — is untouched. The bare-grid equivalent radius \(r_0 = \kappa_0\,\delta\) with \(\kappa_0 = e^{-\gamma}/2^{3/2} \approx 0.1985\) comes from the square-lattice Green’s function, as given in the thin-wire literature [11].

Where a wire ends#

A wire’s end is a grid node, and what it connects to is decided by that node’s other edges:

  • On a PEC solid the shared node joins the wire’s masked edge chain to the solid’s masked edges; current continuity is topological. The solid’s conformal faces keep their values in the last ring, so the foot segment carries the bare-grid inductance — an error of one cell at each such end.

  • On a thin sheet (the thin-metallisation path above) the sheet is one grid plane at its substrate-side face, and its far face is not in the grid. A wire vertex anywhere inside the metal’s thickness — the natural way to draw a bond is on the metal’s top face — collapses onto the sheet plane, so the wire ends on the sheet’s own node and the junction is the same topological contact as on a solid. The ring faces of the foot segment cross the metal layer; there the wire’s factor \(m\) composes with the sub-cell value of the face, as it does with a dielectric, so the free part of the segment keeps the wire’s inductance. Gated against the same monopole on a solid built on one grid: resonance within 1 %, feed resistance within 3 % for a sheet of a fifth of a cell. The comparison is in-house.

  • Open ends and ends on a PMC wall are the ideal open; a PEC wall is the electric mirror (a monopole).

  • A lumped port drives exactly one edge between two wires (or a chain with a skipped edge); the port edge must not be part of a wire.

A wire that lands beside the metal, or crosses a sheet, follows the same rule: only coordinates inside a sheet’s thickness band move, by less than the sheet thickness, itself below the cell floor. Wire–wire junctions away from a conductor, oblique in-cell segments and insulated wires are outside the model.

Mesh quality safeguards#

Hard minimum cell size with floor-aware refits and a longitudinal series-\(\varepsilon\) correction (DD-060), per-axis fine resolution (DD-061), a permanent 30-case stress sentinel (DD-062) and the reported edge floor (DD-191) are in-house engineering.