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Structured hex: extrude, revolve, sweep and mapped all-hex meshing (M5) (#40)

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* Structured hex meshing: multi-level extrude and revolve, one-to-one sweep, mapped blocks (M5)

modelcraft_mesh::hex::hex_mesh meshes a face-tagged boundary surface with
hexahedra by the first structured method that fits: a straight or
rotational multi-level sweep (imprinted cross-section arrangement, cuts at
reflex corners, mapped/annulus/free regions), a one-to-one sweep through
rings of linking faces (least-squares affine layers with residual
correction), or a mapped logical cube (trivariate transfinite
interpolation). Interval assignment couples shared curves; boundary quads
carry B-rep face tags; the certified validity gate, a tiling check and a
fidelity check guard the output; hex20 mid-edge nodes go onto the faces.
Shape::face_surfaces exposes the analytic face surfaces; the corpus runner
gets a structured hex column.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* mesh.hex.generate: structured hex meshing as a command

Meshes one body, every body of the open design or an imported surface,
replaces the mesh, tags element faces with the B-rep faces (so sets by
geometry resolve) and reports per body the method, counts and quality.
The quality pass certifies each element once and feeds the validity gate
(quality::validity_gate_checks). Shape::size gives the cached body size
for defaults instead of a full measure.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Hex: disk O-grids, convex interval cost, early size guard, hex20 corpus column

Smooth single-loop sections (cylinders, tori, vases) get a butterfly O-grid
instead of the free mesher; interval assignment raises counts by the
squared relative error so equal sides share increments evenly; a mesh far
beyond the element limit is refused before any work and mapped limits are
no longer swallowed. The corpus reports hex8 and hex20 quality.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Catalogue: structured hex commands ready (mesh.hex.sweep, mesh.hex.mapped, mesh.hex.generate)

mesh.hex.sweep and mesh.hex.mapped run mesh.hex.generate with the sweeping
methods or the mapped one; mesh.hex.generate is the sweep entry's related
command. The planned M6 octree entry moves to mesh.hex.octree (its icon
renamed with it). docs/features.md regenerated.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Docs: structured hex literature (B8), architecture, roadmap, parity; corpus hex column

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Hex: simplify review fixes

Prepare each sweep frame's cross-section once (arrangement, cuts, fill)
and reuse it across refinement retries; candidate frames come from one
setup pass. Edge triangles come from the feature edges, layer counts go
through size::edge_divisions, sweep residual weights are computed once
per node, flux shape functions live on the stack, and the arrangement's
exact side test and distances serve the region probe. The engine takes
the boundary options mesh.tet.generate takes and names unnamed faces by
index like it; the corpus hex20 run reuses the winning method; the
mapped entry lists mesh.hex.generate too; the octree icon is redrawn
under its new id.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md and docs/meshing/corpus.md after rebasing

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md and docs/meshing/corpus.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md and docs/meshing/corpus.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md and docs/meshing/corpus.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

* Regenerate docs/features.md after rebasing onto main

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm

---------

Co-authored-by: jyviko <2501648+jyviko@users.noreply.github.com>
Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
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docs/meshing/corpus.md
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## Results
| part | categories | bodies | faces | triangles | tol. mm | build ms | enclosed mm³ | SolidCraft mm³ | rel. diff | exact mm³ | boundary | genus | feature edges | check ms | CDT tets | Steiner | cavities | tet gate | CDT ms |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| bent-tube | sweepable, thin-wall, genus | 1 | 18 | 4296 | 0.121 | 435 | 4213.1 | 4159.4 | 1.3e-2 | 4159.8 | ok, 2148 vertices | 1 | 268 | 10.9 | 15550 | 2338 | 0 | pass | 550 |
| bolted-assembly | assembly, blocky | 3 | 22 | 284 | 0.117 | 54 | 33004.7 | 32999.0 | 1.7e-4 | 32998.9 | ok, 144 vertices | 0, 1, 1 | 104 | 0.7 | 690 | 181 | 0 | pass | 31 |
| cross-cylinders | boolean, sharp | 1 | 10 | 1395 | 0.087 | 7682 | 32197.4 | 32357.8 | -5.0e-3 | 32365.8 | error: Cross: invalid boundary surface: 28 defects; triangles 142 and 184 are oriented inconsistently across edge 0–396 at (-15.488656021315903, 4.577998041166734, 7.853092194999726); triangles 184 and 239 are oriented inconsistently across edge 0–554 at (-18.75000000000002, 5.634596901641276, 7.632255089815496); edge 103–108 at (-0.16007784063084018, 0.03984236936862052, 9.998639270596865) is shared by 3 triangles; … | | | | error: Cross: invalid boundary surface: 28 defects; triangles 142 and 184 are oriented inconsistently across edge 0–396 at (-15.488656021315903, 4.577998041166734, 7.853092194999726); triangles 184 and 239 are oriented inconsistently across edge 0–554 at (-18.75000000000002, 5.634596901641276, 7.632255089815496); edge 103–108 at (-0.16007784063084018, 0.03984236936862052, 9.998639270596865) is shared by 3 triangles; … | | | | |
| filleted-bracket | fillet, genus, sweepable | 1 | 13 | 540 | 0.088 | 381 | 32162.3 | 32145.0 | 5.4e-4 | 32144.8 | ok, 268 vertices | 2 | 135 | 1.4 | 5582 | 1613 | 0 | pass | 187 |
| i-beam | sweepable, thin-wall | 1 | 14 | 44 | 0.208 | 4 | 61600.0 | 61600.0 | 0.0e0 | 61600.0 | ok, 24 vertices | 0 | 36 | 0.1 | 94 | 24 | 0 | pass | 3 |
| loft-transition | organic | 1 | 6 | 292 | 0.075 | 55 | 56390.2 | 56450.6 | -1.1e-3 | – | ok, 148 vertices | 0 | 40 | 0.7 | 1329 | 256 | 1 | pass | 33 |
| perforated-plate | genus, thin-wall, sweepable | 1 | 54 | 1644 | 0.144 | 265 | 26822.2 | 26764.6 | 2.2e-3 | 26764.2 | ok, 776 vertices | 24 | 828 | 4.6 | 33611 | 9798 | 0 | pass | 1195 |
| pinhole-block | small-feature, genus | 1 | 38 | 4044 | 0.073 | 202 | 49919.4 | 49909.6 | 2.0e-4 | 49909.5 | ok, 1992 vertices | 16 | 1548 | 12.4 | 96411 | 30221 | 0 | pass | 4408 |
| rounded-box | fillet | 1 | 26 | 2540 | 0.054 | 156 | 23325.7 | 23340.5 | -6.3e-4 | 23340.8 | ok, 1272 vertices | 0 | 216 | 6.3 | 11666 | 1913 | 0 | pass | 371 |
| shallow-groove | near-tangent, boolean | 1 | 8 | 62 | 0.075 | 11 | 47795.2 | 47782.3 | 2.7e-4 | 47781.5 | ok, 33 vertices | 0 | 24 | 0.2 | 267 | 87 | 0 | pass | 11 |
| sharp-wedge | sharp, thin-wall, sweepable | 1 | 5 | 8 | 0.083 | 2 | 3200.0 | 3200.0 | 0.0e0 | 3200.0 | ok, 6 vertices | 0 | 9 | 0.0 | 11 | 3 | 0 | pass | 0 |
| sphere-on-plate | near-tangent, boolean | 1 | 12 | 2826 | 0.098 | 703 | 69353.3 | 69509.0 | -2.2e-3 | 69509.7 | ok, 1415 vertices | 0 | 124 | 5.7 | 4898 | 309 | 9 | pass | 214 |
| spline-vase | organic | 1 | 11 | 3890 | 0.106 | 230 | 94891.2 | 94948.5 | -6.0e-4 | – | ok, 1947 vertices | 0 | 243 | 8.5 | 12461 | 1512 | 0 | pass | 419 |
| stepped-block | blocky, sharp | 1 | 14 | 96 | 0.078 | 30 | 47000.0 | 47000.0 | 0.0e0 | 47000.0 | ok, 48 vertices | 1 | 60 | 0.4 | 223 | 41 | 0 | pass | 6 |
| thin-ribs | thin-wall, blocky | 1 | 26 | 92 | 0.119 | 62 | 32000.0 | 32000.0 | -4.5e-16 | 32000.0 | ok, 48 vertices | 0 | 72 | 0.5 | 171 | 33 | 0 | pass | 6 |
| thin-wall-enclosure | thin-wall, blocky | 1 | 11 | 36 | 0.108 | 8 | 15564.0 | 15564.0 | 0.0e0 | 15564.0 | ok, 20 vertices | 0 | 28 | 0.1 | 191 | 47 | 2 | pass | 5 |
| analytic-sphere | analytic | 1 | 6 | 2784 | 0.069 | 263 | 33353.4 | 33507.9 | -4.6e-3 | 33510.3 | ok, 1394 vertices | 0 | 96 | 5.8 | 4211 | 58 | 0 | pass | 258 |
| analytic-torus | analytic, genus | 1 | 6 | 7230 | 0.109 | 345 | 37644.1 | 37893.7 | -6.6e-3 | 37899.3 | ok, 3615 vertices | 1 | 180 | 15.4 | 19790 | 466 | 0 | pass | 717 |
| analytic-cylinder | analytic, sweepable | 1 | 4 | 1004 | 0.049 | 25 | 12533.1 | 12564.7 | -2.5e-3 | 12566.4 | ok, 504 vertices | 0 | 106 | 2.7 | 7437 | 1568 | 0 | pass | 259 |
| analytic-pipe | analytic, genus, sweepable | 1 | 6 | 1448 | 0.104 | 43 | 94150.9 | 94238.0 | -9.3e-4 | 94247.8 | ok, 724 vertices | 1 | 220 | 3.7 | 6238 | 1340 | 0 | pass | 264 |
| analytic-plate-hole | analytic, genus, sweepable | 1 | 8 | 112 | 0.224 | 11 | 98449.1 | 98429.5 | 2.0e-4 | 98429.2 | ok, 56 vertices | 1 | 62 | 0.4 | 1868 | 543 | 0 | pass | 53 |
| part | categories | bodies | faces | triangles | tol. mm | build ms | enclosed mm³ | SolidCraft mm³ | rel. diff | exact mm³ | boundary | genus | feature edges | check ms | CDT tets | Steiner | cavities | tet gate | CDT ms | hex | hex elements | hex8 min SJ | hex20 min SJ | hex ms |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| bent-tube | sweepable, thin-wall, genus | 1 | 18 | 4296 | 0.121 | 454 | 4213.1 | 4159.4 | 1.3e-2 | 4159.8 | ok, 2148 vertices | 1 | 268 | 11.5 | 15550 | 2338 | 0 | pass | 606 | sweep | 192 | 0.859 | 0.997 | 63 |
| bolted-assembly | assembly, blocky | 3 | 22 | 284 | 0.117 | 206 | 33004.7 | 32999.0 | 1.7e-4 | 32998.9 | ok, 144 vertices | 0, 1, 1 | 104 | 0.7 | 690 | 181 | 0 | pass | 34 | extrude | 357 | 0.398 | 0.222 | 40 |
| cross-cylinders | boolean, sharp | 1 | 10 | 1395 | 0.087 | 7920 | 32197.4 | 32357.8 | -5.0e-3 | 32365.8 | error: Cross: invalid boundary surface: 28 defects; triangles 142 and 184 are oriented inconsistently across edge 0–396 at (-15.488656021315903, 4.577998041166734, 7.853092194999726); triangles 184 and 239 are oriented inconsistently across edge 0–554 at (-18.75000000000002, 5.634596901641276, 7.632255089815496); edge 103–108 at (-0.16007784063084018, 0.03984236936862052, 9.998639270596865) is shared by 3 triangles; … | | | | error: Cross: invalid boundary surface: 28 defects; triangles 142 and 184 are oriented inconsistently across edge 0–396 at (-15.488656021315903, 4.577998041166734, 7.853092194999726); triangles 184 and 239 are oriented inconsistently across edge 0–554 at (-18.75000000000002, 5.634596901641276, 7.632255089815496); edge 103–108 at (-0.16007784063084018, 0.03984236936862052, 9.998639270596865) is shared by 3 triangles; … | | | | | error: Cross: invalid boundary surface: 28 defects; triangles 142 and 184 are oriented inconsistently across edge 0–396 at (-15.488656021315903, 4.577998041166734, 7.853092194999726); triangles 184 and 239 are oriented inconsistently across edge 0–554 at (-18.75000000000002, 5.634596901641276, 7.632255089815496); edge 103–108 at (-0.16007784063084018, 0.03984236936862052, 9.998639270596865) is shared by 3 triangles; … | | | | |
| filleted-bracket | fillet, genus, sweepable | 1 | 13 | 540 | 0.088 | 388 | 32162.3 | 32145.0 | 5.4e-4 | 32144.8 | ok, 268 vertices | 2 | 135 | 1.4 | 5582 | 1613 | 0 | pass | 208 | no: Bracket: not supported yet: no structured hex method fits this body (extrude: cylindrical face 8 is not parallel to the direction; sweep: from face 1: linking face 6 has 2 boundary loops (periodic or holed linking faces are not swept); from face 6: | | | | 5 |
| i-beam | sweepable, thin-wall | 1 | 14 | 44 | 0.208 | 3 | 61600.0 | 61600.0 | 0.0e0 | 61600.0 | ok, 24 vertices | 0 | 36 | 0.1 | 94 | 24 | 0 | pass | 4 | extrude | 336 | 0.707 | 0.707 | 34 |
| loft-transition | organic | 1 | 6 | 292 | 0.075 | 57 | 56390.2 | 56450.6 | -1.1e-3 | – | ok, 148 vertices | 0 | 40 | 0.7 | 1329 | 256 | 1 | pass | 38 | sweep | 2873 | 0.189 | 0.189 | 328 |
| perforated-plate | genus, thin-wall, sweepable | 1 | 54 | 1644 | 0.144 | 273 | 26822.2 | 26764.6 | 2.2e-3 | 26764.2 | ok, 776 vertices | 24 | 828 | 4.8 | 33611 | 9798 | 0 | pass | 1345 | extrude | 467 | 0.304 | 0.033 | 98 |
| pinhole-block | small-feature, genus | 1 | 38 | 4044 | 0.073 | 201 | 49919.4 | 49909.6 | 2.0e-4 | 49909.5 | ok, 1992 vertices | 16 | 1548 | 12.9 | 96411 | 30221 | 0 | pass | 8142 | extrude | 5236 | 0.408 | 0.392 | 735 |
| rounded-box | fillet | 1 | 26 | 2540 | 0.054 | 235 | 23325.7 | 23340.5 | -6.3e-4 | 23340.8 | ok, 1272 vertices | 0 | 216 | 8.4 | 11663 | 1916 | 0 | pass | 469 | no: Box: not supported yet: no structured hex method fits this body (extrude: cylindrical face 8 is not parallel to the direction; sweep: from face 0: 12 faces are not on the sweep path; from face 5: 12 faces are not on the sweep path; revolve: plan | | | | 15 |
| shallow-groove | near-tangent, boolean | 1 | 8 | 62 | 0.075 | 11 | 47795.2 | 47782.3 | 2.7e-4 | 47781.5 | ok, 33 vertices | 0 | 24 | 0.2 | 267 | 87 | 0 | pass | 13 | extrude | 1960 | 0.991 | 0.986 | 201 |
| sharp-wedge | sharp, thin-wall, sweepable | 1 | 5 | 8 | 0.083 | 2 | 3200.0 | 3200.0 | 0.0e0 | 3200.0 | ok, 6 vertices | 0 | 9 | 0.0 | 11 | 3 | 0 | pass | 0 | extrude | 294 | 0.050 | 0.050 | 30 |
| sphere-on-plate | near-tangent, boolean | 1 | 12 | 2826 | 0.098 | 974 | 69353.3 | 69509.0 | -2.2e-3 | 69509.7 | ok, 1415 vertices | 0 | 124 | 7.2 | 4898 | 309 | 9 | pass | 260 | no: Plate: not supported yet: no structured hex method fits this body (extrude: face 6 is curved across the direction; sweep: from face 0: 6 faces are not on the sweep path; from face 5: linking face 11 has 2 boundary loops (periodic or holed linking | | | | 11 |
| spline-vase | organic | 1 | 11 | 3890 | 0.106 | 226 | 94891.2 | 94948.5 | -6.0e-4 | – | ok, 1947 vertices | 0 | 243 | 9.0 | 12461 | 1512 | 0 | pass | 456 | sweep | 4554 | 0.649 | 0.649 | 547 |
| stepped-block | blocky, sharp | 1 | 14 | 96 | 0.078 | 264 | 47000.0 | 47000.0 | 0.0e0 | 47000.0 | ok, 48 vertices | 1 | 60 | 0.4 | 223 | 41 | 0 | pass | 7 | extrude | 1416 | 0.999 | 0.999 | 142 |
| thin-ribs | thin-wall, blocky | 1 | 26 | 92 | 0.119 | 1004 | 32000.0 | 32000.0 | -4.5e-16 | 32000.0 | ok, 48 vertices | 0 | 72 | 0.6 | 171 | 33 | 0 | pass | 8 | extrude | 777 | 0.844 | 0.844 | 100 |
| thin-wall-enclosure | thin-wall, blocky | 1 | 11 | 36 | 0.108 | 340 | 15564.0 | 15564.0 | 0.0e0 | 15564.0 | ok, 20 vertices | 0 | 28 | 0.2 | 191 | 47 | 2 | pass | 11 | extrude | 894 | 0.297 | 0.297 | 112 |
| analytic-sphere | analytic | 1 | 6 | 2784 | 0.069 | 719 | 33353.4 | 33507.9 | -4.6e-3 | 33510.3 | ok, 1394 vertices | 0 | 96 | 12.6 | 4211 | 58 | 0 | pass | 440 | no: Sphere: not supported yet: no structured hex method fits this body (extrude: no direction to try; sweep: no planar face to sweep from; revolve: no axis to try; mapped: meshing failed: meshing failed: mesh validity gate: 8 inverted or degenerate ele | | | | 31 |
| analytic-torus | analytic, genus | 1 | 6 | 7230 | 0.109 | 944 | 37644.1 | 37893.7 | -6.6e-3 | 37899.3 | ok, 3615 vertices | 1 | 180 | 25.6 | 19790 | 466 | 0 | pass | 1412 | revolve | 1134 | 0.706 | 0.706 | 280 |
| analytic-cylinder | analytic, sweepable | 1 | 4 | 1004 | 0.049 | 190 | 12533.1 | 12564.7 | -2.5e-3 | 12566.4 | ok, 504 vertices | 0 | 106 | 3.4 | 7437 | 1568 | 0 | pass | 452 | extrude | 3200 | 0.707 | 0.707 | 585 |
| analytic-pipe | analytic, genus, sweepable | 1 | 6 | 1448 | 0.104 | 336 | 94150.9 | 94238.0 | -9.3e-4 | 94247.8 | ok, 724 vertices | 1 | 220 | 9.0 | 6238 | 1340 | 0 | pass | 479 | extrude | 1288 | 0.998 | 0.998 | 290 |
| analytic-plate-hole | analytic, genus, sweepable | 1 | 8 | 112 | 0.224 | 231 | 98449.1 | 98429.5 | 2.0e-4 | 98429.2 | ok, 56 vertices | 1 | 62 | 0.9 | 1868 | 543 | 0 | pass | 72 | extrude | 321 | 0.320 | 0.142 | 55 |
## Categories
docs/meshing/literature.md
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---
### B8. Mapped and swept hexahedral meshing (implemented in `modelcraft-mesh::hex`)
`mesh.hex.generate` (and `mesh.hex.sweep`, `mesh.hex.mapped`). Every method below is implemented from
its paper; no code was read. DOIs marked "checked" were confirmed on 2026-10-13 through a repository
record or the publisher; the others are cited by venue only.
**B8.1 Blacker 1996, the Cooper tool.** **Verdict: BUILD ON (sweep recognition).**
- **Citation.** T. D. Blacker. *The Cooper tool.* Proc. 5th International Meshing Roundtable, 1996, pp. 13–29.
- **Use.** Source, target and linking faces from B-rep topology: rings of logically rectangular
linking faces followed from the source loops until they close on one target (`hex::sweep`).
**B8.2 Lai, Benzley, White 2000, many-to-many sweeping.** **Verdict: BUILD ON (multi-level extrusions).**
- **Citation.** M. Lai, S. Benzley, D. White. *Automated hexahedral mesh generation by generalized multiple
source to multiple target sweeping.* IJNME 49(1–2):261–275, 2000.
- **Use.** Imprinting every level's outline into one projection plane so each region of the arrangement
is a one-to-one column (`hex::layered`, `hex::arrangement`); here done in a straight or rotating
frame, with the regions' fill decided by the generalized winding number (Jacobson et al. 2013).
**B8.3 Knupp 1998, next-generation sweep tool.** **Verdict: BUILD ON (residual correction).**
- **Citation.** P. M. Knupp. *Next-generation sweep tool: a method for generating all-hex meshes on
two-and-one-half dimensional geometries.* Proc. 7th International Meshing Roundtable, 1998, pp. 505–513.
- **Use.** The boundary residual of each layer's map interpolated into the interior.
**B8.4 Roca and Sarrate 2010, least-squares sweeping.** **Verdict: BUILD ON (inner nodes).**
- **Citation.** X. Roca, J. Sarrate. *An automatic and general least-squares projection procedure for sweep
meshing.* Engineering with Computers 26(4):391–406, 2010. DOI 10.1007/s00366-009-0172-z (checked, repository
record). Conference version: Proc. 15th IMR, 2006, pp. 487–506.
- **Use.** Each layer's interior nodes from the source mesh by the least-squares affine map of the source
ring onto the layer's ring (`hex::sweep::Affine`). Related: Roca, Sarrate, Huerta, *A new least-squares
approximation of affine mappings for sweep algorithms*, Eng. Comput. 26(3):327–337, 2010.
**B8.5 Interval assignment: Tam and Armstrong 1993; Mitchell 1997/2021.** **Verdict: BUILD ON.**
- **Citations.** T. K. H. Tam, C. G. Armstrong. *Finite element mesh control by integer programming.* IJNME
36:2581–2605, 1993. S. A. Mitchell. *High fidelity interval assignment.* Proc. 6th IMR, 1997 (IJCGA
10(4):399–415, 2000). S. A. Mitchell. *Incremental interval assignment by integer linear algebra.*
International Meshing Roundtable 2021 (Zenodo 5559025); extended in Computer-Aided Design 158:103485, 2023.
- **Use.** Division counts of all curves together: side equalities of mapped faces and linking faces,
even (or multiple-of-four) loop sums, minimum loop sums. `hex::interval` uses incremental repair from the
rounded goals with increments only, chosen by squared relative error; it is bounded and reports an
infeasible system instead of looping. Mitchell's integer-linear-algebra solver (Hermite normal form plus
nullspace improvement) is the upgrade if the increment-only scheme proves too coarse.
**B8.6 Whiteley, White, Benzley, Blacker 1996, submapping.** **Verdict: BUILD ON (vertex classification).**
- **Citation.** M. Whiteley, D. White, S. Benzley, T. Blacker. *Two and three-quarter dimensional meshing
facilitators.* Engineering with Computers 12:144–154, 1996.
- **Use.** Corners by interior angle (≤ 135° end, ≥ 225° reflex); cross-section regions are cut from
their reflex corners to the boundary, which decomposes rectilinear sections into rectangles (an explicit
form of submapping's virtual decomposition), then mapped.
**B8.7 Gordon and Thiel 1982, trivariate transfinite interpolation.** **Verdict: BUILD ON (mapped blocks).**
- **Citation.** W. J. Gordon, L. C. Thiel. *Transfinite mappings and their application to grid generation.*
Applied Mathematics and Computation 10–11:171–233, 1982.
- **Use.** Interior nodes of a logical cube from its six face grids (`hex::mapped`).
**B8.8 Support.** Generalized winding number for the inside test (A. Jacobson, L. Kavan, O. Sorkine-Hornung,
*Robust inside-outside segmentation using generalized winding numbers*, ACM TOG 32(4):33, 2013; solid angle
of a triangle: Van Oosterom and Strackee 1983); closest point on a triangle (C. Ericson, *Real-Time
Collision Detection*, 2005, §5.1.5).
## Part C: Datasets and benchmarks
| Dataset | Content | Licence | Usable for ModelCraft? |